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+{"text":"\\section{Introduction} \\label{S:Introduction} \nThe Einstein field equations of general relativity connect the \\emph{Einstein tensor} $R_{\\mu \\nu} - \\frac{1}{2}g_{\\mu \\nu} R,$\nwhich contains information about the curvature of spacetime\\footnote{By spacetime, we mean a four-dimensional time-oriented Lorentzian manifold $\\mathfrak{M}$ together with a Lorentzian metric $g_{\\mu \\nu}$ of signature $(-,+,+,+).$} $(\\mathfrak{M},g_{\\mu \\nu}),$ to the energy-momentum-stress-density tensor (energy-momentum tensor for short) $T_{\\mu \\nu},$ which contains information about the matter present in $\\mathfrak{M}.$ Here, $g_{\\mu \\nu}$ is the \\emph{spacetime metric}, $R_{\\mu \\nu}$ is the \\emph{Ricci curvature tensor}, and $R = (g^{-1})^{\\kappa \\lambda} R_{\\kappa \\lambda}$ is the \\emph{scalar curvature}. In this article, we show the stability of the $1 + 3-$dimensional vacuum Minkowski spacetime solution of the Einstein-nonlinear electromagnetic system\n\n\\begin{subequations}\n\\begin{align} \n\tR_{\\mu \\nu} - \\frac{1}{2}g_{\\mu \\nu} R & = T_{\\mu \\nu}, && (\\mu, \\nu = 0,1,2,3), \n\t\t\\label{E:IntroEinstein} \\\\\n\t(d \\Far)_{\\lambda \\mu \\nu} & = 0, && (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:IntrodFaris0} \\\\\n\t(d \\Max)_{\\lambda \\mu \\nu} & = 0, && (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:IntrodMis0}\n\\end{align}\n\\end{subequations}\nwhere $T_{\\mu \\nu}$ (see \\eqref{E:electromagnetictensorloweroinTermsofLagrangian}) is one of the energy-momentum tensors corresponding to a family of nonlinear models of electromagnetism, $d$ denotes the exterior derivative operator, the two-form $\\Far_{\\mu \\nu}$ denotes the \\emph{Faraday tensor}, the two-form $\\Max$ denotes the \\emph{Maxwell tensor}, and $\\Max_{\\mu \\nu}$ is connected to $(g_{\\mu \\nu},\\Far_{\\mu \\nu})$ through a constitutive relation. We make the following three assumptions concerning the electromagnetic matter model: $(i)$ its Lagrangian $\\Ldual$ is a scalar-valued function of the two electromagnetic invariants\\footnote{Throughout the article, we use Einstein's summation convention in that repeated indices are summed over.} $\\Farinvariant_{(1)} \\eqdef \\frac{1}{2} (g^{-1})^{\\kappa \\mu} (g^{-1})^{\\lambda \\nu} \\Far_{\\kappa \\lambda} \\Far_{\\mu \\nu},$ $\\Farinvariant_{(2)} \\eqdef \\frac{1}{4} (g^{-1})^{\\kappa \\mu} (g^{-1})^{\\lambda \\nu} \\Far_{\\kappa \\lambda} \\Fardual_{\\mu \\nu},$ where $\\star$ denotes the Hodge duality operator corresponding to $g_{\\mu \\nu};$ $(ii)$ the energy-momentum tensor $T_{\\mu \\nu}$ corresponding to $\\Ldual$ satisfies the \\emph{dominant energy condition} (sufficient conditions on $\\Ldual$ are given in \\eqref{E:DECL1} - \\eqref{E:DECTrace} below);\n$(iii)$ $\\Ldual$ is a sufficiently differentiable function of $(\\Farinvariant_{(1)},$ $\\Farinvariant_{(2)}),$ and its Taylor expansion around $(0,0)$ agrees with that of the linear Maxwell-Maxwell equations to first order; i.e., $\\Ldual(\\Farinvariant_{(1)}, \\Farinvariant_{(2)}) = - \\frac{1}{2} \\Farinvariant_{(1)} + O^{\\dParameter + 2}\\big(|(\\Farinvariant_{(1)}, \\Farinvariant_{(2)})|^2 \\big),$ where $\\dParameter \\geq 8$ is an integer; see Section \\ref{SS:Oando} regarding the notation $O^{\\dParameter + 2}(\\cdots)$. We briefly summarize our main results here. They are rigorously stated and proved in Section \\ref{S:GlobalExistence}.\n\n\\begin{changemargin}{.25in}{.25in} \n\\textbf{Main Results.} \\\nThe vacuum Minkowski spacetime background solution to the system \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0} is globally stable. In particular, small perturbations of the trivial initial data corresponding to the background solution have maximal globally hyperbolic developments that are geodesically complete. Furthermore, the perturbed solution converges to the vacuum Minkowski spacetime solution. These conclusions are consequences of a small-data global existence result for the \\emph{reduced} system \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}, which is equivalent to the study of \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0} in a wave coordinate system (i.e., a coordinate system \n$\\lbrace x^{\\mu} \\rbrace_{\\mu = 0,1,2,3}$ on $\\mathbb{R}^{1+3}$ satisfying \n$(g^{-1})^{\\kappa \\lambda} \\mathscr{D}_{\\kappa} \\mathscr{D}_{\\lambda} x^{\\mu} = 0, (\\mu = 0,1,2,3),$ where \n$\\mathscr{D}$ is the Levi-Civita connection corresponding to $g_{\\mu \\nu}$).\n\\end{changemargin}\n\nWe recall the following standard facts (see e.g. \\cite{dC2008}, \\cite{rW1984}) concerning the initial data for the system \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0}, which we refer to as ``abstract'' initial data. The abstract initial data consist of a $3-$dimensional manifold $\\Sigma_0,$ together with the following fields on $\\Sigma_0:$ a Riemannian metric $\\mathring{\\underline{g}}_{jk},$ a symmetric type $\\binom{0}{2}$ tensorfield $\\mathring{K}_{jk},$ and a pair of electromagnetic one-forms $\\mathring{\\mathfrak{D}}_j, \\mathring{\\mathfrak{B}}_j,$ $(j,k =1,2,3).$ Furthermore, they must satisfy the \\emph{Gauss}, \\emph{Codzazzi}, and \\emph{electromagnetic} constraint equations, which are respectively given by\n\n\\begin{subequations}\n\\begin{align}\n\t\\underline{\\mathring{R}} - \\mathring{K}_{ab} \\mathring{K}^{ab} + \\big[(\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{K}_{ab}\\big]^2 & = \n\t\t2T(\\hat{N},\\hat{N})|_{\\Sigma_0}, && \\label{E:GaussIntro} \\\\\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\underline{\\mathring{\\mathscr{D}}}_a \\mathring{K}_{bj} - (\\mathring{\\underline{g}}^{-1})^{ab}  \\underline{\\mathring{\\mathscr{D}}}_j \\mathring{K}_{ab} & = \n\t\tT(\\hat{N},\\frac{\\partial}{\\partial x^j})|_{\\Sigma_0}, && (j=1,2,3), \\label{E:CodazziIntro} \\\\\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\underline{\\mathring{\\mathscr{D}}}_a \\mathring{\\mathfrak{\\Displacement}}_b & = 0,&& \n\t\t\\label{E:DivergenceD0Intro} \\\\\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\underline{\\mathring{\\mathscr{D}}}_a \\mathring{\\mathfrak{\\Magneticinduction}}_b & = 0.&& \n\t\t\\label{E:DivergenceB0Intro}\n\\end{align}\n\\end{subequations}\nIn the above expressions, the indices are lowered and raised with $\\mathring{\\underline{g}}_{jk}$ and $(\\mathring{\\underline{g}}^{-1})^{jk},$ $\\underline{\\mathring{R}}$ denotes the scalar curvature of \n$\\mathring{\\underline{g}}_{jk},$ $\\underline{\\mathring{\\mathscr{D}}}$ denotes the Levi-Civita connection corresponding to $\\mathring{\\underline{g}}_{jk},$ and $\\hat{N}^{\\mu}$ is the future-directed unit $g-$normal to $\\Sigma_0$ (viewed as an embedded submanifold of $(\\mathfrak{M},g_{\\mu \\nu})$). The one-forms $\\mathring{\\mathfrak{D}}_j$ and $\\mathring{\\mathfrak{B}}_j$\ntogether form a geometric decomposition of $\\Far_{\\mu \\nu}|_{\\Sigma_0},$ and the right-hand sides of \\eqref{E:GaussIntro} - \\eqref{E:CodazziIntro} can be computed (in principle) in terms of $\\mathring{\\underline{g}}_{jk},$ $\\mathring{\\mathfrak{\\Displacement}}_j,$ and $\\mathring{\\mathfrak{\\Magneticinduction}}_j$ alone; see Section \\ref{SS:EBDH} for more details concerning the relationship of $\\mathring{\\mathfrak{D}}_j$ and $\\mathring{\\mathfrak{B}}_j$ to $\\Far_{\\mu \\nu}|_{\\Sigma_0}.$ The dominant energy condition manifests itself along $\\Sigma_0$ as the inequalities $T(\\hat{N},\\hat{N}) \\geq 0$ and $T(\\hat{N},\\hat{N})^2 - (\\mathring{\\underline{g}}^{-1})^{ab} T(\\hat{N},\\frac{\\partial}{\\partial x^a})T(\\hat{N},\\frac{\\partial}{\\partial x^b}) \\geq 0.$ \n\nIn this article, we consider the case $\\Sigma_0 = \\mathbb{R}^3.$ We will construct spacetimes of the form $\\mathfrak{M} = I \\times \\mathbb{R}^3,$ where $I$ will be a time interval, and $\\Sigma_0$ will be a spacelike Cauchy hypersurface in $(\\mathfrak{M},g_{\\mu \\nu}).$ The constraints \\eqref{E:GaussIntro} - \\eqref{E:CodazziIntro} are necessary to ensure that \\eqref{E:IntroEinstein} can be satisfied along $\\Sigma_0,$ while the constraints \\eqref{E:DivergenceD0Intro} - \\eqref{E:DivergenceB0Intro} are necessary to ensure that the electromagnetic equations \\eqref{E:IntrodFaris0} - \\eqref{E:IntrodMis0} can be satisfied along $\\Sigma_0.$ Our stability criteria include both decay assumptions at $\\infty$ and smallness assumptions for the abstract initial data. We provide here a description of our decay assumptions at $\\infty,$ which are based on the assumptions of \\cite{hLiR2010}; our smallness assumptions will be addressed in detail in Section \\ref{S:SmallDataAssumptions}.\n\n\n \n\n\\textbf{Assumptions on the abstract initial data:}\nWe assume that there exists a global coordinate chart $x = (x^1,x^2,x^3)$ on $\\Sigma_0 = \\mathbb{R}^3,$ a real number $\\decayparameter > 0,$ and an integer $\\dParameter \\geq 8$ such that (with $r \\eqdef |x| \\eqdef \\big[(x^1)^2 + (x^2)^2 + (x^3)^2 \\big]^{1\/2}$ and $j,k = 1,2,3$)\n\n\\begin{subequations}\n\\begin{align}\n\t\\mathring{\\underline{g}}_{jk} & = \\delta_{jk} + \\mathring{\\underline{h}}_{jk}^{(0)} + \\mathring{\\underline{h}}_{jk}^{(1)},\n\t&& \\label{E:metricdataexpansion} \\\\\n\t\\mathring{\\underline{h}}_{jk}^{(0)} & = \\chi(r) \\frac{2M}{r} \\delta_{jk}, && \\chi(r) \\ \\mbox{is defined in} \\ \n\t\t\\eqref{E:chidef}, \\label{E:h0AbstractDataAsymptotics} \\\\\n\t\\mathring{\\underline{h}}_{jk}^{(1)} & =  o^{\\dParameter+1}(r^{-1 - \\decayparameter}), && \\mbox{as} \\ r \\to \\infty, \\label{E:h1AbstractDataAsymptotics} \\\\\n\t\\mathring{K}_{jk} & = o^{\\dParameter}(r^{-2 - \\decayparameter}), && \\mbox{as} \\ r \\to \\infty, \\label{E:KAbstractDataAsymptotics} \\\\\n\t\\mathring{\\mathfrak{\\Displacement}}_j & = o^{\\dParameter}(r^{-2 - \\decayparameter}), && \\mbox{as} \\ r \\to \\infty, \\\\\n\t\\mathring{\\mathfrak{\\Magneticinduction}}_j & = o^{\\dParameter}(r^{-2 - \\decayparameter}),&& \\mbox{as} \\ r \\to \\infty,\n\t\\label{E:BdecayAssumption}\n\\end{align}\n\\end{subequations}\nwhere the meaning of $o^{\\dParameter}(\\cdots)$ is described in Section \\ref{SS:Oando}.\n\nThe parameter $M$ in \\eqref{E:metricdataexpansion}, which is known as the \\emph{ADM mass}, is constrained by the following requirements: according to the \\emph{positive mass theorem} of Schoen-Yau \\cite{rSstY1979}, \\cite{rSstY1981}, and Witten \\cite{eW1981}, under the assumption that $T_{\\mu \\nu}$ satisfies the dominant energy condition, the only solutions $\\mathring{\\underline{g}}_{jk}$ to the constraint equations \\eqref{E:GaussIntro} - \\eqref{E:DivergenceB0Intro} that have an expansion of the form \\eqref{E:metricdataexpansion} with the asymptotic behavior \\eqref{E:h0AbstractDataAsymptotics} - \\eqref{E:KAbstractDataAsymptotics} either have $M > 0,$ or have $M = 0$ and $\\mathring{\\underline{g}}_{jk} = \\delta_{jk}.$ The groundbreaking work \\cite{dCsK1993} of Christodoulou and Klainerman (which is discussed further in Section \\ref{SSS:MathematicalComparisons}) demonstrated the stability of the Minkowski spacetime solution to the Einstein-vacuum equations in the case that the initial data are \\emph{strongly asymptotically flat}, which corresponds to the parameter range $\\decayparameter \\geq 1\/2$ in the above expansions. Our work here, which relies on the framework developed by Lindblad and Rodnianski in \\cite{hLiR2010} (see Section \\ref{SSS:MathematicalComparisons}), allows for the parameter range $\\kappa > 0.$\n\nIn this article, we do not consider the issue of solving the constraint equations. \nTo the best of our knowledge, under the restrictions on $\\Ldual$ described at the beginning of Section \\ref{S:Introduction},\nthere are presently no rigorous results concerning the construction of initial data on the manifold $\\mathbb{R}^3$ that satisfy the constraints. However, we remark that for the Einstein-vacuum equations $T_{\\mu \\nu} \\equiv 0,$ initial data that satisfy the constraints and that coincide with the standard Schwarzschild data\n\n\\begin{subequations}\n\\begin{align}\n\t\\mathring{\\underline{g}}_{jk} & = \\big(1 + \\frac{2M}{r} \\big) \\delta_{jk},\\\\\n\t\\mathring{K}_{jk} & = 0\n\\end{align}\n\\end{subequations}\noutside of the unit ball centered at the origin were shown to exist in \\cite{pCeD2002erratum} - \\cite{pCeD2002} and \\cite{jC2000}. The stability of the Minkowski spacetime solution to the Einstein-vacuum equations for such data follows from the methods of the aforementioned works \\cite{dCsK1993}, \\cite{hLiR2010} (and its precursor \\cite{hLiR2005}), and also from the \\emph{conformal method} approach of Friedrich \\cite{hF1986a}.\n\n\\begin{remark}\n\tThe only role of the dominant energy condition in this article is to ensure the physical condition $M \\geq 0;$ \n\twe assume this physical condition throughout the article. However, although the \n\tsmallness of $|M|$ is needed to prove our global stability result, the sign of $M$ does not enter into the stability \n\tanalysis. In particular, if there existed small initial data with small negative ADM mass, we would still be able to prove \n\tthat the corresponding solution to the equations exists globally. Similarly, if we made the replacement\n\t$T_{\\mu \\nu} \\rightarrow - T_{\\mu \\nu}$ in the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary},\n\twe could still prove a small-data global existence result.\n\\end{remark}\n\n\n\n\\subsection{Comparison with previous work}\n\n\\subsubsection{Mathematical comparisons} \\label{SSS:MathematicalComparisons}\n\nOur result is an extension of a large and growing hierarchy of stability results for the ${1 + 3-}$dimensional\nMinkowski spacetime solution to the Einstein equations, which began with the celebrated work \\cite{dCsK1993} of Christodoulou and Klainerman, and which was later replicated by Klainerman and Nicol{\\`o} in \\cite{sKfN2003} using alternate techniques. Both of these proofs used a manifestly covariant framework for both the formulation of the problem and the derivation of the estimates. However, mathematically speaking, the closest relatives to the present article are the seminal works \\cite{hLiR2005} and \\cite{hLiR2010}, in which Lindblad and Rodnianski developed a technically simpler framework for showing the stability of the vacuum Minkowski spacetime solution of the Einstein-scalar field system using a \\emph{wave coordinate} gauge. Although their decay estimates are not as precise as those of \\cite{dCsK1993} and \\cite{sKfN2003}, their work was \\emph{much} shorter than its predecessors, yet is robust enough to allow for modifications, including the presence of the nonlinear electromagnetic fields examined in this article. We remark that many of the technical results we need are contained in \\cite{hLiR2005} and \\cite{hLiR2010} and we will often direct the reader to these works for their proofs. \n\nOther stability results in this vein include \\cite{nZ2000}, in which Zipser extended the framework of \\cite{dCsK1993} to show the stability of the vacuum Minkowski spacetime solution to the Einstein-Maxwell system, and \\cite{lBnZ2009}, in which Bieri weakened the assumptions of \\cite{dCsK1993} on the decay of the initial data at infinity. We also mention the works \\cite{jL2008} (see also \\cite{jL2006}, \\cite{jL2009}), in which Loizelet used the framework of \\cite{hLiR2005} and \\cite{hLiR2010} to demonstrate the stability of the vacuum Minkowski spacetime solution of the Einstein-scalar field-Maxwell system in $1 + n,$ $n \\geq 3,$ dimensions. Moreover, in spacetimes of dimension $1 + n,$ with $n \\geq 5$ odd, it has been shown \\cite{yCBpCjL2006} that the \\emph{conformal method} can be used to show the stability of the Einstein-Maxwell system for initial data that coincide with the standard Schwarzschild data outside of a compact set. Roughly speaking, the conformal method is a way of mapping a global existence problem into a local existence problem. Whenever it is available, the method tends to give very precise information concerning the asymptotics of the global solutions. In particular, the results of \\cite{yCBpCjL2006} provide a more detailed description of the asymptotics than the results of \\cite{jL2008}.\n\nWe state with emphasis that the techniques used in this article differ in a fundamental way from those used by Loizelet in \\cite{jL2008}. More specifically, in \\cite{jL2008}, Loizelet analyzed the familiar linear Maxwell-Maxwell\\footnote{Our use of the terminology ``Maxwell-Maxwell'' equations, which are commonly referred to as the ``Maxwell'' equations,\nis explained in \\cite{jS2010a}.} equations through the use of a four-potential\\footnote{Recall that a four-potential is a one-form $A_{\\mu}$ such that $\\Far_{\\mu \\nu} = (dA)_{\\mu \\nu}.$} $A_{\\mu}$ satisfying the \\emph{Lorenz gauge} condition $(g^{-1})^{\\kappa \\lambda} \\mathscr{D}_{\\kappa} A_{\\lambda} = 0,$ where $\\mathscr{D}$ is the Levi-Civita connection corresponding to $g_{\\mu \\nu}.$ In Loizelet's analysis of the linear Maxwell-Maxwell equations, the Lorenz gauge leads to a system of linear wave equations for the components $A_{\\mu}.$ Furthermore, these equations can be analyzed using the same techniques that are used in the study of the components of the metric (see equation \\eqref{E:Reducedh1Summary}) and the scalar field. In particular, in Loizelet's case, Lemma \\ref{L:weightedenergy} can be used to deduce suitable weighted energy estimates for the components $\\nabla_{\\mu} A_{\\nu}.$ In contrast, as discussed in \\cite{jS2010a}, it is not clear that Lorenz gauge can be used for the kinds of quasilinear electromagnetic field equations \\eqref{E:IntrodMis0} studied in this article. More specifically, it is not clear that the Lorenz gauge in general leads to a hyperbolic formulation of the electromagnetic equations that is suitable for deriving the kinds of $L^2$ energy estimates needed for our analysis. For this reason, throughout this article, we work directly with the Faraday tensor. In particular, as described in detail in Section \\ref{E:EOVandStress}, we use Christodoulou's geometric framework \\cite{dC2000} to generate \\emph{energy currents} that can be used to derive the kinds of $L^2$ estimates needed in our analysis. In this way, we prove Lemma \\ref{L:weightedenergyFar}, which compensates for the fact that Lemma \\ref{L:weightedenergy} is not generally available for controlling the electromagnetic quantities. We remark that there is another advantage to working directly with the Faraday tensor: \\emph{our smallness condition for stability depends only on the physical field variables, and not on auxiliary mathematical quantities such as the values achieved by the components $\\nabla_{\\mu} A_{\\nu}.$}\n\nNow roughly speaking, the reason that we are able to prove our main stability result is because in our wave coordinate gauge\n(see the discussion in Section \\ref{SSS:Settinguptheequations}), the nonlinear terms have a special algebraic structure, which Lindblad and Rodnianski have labeled \\cite{hLiR2003} \\emph{the weak null condition}. We remark that in order for small-data global existence to hold, it is essential that the quadratic nonlinearities have special structure: John's blow-up result \\cite{fJ1981} shows that quadratic perturbations of the linear wave equation in $1 + 3$ dimensional Minkowski space (of which our equations \\eqref{E:Reducedh1Intro} below are an example), \\emph{do not necessarily} have small-data global existence. Now by definition, a system of PDEs satisfies the weak null condition if the corresponding \\emph{asymptotic system} has \nsmall-data global solutions. The asymptotic system is obtained by discarding cubic and higher order terms, and also derivatives that are tangential to the outgoing Minkowskian null cones (see the discussion in Section \\ref{SSS:GeometryandNullDecompositions}); the discarded terms are expected to decay faster than the remaining terms. The general philosophy is that if the asymptotic system has small-data global existence, then one should be hopeful that the original system does too. In \\cite{hLiR2010}, Lindblad and Rodnianski showed that the asymptotic system corresponding to the Einstein-scalar field system in wave coordinates has global solutions. Although we do not carry out such an analysis in this article, we remark that it can be checked that the asymptotic system\\footnote{To obtain this asymptotic system, one also discards the quadratic terms containing the fast-decaying null components $\\alpha[\\Far], \\rho[\\Far]$ and $\\sigma[\\Far]$ of the Faraday tensor; see Section \\ref{SSS:GeometryandNullDecompositions}.} corresponding to the Einstein-nonlinear electromagnetic system in wave coordinates also has global solutions. This was our original motivation for pursuing the present work.\n\nThe aforementioned weak null condition is a generalization of the classic \\emph{null condition} of Klainerman \\cite{sK1986} (see also Christodoulou's work \\cite{dC1986}), in which the quadratic nonlinearities are \\emph{standard null forms} (which are defined below in the statement of Lemma \\ref{L:RicciInWave}). By now, there is a very large body of global existence and almost-global existence results that are based on the analysis of quadratic nonlinearities that satisfy generalizations of Klainerman's null condition. This includes the stability results for the Einstein equations mentioned above, but also many other results; there are far too many to list exhaustively, but we mention the following as examples: \\cite{sK2005}, \\cite{sKtS1996}, \\cite{hL2004}, \\cite{hL2008}, \\cite{jMcS2007}, \\cite{jMmNcS2005b}, \\cite{tS1996}, \\cite{jS2010a}. \n\n\n\n\\subsubsection{Connections to the ``divergence'' problem}\n\nOne of the most important unresolved issues in physics is that of the so-called ``divergence problem.'' In the setting of classical electrodynamics, this problem manifests itself as the unhappy fact that the familiar linear Maxwell-Maxwell equations with \\emph{point charge} sources (i.e., delta function source terms), together with the \\emph{Lorentz force law}\\footnote{Recall that the Lorentz force is $F_{Lorentz} = q [\\Electricfield + v \\times \\Magneticinduction],$ where $q$ is the charge associated to the point charge, $\\Electricfield$ is the electric field, $v$ is the instantaneous point charge velocity, and $\\Magneticinduction$ is the magnetic induction field.}, do not comprise a well-defined system of equations. This is because the theory dictates that the Lorentz force at the location of a point charge is ``infinite in all directions,'' so that the charge's motion is ill-defined. A further symptom of the divergence problem in this theory is that the energy of a static point charge is infinite. Moreover, our present-day flagship model of quantum electrodynamics (QED), which is based on a quantization of the classical Maxwell-Dirac field equations, has not yet fixed the crux of the problem; similar manifestations of the divergence problem arise in QED; see \\cite{mK2004a}, \\cite{mK2004b} for a detailed discussion of these issues.\n\nNow in \\cite{mK2004a}, \\cite{mK2004b}, Kiessling has taken a preliminary step in the direction of resolving the divergence problem by reconsidering classical electrodynamics. One of Kiessling's primary strategies is to follow the lead of of Max Born \\cite{mB1933} by replacing the linear Maxwell-Maxwell equations with a suitable nonlinear system, the hope being that it will be possible to make rigorous mathematical sense of the motion of point charges in the nonlinear theory. As is discussed below, Kiessling's leading candidate is the Maxwell-Born-Infeld (MBI) model \\cite{mBlI1934} of classical electromagnetism, a model put forth by Born and Infeld in $1934$ based on Born's earlier ideas. The electromagnetic Lagrangian for this model is \n\\begin{align} \\label{E:LMBI}\n\t\\Ldual_{(MBI)}  \\eqdef \\frac{1}{\\upbeta^4} - \\frac{1}{\\upbeta^4} \\big(1 + \\upbeta^4 \\Farinvariant_{(1)} - \\upbeta^8 \n\t\t\\Farinvariant_{(2)}^2 \\big)^{1\/2} = \\frac{1}{\\upbeta^4} - \\frac{1}{\\upbeta^4} \\big(\\mbox{det}_g(g + \\Far) \\big)^{1\/2},\n\\end{align}\nwhere $\\upbeta > 0$ denotes \\emph{Born's ``aether'' constant.} We point out that as verified in e.g. \\cite{jS2010a}, this Lagrangian satisfies the assumptions \\eqref{E:Ldualassumptions} and \\eqref{E:DECL1} - \\eqref{E:DECTrace} below, so that the main results of this article apply to the MBI model. Now it turns out that it was not enough for Kiessling to simply replace the linear Maxwell-Maxwell equations with the Maxwell-Born-Infeld equations, for such a modification fails to fix the problem of the Lorentz force being ill-defined at the location of the point charge. On the other hand, in MBI theory on the Minkowski spacetime background, there exist \\emph{Lipschitz-continuous} electromagnetic potentials corresponding to single static point charge solutions to the field equations. Kiessling observed that this level of regularity is (just barely) sufficient for a relativistic version of Hamilton-Jacobi theory to be well-defined; he thus proposed a new relativistic Hamilton-Jacobi ``guiding law'' of motion for the point charges (see \\cite{mK2004a} for the details).\n\nKiessling's interest in the Maxwell-Born-Infeld system was further motivated by results contained in \\cite{gB1969} and \\cite{jP1970}, which show that it is the unique\\footnote{More precisely, there is a one-parameter family of such theories indexed by $\\upbeta > 0.$} theory of classical electromagnetism that is derivable from an action principle and that satisfies the following $5$ postulates (see also the discussion in \\cite{iBB1983}, \\cite{mK2004a}):\n\n\\begin{enumerate}\n\t\\item The field equations transform covariantly under the Poincar\\'e group.\n\t\\item The field equations are covariant under a Weyl (gauge) group.\n\t\\item The electromagnetic energy surrounding a stationary point charge is finite.\n\t\\item The field equations reduce to the linear Maxwell-Maxwell equations in the weak field limit.\n\t\\item The solutions to the field equations are not birefringent.\n\\end{enumerate}\nWe remark that the linear Maxwell-Maxwell system satisfies all of the above postulates except for (iii), and that\nthe MBI system was shown to satisfy (iii) by Born in \\cite{mB1933}. Physically, postulate (v) is equivalent to the statement that the ``speed of light propagation'' is independent of the polarization of the wave fields. Mathematically, this is the postulate that there is only a single \\emph{null cone}\\footnote{In general this ``light cone'' does not have to coincide with the gravitational null cone, although it \\emph{does} in the case of the linear Maxwell-Maxwell equations.} associated to the electromagnetic equations; in a typical theory of classical electromagnetism, the causal structure of the electromagnetic equations is more complicated than the structure corresponding to a single null cone (see \\cite{jS2010a} for a detailed discussion of this issue in the context of the Maxwell-Born-Infeld equations on the Minkowski spacetime background).\n\nIt is here that we can mention the connection of the present article to Kiessling's work. First, as noted in \\cite{mK2004a}, \nKiessling expects that his theory can be generalized to the case of a curved spacetime through a coupling to the Einstein equations. Next, we mention that although the Maxwell-Born-Infeld system is Kiessling's leading candidate for an electromagnetic model, he is also considering other models. In particular, by relaxing postulate (v) above, a relaxation that in principle could be supported by experimental evidence, one is led to consider a larger family of electromagnetic models. Now one basic criterion for any viable electromagnetic model is that small, nearly linear-Maxwellian electromagnetic fields in near-Minkowski vacuums should not lead to a severe breakdown in the structure of spacetime or other degenerate behavior. The present work confirms this criterion for a large family of electromagnetic models coupled to the Einstein equations, including the Maxwell-Born-Infeld system and many other models that fall under the scope of Kiessling's program.\n\n\n\\subsection{Discussion of the analysis}\n\\subsubsection{The splitting of the spacetime metric and setting up the equations} \\label{SSS:Settinguptheequations}\n\nAs in the works \\cite{hLiR2005} and \\cite{hLiR2010}, in order to analyze the spacetime metric,\nwe split it into the following three pieces:\n\n\\begin{subequations}\n\\begin{align} \n\tg_{\\mu \\nu} & = m_{\\mu \\nu} + h_{\\mu \\nu}, && (\\mu, \\nu = 0,1,2,3), \\label{E:gmhexpansion} \\\\\n\th_{\\mu \\nu} & = h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)}, && (\\mu, \\nu = 0,1,2,3), \\label{E:hdefIntro} \\\\\n\th_{\\mu \\nu}^{(0)} & \\eqdef \\chi\\big(\\frac{r}{t}\\big)\\chi(r)\\frac{2M}{r} \\delta_{\\mu \\nu}, \\ \\Big(h_{\\mu \\nu}^{(0)}|_{t = 0} = \t\n\t\\chi(r)\\frac{2M}{r} \\delta_{\\mu \\nu}, \\ \\partial_t h_{\\mu \\nu}^{(0)}|_{t = 0} = 0 \\Big), && (\\mu, \\nu = 0,1,2,3), \t\n\t\t\\label{E:h0defIntro}\n\\end{align} \n\\end{subequations}\nwhere $m_{\\mu \\nu} = \\mbox{diag}(-1,1,1,1)$ is the Minkowski metric, and the function\n$\\chi$ plays several roles that will be discussed in Section \\ref{SSS:h0}. Above and throughout,\n$\\chi(z)$ is a fixed cut-off function that satisfies\n\n\\begin{align}\n\t\\chi \\in C^{\\infty}, \\qquad \\chi \\equiv 1 \\ \\mbox{for} \\ z \\geq 3\/4, \\qquad \\chi \\equiv 0 \\ \\mbox{for} \\ z \\leq 1\/2.\n\\end{align}\n\\textbf{We remark that here and throughout the rest of the article, unless we explicitly indicate otherwise \n(which, as is explained in Section \\ref{SS:Indices}, we sometimes do with the use of the symbol $\\#$), all indices on all tensors are lowered and raised with the Minkowski metric $m_{\\mu \\nu} = \\mbox{diag}(-1,1,1,1)$ and its inverse $(m^{-1})^{\\mu \\nu} = \\mbox{diag}(-1,1,1,1)$.} Furthermore, as in \\cite{hLiR2005} and \\cite{hLiR2010}, we work in a wave coordinate system, which is a coordinate system in which the contracted Christoffel symbols \n$\\Gamma^{\\mu} \\eqdef (g^{-1})^{\\kappa \\lambda} \\Gamma_{\\kappa \\ \\lambda}^{\\ \\mu}$ (see \\eqref{E:EMBIChristoffeldef}) of the metric $g_{\\mu \\nu}$ satisfy\n\n\\begin{align} \\label{E:Wavecoordinateintro}\n\t\\Gamma^{\\mu} & = 0,&& (\\mu = 0,1,2,3).\n\\end{align}\nWe remark that several equivalent definitions of the wave coordinate condition \\eqref{E:Wavecoordinateintro} are discussed in Section \\ref{SS:WaveCoordinates}, and that the viability of the wave coordinate gauge for the system \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0} (which is a rather standard result based on the ideas of \\cite{CB1952}) is discussed in Section \\ref{SS:WaveCoordinatesPreserved}.\n\nAs is discussed in detail in Section \\ref{SS:ReducedEquations}, in a wave coordinate system $(t,x),$ the equations \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0} are equivalent to the \\emph{reduced equations} \n\n\\begin{subequations}\n\\begin{align}\n\t\\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)} & = \\mathfrak{H}_{\\mu \\nu} - \\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(0)},&& \n\t\t(\\mu, \\nu = 0,1,2,3), \\label{E:Reducedh1Intro} \\\\\n\t\\nabla_{\\lambda} \\Far_{\\mu \\nu} + \\nabla_{\\mu} \\Far_{\\nu \\lambda} + \\nabla_{\\nu} \\Far_{\\lambda \\mu} & = 0,&&\n\t\t(\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:ReduceddFis0Intro} \\\\\n\tN^{\\#\\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda} & = \\mathfrak{F}^{\\nu},&& (\\nu = 0,1,2,3),\n\t\t\\label{E:ReduceddMis0Intro} \n\\end{align}\n\\end{subequations}\nwhere $\\widetilde{\\Square}_{g} = (g^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}$ is the reduced wave operator corresponding \nto $g_{\\mu \\nu},$ $\\nabla$ is the Levi-Civita connection corresponding to the \\emph{Minkowski metric} $m_{\\mu \\nu},$\n$N^{\\#\\mu \\nu \\kappa \\lambda} \\eqdef \\frac{1}{2} \\big\\lbrace (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa} - h^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} + h^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa} - (m^{-1})^{\\mu \\kappa} h^{\\nu \\lambda} + (m^{-1})^{\\mu \\lambda} h^{\\nu \\kappa} \\big\\rbrace + N_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda},$ $N_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda} = O^{\\dParameter}\\big(|(h,\\Far)|^2\\big)$ is a quadratic error term that depends on the chosen model of nonlinear electromagnetism, and $\\mathfrak{H}_{\\mu \\nu},$ $\\mathfrak{F}^{\\nu}$ are inhomogeneous terms that depend in part on on the chosen model of nonlinear electromagnetism.\n \n\nThe question of the stability of the Minkowski spacetime solution to \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0}\nhas thus been reduced to two subquestions: i) show that the reduced system \\eqref{E:Reducedh1Intro} - \\eqref{E:ReduceddMis0Intro}, where the unknowns are viewed to be $(h_{\\mu \\nu}^{(1)},\\Far_{\\mu \\nu}),$ has small-data global existence (if the ADM mass $M$ is sufficiently small); ii) show that the resulting spacetime $(\\mathbb{R}^{1+3}, g_{\\mu \\nu} = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)})$ is geodesically complete. The second question is very much related to the first, for as in \\cite[Section 16]{hLiR2005}, \\cite[Section 9]{jL2008}, \nthe question of geodesic completeness can be answered if one has sufficiently detailed information about the asymptotic behavior of $h_{\\mu \\nu}^{(1)};$ our stability theorem (see Section \\ref{S:GlobalExistence}) provides sufficient information.\n\n\\subsubsection{The smallness condition}\nOur smallness condition the abstract initial data is stated in terms of the ADM mass $M$ and a weighted Sobolev norm\nof the abstract initial data $\\underline{\\nabla}_i \\mathring{\\underline{h}}^{(1)}_{jk},$ $\\mathring{K}_{jk},$\n$\\mathring{\\mathfrak{\\Displacement}}_j,$ and $\\mathring{\\mathfrak{\\Magneticinduction}}_k.$ More specifically, in order to deduce global existence, we will require that\n\n\\begin{align}\n\tE_{\\dParameter;\\upgamma}(0) + M < \\varepsilon_{\\dParameter},\n\\end{align}\t\nwhere $\\varepsilon_{\\dParameter} > 0$ is a sufficiently small positive number, $E_{\\dParameter;\\upgamma}(0) \\geq 0$ is defined by\n\n\\begin{align}   \\label{E:DataNormIntro}\n\tE_{\\dParameter;\\upgamma}^2(0) \n\t& \\eqdef \\| \\underline{\\nabla} \\mathring{\\underline{h}}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{K} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{\\mathfrak{\\Displacement}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{\\mathfrak{\\Magneticinduction}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2,\n\\end{align}\nthe weighted Sobolev norm $\\| \\cdot \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}$ is defined in \nDefinition \\ref{D:HNdeltanorm} below, $0 < \\upgamma < 1\/2$ is a constant, and $\\dParameter \\geq 8$ is an integer. The condition $\\dParameter \\geq 8$ is needed for various weighted Sobolev embedding results, including the weighted Klainerman-Sobolev inequality \\eqref{E:KSIntro}, and the results stated in Appendix \\ref{A:SobolevMoser}. In the above expressions, $\\underline{\\nabla}$ is the Levi-Civita connection corresponding to the Euclidean metric\\footnote{Throughout the article, we use the symbol \n$\\um$ to denote both the Euclidean metric $\\um_{jk} \\eqdef \\mbox{diag}(1,1,1)$ on $\\mathbb{R}^3,$ and the \nfirst fundamental form $\\um_{\\mu \\nu} \\eqdef \\mbox{diag}(0,1,1,1)$ of the constant time hypersurfaces\n$\\Sigma_t$ viewed as embedded hypersurfaces of Minkowski spacetime; this double-use of notation should not cause any confusion.} $\\um_{jk} \\eqdef \\mbox{diag}(1,1,1).$ \nNote that the assumed fall-off conditions \\eqref{E:h1AbstractDataAsymptotics} - \\eqref{E:BdecayAssumption} \nguarantee the existence of a constant $0 < \\upgamma < 1\/2$ such that $E_{\\dParameter;\\upgamma}(0) < \\infty.$\n\nAlthough the norm \\eqref{E:DataNormIntro} is useful for expressing the small-data global existence condition\nin terms of quantities inherent to the data, from the perspective of analysis, a more useful quantity is\nthe energy ${\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\geq 0},$ which is defined by\n\n\\begin{align} \\label{E:EnergyIntro}\n\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}^2(t) & \\eqdef \\underset{0 \\leq \\tau \\leq t}{\\mbox{sup}} \n\t\t\\sum_{|I| \\leq \\dParameter } \\int_{\\Sigma_{\\tau}} \n\t\t\\Big\\lbrace |\\nabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|^2 \\Big\\rbrace w(q) \\, d^3 x,\n\\end{align}\nwhere $\\nabla$ denotes the Levi-Civita connection corresponding to the \\emph{full Minkowski spacetime metric}, \n$q \\eqdef |x| - t$ is a null coordinate, the weight function $w(q)$ is defined by\n\n\\begin{align} \\label{E:weightintro}\n\tw = w(q) = \\left \\lbrace\n\t\t\\begin{array}{lr}\n    \t1 \\ + \\ (1 + |q|)^{1 + 2 \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      1 \\ + \\ (1 + |q|)^{-2 \\upmu}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \\right.,\n\\end{align}\n$\\upgamma$ is from \\eqref{E:DataNormIntro}, and $0 < \\upmu < 1\/2$ is a fixed constant. In the above expression, \n$\\mathcal{Z} \\eqdef \\big\\lbrace \\partial_{\\mu}, x_{\\mu} \\partial_{\\nu} - x_{\\nu} \\partial_{\\mu}, \nx^{\\kappa} \\partial_{\\kappa} \\big\\rbrace_{0 \\leq \\mu < \\nu \\leq 3}$ is a subset of the conformal Killing fields of\nMinkowski space, $I$ is a vectorfield multi-index, $\\nabla_{\\mathcal{Z}}^I$ represents iterated Minkowski covariant differentiation with respect to vectorfields in $\\mathcal{Z},$ and $\\Lie_{\\mathcal{Z}}^I$ represents iterated Lie differentiation with respect to vectorfields in $\\mathcal{Z}.$ The significance of the set $\\mathcal{Z}$ is that it is needed\nfor the weighted Klainerman-Sobolev inequality \\eqref{E:KSIntro}, which is discussed below.\n\n\\begin{remark}\\label{R:Roleofmu}\n\tThe presence of the parameter $\\upmu > 0$ in \\eqref{E:weightintro} might seem unnecessary, since \n\t$1 \\ + \\ (1 + |q|)^{-2 \\upmu} \\approx 1.$ However, as is explained in Section \\ref{SSS:EnergyandStress}, \n\tthe presence of $\\upmu > 0$ ensures that $w'(q) > 0,$ a fact that plays a key role in our energy estimates.\n\\end{remark}\n\n\\subsubsection{Overall strategy of the proof} \\label{SS:DiscussionofProof}\n\nThe overall strategy is to deduce a hierarchy of Gronwall-amenable inequalities for the energies $\\mathcal{E}_{k;\\upgamma;\\upmu}(t),$ $(0 \\leq k \\leq \\dParameter );$ this is accomplished in \\eqref{E:Mainenergyinequalityreexpressed} below. The net effect is that under the assumption $E_{\\dParameter;\\upgamma}(0) + M \\leq \\varepsilon,$ we are able to deduce the following a-priori estimate for the solution, which is valid during its classical lifetime:\n\n\\begin{align} \\label{E:EnergyaprioriIntro}\n\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\leq c_{\\dParameter} \\varepsilon (1 + t)^{\\widetilde{c}_{\\dParameter} \\varepsilon}.\n\\end{align}\nIn the above inequality, $c_{\\dParameter}$ and $\\widetilde{c}_{\\dParameter}$ are positive constants. Now it is a standard result in the theory\nof hyperbolic PDEs that if $\\varepsilon$ is sufficiently small, then an a-priori estimate of the form \\eqref{E:EnergyaprioriIntro} implies that the solution exists for $(t,x) \\in (-\\infty, \\infty) \\times \\mathbb{R}^3;$ see Proposition \\ref{P:LocalExistence} for more details. Furthermore, as shown in \\cite{hLiR2005} and \\cite{jL2008}, if $\\varepsilon$ is sufficiently small, then it also follows that the spacetime $(\\mathbb{R}^{1+3}, g_{\\mu \\nu} = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)})$ is geodesically complete. \\textbf{The main goal of this article is therefore to derive \\eqref{E:EnergyaprioriIntro}}.\n\n\n\\subsubsection{Geometry and null decompositions} \\label{SSS:GeometryandNullDecompositions}\n\nLet us now describe the tools used to derive \\eqref{E:EnergyaprioriIntro}. First and foremost, as mentioned above in Section \\ref{SSS:MathematicalComparisons}, the reason we are able to prove our stability result is that the reduced equations \\eqref{E:Reducedh1Intro} - \\eqref{E:ReduceddMis0Intro} have special algebraic structure, and satisfy (in the language of Lindblad and Rodnianski) the \\emph{weak null condition}. Now in order to see the special structure of the terms in the reduced equations, we follow the strategy of Lindblad and Rodnianski and decompose them into their \\emph{Minkowskian null components}; we refer to this as a \\emph{Minkowskian null decomposition}. We emphasize the following point: \\textbf{the Minkowskian geometry is not the ``correct'' geometry to use for analyzing the equations, for the actual characteristics of the system \ncorrespond to the null cones of the spacetime metric $g_{\\mu \\nu}$ and the characteristics of the nonlinear electromagnetic equations (which in general do not have to coincide with the gravitational null cones). However, the errors that we make in using the Minkowskian geometry (which has the advantage of being simple) for our analysis are controllable.} Let us briefly recall the meaning of a Minkowskian null decomposition; a more detailed description is offered in Section \\ref{S:NullFrame}. The notion of a Minkowskian null decomposition is intimately connected to the following spacetime subsets: the \\emph{outgoing Minkowskian null cones} $C_{q}^+ \\eqdef \\lbrace (\\tau,y) \\ | \\ |y| - \\tau = q \\rbrace,$ the \\emph{ingoing Minkowskian null cones} $C_{s}^- \\eqdef \\lbrace (\\tau,y) \\ | \\ |y| + \\tau = s \\rbrace,$ the \\emph{constant time slices} $\\Sigma_t \\eqdef \\lbrace (\\tau,y) \\ | \\ \\tau = t \\rbrace,$ and the \\emph{Euclidean spheres} $S_{r,t} \\eqdef \\lbrace (\\tau,y) \\ | \\ t = \\tau, |y| = r \\rbrace.$ Observe that the \\emph{null coordinate} $q \\eqdef |x| - t$ associated to the spacetime point with coordinates $(t,x)$ is constant on the outgoing cones, and the null coordinate $s \\eqdef |x| + t$ is constant on the ingoing cones. These coordinates will be used throughout the article to discuss the rates of decay of various quantities. With $\\omega^j \\eqdef x^j\/r,$ $(j=1,2,3),$ we also define the \\emph{ingoing Minkowskian null geodesic vectorfield} $\\uL^{\\mu} \\eqdef (1,-\\omega^1,-\\omega^2,-\\omega^3),$ which satisfies $m_{\\kappa \\lambda}\\uL^{\\kappa} \\uL^{\\lambda} = 0$ and is tangent to the $C_{s}^-,$ and the \\emph{outgoing Minkowskian null geodesic vectorfield} $L^{\\mu} \\eqdef (1,\\omega^1,\\omega^2,\\omega^3),$ which satisfies $m_{\\kappa \\lambda} L^{\\kappa} L^{\\lambda} = 0,$ $m_{\\kappa \\lambda} \\uL^{\\kappa} L^{\\lambda} = -2$ and is tangent to the $C_{q}^+.$ Furthermore, in a neighborhood of each non-zero spacetime point $p,$ there exists a locally defined pair of $m-$orthonormal vectorfields $e_1, e_2$ that are tangent to the family of Euclidean spheres, and $m-$orthogonal to $\\uL$ and $L.$ The set $\\mathcal{N} \\eqdef \\lbrace \\uL, L, e_1, e_2 \\rbrace,$ which spans the tangent space, is known as a \\emph{Minkowskian null frame}. In the discussion that follows, we will also make use of the set $\\mathcal{T} \\eqdef \\lbrace L, e_1, e_2 \\rbrace,$ which is the subset consisting of only those frame vectors tangent to the $C_{q}^+,$ and the set $\\mathcal{L} \\eqdef \\lbrace L \\rbrace.$\n\nGiven any two form $\\Far,$ we can decompose it into its Minkowskian null components $\\ualpha[\\Far],$ $\\alpha[\\Far],$\n$\\rho[\\Far],$ and $\\sigma[\\Far],$ where $\\ualpha,$ $\\alpha$ are two-forms $m-$tangent\\footnote{By $m-$tangent, we mean that their vector duals relative to the Minkowski metric are tangent to the $S_{r,t}.$} to the spheres $S_{r,t}$, and $\\rho,$ $\\sigma$ are scalars. More specifically, we define $\\ualpha_A = \\Far_{A \\uL},$ $\\alpha_A = \\Far_{AL},$\n$\\rho = \\frac{1}{2} \\Far_{\\uL L},$ and $\\sigma = \\Far_{12},$ where $A \\in \\lbrace 1,2 \\rbrace,$ and we have abbreviated\n$\\Far_{A \\uL} \\eqdef e_{A}^{\\kappa} \\uL^{\\lambda} \\Far_{\\kappa \\lambda},$ etc. Similarly, we can decompose the tensor\n$h_{\\mu \\nu}$ into its null components $h_{LL},$ $h_{\\uL L},$ $h_{LT},$ etc., where $T$ stands for any of the vectors in\n$\\mathcal{T}.$ We are now ready to discuss one of the major themes running throughout this article: the rates of decay of the various null components of $\\Far$ and $h$ are distinguished by the kinds of contractions taken against the null frame vectors. In particular, contractions against $L,e_1,e_2$ are associated with favorable decay, with $L$ being the most favorable, while contractions against $\\uL$ are associated with unfavorable decay. Similarly, differentiation in the directions $L,e_1,e_2$ are associated with creating \\emph{additional favorable decay} in the null coordinate $s,$ while differentiation in the direction $\\uL$ is associated with creating less favorable additional decay in $q$  (see Lemma \\ref{L:PointwisetandqWeightedNablainTermsofZestiamtes} for a precise version of this claim).\nEquivalently, the operator $\\conenabla$ creates favorable decay in $s,$ while $\\nabla$ only creates decay in $q.$ Here and throughout, $\\conenabla$ is the projection (of the derivative component only) of the Minkowski connection $\\nabla$ onto the outgoing Minkowski null cones. From this point of view, the most dangerous terms in the equations are $\\ualpha$ and $h_{\\uL \\uL},$ and the $\\partial_q \\sim \\nabla_{\\uL}$ derivatives (see Section \\ref{SS:Derivatives}) of these quantities. We recommend that at this point, the reader examine the conclusions of Propositions \\ref{P:UpgradedDecayhA} and \\ref{P:UpgradedDecayh1A} to get a feel for the kind of decay properties possessed by the various null components.\n\nThe main idea behind the Minkowskian null decomposition is that it can be used to show the following fact: \\emph{the worst possible combinations of terms, from the point of view of decay rates, are not present in the reduced equations} \\eqref{E:Reducedh1Intro} - \\eqref{E:ReduceddMis0Intro}. This special algebraic structure, which is of central importance in our small-data global existence proof, is examined in detail in Propositions \\ref{P:AlgebraicInhomogeneous} - \\ref{P:harmonicgauge} of Section \\ref{S:AlgebraicEstimates}. We remark that as revealed in \\cite{hLiR2003}, \\cite{hLiR2005}, and \\cite{hLiR2010}, this special algebraic structure is tensorial in nature.\n\n\\subsubsection{Energy inequalities and the canonical stress} \\label{SSS:EnergyandStress}\n\nThe first major analytical step in deriving the all-important Gronwall-amenable estimate \\eqref{E:Mainenergyinequalityreexpressed} is to deduce the energy inequalities of Lemma \\ref{L:weightedenergyFar} and Lemma \\ref{L:weightedenergy}, which respectively provide $L^2$ estimates for solutions to the electromagnetic \\emph{equations of variation} (which are the linearized equations satisfied by the derivatives of solutions $\\Far$ to \\eqref{E:ReduceddFis0Intro} - \\eqref{E:ReduceddMis0Intro}), and $L^2$ estimates for solutions to quasilinear wave equations whose principal operator agrees with that of \\eqref{E:Reducedh1Intro} (i.e., $\\widetilde{\\Square}_g$). As is explained below, such equations come into play because we require $L^2$ estimates for derivatives of $h^{(1)}$ and $\\Far$ in order to close our global existence argument. We will comment mainly on the estimates for the electromagnetic equations of variation, since the estimates of Lemma \\ref{L:weightedenergy} are perhaps more familiar to the reader, and in any case are explained in detail in \\cite[Lemma 6.1 and Proposition 6.2]{hLiR2010}. Our proof of Lemma \\ref{L:weightedenergyFar} is based on the construction of a suitable \\emph{energy current} $\\dot{J}^{\\mu} \\eqdef - \\Stress_{\\ \\nu}^{\\mu}X^{\\nu},$ where $\\Stress_{\\ \\nu}^{\\mu}$ is the \\emph{canonical stress}, which is a tensorfield that depends quadratically on the linearized variables $\\dot{\\Far}_{\\mu \\nu},$ $X^{\\nu} \\eqdef w(q) \\delta_0^{\\nu},$ $(\\nu=0,1,2,3)$ is a ``multiplier vectorfield,'' and $w(q)$ is the weight function defined in \\eqref{E:weightintro}. The end result is provided by inequality \\eqref{E:FirstweightedenergyFar} below. Although at first glance inequality \\eqref{E:FirstweightedenergyFar} below may appear to be a standard energy inequality, one of the most important features of this particular energy current is that it provides the \\emph{additional positive} spacetime integral $\\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \\big(\\dot{\\alpha}^2 + \\dot{\\rho}^2 + \\dot{\\sigma}^2 \\big) w'(q) \\,d^3x \\, d \\tau$ on the left-hand side of \\eqref{E:FirstweightedenergyFar}; here, $\\dot{\\alpha},$ $\\dot{\\rho},$ and $\\dot{\\sigma}$ are the ``favorable'' null components of the two-form $\\dot{\\Far}.$ This additional positive quantity, which is analogous to the quantity $\\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} |\\conenabla \\phi|^2 w'(q) \\,d^3x \\, d \\tau$ on the left-hand side of \\eqref{E:Firstweightedenergyscalar} that was exploited by Lindblad and Rodnianski, is one of the key advantages afforded by our use of a weight function of the form \\eqref{E:weightintro}. Its availability is directly related to the fact that we have better integrated control over the quadratic terms $\\dot{\\alpha}^2 + \\dot{\\rho}^2 + \\dot{\\sigma}^2$ than we do over the term $\\dot{\\ualpha}^2.$ The quantity plays a key role in the derivation of the energy inequality \\eqref{E:Mainenergyinequalityreexpressed}.\n\nLet us now make a few comments concerning the canonical stress and the construction of the above energy current. A very detailed description is located in \\cite{dC2000} and \\cite{jS2010a}, so we confine ourselves here to its two most salient features. The canonical stress (see \\eqref{E:Stressdef}) plays the role of an energy-momentum-type tensor for the electromagnetic equations of variation. Because these linearized equations depend on the ``background'' $\\Far_{\\mu \\nu}$ in addition to the linearized variables $\\dot{\\Far}_{\\mu \\nu},$ it is \\emph{not} the case that $\\mathscr{D}_{\\mu} \\Stress_{\\ \\nu}^{\\mu} = 0;$ this is in contrast to the property  $(g^{-1})^{\\kappa \\lambda}\\mathscr{D}_{\\kappa} T_{\\lambda \\nu} = 0$ (see \\eqref{E:TisDivergenceFree}) enjoyed by the energy-momentum tensor. However, we now point out the first key property of the canonical stress: $\\nabla_{\\mu} \\Stress_{\\ \\nu}^{\\mu}$ is lower-order in the sense that it does not depend on $\\nabla_{\\lambda} \\dot{\\Far}_{\\mu \\nu};$ by using the equations of variation for substitution, the $\\nabla_{\\lambda} \\dot{\\Far}_{\\mu \\nu}$ terms can be replaced with inhomogeneous terms (see \\ref{E:divergenceofStress}). It is already important to appreciate the availability of this non-trivial quadratic quantity whose divergence can be controlled by the background and inhomogeneous terms. For the availability of such a quantity is not a feature inherent to all systems of equations\\footnote{It is however a feature inherent to all scalar quasilinear wave equations.}, but is instead related to the symmetry properties of the indices of the principal terms (i.e., the terms on the left-hand side) in equations \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0}, which themselves are related to the fact the original nonlinear equations are derivable from a Lagrangian. \n\nThe second key property enjoyed by the canonical stress is that of positivity upon contraction against certain covector\/vector pairs $(\\xi, X).$ That is, for certain choices of $(\\xi, X),$ the quantity $\\Stress_{\\ \\nu}^{\\mu}\\xi_{\\mu}X^{\\nu}$ is a positive definite quadratic form in $\\dot{\\Far}.$ \nThese properties are analogous to (but distinct from) the positivity properties of an energy-momentum tensor satisfying the dominant energy condition, and the positivity properties of the Bel-Robinson tensor (which played a central role in \\cite{dCsK1993}). As is explained in \\cite{dC2000} and \\cite{jS2010a}, the set of pairs leading to integrated positivity is intimately connected to the \\emph{hyperbolicity of and the geometry of the electromagnetic equations}, and to the speeds and directions of propagation in the system. In this article, the only pair $(\\xi, X)$ that we make use of is $\\xi_{\\mu} = - \\delta_{\\mu}^0,$ and $X^{\\nu} = w(q) \\delta_0^{\\nu}.$ The special positivity properties stemming from this choice of $(\\xi, X)$ are derived in Lemma \\ref{L:weightedenergyFar}.\n\n\n\\subsubsection{Weighted Klainerman-Sobolev inequalities}\n\nBased on the energy inequalities of Proposition \\ref{P:weightedenergy}, which are relatively straightforward consequences of\nLemmas \\ref{L:weightedenergyFar} and Lemmas \\ref{L:weightedenergy}, it is clear that most of the hard work in deriving the estimate \\eqref{E:Mainenergyinequalityreexpressed} goes into estimating the integrals of the inhomogeneous terms on the\nright-hand sides of \\eqref{E:Secondweightedenergyscalar} and \\eqref{E:SecondweightedenergyFar}. In particular, we attempt to summarize here the origin of the factors $(1 + \\tau)^{-1}$ and $(1 + \\tau)^{-1 + C \\varepsilon}$ that appear in \n\\eqref{E:Mainenergyinequalityreexpressed}, and that are of central importance in our derivation of the fundamental energy inequality \\eqref{E:EnergyaprioriIntro}. Roughly speaking, these factors arise from a variety of pointwise decay estimates that we will soon explain. The first tools of interest to us along these lines are the weighted Klainerman-Sobolev inequalities, which allow us to deduce pointwise decay estimates for functions $\\phi \\in C_0^{\\infty}(\\mathbb{R}^3),$ in terms of weighted $L^2$ estimates for $\\phi$ and its Minkowskian covariant derivatives with respect to vectorfields $Z \\in \\mathcal{Z}.$ More specifically (see also Appendix \\ref{A:WeightedKS}), the weighted Klainerman-Sobolev inequalities state that \n\n\\begin{align} \\label{E:KSIntro}\n\t(1 + t + |x|)[(1 + |q|) w(q)]^{1\/2} |\\phi(t,x)| & \\leq C \\sum_{|I| \\leq 2} \n\t\\big\\| w^{1\/2}(q) \\nabla_{\\mathcal{Z}}^I \\phi(t, \\cdot) \\big \\|_{L^2}, && q \\eqdef |x| - t.\n\\end{align}\nWe refer to these estimates as ``weak pointwise decay estimates,'' since they have nothing to do\nwith the special structure of the Einstein-nonlinear electromagnetic equations; a major theme permeating this article is that\nin order to close our global existence bootstrap argument, the estimate \\eqref{E:KSIntro} need to be upgraded using the special structure of the equations. Inequality \\eqref{E:KSIntro} can therefore be viewed as a preliminary estimate that will play a role in the proof of the upgraded estimates.\n\nThe form of the inequalities \\eqref{E:KSIntro} raises several important issues. First, in order to apply the \nweighted Klainerman-Sobolev inequalities to $h^{(1)},$ we have to achieve $L^2$ control over the quantities $w^{1\/2}(q) \\nabla_{\\mathcal{Z}}^I h^{(1)}.$ To this end, we have to study the equations satisfied by the quantities $\\nabla_{\\mathcal{Z}}^I h^{(1)}.$ In order to derive these equations, we have to commute the operator $\\nabla_{\\mathcal{Z}}^I$ through the reduced wave operator term $\\widetilde{\\Square}_{g} h^{(1)}.$ Lindblad and Rodnianski accomplished this commutation through the use of \\emph{modified covariant derivatives} $\\hat{\\nabla}_Z,$ which are equal to ordinary covariant derivatives plus a scalar multiple (depending on $Z \\in \\mathcal{Z}$) of the identity; see Definition \\ref{D:ModifiedDerivatives}. The main advantage of these operators is that $\\hat{\\nabla}_Z \\Square_{m} - \\Square_{m} \\nabla_Z = 0,$ where $\\Square_{m} \\eqdef (m^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}$ denotes the wave operator of the Minkowski metric; see Lemma \\ref{L:NablaModZLiemodMinkowskiWaveOperatorCommutator}. Therefore, $\\nabla_{\\mathcal{Z}}^I h^{(1)}$ is a solution to the equation $\\widetilde{\\Square}_{g} \\nabla_{\\mathcal{Z}}^I h^{(1)}$ $= \\hat{\\nabla}_{\\mathcal{Z}}^I \\widetilde{\\Square}_{g} h^{(1)}$ $+ H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda} \\nabla_{\\mathcal{Z}}^I h^{(1)}$ $- \\hat{\\nabla}_{\\mathcal{Z}}^I \\big(H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda} h^{(1)}\\big),$ where $\\widetilde{\\Square}_{g} h^{(1)}$ is equal to the inhomogeneous term on the right-hand side of \\eqref{E:Reducedh1Intro} above, and $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu} = - h^{\\mu \\nu} + O(|h|^2).$ We remark that the analysis of the commutator term $H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda} \\nabla_{\\mathcal{Z}}^I h^{(1)}$ \n$- \\hat{\\nabla}_{\\mathcal{Z}}^I \\big(H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda} h^{(1)}\\big),$ which was performed in \\cite{hLiR2010} (see also Propositions \\ref{P:InhomogeneousTermsNablaZIh1} and Lemmas \\ref{L:NablaZIBoxCommutatorIntegrated}), is among the most challenging work encountered. Rather than repeat this analysis and the discussion behind it, which is throughly explained and carried out in \\cite{hLiR2010}, we will instead focus on the analogous difficulties that arise in our analysis of $\\Far.$ We do, however point out the role that Hardy inequalities of Proposition \\ref{P:Hardy} play in the analysis of $h^{(1)}:$ they are used to estimate a weighted $L^2$ norm of $\\nabla_{\\mathcal{Z}}^I h^{(1)},$ which is \\emph{not} directly controlled in $L^2$ by the energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t),$ by a weighted $L^2$ norm of $\\nabla \\nabla_{\\mathcal{Z}}^I h^{(1)},$ which \\emph{is} controlled in $L^2$ by the energy. The cost of applying this inequality is powers of $1 + |q|,$ which are always sufficiently available thanks to our use of the weight $w(q).$\n\n\n\\subsubsection{The role of Lie derivatives}\n\nThe next important issue concerning inequality \\eqref{E:KSIntro} is that it is more convenient to work with\nLie derivatives of $\\Far$ rather than covariant derivatives of $\\Far;$ this claim has already been suggested by the\ndefinition \\eqref{E:EnergyIntro} of our energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t).$ According to inequality \\eqref{E:LieZIinTermsofNablaZI} below, inequality \\eqref{E:KSIntro} remains valid if we replace the operators $\\nabla_{\\mathcal{Z}}^I$ with $\\Lie_{\\mathcal{Z}}^I.$\nHowever, as in the case of covariant derivatives, we have to study the equations satisfied by the $\\Lie_{\\mathcal{Z}}^I \\Far.$\nNow on the one hand, Lemma \\ref{L:Liecommuteswithcoordinatederivatives} shows that the operator $\\Lie_Z$ can be commuted through the Minkowski connection $\\nabla$ in equation \\eqref{E:ReduceddFis0Intro}. On the other hand, to commute Lie derivatives through equation \\eqref{E:ReduceddMis0Intro}, it is convenient to work with \\emph{modified Lie derivatives} $\\Liemod_Z,$ which are equal to ordinary Lie derivatives plus a scalar multiple\\footnote{The multiple is $2 c_Z,$ where $c_Z$ is the multiple corresponding to the modified covariant derivative $\\hat{\\nabla}_Z.$} (depending on $Z \\in \\mathcal{Z}$) of the identity; see Definition \\ref{D:ModifiedDerivatives}. Unlike covariant derivatives, these operators have favorable commutation properties with the linear Maxwell term $\\nabla_{\\mu} \\Far^{\\mu \\nu},$ which is the leading term in \\eqref{E:ReduceddMis0Intro}.\nMore specifically, $\\Liemod_{Z} \\Big\\lbrace \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \\Big\\rbrace = \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Lie_Z \\Far_{\\kappa \\lambda};$ see Lemma \\ref{L:LiemodZLiemodMaxwellCommutator}. As is captured by Proposition \\ref{P:InhomogeneoustermsLieZIFar}, these operators are also useful for differentiating equation \\eqref{E:ReduceddMis0Intro}; the error terms generated have a favorable null structure that is captured in Proposition \\ref{P:EnergyInhomogeneousTermAlgebraicEstimate}.\n\n\n\\subsubsection{The tensorfield $h_{\\mu \\nu}^{(0)}$} \\label{SSS:h0}\n\nLet us now discuss the ideas behind the Lindblad-Rodnianski splitting of the metric defined in \n\\eqref{E:gmhexpansion} - \\eqref{E:h0defIntro}. We first note that because of \nthe $2M\/r$ ADM mass term present in $h_{\\mu \\nu}^{(0)},$ substituting the tensorfield \n$h_{\\mu \\nu} \\eqdef h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)}$ in place of $h_{\\mu \\nu}^{(1)}$ in the definition of the energy would lead to $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0) = \\infty.$ Thus, as a practical matter, the introduction of $h_{\\mu \\nu}^{(1)}$ allows us to work with a quantity of finite energy. Now according to the discussion in \\cite{hLiR2010}, the precise form $h_{\\mu \\nu}^{(0)} = \\chi\\big(\\frac{r}{t}\\big)\\chi(r)\\frac{2M}{r} \\delta_{\\mu \\nu}$ was determined by making an ``educated'' guess concerning the contribution of the ADM mass term $(2M\/r) \\delta_{\\mu \\nu}$ to the solution. The term $h_{\\mu \\nu}^{(0)}$ manifests itself in the reduced equations as the $\\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(0)}$ inhomogeneous term on the right-hand side of the reduced equation \\eqref{E:Reducedh1Summary}. Because of the identity $\\Square_m (1\/r ) = 0$ for $r > 0,$ where $\\Square_m = (m^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa}\\nabla_{\\lambda}$ is the Minkowski wave operator, it follows that the main contribution of the term $\\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(0)}$ comes from the ``interior'' region $\\lbrace (t,x) \\mid 1\/2 < r\/t < 3\/4 \\rbrace;$ this is because the derivatives of $\\chi(z)$ are supported in the interval $[1\/2, 3\/4].$ Now in the interior region, the quantities $1 + |q|$ and $1 + s$ are uniformly comparable. Thus, the weighted Klainerman-Sobolev inequality \\eqref{E:KSIntro} predicts strong decay for the solution in this region, and consequently, one can derive suitable weighted Sobolev bounds for the inhomogeneity $\\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(0)};$ see Lemma \\ref{L:mathfrakH0partialZIhAproductestimate} for a precise statement of this estimate.\n\n\n\\subsubsection{The wave coordinate condition}\n\nBefore expanding our discussion of the pointwise decay estimates, we will discuss the analytic role of the \\emph{wave coordinate} condition $\\nabla_{\\nu} [\\sqrt{|\\mbox{det}\\, g|}(g^{-1})^{\\mu \\nu}] = 0,$ $(\\mu = 0,1,2,3),$ which plays \\emph{multiple} roles in this article. First, it hyperbolizes the Einstein equations and allows us to replace\ncertain unfavorable terms from the equations \\eqref{E:IntroEinstein} - \\eqref{E:IntroEinstein}\nwith more favorable ones; the culmination of this procedure is exactly the reduced system \\eqref{E:Reducedh1Intro} - \\eqref{E:ReduceddMis0Intro}. In addition, the wave coordinate condition allows us to deduce several \\emph{independent and improved estimates}, both at both the pointwise level and the $L^2$ level, for the components $h_{LL}$ and $h_{LT}.$ As we will see, these improved estimates are central to the structure of the proof of \nTheorem \\ref{T:ImprovedDecay}, and our stability argument would not close without them. More specifically, as shown in \\cite{hLiR2010}, a null decomposition of the wave coordinate condition leads to the algebraic inequality\n\n\\begin{align} \\label{E:wavecoordinatenulldecompintro}\n\t|\\nabla h|_{\\mathcal{L}\\mathcal{T}} + |\\nabla \\nabla_Z h|_{\\mathcal{L}\\mathcal{L}}\n\t& \\lesssim |\\conenabla h| + |h||\\nabla h|,\n\\end{align}\nwhere $\\conenabla$ is the projection of $\\nabla$ (the derivative component only) onto the outgoing Minkowski cones. Note that the right-hand side of \\eqref{E:wavecoordinatenulldecompintro} involves only favorable derivatives of $h$ and quadratic error terms, while the left-hand side involves \\emph{all} derivatives of $h,$ including the dangerous $\\nabla_{\\uL}$ derivative. Generalizations of \\eqref{E:wavecoordinatenulldecompintro} for $\\nabla_{\\mathcal{Z}}^I h$ are stated in Proposition \\ref{P:harmonicgauge}. We remark that it is important to note in these generalizations that the estimates for \n$|\\nabla \\nabla_Z h|_{\\mathcal{L}\\mathcal{L}}$ are stronger than what can be proved for $|\\nabla \\nabla_Z h|_{\\mathcal{L}\\mathcal{T}}.$ \n\n\n\n\\subsubsection{Upgraded pointwise decay estimates}\n\nWe now discuss the full collection of \\emph{upgraded pointwise decay estimates} (see Propositions \\ref{P:FLUTTimproveddecay} - \\ref{P:UpgradedDecayh1A} below), which are of central importance in closing the global existence bootstrap argument. For, as mentioned above, the pointwise decay estimates \\eqref{E:KSIntro} are not sufficient to close the argument. Aside from the components $h_{LL}$ and $h_{LT},$ which are controlled by the wave coordinate condition,\nthere is a relatively strong coupling between the remaining components of $h$ and the dangerous $\\ualpha[\\Far]$ component of the Faraday tensor. Therefore, our proofs of the upgraded estimates (and Proposition \\ref{P:UpgradedDecayh1A} in particular) have a hierarchal structure; i.e., the order in which they are proved is very important. Although we don't provide a complete description of all of the subtleties of this hierarchy in this introduction, we do provide a preliminary description of some of its salient features. We first emphasize the following important feature: most null components of $h,$ all null components of $\\Far,$ and the components $\\nabla_Z h_{LL}$ (for $Z \\in \\mathcal{Z}$) have better $t-$decay properties than their higher-order-derivative counterparts; this is the content of Proposition \\ref{P:UpgradedDecayhA}. Roughly speaking, the reason for this discrepancy is that the un-differentiated reduced equations have a more favorable algebraic structure than the differentiated reduced equations. This feature will be particularly important during our global existence argument, for the principal terms (from the point of view of differentiability) in the Leibniz expansion of the operator $\\nabla_{\\mathcal{Z}}^I$ acting on a quadratic term are of the form $u \\nabla_{\\mathcal{Z}}^I v,$ and similarly for the operator $\\Lie_{\\mathcal{Z}}^I.$ Consequently, the strong pointwise decay property of the un-differentiated quantity, which is represented by $u,$ \nis an important ingredient the derivation of the $C\\varepsilon \\int_{0}^{t} (1 + \\tau)^{-1} \\mathcal{E}_{k;\\upgamma;\\upmu}^2(\\tau) \\, d \\tau$ term on the right-hand side of \\eqref{E:Mainenergyinequalityreexpressed}. We emphasize that our stability proof would not go through if this term were replaced with $C \\varepsilon \\int_{0}^{t} (1 + \\tau)^{-1 + C \\varepsilon} \\mathcal{E}_{k;\\upgamma;\\upmu}^2(\\tau) \\, d \\tau.$\n\n\nThe derivation of the upgraded pointwise decay estimates for the Faraday tensor begins with Proposition \\ref{P:EOVNullDecomposition}, which provides a null decomposition of the electromagnetic equations of variation, and \nProposition \\ref{P:EnergyInhomogeneousTermAlgebraicEstimate}, which provides a null decomposition of the inhomogeneous terms\nthat result when differentiating the reduced electromagnetic equations with modified Lie derivatives.\nThe net effect is that the null components of the \\emph{lower-order} Lie derivatives of $\\Far$ satisfy ODEs along ingoing and outgoing cones (see Proposition \\ref{P:ODEsNullComponentsLieZIFar}), and furthermore, the inhomogeneous terms appearing on the right-hand side of the ODEs can be inductively controlled (see Proposition \\ref{P:UpgradedDecayh1A}). It is important to distinguish between two classes of ODEs that play a role in this analysis. The first class consists of ODEs for the null components $(\\dot{\\alpha}, \\dot{\\rho}, \\dot{\\sigma}) \\eqdef (\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far], \\rho[\\Lie_{\\mathcal{Z}}^I \\Far], \\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]),$ and involves differentiation in the direction of the null generators of the \\emph{ingoing Minkowskian cones}; i.e., the principal part of the ODEs is $\\nabla_{\\uL}.$ We remark that this point of view represents a rather crude treatment of equations \\eqref{E:uLdotalphaEOVnulldecomp} - \\eqref{E:uLdotnablasigmaEOVnulldecomp}, but because of the favorable decay properties of the inhomogeneities, this approach is sufficient to conclude the desired estimates: by integrating back towards the Cauchy hypersurface $\\Sigma_0,$ we are able to deduce $t-$decay for $\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far],$ $\\rho[\\Lie_{\\mathcal{Z}}^I \\Far],$ and $\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]$ from $t-$decay of the inhomogeneous terms at the expense of a loss of decay in $q.$ We remark that the proof of the upgraded estimates for these components happens in two stages. We refer to the first stage, which are proved in Proposition \\ref{P:FLUTTimproveddecay}, as the ``initial upgraded'' pointwise decay estimates. These first-stage estimates follow from using the weighted Klainerman-Sobolev estimates to bound the inhomogeneous terms in the ODEs. The second-stage upgraded estimates, which we refer to as ``fully upgraded'' pointwise decay estimates, are proved at the end of\nSection \\ref{P:UpgradedDecayh1A}, after all of the other upgraded pointwise decay estimates for the remaining components of \nthe lower-order derivatives of $h$ and $\\Far$ have been proved. For at this point in the upgraded hierarchy, we will have better pointwise control over the inhomogeneous terms in the ODEs than that afforded by the weighted Klainerman-Sobolev estimates.\n\n\nThe next class consists of ODEs for the null components $\\dot{\\ualpha} \\eqdef \\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far].$ Notice that (see equation \\eqref{E:dotualphaEOVnulldecomp}) unlike the other null components, the $\\dot{\\ualpha}$ do \\emph{not} satisfy an ODE that to $0-$th order involves differentiation in the direction of $\\uL.$ Instead, at first sight, it might appear that one should reason in analogy with the first class and view equation \\eqref{E:dotualphaEOVnulldecomp} as ODE in the direction of $L$ with inhomogeneous terms. However, the desired decay estimates do \\emph{not} close at this level. Instead, one must also consider the effect of the quadratic term $-\\angm_{\\nu}^{\\ \\lambda} h^{\\mu \\kappa} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}.$ A null decomposition of this term reveals that it contains the dangerous term $\\frac{1}{4} h_{LL} \\nabla_{\\uL} \\dot{\\ualpha}_{\\nu},$ which decays too slowly to be treated as an inhomogeneous term in the ODE satisfied by $\\dot{\\ualpha}.$ To remedy this difficulty, we introduce the vectorfield $\\Lambda = L + \\frac{1}{4} h_{LL} \\uL,$ which can be viewed as a first-order correction to the Minkowski outgoing null direction arising from the presence of a non-zero tensorfield $h$ in the expansion $g_{\\mu \\nu} = m_{\\mu \\nu} + h_{\\mu \\nu}.$ Note that for these upgraded pointwise decay estimates for the lower-order Lie derivatives, we do not bother to correct for the fact that the electromagnetic model is not necessarily the linear Maxwell model; the deviation from the linear Maxwell model comprises cubic terms, which we can treat as small inhomogeneities. We may thus view equation \\eqref{E:dotualphaEOVnulldecomp} as ODE in the direction of $\\Lambda$ with inhomogeneous terms; this is exactly the point of view emphasized in Proposition \\ref{P:ODEsNullComponentsLieZIFar}. Because we have a sufficiently strong independent decay estimates for $h_{LL}$ (this is yet another example of the special role played by the component $h_{LL}$) and also for the inhomogeneities, this approach is sufficient to achieve the desired estimates.\n\n\nOur analysis of the upgraded pointwise decay estimates for the metric-related quantities $h$ and $h^{(1)}$ closely mirrors the analysis in \\cite{hLiR2010}. Hence, we will not discuss them in full detail here, but instead refer the reader to the discussion in \\cite{hLiR2010}. The estimates can be divided into three classes, the first one being the estimates \\eqref{E:partialhLTpartialZhLLpointwise} and \\eqref{E:hLTZhLLpointwise} for \n$|\\nabla h|_{\\mathcal{L} \\mathcal{T}},$ $|\\nabla \\nabla_{Z} h|_{\\mathcal{L} \\mathcal{L}},$\n$|h|_{\\mathcal{L} \\mathcal{T}},$  and $|\\nabla_{Z} h|_{\\mathcal{L} \\mathcal{L}}.$ As was suggested above, the first-class estimates are consequences of the additional special algebraic structure that follows from the wave coordinate condition, together with the weighted Klainerman-Sobolev inequality. The second class consists of the estimates \\eqref{E:partialhTUpointwise} and \\eqref{E:partialhpointwise} for $|\\nabla h|_{\\mathcal{T} \\mathcal{N}}$ and $|\\nabla h|.$ These estimates heavily rely on the decay estimates of Lemma \\ref{L:scalardecay} and Corollary \\ref{C:systemdecay} below, which\nwere proved in \\cite{hLiR2010} and which are independent of the specific structure of the Einstein equations. The lemma and its corollary can be viewed as a second-order counterpart to the ODE estimates for the Faraday tensor discussed in the previous paragraphs. It is important to note that the hypotheses of the lemma and its corollary are satisfied \\emph{as a consequence} of the independent upgraded pointwise decay estimates provided by the wave coordinate condition. The third class consists of the estimates \\eqref{E:partialZIh1Aupgraded}, \\eqref{E:ZIh1Aupgraded}, and \\eqref{E:barpartialZIh1Aupgraded} for $|\\nabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|,$ $|\\nabla_{\\mathcal{Z}}^I h^{(1)}|,$ and $|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|$ (related estimates for the tensorfield $h$ also hold). Their derivation is similar in spirit to the derivation of the second-class estimates, but the inductive proof we give is highly coupled to the simultaneous derivation of analogous upgraded pointwise decay estimates for $|\\Lie_{\\mathcal{Z}}^I\\Far|,$ which were discussed two paragraphs ago. \n\n\\subsubsection{The geometry of Lie derivatives}\n\nWe make some final comments concerning the relationship between Lie derivatives and covariant derivatives. On the one hand, \nsince we differentiate the equations satisfied by $h^{(1)}$ with the operators $\\nabla_{\\mathcal{Z}}^I,$\nour analysis of $h^{(1)}$ naturally allows us to estimate the quantities $|\\nabla_{\\mathcal{Z}}^I h|,$ \n$|\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{L}}$ and $|\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{T}},$ etc. Furthermore, as discussed above, the quantities $|\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{L}}$ and $|\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{T}}$ have a distinguished role in view of their connection to the wave coordinate condition. One the other hand, because we use modified Lie derivatives to differentiate the electromagnetic equations, we will have to confront the terms $|\\Lie_{\\mathcal{Z}}^I h|,$ $|\\Lie_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{L}},$ and $|\\Lie_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{T}},$ etc. In order to bridge the gap between Lie derivative estimates and covariant derivative estimates, we provide Proposition \\ref{P:LievsCovariantLContractionRelation}, the proof of which relies on the special algebraic structure of the vectorfields in $\\mathcal{Z}.$ Proposition \\ref{P:LievsCovariantLContractionRelation} is an especially important ingredient in the null decomposition estimate \\eqref{E:EnergyInhomogeneousTermAlgebraicEstimate}. As an example of the role played by this proposition, we cite the estimate \\eqref{E:LieZILLinTermsofNablaZILLLieZJLTPlusJunk}, which reads $|\\Lie_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{L}} \\lesssim |\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{L}} + \\underbrace{\\sum_{|J| \\leq |I|-1} |\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L} \\mathcal{T}}}_{\\mbox{absent if $|I| = 0$}}\n+ \\underbrace{\\sum_{|J'| \\leq |I|-2} |\\nabla_{\\mathcal{Z}}^{J'} h|}_{\\mbox{absent if $|I| \\leq 1$}}.$ This shows that in the translation from Lie derivatives to covariant derivatives, the error terms that arise in the analysis of the $|\\cdot|_{\\mathcal{L}\\mathcal{L}}$ seminorm are either $1$ degree lower in order \\emph{and} controllable by the wave coordinate condition (i.e. the terms with $|J| \\leq |I|-1$), or are $2$ degrees lower in order (i.e. the terms with $|J'| \\leq |I|-2$). This fact, and others similar to it, play a role in allowing our hierarchy of estimates unfold in a viable order.\n\n\\subsection{Outline of the article}\n\nThe remainder of the article is organized as follows.\n\n\\begin{itemize}\n\t\\item In Section \\ref{S:Notation}, we provide for convenience a summary of the notation that is used throughout \n\t\tthe article.\n\t\\item In Section \\ref{S:ENESinWaveCoordinates}, we discuss the Einstein-nonlinear electromagnetic equations in detail.\n\t\tWe also introduce our wave coordinate condition and our assumptions on the electromagnetic Lagrangian.\n\t\tNext, we derive a reduced system of equations, which is equivalent to the system of interest in our wave coordinate gauge.\n\t\tIn Section \\ref{SS:ReducedEquations}, we summarize the version of the reduced equations that we work with for most of the \n\t\tarticle.\n\t\\item In Section \\ref{S:IVP}, we construct initial data for the reduced system from the \n\t\tabstract initial data in a manner compatible with the wave coordinate condition. We also sketch a proof of the fact\n\t\tthat the wave coordinate condition is preserved by the flow of the reduced equations.\n\t\\item In Section \\ref{S:NullFrame}, we introduce the notion of a Minkowskian null frame and discuss the \n\t\tcorresponding null decomposition of various tensorfields. \n\t\\item In Section \\ref{S:DifferentialOperators}, we introduce the differential operators that will be used throughout \n\t\tthe remainder of the article, including modified Lie derivatives and modified covariant derivatives with respect to a\n\t\tspecial subset $\\mathcal{Z}$ of Minkowskian conformal Killing fields. We also provide a collection of lemmas that\n\t\trelate the various operators.\n\t\\item In Section \\ref{S:EquationSatisfiedbyNablaZIh1}, we provide a preliminary algebraic expression for the equations\n\t\tsatisfied by $\\nabla_{\\mathcal{Z}}^I h^{(1)},$ where $h^{(1)}$ is a solution to the reduced equations.\n\t\\item In Section \\ref{E:EOVandStress}, we introduce the electromagnetic equations of variation, which \n\t\tare a linearized version of the electromagnetic equations. We also provide a preliminary algebraic expression \n\t\tfor the inhomogeneous terms in the equations of variation\n\t\tsatisfied by $\\Lie_{\\mathcal{Z}}^I \\Far,$ where $\\Far$ is a solution to the reduced equations.\n\t\tWe then introduce the canonical stress tensor and use it to construct an energy current that will be used to control \n\t\tweighted Sobolev norms of $\\Lie_{\\mathcal{Z}}^I \\Far.$\n\t\\item In Section \\ref{S:DecompositionsofElectromagneticEquations}, we perform two decompositions of the electromagnetic \n\t\tequations, including a null decomposition of the electromagnetic equations of variation, and a decomposition of the \n\t\telectromagnetic equations into constraint equations and evolution equations for the Minkowskian one-forms \n\t\t$\\Electricfield,$ $\\Displacement,$ $\\Magneticinduction,$ and $\\Magneticfield.$ In order to connect these one-forms \n\t\tto the abstract initial data, we also introduce the geometric electromagnetic one-forms \n\t\t$\\mathfrak{\\Electricfield},$ $\\mathfrak{\\Displacement},$ \n\t\t$\\mathfrak{\\Magneticinduction},$ and $\\mathfrak{\\Magneticfield}.$\n\t\\item In Section \\ref{S:SmallDataAssumptions}, we introduce our smallness condition on the abstract initial data. \n\t\tWe then prove that this smallness condition guarantees that the energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t)$ of the \n\t\tcorresponding solution to the reduced equations is small at $t = 0;$ it is this smallness of \n\t\t$\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0)$ that will lead to a global solution of the reduced equations. \n\t\\item In Section \\ref{S:AlgebraicEstimates}, we provide algebraic estimates for the inhomogeneities in the reduced\n\t\tequations under the assumption that the wave coordinate condition holds. We also derive differential inequalities for the \n\t\tnull components of $\\Lie_{\\mathcal{Z}}^I \\Far,$ and provide algebraic estimates for the corresponding inhomogeneities.\n\t\\item In Section \\ref{S:WeightedEnergy}, we prove weighted energy estimates for solutions to the electromagnetic\n\t\tequations of variation. We also recall some results of \\cite{hLiR2010} that provide analogous weighted energy estimates for \n\t\tboth scalar wave equations and tensorial systems of wave equations with principal part $(g^{-1})^{\\kappa \\lambda} \n\t\t\\nabla_{\\kappa} \\nabla_{\\lambda}.$\n\t\\item In Section \\ref{S:WaveEquationDecay}, we recall some results of \\cite{hLiR2010} that\n\t\tprovide pointwise decay estimates for both scalar wave equations and tensorial systems of wave equations with principal \n\t\tpart $(g^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}.$\n\t\\item In Section \\ref{S:LocalExistence}, we state a basic local existence result and continuation principle for the\n\t\treduced equations. The continuation principle will be used in Section \\ref{S:GlobalExistence} in order to\n\t\tdeduce small-data global existence for the reduced equations.\n\t\\item In Section \\ref{S:DecayFortheReducedEquations}, we introduce our bootstrap assumption on the energy \n\t\t$\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t).$ We then use this assumption to deduce a collection of pointwise decay estimates for \n\t\tsolutions to the reduced equations under the assumption that the wave coordinate condition holds.\n\t\\item In Section \\ref{S:GlobalExistence}, we prove our main stability results. The results are separated into two theorems.\n\t\tIn Theorem \\ref{T:ImprovedDecay}, we use the decay estimates proved in Section \\ref{S:DecayFortheReducedEquations} to\n\t\tderive a ``strong'' inequality for the energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t);$ \n\t\t\\textbf{the proof of this theorem is the centerpiece of the article}. Theorem \\ref{T:MainTheorem}, which is our main \n\t\ttheorem, is then an easy consequence of Theorem \\ref{T:ImprovedDecay} and the continuation principle of Section \n\t\t\\ref{S:LocalExistence}. Both of these theorems rely upon the assumption that the wave coordinate condition holds.\n\\end{itemize}\n\n\n\n\\section{Notation}\\label{S:Notation}\nFor convenience, in this section we collect together some of the important notation that is introduced throughout the article.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{Constants}\nWe use the symbols $c,$ $\\widetilde{c},$ $C,$ and $\\widetilde{C}$ to denote generic \\emph{positive} constants that are free to vary from line to line. In general, they can depend on many quantities, but in the small-solution regime that we consider in this article, they can be chosen uniformly. Sometimes it is illuminating to explicitly indicate one of the quantities $\\mathfrak{Q}$ that a constant depends on; we do by writing e.g. $C_{\\mathfrak{Q}}.$ If $A$ and $B$ are two quantities, then we often write \n\\begin{align*}\n\tA \\lesssim B\n\\end{align*}\nto mean that ``there exists a $C > 0$ such that $A \\leq C B.$'' Furthermore, if $A \\lesssim B$ and $B \\lesssim A,$ then we \noften write\n\n\\begin{align*}\n\tA \\approx B.\n\\end{align*}\n\n\n\\subsection{Indices} \\label{SS:Indices}\n\\begin{itemize}\n\t\\item Lowercase Latin indices $a,b,j,k,$ etc. take on the values $1,2,3.$\n\t\\item Greek indices $\\kappa, \\lambda, \\mu, \\nu,$ etc. take on the values $0,1,2,3.$\n\t\\item Uppercase Latin indices $A,B$ etc. take on the values $1,2$ and are used to enumerate\n\t\tthe two Minkowski-orthogonal null frame vectors tangent to the spheres $S_{r,t}.$\n\t\\item As a convention, the tensorfields $\\Far_{\\mu \\nu},$ $\\Max_{\\mu \\nu},$ $R_{\\mu \\nu},$ $T_{\\mu \\nu},$ \n\t\t$\\epsilon_{\\mu \\nu \\kappa \\lambda\t},$ and $N_{\\mu \\nu \\kappa \\lambda}$ are assumed to ``naturally'' have all of their \n\t\tindices downstairs, and unless indicated otherwise, all indices on all tensors \n\t\tare lowered and raised with the Minkowski metric $m_{\\mu \\nu}$ \n\t\tand its inverse $(m^{-1})^{\\mu \\nu};$ e.g. $T^{\\mu \\nu} \\eqdef (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda}T_{\\kappa \n\t\t\\lambda}.$\n\t\\item The symbol $\\#$ is used to indicate that all indices of a given tensorfield have been raised with $g^{-1};$ e.g.\n\t\t$T^{\\#\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\kappa} (g^{-1})^{\\nu \\lambda}T_{\\kappa \\lambda}.$\n\t\\item Repeated indices are summed over.\n\\end{itemize}\n\n\n\n\\subsection{Coordinates}\n\n\\begin{itemize}\n\t\\item $\\lbrace x^{\\mu} \\rbrace_{\\mu = 0,1,2,3}$ denotes the wave coordinate system.\n\t\\item $t = x^0,$ $x = (x^1,x^2,x^3).$\n\t\\item $q = r - t,$ $s = r + t$ are the null coordinates of the spacetime point $(t,x),$ where $r = |x|.$\n\t\\item $q_{-} = 0$ if $q \\geq 0$ and $q_{-} = |q|$ if $q < 0.$ \n\t\\item $\\omega^j = x^j\/r,$ $(j = 1,2,3).$\n\\end{itemize}\n\n\n\n\\subsection{Surfaces}\n\tRelative to the wave coordinate system:\n\\begin{itemize}\n\t\\item $C_{s}^- \\eqdef \\lbrace (\\tau,y) \\ | \\ |y| + \\tau = s \\rbrace$\n\t\tare the ingoing Minkowskian null cones.\n\t\\item $C_{q}^+ \\eqdef \\lbrace (\\tau,y) \\ | \\ |y| - \\tau = q \\rbrace$ are the outgoing Minkowskian null cones. \n\t\\item $\\Sigma_t \\eqdef \\lbrace (\\tau,y) \\ | \\ \\tau = t \\rbrace$ are the constant Minkowskian time slices.\n\t\\item $S_{r,t} \\eqdef \\lbrace (\\tau,y) \\ | \\ \\tau = t, |y| = r \\rbrace$ are the Euclidean spheres.\n\\end{itemize}\n\n\n\\subsection{Metrics and volume forms}\n\\begin{itemize}\n\t\\item $m_{\\mu \\nu}$ denotes the standard Minkowski metric on $\\mathbb{R}^{1+3};$ in our wave coordinate system,\n\t\t$m_{\\mu \\nu} = \\mbox{diag}(-1,1,1,1).$\n\t\\item $\\underline{m}$ denotes the Minkowskian first fundamental form of $\\Sigma_t;$ in our wave coordinate system, \n\t\t$\\underline{m}_{\\mu \\nu} = \\mbox{diag}(0,1,1,1).$\n\t\\item $\\angm$ denotes the Minkowskian first fundamental form of $S_{r,t};$ relative to an arbitrary coordinate system, \\\\\n\t\t$\\angm_{\\mu \\nu} = m_{\\mu \\nu} + \\frac{1}{2}\\big(L_{\\mu} \\uL_{\\nu} + \\uL_{\\mu} L_{\\nu} \\big),$\n\t\twhere $\\uL, L$ are defined in Section \\ref{SS:NullFrames}.\n\t\\item $g_{\\mu \\nu}$ denotes the spacetime metric.\n\t\\item $g_{\\mu \\nu} = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)}$ is the splitting of the spacetime metric into\n\t\tthe Minkowski metric $m_{\\mu \\nu},$ the Schwarzschild tail $h_{\\mu \\nu}^{(0)} \n\t\t= \\chi\\big(\\frac{r}{t}\\big)\\chi(r)\\frac{2M}{r} \\delta_{\\mu \\nu},$ and the remainder $h_{\\mu \\nu}^{(1)}.$ \n\t\\item $h_{\\mu \\nu} = h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)}.$\n\t\\item $(g^{-1})^{\\mu \\nu} = (m^{-1})^{\\mu \\nu} + H_{(0)}^{\\mu \\nu} + H_{(1)}^{\\mu \\nu}$ \n\t\tis the splitting of the inverse spacetime metric into the inverse Minkowski metric \n\t\t$(m^{-1})^{\\mu \\nu},$ the Schwarzschild tail $H_{(0)}^{\\mu \\nu} = - \\chi\\big(\\frac{r}{t}\\big) \\chi(r) \\frac{2M}{r} \n\t\t\\delta^{\\mu \\nu},$ and the remainder $H_{(1)}^{\\mu \\nu}.$ \n\t\\item $H^{\\mu \\nu} = H_{(0)}^{\\mu \\nu} + H_{(1)}^{\\mu \\nu}.$\n\t\\item $\\mathring{\\underline{g}}$ denotes the first fundamental form of the Cauchy hypersurface $\\Sigma_0$\n\t\trelative to the spacetime metric $g.$\n\t\\item $\\mathring{\\underline{g}}_{jk} = \\delta_{jk} + \\chi(r)\\frac{2M}{r} \\delta_{jk} + \\mathring{\\underline{h}}_{jk}^{(1)}$\n\t\tis the splitting of $\\mathring{\\underline{g}}_{jk}$ into the Schwarzschild tail $\\chi(r)\\frac{2M}{r} \\delta_{jk}$\n\t\tand the remainder $\\mathring{\\underline{h}}_{jk}^{(1)}.$\n\t\\item $\\Minkvolume_{\\mu \\nu \\kappa \\lambda} = |\\mbox{det} \\, m|^{1\/2} [\\mu \\nu \\kappa \\lambda]$ denotes the volume form of\n\t\tthe Minkowski metric $m;$ $[\\mu \\nu \\kappa \\lambda]$ is totally antisymmetric with normalization\n\t\t$[0123] = 1;$ $|\\mbox{det} \\, m|^{1\/2} = 1$ in our wave coordinate system.\n\t\\item $\\epsilon_{\\mu \\nu \\kappa \\lambda} = |\\mbox{det} \\, g|^{1\/2} [\\mu \\nu \\kappa \\lambda]$ denotes the volume form of\n\t\tthe spacetime metric $g.$\n\t\\item $\\epsilon^{\\# \\mu \\nu \\kappa \\lambda} = -|\\mbox{det} \\, g|^{-1\/2} [\\mu \\nu \\kappa \\lambda]$ denotes the volume form of\n\t\tthe spacetime metric $g$ with all of the indices raised with $g^{-1}.$\n\t\\item $\\uvolume_{\\nu \\kappa \\lambda} = [0 \\nu \\kappa \\lambda]$ denotes the Euclidean volume\n\t\tform of the surfaces $\\Sigma_t$ viewed as embedded submanifolds of Minkowski spacetime equipped with\n\t\tthe wave coordinate system. \n\t\\item $\\uvolume_{ijk}= [ijk]$ denotes the Euclidean volume form of the surfaces $\\Sigma_t$ viewed as abstract $3-$manifolds.\n\t\\item $\\angupsilon_{\\mu \\nu} = \\Minkvolume_{\\mu \\nu \\kappa \\lambda} \\uL^{\\kappa} L^{\\lambda}$ denotes the Euclidean\n\t\tvolume form of the spheres $S_{r,t}.$\n\\end{itemize}\n\n\\subsection{Hodge duals} \\label{SS:Hodge}\n\tFor an arbitrary two-form $\\Far_{\\mu \\nu}:$\n\\begin{itemize}\n\t\\item $\\Fardual_{\\mu \\nu} = \\frac{1}{2} g_{\\mu \\mu'} g_{\\nu \\nu'} \n\t\t\\epsilon^{\\# \\mu' \\nu' \\kappa \\lambda} \\Far_{\\kappa \\lambda}\n\t\t= - \\frac{1}{2} |\\mbox{det} \\, g|^{-1\/2} g_{\\mu \\mu'} g_{\\nu \\nu'}\n\t\t[\\mu' \\nu' \\kappa \\lambda] \\Far_{\\kappa \\lambda}$ denotes the Hodge dual\n\t\tof $\\Far_{\\mu \\nu}$ with respect to the spacetime metric $g_{\\mu \\nu}.$\n\t\\item $\\FarMinkdual_{\\mu \\nu} = \\frac{1}{2} \\Minkvolume_{\\mu \\nu}^{\\ \\ \\kappa \\lambda} \\Far_{\\kappa \\lambda}\n\t = - \\frac{1}{2} |\\mbox{det} \\, m|^{-1\/2} \n\t\tm_{\\mu \\mu'} m_{\\nu \\nu'} [\\mu' \\nu' \\kappa \\lambda]\\Far_{\\kappa \\lambda}$ \n\t\tdenotes the Hodge dual of $\\Far_{\\mu \\nu}$ with respect to the Minkowski metric $m_{\\mu \\nu}.$ In our wave coordinate \n\t\tsystem, $|\\mbox{det} \\, m|^{-1\/2} = 1.$\n\\end{itemize}\n\n\n\\subsection{Derivatives} \\label{SS:Derivatives}\n\\begin{itemize}\n\t\\item $\\nabla$ denotes the Levi-Civita connection corresponding to $m.$\n\t\\item $\\mathscr{D}$ denotes the Levi-Civita connection corresponding to $g.$\n\t\\item $\\mathring{\\underline{\\mathscr{D}}}$ denotes the Levi-Civita connection corresponding to $\\mathring{\\underline{g}}.$\n\t\\item $\\unabla$ denotes the Levi-Civita connection corresponding to $\\underline{m}.$\n\t\\item $\\angn$ denotes the Levi-Civita connection corresponding to $\\angm.$\n\t\\item $\\conenabla$ denotes the projection of $\\nabla$ onto the outgoing Minkowski null cones; \n\t\ti.e., $\\conenabla_{\\mu} = \\coneproject_{\\mu}^{\\ \\kappa} \\nabla_{\\kappa},$ where\n\t\t$\\coneproject_{\\mu}^{\\ \\nu} = \\delta_{\\mu}^{\\nu} + \\frac{1}{2}L_{\\mu} \\uL^{\\nu}$ projects vectors $X^{\\mu}$ onto\n\t\tthe outgoing Minkowski null cones.\n\t\\item In our wave coordinate system $\\lbrace x^{\\mu} \\rbrace_{\\mu=0,1,2,3},$ \n\t\t$\\partial_{\\mu} = \\frac{\\partial}{\\partial x^{\\mu}},$ $\\nabla_{\\mu} = \\nabla_{\\frac{\\partial}{\\partial x^{\\mu}}}.$\n\t\\item In our wave coordinate system, $\\partial_r = \\omega^a \\partial_a$ denotes the radial \n\t\tderivative, where $\\omega^j = x^j\/r.$\n\n\n\t\n\t\\item In our wave coordinate system,\n\t\t$\\partial_s \\eqdef \\frac{1}{2}(\\partial_r + \\partial_t), \\partial_q \\eqdef \\frac{1}{2}(\\partial_r - \\partial_t)$\n\t\tdenote the null derivatives; $\\partial_q$ denotes partial differentiation at fixed $s$ and fixed angle $x\/|x|,$\n\t\twhile $\\partial_s$ denotes partial differentiation at fixed $q$ and fixed angle $x\/|x|,$\n\t\\item If $X$ is a vectorfield and $\\phi$ is a function, then $X \\phi = X^{\\kappa} \\partial_{\\kappa} \\phi.$\n\t\\item $\\nabla_X$ denotes the differential operator $X^{\\kappa} \\nabla_{\\kappa}.$\n\t\\item $\\underline{\\nabla}_X$ denotes the differential operator $X^{\\kappa} \n\t\t\\underline{\\nabla}_{\\kappa}.$\n\t\\item $\\angn_X$ denotes the differential operator $X^{\\kappa} \\angn_{\\kappa}.$\n\n\t\n\n\n\n\n\n\t\n\n\t\n\t\\item $\\Lie_X$ denotes the Lie derivative with respect to the vectorfield $X.$\n\t\\item $[X,Y]^{\\mu} = (\\Lie_X Y)^{\\mu} = X^{\\kappa}\\partial_{\\kappa}Y^{\\mu} - Y^{\\kappa}\\partial_{\\kappa} X^{\\mu}$\n\t\tdenotes the Lie bracket of the vectorfields $X$ and $Y.$\n\t\\item For $Z \\in \\mathcal{Z},$ $\\hat{\\nabla}_Z = \\nabla_Z + c_Z$ denotes the modified covariant derivative,\n\t\twhere the constant $c_Z$ is defined in Section \\ref{SS:Killingnotation}.\n\t\\item For $Z \\in \\mathcal{Z},$ $\\Liemod_Z = \\Lie_Z + 2c_Z$ denotes the modified Lie derivative,\n\t\twhere the constant $c_Z$ is defined in Section \\ref{SS:Killingnotation}.\n\t\\item $\\nabla^I U,$ $\\unabla^I U,$ $\\nabla_{\\mathcal{Z}}^I U,$ $\\hat{\\nabla}_{\\mathcal{Z}}^I U,$ \n\t\t$\\Lie_{\\mathcal{Z}}^I U,$ and $\\Liemod_{\\mathcal{Z}}^I U$ respectively \n\t\tdenote an $|I|^{th}$ order iterated Minkowski covariant derivative, iterated Euclidean (spatial) covariant derivative,\n\t\titerated Minkowski $\\mathcal{Z}-$covariant derivative, iterated modified Minkowski $\\mathcal{Z}-$covariant derivative,\n\t\titerated $\\mathcal{Z}-$Lie  derivative, and iterated modified $\\mathcal{Z}-$Lie derivative of the tensorfield $U.$\n\t\\item $\\Square_m = (m^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}$ denotes the standard\n\t\tMinkowski wave operator.\n\t\\item $\\widetilde{\\Square}_g = (g^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}$ denotes the \n\t\treduced wave operator corresponding to the spacetime metric $g.$ Note that $\\nabla$ is the \\emph{Minkowskian} connection.\n\\end{itemize}\n\n\n\\subsection{Minkowskian conformal Killing fields} \\label{SS:Killingnotation} \\ \\\\\nRelative to the wave coordinate system $\\lbrace x^{\\mu} \\rbrace_{\\mu=0,1,2,3} = (t,x):$\n\\begin{itemize}\n\t\\item $\\partial_{\\mu} = \\frac{\\partial}{\\partial x^{\\mu}},$ $(\\mu=0,1,2,3),$ denotes a translation vectorfield.\n\t\\item $\\Omega_{jk} = x_j \\frac{\\partial}{\\partial x^k} - x_k \\frac{\\partial}{\\partial x^j},$ \n\t\t$(1 \\leq j < k \\leq 3),$ denotes a rotation vectorfield.\n\t\\item $\\Omega_{0j} = -t \\frac{\\partial}{\\partial x^j} - x_j \\frac{\\partial}{\\partial t},$ \n\t\t$(j=1,2,3),$ denotes a Lorentz boost vectorfield.\n\t\\item $S = x^{\\kappa} \\frac{\\partial}{\\partial x^{\\kappa}}$ denotes the scaling vectorfield.\n\n\t\\item $\\mathcal{O} = \\big\\lbrace \\Omega_{jk} \\big\\rbrace_{1 \\leq j < k \\leq 3}$ are the rotational\n\t\tMinkowskian Killing fields.\n\t\\item $\\mathcal{Z} = \\big\\lbrace \\frac{\\partial}{\\partial x^{\\mu}}, \n\t\t\\Omega_{\\mu \\nu}, S \\big\\rbrace_{0 \\leq \\mu < \\nu \\leq 3}.$\n\n\t\n\t\\item For $Z \\in \\mathcal{Z},$ $^{(Z)}\\pi_{\\mu \\nu} = \\nabla_{\\mu} Z_{\\nu} + \\nabla_{\\nu} Z_{\\mu} = c_Z m_{\\mu \\nu}$\n\t\tis the Minkowskian deformation tensor of $Z,$\twhere $c_Z$ is a constant. \n\t\\item Commutation properties with the Maxwell-Maxwell term: \\\\\n\t\t$\\Liemod_{\\mathcal{Z}}^I \\Big\\lbrace \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} \n\t\t\t(m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \\Big\\rbrace\n\t\t= \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\\big] \n\t\t\t\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}.$\n\t\\item Commutation properties with the Minkowski wave operator \\\\\n\t\t$\\Square_m = (m^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}:$ \\\\\n\t\t$[\\Square_m, \\partial_{\\mu}] = [\\Square_m,$ $\\Omega_{\\mu \\nu}] = 0,$ \\ $[\\Square_m, S] = 2 \\Square_m,$ \\\n\t\t$[\\nabla_Z, \\Square_m] = - c_Z \\Square_m,$ \\ $\\Square_m \\nabla_Z \\phi = \\nablamod \\Square_m \\phi.$ \n\\end{itemize}\n\n\\subsection{Minkowskian null frames} \\label{SS:NullFrames}\n\\begin{itemize}\n\t\\item $\\uL = \\partial_t - \\partial_r$ denotes the Minkowskian null geodesic vectorfield transversal to the $C_{q}^+;$ it \t\n\t\tgenerates the cones $C_{s}^-.$\n\t\\item $L = \\partial_t + \\partial_r$ denotes the Minkowskian null geodesic vectorfield generating the cones $C_q^+.$\n\t\\item $e_A, \\ A = 1,2$ denotes Minkowski-orthonormal vectorfields spanning the tangent space of the spheres $S_{r,t}.$  \n\t\\item The set $\\mathcal{L} \\eqdef \\lbrace L \\rbrace$ contains only $L.$\n\t\\item The set $\\mathcal{T} \\eqdef \\lbrace L, e_1, e_2 \\rbrace$ denotes the frame vector fields tangent\n\t\tto the $C_q^+.$\n\t\\item The set $\\mathcal{N} \\eqdef \\lbrace \\uL,L, e_1, e_2 \\rbrace$ denotes the entire Minkowski null frame.\n\\end{itemize}\n\n\n\n\\subsection{Minkowskian null frame decomposition}\n\n\\begin{itemize}\n\t\\item For an arbitrary vectorfield $X$ and frame vector $N \\in \\mathcal{N},$ we define\n\t\t$X_N = X_{\\kappa} N^{\\kappa},$ where $X_{\\mu} = m_{\\mu \\kappa} X^{\\kappa}.$\n\t\\item For an arbitrary vectorfield $X = X^{ \\kappa}\\partial_{\\kappa} = X^{L} L + X^{\\uL} \\uL\n\t\t+ X^{A} e_A,$ where \\\\\n\t\t$X^{L} = - \\frac{1}{2}X_{\\uL},$ $X^{\\uL} = - \\frac{1}{2}X_{L},$ $X^A = X_A.$\n\t\\item For an arbitrary pair of vectorfields $X,Y:$ \\\\\n\t\t$m(X,Y) = m_{\\kappa \\lambda} X^{\\kappa} X^{\\lambda} \n\t\t= X^{\\kappa}Y_{\\kappa} = -\\frac{1}{2}X_{L}Y_{\\uL} - \\frac{1}{2}X_{\\uL}Y_{L} + X_A Y_A.$\n\\end{itemize}\n\nIf $\\Far_{\\mu \\nu}$ is any two-form, its Minkowskian null components are:\n\n\\begin{itemize}\n\t\\item $\\ualpha_{\\mu} = \\angm_{\\mu}^{\\ \\nu} \\Far_{\\nu \\lambda} \\uL^{\\lambda}.$\n\t\\item $\\alpha_{\\mu} = \\angm_{\\mu}^{\\ \\nu} \\Far_{\\nu \\lambda} L^{\\lambda}.$\n\t\\item $\\rho = \\frac{1}{2} \\Far_{\\kappa \\lambda}\\uL^{\\kappa} L^{\\lambda}.$\n\t\\item $\\sigma = \\frac{1}{2} \\angupsilon^{\\kappa \\lambda} \\Far_{\\kappa \\lambda}.$\n\\end{itemize}\n\n\n\n\\subsection{Electromagnetic decompositions}\n\nIf $\\Far_{\\mu \\nu}$ is any two-form, $\\Maxdual_{\\mu \\nu} = g_{\\mu \\kappa} g_{\\nu \\lambda} \n\\frac{\\partial \\Ldual}{\\partial \\Far_{\\kappa \\lambda}},$\nand $\\hat{N}^{\\mu}$ is the future-directed unit $g-$normal to $\\Sigma_t,$ then its electromagnetic components are:\n\n\\begin{itemize}\n\t\\item $\\mathfrak{\\Electricfield}_{\\mu} = \\Far_{\\mu \\kappa}\\hat{N}^{\\kappa}.$ \n\t\\item $\\mathfrak{\\Magneticinduction}_{\\mu} = - \\Fardual_{\\mu \\kappa}\\hat{N}^{\\kappa}.$ \n\t\\item $\\mathfrak{\\Displacement}_{\\mu} = - \\Maxdual_{\\mu \\kappa} \\hat{N}^{\\kappa}.$  \t\n\t\\item $ \\mathfrak{\\Magneticfield}_{\\mu} = - \\Max_{\\mu \\kappa}\\hat{N}^{\\kappa}.$\n\\end{itemize}\n\nIf $\\Far_{\\mu \\nu}$ is any two-form, then relative to the wave coordinate system, its Minkowskian\nelectromagnetic components are:\n\n\\begin{itemize}\n\t\\item $\\Electricfield_{\\mu} = \\Far_{\\mu 0}.$  \n\t\\item $\\Magneticinduction_{\\mu} = - \\FarMinkdual_{\\mu 0}.$ \n\t\\item $\\Displacement_{\\mu} = - \t\\MaxMinkdual_{\\mu 0}.$  \t\n\t\\item $\\Magneticfield_{\\mu} = - \\Max_{\\mu 0}.$\n\\end{itemize}\n\n\n\\subsection{Seminorms and energies}\n\nFor an arbitrary type $\\binom{0}{2}$ tensorfield $P_{\\mu \\nu},$ and $\\mathcal{V}, \\mathcal{W} \\in \\lbrace \\mathcal{L}, \\mathcal{T},\\mathcal{N} \\rbrace:$\n\n\\begin{itemize}\n\t\\item $|P|_{\\mathcal{V} \\mathcal{W}} = \\sum_{V \\in \\mathcal{V}, W \\in \\mathcal{W}} |V^{\\kappa} W^{\\lambda} \n\t\tP_{\\kappa \\lambda}|.$\n\t\\item $|\\nabla P|_{\\mathcal{V} \\mathcal{W}} = \\sum_{N \\in \\mathcal{N}, V \\in \\mathcal{V}, W \\in \\mathcal{W}} \n\t\t|V^{\\kappa} W^{\\lambda} N^{\\gamma} \\nabla_{\\gamma} P_{\\kappa \\lambda}|.$\n\t\\item $|\\conenabla P|_{\\mathcal{V} \\mathcal{W}} = \\sum_{T \\in \\mathcal{T}, V \\in \\mathcal{V}, W \\in \\mathcal{W}} \n\t\t|V^{\\kappa} W^{\\lambda} T^{\\gamma} \\nabla_{\\gamma} P_{\\kappa \\lambda}|.$\n\t\\item $|P| = |P|_{\\mathcal{N} \\mathcal{N}}.$\n\t\\item $|\\nabla P| = |\\nabla P|_{\\mathcal{N} \\mathcal{N}}.$\n\t\\item $|\\conenabla P| = |\\conenabla P|_{\\mathcal{N} \\mathcal{N}}.$\n\t\\item We use similar notation for an arbitrary tensorfield $U$ of type $\\binom{n}{m}.$\n\\end{itemize}\n\n\nFor an arbitrary tensorfield $U$ defined on the Euclidean space $\\Sigma_0$ with Euclidean coordinate system \n$x = (x^1,x^2,x^3):$\n\n\\begin{itemize}\n\t\\item $\\| U \\|_{L^2}^2 = \\int_{x \\in \\mathbb{R}^3} |U(x)|^2 \\, d^3 x$ is the square of the standard spatial $L^2$ norm of $U.$\n\t\\item $\\| U \\|_{L^{\\infty}} = \\mbox{ess} \\sup_{x \\in \\mathbb{R}^3} |U(x)|$ is the standard spatial $L^{\\infty}$ norm of $U.$\n\t\\item $\\| U \\|_{H_{\\eta}^{\\dParameter}}^2 = \\sum_{|I| \\leq \\dParameter } \\int_{x \\in \\mathbb{R}^3} \n\t\t(1 + |x|^2)^{(\\eta + |I|)} |\\unabla^I U(x)|^2 \\, d^3 x$ is the square of a weighted Sobolev norm of $U.$ \n\t\\item $\\| U \\|_{C_{\\eta}^{\\dParameter}}^2 =  \n\t\t\\sum_{|I| \\leq \\dParameter } \\mbox{ess} \\sup_{x \\in \\mathbb{R}^3} (1 + |x|^2)^{(\\eta + |I|)} |\\unabla^I U(x)|^2$ \n\t\tis the square of a weighted pointwise norm of $U.$\n\\end{itemize}\n\n\nFor arbitrary abstract initial data \n$(\\mathring{\\underline{h}}^{(1)}_{jk}, \\mathring{K}_{jk}, \\mathring{\\mathfrak{\\Displacement}}_j, \\mathring{\\mathfrak{\\Magneticinduction}}_j)$ on the manifold $\\mathbb{R}^3:$\n\n\\begin{itemize}\n\t\\item $E_{\\dParameter;\\upgamma}^2(0) \n\t\t= \\| \\underline{\\nabla} \\mathring{\\underline{h}}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{K} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{\\mathfrak{\\Displacement}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{\\mathfrak{\\Magneticinduction}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2$\n\t\tis the square of the norm of the abstract initial data.\n\\end{itemize}\n\nFor an arbitrary symmetric type $\\binom{0}{2}$ tensorfield $h_{\\mu \\nu}^{(1)}$ and an arbitrary two-form $\\Far_{\\mu \\nu}:$\n\n\\begin{itemize}\n\t\\item $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}^2(t) = \\underset{0 \\leq \\tau \\leq t}{\\mbox{sup}} \n\t\t\\sum_{|I| \\leq \\dParameter } \\int_{\\Sigma_{\\tau}} \n\t\t\\Big\\lbrace |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|^2 \\Big\\rbrace w(q) \\, d^3 x$\n\t\tis the square of the energy of the pair $(h_{\\mu \\nu}^{(1)}, \\Far_{\\mu \\nu}).$ \n\\end{itemize}\n\n\n\\subsection{$O^{\\dParameter}()$ and $o^{\\dParameter}()$} \\label{SS:Oando}\n\t\n\\begin{itemize}\n\t\\item Given an $\\dParameter -$times continuously differentiable \n\tfunction $f(\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_m)$ depending on the tensorial quantities\n\t$\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_m,$ we write \n\t$f(\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_m)= O^{\\dParameter}\\big(|\\mathfrak{Q}_1|^{p_1} \\cdots |\\mathfrak{Q}_k|^{p_k};\n\t\\mathfrak{Q}_{k+1},\\cdots,\\mathfrak{Q}_m\\big)$ if we can decompose \\\\\n\t$f(\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_m) \n\t= \\sum_{i = 1}^n p_i(\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_k) \\widetilde{f}_i(\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_m),$\n\twhere $n$ is a positive integer, each $p_i(\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_k)$ is a polynomial in the components of \n\t$\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_k$\n\tthat satisfies $|p_i(\\mathfrak{Q}_1,\\cdots,\\mathfrak{Q}_k)| \\lesssim |\\mathfrak{Q}_1|^{p_1} \\cdots |\\mathfrak{Q}_k|^{p_k}$\n\tin a neighborhood of the origin, and $\\widetilde{f}_i(\\cdot)$ is $\\dParameter -$times continuously \n\tdifferentiable in a neighborhood of the origin. \n\t\\item Given an $\\dParameter -$times continuously differentiable function $f(x),$ we write $f(x) = o^{\\dParameter}(r^{-a})$ if \n\t$\\lim_{r \\to \\infty} \\frac{|\\unabla^I f(x)|}{r^{a + |I|}} = 0$ for $|I| \\leq \\dParameter .$ \n\\end{itemize}\n\n\n\\subsection{Fixed constants} \\label{SS:FixedConstants}\nThe fixed constants $\\dParameter,$ $\\updelta,$ $\\upgamma,$ $\\upmu,$ $\\upgamma',$ $\\upmu'$ are subject to the following constraints:\n\n\\begin{itemize}\n\t\\item To prove our global stability theorem, we assume that $\\dParameter$ is an integer satisfying $\\dParameter \\geq 8.$ \n\t\\item $0 < \\updelta < \\frac{1}{4}.$\n\t\\item $0 < \\updelta < \\upgamma < 1\/2.$\n\t\\item $0 < \\upgamma' < \\upgamma - \\updelta.$\n\t\\item $0 < \\updelta < \\upmu' < \\frac{1}{2}.$\n\t\\item $0 < \\upmu < \\frac{1}{2} - \\upmu'.$\n\\end{itemize}\n\n\n\\subsection{Weights} \\label{SS:Weights}\n\n\\begin{itemize}\n\t\\item $w = w(q) = \\left \\lbrace\n\t\t\\begin{array}{lr}\n    \t1 \\ + \\ (1 + |q|)^{1 + 2 \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      1 \\ + \\ (1 + |q|)^{-2 \\upmu}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \\right.$ is the energy estimate weight function.\n \t\\item $\\varpi = \\varpi(q) = \\left \\lbrace\n\t\t\\begin{array}{lr}\n    \t(1 + |q|)^{1 + \\upgamma'}, &  \\mbox{if} \\ q > 0, \\\\\n      (1 + |q|)^{1\/2 - \\upmu'}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \\right.$ is the decay estimate weight function.\n\\end{itemize}\n\n\n\n\n\\section{The Einstein-Nonlinear Electromagnetic System in Wave Coordinates} \\label{S:ENESinWaveCoordinates}\nIn this section, we discuss equations \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0} in detail. We also discuss our assumptions on the electromagnetic Lagrangian and introduce our wave coordinate gauge. We then derive a reduced system of equations, which are equivalent to \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0} in the wave coordinate gauge. Finally, we summarize the results by providing a version of the reduced equations that will be used throughout the remainder of the article. In particular, in this version, we distinguish between principal terms, which require a careful treatment, and ``error terms,'' which are, from the point of view of decay rates, relatively easy to estimate.\n\n\n\\noindent \\hrulefill\n\\ \\\\\n\nIn this article, we consider the $1+3-$dimensional electro-gravitational system \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0}, which we restate here for convenience:\n\n\\begin{subequations}\n\\begin{align} \n\tR_{\\mu \\nu} - \\frac{1}{2}g_{\\mu \\nu} R & = T_{\\mu \\nu}, && (\\mu, \\nu = 0,1,2,3), \n\t\t\\label{E:IntroEinsteinagain} \\\\\n\t(d \\Far)_{\\lambda \\mu \\nu} & = 0, && (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:IntrodFaris0again} \\\\\n\t(d \\Max)_{\\lambda \\mu \\nu} & = 0, && (\\lambda, \\mu, \\nu = 0,1,2,3). \\label{E:IntrodMis0again}\n\\end{align}\n\\end{subequations}\nWe remark that the spacetimes we consider will always have the manifold structure $I \\times \\mathbb{R}^3$ for some ``time'' interval $I.$ The energy-momentum tensor $T_{\\mu \\nu}$ is given below in \\eqref{E:electromagnetictensorloweroinTermsofLagrangian}, while $\\Max_{\\mu \\nu}$ is related to $(g_{\\mu \\nu}, \\Far_{\\mu \\nu})$ via the constitutive relation \\eqref{E:Maxdualdef}. The precise forms of $T_{\\mu \\nu}$ and $\\Max_{\\mu \\nu}$ depend on the chosen model of electromagnetism, which, as discussed in detail in Section \\ref{SS:Lagrangianformluationofnonlinearelectromagnetism}, we assume is a Lagrangian-derived model subject to the restrictions \\eqref{E:Ldualassumptions}, \\eqref{E:DECL1} - \\eqref{E:DECTrace} below. We recall (see e.g. \\cite{dC2008}, \\cite{rW1984}) the following relationships between the \\emph{spacetime metric} $g_{\\mu \\nu},$ the \\emph{Riemann curvature tensor}\\footnote{Under our sign convention, $\\mathscr{D}_{\\mu} \\mathscr{D}_{\\nu} X_{\\kappa} - \\mathscr{D}_{\\nu} \\mathscr{D}_{\\mu} X_{\\kappa} = R_{\\mu \\nu \\kappa}^{\\ \\ \\ \\ \\lambda} X_{\\lambda}.$}, $R_{\\mu \\kappa \\nu}^{\\ \\ \\ \\ \\lambda},$ the \\emph{Ricci tensor} $R_{\\mu \\nu},$ the \\emph{scalar curvature} $R,$ and the \\emph{Christoffel symbols} $\\Gamma_{\\mu \\ \\nu}^{\\ \\kappa},$ which are valid in an arbitrary coordinate system on $\\mathbb{R}^{1+3}:$\n\n\\begin{subequations}\n\\begin{align}\nR_{\\mu \\kappa \\nu}^{\\ \\ \\ \\ \\lambda} & \\eqdef \n\t\\partial_{\\kappa} \\Gamma_{\\mu \\ \\nu}^{\\ \\lambda}\n\t- \\partial_{\\mu} \\Gamma_{\\kappa \\ \\nu}^{\\ \\lambda}\n  + \\Gamma_{\\kappa \\ \\beta}^{\\ \\lambda} \\Gamma_{\\mu \\ \\nu}^{\\ \\beta}\n\t- \\Gamma_{\\mu \\ \\beta}^{\\ \\lambda} \\Gamma_{\\kappa \\ \\nu}^{\\ \\beta},\t\n\t\\label{E:EMBIRiemanndef} \\\\\nR_{\\mu \\nu} & \\eqdef R_{\\mu \\kappa \\nu}^{\\ \\ \\ \\ \\kappa} \n\t= \\partial_{\\kappa} \\Gamma_{\\mu \\ \\nu}^{\\ \\kappa}\n\t- \\partial_{\\mu} \\Gamma_{\\kappa \\ \\nu}^{\\ \\kappa}\n  + \\Gamma_{\\kappa \\ \\lambda}^{\\ \\kappa} \\Gamma_{\\mu \\ \\nu}^{\\ \\lambda}\n\t- \\Gamma_{\\mu \\ \\kappa}^{\\ \\lambda} \\Gamma_{\\lambda \\ \\nu}^{\\ \\kappa},\t\n\t\\label{E:Riccidef} \\\\\nR & \\eqdef (g^{-1})^{\\kappa \\lambda} R_{\\kappa \\lambda}, \\label{E:EMBIRdef} \\\\\n\\Gamma_{\\mu \\ \\nu}^{\\ \\kappa} & \\eqdef \\frac{1}{2} (g^{-1})^{\\kappa \\lambda}(\\partial_{\\mu} g_{\\lambda \\nu} \n\t+ \\partial_{\\nu} g_{\\mu \\lambda} - \\partial_{\\lambda} g_{\\mu \\nu}). \\label{E:EMBIChristoffeldef}\n\\end{align}\n\\end{subequations}\nWe also recall the following symmetry properties:\n\n\\begin{align}\n\tR_{\\mu \\nu} = R_{\\nu \\mu}, \\\\ \n\t\\Gamma_{\\mu \\ \\nu}^{\\ \\kappa}= \\Gamma_{\\nu \\ \\mu}^{\\ \\kappa}.\n\\end{align}\n\nWe note for future use that taking the trace with respect to $g$ of each side of \\eqref{E:IntroEinsteinagain} implies that\n\n\\begin{align}\n\tR = -(g^{-1})^{\\kappa \\lambda} T_{\\kappa \\lambda}.\n\\end{align}\nHence, \\eqref{E:IntroEinsteinagain} is equivalent to\n\n\\begin{align}\n\tR_{\\mu \\nu} & = T_{\\mu \\nu} - \\frac{1}{2} g_{\\mu \\nu} (g^{-1})^{\\kappa \\lambda} T_{\\kappa \\lambda}  \t\n\t\\tag{\\ref{E:IntroEinsteinagain}'}.\n\\end{align}\nFurthermore, we note that the twice-contracted Bianchi identities (see e.g. \\cite{rW1984}) are the relation (see Section \\ref{SS:Indices}\nconcerning our use of the notation $\\#$)\n\n\\begin{align}\n\t\\mathscr{D}_{\\mu} \\big(R^{\\# \\mu \\nu} - \\frac{1}{2}(g^{-1})^{\\mu \\nu} R \\big) = 0, && (\\nu =0,1,2,3),\n\\end{align}\nso that by \\eqref{E:IntroEinsteinagain}, $T_{\\mu \\nu}$ necessarily satisfies the following divergence-free condition:\n\n\\begin{align} \\label{E:EMBITconservation}\n\t\\mathscr{D}_{\\mu} T^{\\# \\mu \\nu} & = 0, && (\\nu =0,1,2,3).\n\\end{align}\nIn the above expressions, $\\mathscr{D}$ denotes the Levi-Civita connection corresponding to $g_{\\mu \\nu}.$\n\n\n\n\\subsection{Wave coordinates} \\label{SS:WaveCoordinates}\n\nIn this article, we use the framework developed in \\cite{hLiR2005}, \\cite{hLiR2010} and work in a \\emph{wave coordinate} system, which is defined to be a coordinate system in which\n\n\\begin{subequations}\n\\begin{align} \\label{E:wavecoordinategauge1}\n\t\\Gamma^{\\mu} \\eqdef (g^{-1})^{\\kappa \\lambda} \\Gamma_{\\kappa \\ \\lambda}^{\\ \\mu} = 0, && (\\mu = 0,1,2,3).\n\\end{align}\nThe condition \\eqref{E:wavecoordinategauge1} is also known as \\emph{harmonic gauge} or \\emph{de Donder gauge}. It is easy to check that the condition \\eqref{E:wavecoordinategauge1} is equivalent to the conditions\n\n\\begin{align}\n\tg_{\\mu \\nu} (g^{-1})^{\\kappa \\lambda} \\Gamma_{\\kappa \\ \\lambda}^{\\ \\nu} & = 0, && (\\mu = 0,1,2,3), \n\t\t\\label{E:wavecoordinategauge2} \\\\\n\t(g^{-1})^{\\kappa \\lambda} \\partial_{\\kappa} g_{\\lambda \\mu} - \\frac{1}{2} (g^{-1})^{\\kappa \\lambda} \\partial_{\\mu} g_{\\kappa \n\t\t\\lambda} & = 0, && (\\mu = 0,1,2,3), \\label{E:wavecoordinategauge3} \\\\\n\t\\partial_{\\nu} [\\sqrt{|\\mbox{det} \\, g|}(g^{-1})^{\\mu \\nu}] & = 0, && (\\mu = 0,1,2,3). \\label{E:wavecoordinategauge4}\n\\end{align}\n\\end{subequations}\nWe also note that condition \\eqref{E:wavecoordinategauge4} follows from the identity\n\n\\begin{align} \\label{E:ContractedChristoffelIdendity}\n\t\\Gamma^{\\mu} \\eqdef (g^{-1})^{\\kappa \\lambda} \\Gamma_{\\kappa \\ \\lambda}^{\\ \\mu}\n\t= - \\frac{1}{\\sqrt{|\\mbox{det} \\, g|}} \\partial_{\\nu} [\\sqrt{|\\mbox{det} \\, g|}(g^{-1})^{\\mu \\nu}], && (\\mu = 0,1,2,3),\n\\end{align}\nwhich holds in any coordinate system. Furthermore, if the wave coordinate system is also interpreted to be a coordinate system in which the Minkowski metric takes the form $m_{\\mu \\nu} = \\mbox{diag}(-1,1,1,1),$ then all coordinate derivatives $\\partial$ can be interpreted as covariant\nderivatives $\\nabla,$ where $\\nabla$ is the Levi-Civita connection corresponding to the Minkowski metric. \\textbf{Throughout the article, we will often take this point of view, because it allows for a covariant interpretation of all of our equations}.\n\nWe remark that the use of wave coordinates in the context of the Einstein equations goes back at least to the work \\cite{tD1921} of de Donder. However, the role of wave coordinates in the context of the local aspects of the initial-value problem formulation of the Einstein equations was realized to its fullest extent by Choquet-Bruhat in \\cite{CB1952}. See Section \\ref{SS:WaveCoordinatesPreserved} for further discussion on the viability of using wave coordinates to analyze the system \\eqref{E:IntroEinsteinagain} - \\eqref{E:IntrodMis0again}.\n\n\n\n\\subsection{The Lagrangian formulation of nonlinear electromagnetism} \\label{SS:Lagrangianformluationofnonlinearelectromagnetism}\n\nIn this section, we recall some standard facts concerning a classical electromagnetic field theory in a Lorentzian spacetime \n$(\\mathbb{R}^{1+3},g_{\\mu \\nu}).$ Our goal is to explain the origin of the equations \\eqref{E:IntrodFaris0again} - \\eqref{E:IntrodMis0again}. We remark that for our purposes in this section, we may assume that the spacetime is known. The fundamental quantity in such a classical electromagnetic field theory is the \\emph{Faraday tensor} $\\Far_{\\mu \\nu},$ an anti-symmetric type $\\binom{0}{2}$ tensorfield (i.e., a two-form). We assume the \\emph{Faraday-Maxwell law}, which is the postulate that $\\Far_{\\mu \\nu}$ is closed:\n\n\\begin{align} \\label{E:dFis0}\n\t(d \\Far)_{\\lambda \\mu \\nu} = 0,&& (\\lambda, \\mu, \\nu = 0,1,2,3),\n\\end{align}\nwhere $d$ denotes the exterior derivative operator. \n\nWe restrict our attention to covariant theories of nonlinear electromagnetism arising from a Lagrangian $\\mathscr{L}.$  In such a theory, the Hodge dual\\footnote{For brevity, we often refer to $\\Ldual$ as the Lagrangian.} $\\Ldual$ of $\\mathscr{L}$ is a scalar-valued function of the two invariants of the Faraday tensor, which we denote by $\\Farinvariant_{(1)}$ and $\\Farinvariant_{(2)}:$ \n\n\\begin{subequations}\n\\begin{align}\n\t\\Ldual & = \\Ldual(\\Farinvariant_{(1)},\\Farinvariant_{(2)}), \\\\\n\t \\Farinvariant_{(1)} & = \\Farinvariant_{(1)}[\\Far] \\eqdef \\frac{1}{2} (g^{-1})^{\\kappa \\mu} (g^{-1})^{\\lambda \\nu} \n\t \t\\Far_{\\kappa \\lambda} \\Far_{\\mu \\nu}, \\label{E:firstinvariant} \\\\\n\t\\Farinvariant_{(2)} & = \\Farinvariant_{(2)}[\\Far] \\eqdef \\frac{1}{4} (g^{-1})^{\\kappa \\mu} (g^{-1})^{\\lambda \\nu} \n\t\t\\Far_{\\kappa \\lambda} \\Fardual_{\\mu \\nu}\n\t\t= \\frac{1}{8} \\epsilon^{\\# \\kappa \\lambda \\mu \\nu} \\Far_{\\kappa \\lambda} \\Far_{\\mu \\nu}. \\label{E:secondinvariant}\n\\end{align}\n\\end{subequations}\nThroughout the article, we use $\\star$ to denote the Hodge duality operator corresponding to the spacetime metric \n$g_{\\mu \\nu}:$ \n\n\\begin{align}\t\n\t\\Fardual^{\\#\\mu \\nu} & \\eqdef \\frac{1}{2} \\epsilon^{\\#\\mu \\nu \\kappa \\lambda} \\Far_{\\kappa \\lambda}.\n\\end{align}\nHere, $\\epsilon^{\\#\\mu \\nu \\kappa \\lambda}$ is totally anti-symmetric with normalization $\\epsilon^{\\#0123}= \n-|\\mbox{det} \\, g|^{-1\/2},$ while $\\epsilon_{\\mu \\nu \\kappa \\lambda}$ is totally anti-symmetric with normalization $\\epsilon_{0123}= |\\mbox{det} \\, g|^{1\/2}.$ See Section \\ref{SS:Indices} concerning our use of the notation $\\#.$ \n\nWe now introduce the \\emph{Maxwell tensor} $\\Max_{\\mu \\nu},$ a two-form whose Hodge dual $\\Maxdual_{\\mu \\nu}$ is defined by\n\n\\begin{align} \\label{E:Maxdualdef}\n\t\\Maxdual^{\\# \\mu \\nu} & \\eqdef \\frac{\\partial \\Ldual}{\\partial \\Far_{\\mu \\nu}}.\n\\end{align}\nWe also postulate that $\\Max_{\\mu \\nu}$ is closed:\n\n\\begin{align} \\label{E:dMis0}\n\t(d \\Max)_{\\lambda \\mu \\nu} = 0,&& (\\lambda, \\mu, \\nu = 0,1,2,3).\n\\end{align}\nTaken together, \\eqref{E:dFis0} and \\eqref{E:dMis0} are the electromagnetic equations for $\\Far_{\\mu \\nu}$ corresponding to $\\Ldual.$ \n\nWe remark for future use that it can be easily checked that equation \\eqref{E:dFis0} is equivalent to any of\n\n\\begin{subequations}\n\\begin{align} \n\t\\mathscr{D}_{\\lambda} \\Far_{\\mu \\nu} + \\mathscr{D}_{\\nu} \\Far_{\\lambda \\mu} + \\mathscr{D}_{\\mu} \\Far_{\\nu \\lambda} &= 0,\n\t\t&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dFis0Dversion} \\\\\n\t\\nabla_{\\lambda} \\Far_{\\mu \\nu} + \\nabla_{\\nu} \\Far_{\\lambda \\mu} + \\nabla_{\\mu} \\Far_{\\nu \\lambda} &= 0,\n\t\t&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dFis0nablaversion} \\\\\n\t\\mathscr{D}_{\\mu} \\Fardual^{\\# \\mu \\nu} & = 0,&& (\\nu = 0,1,2,3), \\label{E:DdivergenceofFardualis0} \\\\\n\t\\nabla_{\\mu} \\FarMinkdual^{\\mu \\nu} & = 0,&& (\\nu = 0,1,2,3), \\label{E:DdivergenceofFarMinkdualis0}\n\\end{align}\n\\end{subequations}\nand that equation \\eqref{E:dMis0} is equivalent to any of\n\n\n\\begin{subequations}\n\\begin{align} \n\t\\mathscr{D}_{\\lambda} \\Max_{\\mu \\nu} + \\mathscr{D}_{\\nu} \\Max_{\\lambda \\mu} + \\mathscr{D}_{\\mu} \\Max_{\\nu \\lambda} &= 0,\n\t\t&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dMis0Dversion} \\\\\n\t\\nabla_{\\lambda} \\Max_{\\mu \\nu} + \\nabla_{\\nu} \\Max_{\\lambda \\mu} + \\nabla_{\\mu} \\Max_{\\nu \\lambda} &= 0,\n\t\t&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dMis0nablaversion} \\\\\n\t\\mathscr{D}_{\\mu} \\Maxdual^{\\# \\mu \\nu} & = 0,&& (\\nu = 0,1,2,3), \\label{E:Euler-Lagrange} \\\\\n\t\\nabla_{\\mu} \\MaxMinkdual^{\\mu \\nu}& = 0,&& (\\nu = 0,1,2,3). \\label{E:DdivergenceofMaxMinkdualis0}\n\\end{align}\n\\end{subequations}\nIn the above formulas, $\\ostar$ denotes the Hodge duality operator corresponding to the Minkowski metric $m_{\\mu \\nu};$ this operator is defined in Section \\ref{SS:Hodge}.\n\n\nWe state as a lemma the following identities, which will be used for various computations. We leave the proof\nas a simple exercise for the reader.\n\n\\begin{lemma} \\label{L:electromagneticidentities}\n\t\\textbf{(Identities)}\n\tThe following identities hold:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\frac{\\partial |\\mbox{det} \\, g|}{\\partial g_{\\mu \\nu}} & = |\\mbox{det} \\, g| (g^{-1})^{\\mu \\nu}, \\\\\n\t\t\\frac{\\partial (g^{-1})^{\\kappa \\lambda}}{g_{\\mu \\nu}} & = -(g^{-1})^{\\kappa \\mu} (g^{-1})^{\\lambda \\nu}, \\\\\n\t\t\\Farinvariant_{(2)}^2 & = |\\mbox{det} \\, \\Far| |\\mbox{det} \\, g|^{-1}, \\\\\n\t\t(g^{-1})^{\\kappa \\lambda }\\Far_{\\mu \\kappa} \\Fardual_{\\nu \\lambda} & = \\Farinvariant_{(2)} g_{\\mu \\nu}, \\\\\n\t\t\n\t\n\t\t\\frac{\\partial \\Farinvariant_{(1)}}{\\partial g_{\\mu \\nu}} & = \n\t\t\t- g_{\\kappa \\lambda} \\Far^{\\#\\mu \\kappa}\\Far^{\\#\\nu \\lambda}, \\\\\n\t\t\\frac{\\partial \\Farinvariant_{(2)}}{\\partial g_{\\mu \\nu}} & = - \\frac{1}{2} \\Farinvariant_{(2)} (g^{-1})^{\\mu \\nu}, \\\\\n\t\n\t\t\n\t\t\\frac{\\partial \\Farinvariant_{(1)}}{\\partial \\Far_{\\mu \\nu}} & = 2 \\Far^{\\#\\mu \\nu}, \n\t\t\t\\label{E:partialfirstinvariantpartialFarmunu} \\\\\n\t\t\\frac{\\partial \\Farinvariant_{(2)}}{\\partial \\Far_{\\mu \\nu}} & = \\Fardual^{\\#\\mu \\nu}, \\\\\n\t\t\\frac{\\partial \\Far^{\\#\\mu \\nu}}{\\partial \\Far_{\\kappa \\lambda}} & \n\t\t\t= (g^{-1})^{\\mu \\kappa} (g^{-1})^{\\nu \\lambda} - (g^{-1})^{\\mu \\lambda}(g^{-1})^{\\nu \\kappa}, \\\\\n\t\t\\frac{\\partial \\Fardual^{\\#\\mu \\nu}}{\\partial \\Far_{\\kappa \\lambda}} & = \n\t\t\t\\epsilon^{\\#\\mu \\nu \\kappa \\lambda}, \\label{E:partialFardualmunupartialFarkappalambda} \\\\ \n\t\t\\mathscr{D}_{\\mu} \\Farinvariant_{(1)} & = \\Far^{\\# \\kappa \\lambda} \\mathscr{D}_{\\mu} \\Far_{\\kappa \\lambda}, \n\t\t\t\\qquad (\\mu = 0,1,2,3), \\label{E:DmuFarinvariant1} \\\\\n\t\t\\mathscr{D}_{\\mu} \\Farinvariant_{(2)} & = \\frac{1}{2} \\Fardual^{\\# \\kappa \\lambda} \\mathscr{D}_{\\mu} \\Far_{\\kappa \\lambda},\n\t\t\t\\qquad (\\mu = 0,1,2,3). \\label{E:DmuFarinvariant2}\n\t\\end{align}\n\t\\end{subequations}\n\n\\end{lemma}\n\n\\hfill $\\qed$\n\n\n\n\n\\subsection{Assumptions on the electromagnetic Lagrangian}\n\n\nThe familiar linear Maxwell-Maxwell equations correspond to the Lagrangian\n\n\\begin{align} \\label{E:LlinearMaxwell}\n\t\\Ldual_{(Maxwell)} & = - \\frac{1}{2} \\Farinvariant_{(1)},\n\\end{align}\nwhich by \\eqref{E:Maxdualdef} and \\eqref{E:partialfirstinvariantpartialFarmunu} leads to the relationship\n\n\\begin{align}\n\t\\Max_{\\mu \\nu}^{(Maxwell)} = \\Fardual_{\\mu \\nu}.\n\\end{align}\nRoughly speaking, we will assume that our electromagnetic Lagrangian is a covariant perturbation of $\\Ldual_{(Maxwell)}.$\nMore precisely, we make the following assumptions concerning our Lagrangian $\\Ldual.$\\vspace{.5in}\n\n\\begin{center}\n\t\\textbf{\\LARGE Assumptions on the electromagnetic Lagrangian}\n\\end{center}\nWe assume that in a neighborhood of $(0,0),$ $\\Ldual$ is an $\\dParameter + 2-$times (where $\\dParameter \\geq 8$) continuously differentiable function of the invariants $(\\Farinvariant_{(1)},\\Farinvariant_{(2)})$ that can be expanded as follows:\n\n\\begin{subequations}\n\\begin{align} \\label{E:Ldualassumptions} \n\t\\Ldual = \\Ldual_{(Maxwell)} + O^{\\dParameter+2}\\big(|(\\Farinvariant_{(1)},\\Farinvariant_{(2)})|^2\\big).\n\\end{align}\nThe notation $O^{\\dParameter+2}(\\cdots)$ is defined in Section \\ref{SS:Oando}.\n\nWe also assume that the corresponding energy-momentum tensor $T_{\\mu \\nu},$ which is defined below in \\eqref{E:electromagnetictensorupper},\nsatisfies the \\emph{dominant energy condition}, which is the assumption that \n\n\\begin{align} \\label{E:DEC}\n\tT_{\\kappa \\lambda} X^{\\kappa} Y^{\\lambda} \\geq 0\n\\end{align}\n\\end{subequations}\nwhenever the following conditions are satisfied:\n\n\\begin{itemize}\n\t\\item $X,$ $Y$ are both timelike (i.e., $g_{\\kappa \\lambda}X^{\\kappa}X^{\\lambda} < 0,$ $g_{\\kappa \\lambda}Y^{\\kappa}Y^{\\lambda} < 0)$ \n\t\\item $X,$ $Y$ are $g-$future-directed.\n\\end{itemize}\t\n\nAs discussed in e.g. \\cite{gGcH2001}, sufficient conditions for the dominant energy condition to hold are\n\n\\begin{subequations}\n\\begin{align}\n\t\\frac{\\partial \\Ldual}{ \\partial \\Farinvariant_{(1)}} & < 0, \\label{E:DECL1} \\\\\n\t\\Ldual - \\Farinvariant_{(1)}\\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(1)}}\n\t\t- \\Farinvariant_{(2)}\\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(2)}} & \\leq 0. \\label{E:DECTrace}\n\\end{align}\n\\end{subequations}\nWe remark that it is straightforward to verify the sufficiency of these conditions\nusing equation \\eqref{E:AlternateelectromagnetictensorloweroinTermsofLagrangian} below, and that condition\n\\eqref{E:DECTrace} is equivalent to the non-positivity of the trace of the energy-momentum tensor corresponding to $\\Ldual.$\nFurthermore, we recall that the trace vanishes in the case of the linear Maxwell-Maxwell model.\n\n\\vspace{.5in}\t\n\n\\begin{remark}\n\tWe make the $\\dParameter +2-$times differentiability assumption because we will need to differentiate the equations\n\t\\eqref{E:EquationSatisfiedbyMaxdualChainruleExpandedfirstNversion} below $\\dParameter $ times in order to prove our\n\tmain stability theorem.\n\\end{remark}\n\nWe will now derive an equivalent version of the electromagnetic equations that will be used throughout the remainder of the article. The final form, which is valid only in a wave coordinate system, is given below in Lemma \\ref{L:MBIAsystem}. To begin, we use \\eqref{E:Maxdualdef}, Lemma \\ref{L:electromagneticidentities}, and the chain rule to compute that\n\n\\begin{align} \\label{E:MaxdualintermsofLagrangian}\n\t\\Maxdual^{\\#\\mu \\nu} & = 2\\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(1)}} \\Far^{\\#\\mu \\nu}\n\t\t+ \\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(2)}}\\Fardual^{\\#\\mu \\nu}.\n\\end{align}\nWe then use \\eqref{E:DdivergenceofFardualis0}, \\eqref{E:Euler-Lagrange}, and \\eqref{E:MaxdualintermsofLagrangian} to compute that the following equation holds:\n\n\\begin{align} \\label{E:FirstEquationSatisfiedbyMaxdual}\n\t- 2 \\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(1)}} \\mathscr{D}_{\\mu} \\Far^{\\#\\mu \\nu}\n\t- 2 \\Far^{\\#\\mu \\nu} \\mathscr{D}_{\\mu} \\Big(\\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(1)}}\\Big)\n\t- \\Fardual^{\\#\\mu \\nu} \\mathscr{D}_{\\mu} \\Big(\\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(2)}}\\Big) & = 0.\n\\end{align}\nFurthermore, using the chain rule and the fact that $\\mathscr{D}_{\\mu} \\phi = \\nabla_{\\mu} \\phi$ \nfor scalar-valued functions $\\phi,$ it follows from \\eqref{E:FirstEquationSatisfiedbyMaxdual} that\n\n\\begin{align} \\label{E:EquationSatisfiedbyMaxdualChainruleExpanded}\n\t- 2 \\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(1)}} \\mathscr{D}_{\\mu} \\Far^{\\#\\mu \\nu}\n\t& - \\Big(2 \\Far^{\\#\\mu \\nu} \\frac{\\partial^2 \\Ldual}{\\partial \\Farinvariant_{(1)}^2}\n\t\t+ \\Fardual^{\\#\\mu \\nu} \\frac{\\partial^2 \\Ldual}{\\partial \\Farinvariant_{(1)} \\partial \\Farinvariant_{(2)}} \n\t\t\\Big) \\nabla_{\\mu} \\Farinvariant_{(1)}  \\\\\n\t& - \\Big(2 \\Far^{\\#\\mu \\nu} \\frac{\\partial^2 \\Ldual}{\\partial \\Farinvariant_{(1)} \\partial \n\t\t\\Farinvariant_{(2)}}+ \\Fardual^{\\#\\mu \\nu} \\frac{\\partial^2 \\Ldual}{\\partial \\Farinvariant_{(2)}^2} \\Big) \n\t\t\\nabla_{\\mu} \\Farinvariant_{(2)} = 0. \\notag\n\\end{align}\n\nWe note for future use that equation \\eqref{E:EquationSatisfiedbyMaxdualChainruleExpanded} can be expressed as\n\n\\begin{align} \\label{E:EquationSatisfiedbyMaxdualChainruleExpandedfirstNversion}\n\tN^{\\#\\mu \\nu \\kappa \\lambda} \\mathscr{D}_{\\mu} \\Far_{\\kappa \\lambda} = 0,&& (\\nu = 0,1,2,3),\n\\end{align}\nwhere the tensorfield $N^{\\#\\mu \\nu \\kappa \\lambda}$ is defined by\n\n\\begin{align} \\label{E:firstNdef}\n\tN^{\\#\\mu \\nu \\kappa \\lambda} \n\t& \\eqdef - \\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(1)}} \\Big\\lbrace(g^{-1})^{\\mu \\kappa} (g^{-1})^{\\nu \\lambda} - (g^{-1})^{\\mu \\lambda} \n\t\t(g^{-1})^{\\nu \\kappa} \\Big\\rbrace \n\t\t\\ - \\ 2 \\frac{\\partial^2 \\Ldual}{\\partial \\Farinvariant_{(1)}^2} \\Far^{\\#\\mu \\nu} \\Far^{\\# \\kappa \\lambda} \\\\\n\t& \\ \\ - \\ \\frac{\\partial^2 \\Ldual}{\\partial \\Farinvariant_{(1)} \\partial \n\t\t\\Farinvariant_{(2)}} \\Big\\lbrace \\Far^{\\#\\mu \\nu} \\Fardual^{\\# \\kappa \\lambda} \n\t\t\\ + \\ \\Fardual^{\\#\\mu \\nu} \\Fardual^{\\# \\kappa \\lambda} \\Big\\rbrace\n\t\t\\ - \\ \\frac{1}{2} \\frac{\\partial^2 \\Ldual}{\\partial \\Farinvariant_{(2)}^2} \\Fardual^{\\#\\mu \\nu} \\Fardual^{\\# \\kappa \\lambda}. \\notag\n\\end{align}\nWe also note that $N^{\\#\\mu \\nu \\kappa \\lambda}$ has the following symmetry properties, which will play an important role during our construction of suitable energies for $\\Far_{\\mu \\nu}$ (and in particular during our proof of Lemma \\ref{L:DivergenceofStress}):\n\n\n\\begin{subequations}\n\\begin{align}\n\tN^{\\#\\nu \\mu \\kappa \\lambda} & = - N^{\\#\\mu \\nu \\kappa \\lambda},&& (\\kappa, \\lambda, \\mu, \\nu = 0,1,2,3), \t\n\t\t\\label{E:Nminussignproperty1} \\\\\n\tN^{\\#\\mu \\nu \\lambda \\kappa } & = - N^{\\#\\mu \\nu \\kappa \\lambda},&& (\\kappa, \\lambda, \\mu, \\nu = 0,1,2,3), \t\n\t\t\\label{E:Nminussignproperty2} \\\\\n\tN^{\\#\\kappa \\lambda \\mu \\nu } & = N^{\\#\\mu \\nu \\kappa \\lambda},&& (\\kappa, \\lambda, \\mu, \\nu = 0,1,2,3). \t\n\t\t\\label{E:Nsymmetryproperty}\n\\end{align}\n\\end{subequations}\nThe moral reason that the above properties are satisfied is that $N^{\\#\\mu \\nu \\kappa \\lambda}$ is closely related to the Hessian of $\\Ldual:$\n\n\\begin{align} \\label{E:NisHessian}\n\tN^{\\#\\mu \\nu \\kappa \\lambda} = - \\frac{1}{2} \\frac{\\partial^2 \\Ldual}{\\partial \\Far_{\\mu \\nu} \\partial \\Far_{\\kappa \\lambda}}\n\t\t\\ + \\ \\frac{1}{2} \\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(2)}}\\epsilon^{\\#\\mu \\nu \\kappa \\lambda}.\n\\end{align}\nWe have added the last term on the right-hand side of \\eqref{E:NisHessian} in order to cancel a term appearing in the Hessian; this is permissible because equation \\eqref{E:dFis0Dversion} implies that this term does not contribute to equation \\eqref{E:EquationSatisfiedbyMaxdualChainruleExpandedfirstNversion}.\n\n\nOur next goal is to formulate a ``reduced'' electromagnetic equation that is equivalent to equation \\eqref{E:EquationSatisfiedbyMaxdualChainruleExpandedfirstNversion} in a wave coordinate system, and \nto decompose the reduced equation into the principal terms and error terms of an equation involving the Minkowski connection $\\nabla.$ This is accomplished in Lemma \\ref{L:MBIAsystem} below. Before proving this lemma, we first provide the following preliminary lemma, whose simple proof is left to the reader.\n\n\n\\begin{lemma} \\label{L:gmhexpansions} \\textbf{(Expansions)}\n\tAssume that the electromagnetic Lagrangian $\\Ldual$ satisfies \\eqref{E:Ldualassumptions}.\n\tThen in terms of the expansion $h_{\\mu \\nu} \\eqdef g_{\\mu \\nu} - m_{\\mu \\nu}$ from \\eqref{E:gmhexpansion}, and with \n\t$H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu},$ we have that:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\tH^{\\mu \\nu} & = - h^{\\mu \\nu} + O^{\\infty}(|h|^2) \n\t\t\t= - h^{\\mu \\nu} \\ + \\ O^{\\infty}(|H|^2),&& \\label{E:Hintermsofh} \\\\\n\t\t\\nabla_{\\lambda} (g^{-1})^{\\mu \\nu} & = - (g^{-1})^{\\mu \\mu'} (g^{-1})^{\\nu \\nu'} \\nabla_{\\lambda} h_{\\mu' \\nu'}\n\t\t\t= - (m^{-1})^{\\mu \\mu'} (m^{-1})^{\\nu \\nu'} \\nabla_{\\lambda} h_{\\mu' \\nu'} \\ + \\ O^{\\infty}(|h||\\nabla h|),&&\n\t\t\t\\label{E:derivativeginversegmhexpansion} \\\\\n\t\t|\\mbox{det} \\, g| & =  1 \\ + \\ (m^{-1})^{\\kappa \\lambda} h_{\\kappa \\lambda} \\ + \\ O^{\\infty}(|h|^2)\n\t\t\t= 1 \\ - \\ m_{\\kappa \\lambda} H^{\\kappa \\lambda} + O^{\\infty}(|H|^2),&& \\\\\n\t\t|\\mbox{det} \\, g|^{1\/2} & =  1 + \\frac{1}{2} (m^{-1})^{\\kappa \\lambda} h_{\\kappa \\lambda} \\ + \\ O^{\\infty}(|h|^2)\n\t\t\t= 1 \\ - \\ \\frac{1}{2} m_{\\kappa \\lambda} H^{\\kappa \\lambda} + O^{\\infty}(|H|^2),&& \\\\\n\t\t|\\mbox{det} \\, g|^{-1\/2} & =  1 \\ - \\ \\frac{1}{2}(m^{-1})^{\\kappa \\lambda} h_{\\kappa \\lambda} \\ + \\ O^{\\infty}(|h|^2)\n\t\t\t= 1 \\ + \\ \\frac{1}{2} m_{\\kappa \\lambda} H^{\\kappa \\lambda} \\ + \\ O^{\\infty}(|H|^2),&& \\label{E:detgminusonehalfexpansion} \\\\\n\t\t\\epsilon^{\\#\\mu \\nu \\kappa \\lambda} & = - \\big(1 + O^{\\infty}(|h|) \\big) [\\mu \\nu \\kappa \\lambda],&& \n\t\t\t\\label{E:volumeformraisedhexpansion} \\\\\n\t\t\\epsilon_{\\mu \\nu \\kappa \\lambda} & = \\big(1 + O^{\\infty}(|h|) \\big) [\\mu \\nu \\kappa \\lambda],&& \n\t\t\t\\label{E:volumeformloweredhexpansion} \\\\\n\t\t\\Far^{\\#\\mu \\nu} & = \\Far^{\\mu \\nu}\n\t\t\t\\ + \\ O^{\\infty}(|h||\\Far|) \\eqdef (m^{-1})^{\\mu \\kappa}(m^{-1})^{\\nu \\lambda} \\Far_{\\kappa \\lambda} \\ + \\ O^{\\infty}(|h||\\Far|) ,&& \\\\\n\t\t\\Fardual^{\\# \\mu \\nu} & \t= \\FarMinkdual^{\\mu \\nu} \\ + \\ O^{\\infty}(|h||\\Far|)\n\t\t\t\\eqdef - \\frac{1}{2} [\\mu \\nu \\kappa \\lambda] \\Far_{\\kappa \\lambda} \\ + \\ O^{\\infty}(|h||\\Far|), && \n\t\t\t\\label{E:FargDualIntermsofFarmDual}\\\\\n\t\t\\Farinvariant_{(1)} & = \\frac{1}{2} (m^{-1})^{\\kappa \\mu} (m^{-1})^{\\lambda \\nu} \\Far_{\\kappa \\lambda} \\Far_{\\mu \\nu} \n\t\t\t\\ + \\ O^{\\infty}(|h||\\Far|^2),&& \\\\\n\t\t\\Farinvariant_{(2)} & = - \\frac{1}{8} [\\mu \\nu \\kappa \\lambda ] \\Far_{\\mu \\nu} \\Far_{\\kappa \\lambda} \n\t\t\t\\ + \\ O^{\\infty}(|h||\\Far|^2),&& \\\\\n\t\t\\Ldual & = - \\frac{1}{4} (m^{-1})^{\\eta \\kappa} (m^{-1})^{\\zeta \\lambda} \\Far_{\\kappa \\lambda} \\Far_{\\eta \\zeta} \n\t\t\t\\ + \\ O^{\\dParameter+2}(|h||\\Far|^2) \\ + \\ O^{\\dParameter+2}(|\\Far|^3;h),&& \\\\\n\t\t\\nabla \\Farinvariant_{(i)} & = O^{\\infty}(|\\Far||\\nabla \\Far|) \n\t\t\t\\ + \\ O^{\\infty}(|\\nabla h||\\Far|^2;h) \\ + \\ O^{\\infty}(|h||\\Far||\\nabla \\Far|), && (i = 1,2), \\\\\n\t\\Max_{\\mu \\nu} & = \\FarMinkdual_{\\mu \\nu} \\ + \\ O^{\\dParameter+1}(|h||\\Far|) \n\t\t\\ + \\ O^{\\dParameter+1}(|\\Far|^3;h).&& \\label{E:MaxintermsofFarMinkDualPlusError}\n\t\\end{align}\n\t\\end{subequations}\n\t\n\tIn formulas \\eqref{E:volumeformraisedhexpansion} - \\eqref{E:volumeformloweredhexpansion}, \n\t$[\\mu \\nu \\kappa \\lambda]$ is totally anti-symmetric with normalization $[0123] = 1,$ while $\\star$ denotes the Hodge duality \n\toperator corresponding to the spacetime metric $g_{\\mu \\nu},$ and $\\ostar$ denotes the Hodge duality operator corresponding \n\tto the Minkowski metric $m_{\\mu \\nu}.$ \tFurthermore, the notation $O(\\cdots)$ is defined in Section \\ref{SS:Oando}.\n\\end{lemma}\n\n\\hfill $\\qed$\n\n\n\n\\subsection{The reduced electromagnetic equations} \\label{SS:ReducedElectromagnetic}\n\nIn this section, we provide the aforementioned decomposition of the reduced electromagnetic equations. \n\n\n\\begin{lemma} \\label{L:MBIAsystem} \\textbf{(The reduced electromagnetic equations)}\n\tAssume that the wave coordinate condition \\eqref{E:wavecoordinategauge1} holds.\n\tThen in terms of the expansion \\eqref{E:gmhexpansion}, the system of electromagnetic equations  \n\t\\eqref{E:dFis0}, \\eqref{E:EquationSatisfiedbyMaxdualChainruleExpandedfirstNversion} is equivalent to the \n\tfollowing reduced system of equations:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\nabla_{\\lambda} \\Far_{\\mu \\nu} + \\nabla_{\\mu} \\Far_{\\nu \\lambda} + \\nabla_{\\nu} \\Far_{\\lambda \\mu} & = 0, \n\t\t\t&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dFis0Expansion} \\\\\n\t\tN^{\\#\\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda} & =\n\t\t\t\\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla h, \\Far) + O^{\\dParameter}(|h||\\nabla h||\\Far|) + O^{\\dParameter}(|\\nabla h||\\Far|^2;h), && (\\nu = 0,1,2,3), \n\t\t\t\\label{E:dMis0Expansion}\n\t\\end{align}\n\t\\end{subequations}\n\twhere\n\t\n\t\\begin{align}\n\t\tN^{\\#\\mu \\nu \\kappa \\lambda} & = \\frac{1}{2} \\Big\\lbrace (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - \n\t\t\t(m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\n\t\t\t- h^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} + h^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\n\t\t\t- (m^{-1})^{\\mu \\kappa} h^{\\nu \\lambda} + (m^{-1})^{\\mu \\lambda} h^{\\nu \\kappa}\\Big\\rbrace \\label{E:secondNdef}  \\\\\n\t\t\t& \\ \\ + N_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda}, \\notag \\\\\n\t\t\\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla h, \\Far) & = \n\t\t\t(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\nu'} (m^{-1})^{\\lambda \\lambda'} (\\nabla_{\\mu} h_{\\nu' \\lambda'}) \n\t\t\t\\Far_{\\kappa \\lambda}. \\label{E:Q2Farfirstdef}\n\t\\end{align}\n\t\n\tFurthermore, \n\t\n\t\\begin{align} \\label{E:NtriangleErrorProperties}\n\t\tN_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda} = O^{\\dParameter}\\big(|(h,\\Far)|^2\\big),\n\t\\end{align}\n\tand like $N^{\\#\\mu \\nu \\kappa \\lambda},$\n\tthe tensorfield $N_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda}$ also \n\tpossesses the symmetry properties \\eqref{E:Nminussignproperty1} - \\eqref{E:Nsymmetryproperty}.\n\\end{lemma}\n\n\\begin{remark}\n\tEquations \\eqref{E:secondNdef} - \\eqref{E:secondNdef} are valid only in a wave coordinate system. Hence, we\n\trefer to the system \\eqref{E:secondNdef} - \\eqref{E:secondNdef} as the ``reduced'' electromagnetic equations.\n\\end{remark}\t\n\n\\begin{proof}\nWe use the assumption \\eqref{E:Ldualassumptions} and the Leibniz rule to expand \\eqref{E:EquationSatisfiedbyMaxdualChainruleExpanded} and apply the results of Lemma \\ref{L:gmhexpansions}, arriving at the following expansion:\n\n\\begin{align} \\label{E:MBIEuler-Lagrangeexpanded}\n\t\\mathscr{D}_{\\mu} \\Far^{\\#\\mu \\nu} \n\t\t\\ + \\ \\widetilde{N}^{\\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda} =  \n\t\tO^{\\dParameter}(|h||\\nabla h||\\Far|) \\ + \\ O^{\\dParameter}(|\\nabla h||\\Far|^2;h),\n\\end{align}\nwhere $\\widetilde{N}^{\\mu \\nu \\kappa \\lambda} = O^{\\dParameter}\\big(|(h,\\Far)|^2\\big).$ Let us now decompose the $\\mathscr{D}_{\\mu} \\Far^{\\#\\mu \\nu}$ term. Using the anti-symmetry of $\\Far^{\\#\\mu \\nu},$ the symmetry of the Christoffel symbol $\\Gamma_{\\mu \\ \\lambda}^{\\ \\nu}$ under the exchanges $\\mu \\leftrightarrow \\lambda,$ the identity \n$\\Gamma_{\\kappa \\ \\mu}^{\\ \\kappa} = \\frac{1}{\\sqrt{|\\mbox{det $g$}|}} \\nabla_{\\mu} (\\sqrt{|\\mbox{det} \\, g|}),$ and the wave coordinate condition $\\nabla_{\\mu}\\big[\\sqrt{|\\mbox{det} \\, g|} (g^{-1})^{\\mu \\kappa} \\big] = 0,$ $(\\kappa = 0,1,2,3),$ \nwe have that\n\n\\begin{align} \\label{E:divergenceFarardayexpansion1}\n\t\\mathscr{D}_{\\mu} \\Far^{\\#\\mu \\nu} & = \\nabla_{\\mu} \\Far^{\\#\\mu \\nu} \\ + \\ \\Gamma_{\\kappa \\ \\mu}^{\\ \\kappa} \\Far^{\\#\\mu \\nu}\n\t\t\\ + \\ \\Gamma_{\\mu \\ \\lambda}^{\\ \\nu}\\Far^{\\#\\mu \\lambda} \\\\\n\t& = \\nabla_{\\mu} \\big[(g^{-1})^{\\mu \\kappa} (g^{-1})^{\\nu \\lambda} \\Far_{\\kappa \\lambda}\\big] \n\t\t\t\\ + \\ \\Big[\\frac{1}{\\sqrt{|\\mbox{det} \\, g|}} \\nabla_{\\mu} (\\sqrt{|\\mbox{det} \\, g|})\\Big] \n\t\t\t (g^{-1})^{\\mu \\kappa} (g^{-1})^{\\nu \\lambda} \\Far_{\\kappa \\lambda} \\notag \\\\\n\t& = \\frac{1}{\\sqrt{|\\mbox{det} \\, g|}} \n\t\t\\nabla_{\\mu} \\big[\\sqrt{|\\mbox{det} \\, g|} (g^{-1})^{\\mu \\kappa} (g^{-1})^{\\nu \\lambda} \\Far_{\\kappa \\lambda}\\big]\n\t= (g^{-1})^{\\mu \\kappa} (g^{-1})^{\\nu \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \n\t\t\\ + \\ \\big[(g^{-1})^{\\mu \\kappa} \\nabla_{\\mu} (g^{-1})^{\\nu \\lambda} \\big] \\Far_{\\kappa \\lambda}. \\notag\n\\end{align}\nUsing \\eqref{E:Hintermsofh}, we conclude that the term $(g^{-1})^{\\mu \\kappa} (g^{-1})^{\\nu \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda}$ on the right-hand side of \\eqref{E:divergenceFarardayexpansion1} can be expressed as\n\n\\begin{align}\n\t& \\frac{1}{2} \\Big\\lbrace (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\n\t\t- h^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} + h^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa} - (m^{-1})^{\\mu \\kappa} h^{\\nu \\lambda} + \n\t\t(m^{-1})^{\\mu \\lambda} h^{\\nu \\kappa}\\Big\\rbrace \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \\\\ \n\t& \\ \\ + \\ O^{\\dParameter}(|h^2|) \\nabla_{\\mu} \\Far_{\\kappa \\lambda}, \\notag\n\\end{align}\t\nwhere the term in braces is equal to the term in braces on the right-hand side of \\eqref{E:secondNdef}. \n\nSimilarly, using \\eqref{E:derivativeginversegmhexpansion}, we conclude that the term $\\big[(g^{-1})^{\\mu \\kappa} \\nabla_{\\mu} (g^{-1})^{\\nu \\lambda} \\big] \\Far_{\\kappa \\lambda}$ on the right-hand side of \\eqref{E:divergenceFarardayexpansion1} is equal to $-\\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla h, \\Far)$ + $O^{\\dParameter}(|h||\\nabla h||\\Far|),$ where $\\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla h, \\Far)$ is defined in \\eqref{E:Q2Farfirstdef}. Combining these expansions with \\eqref{E:MBIEuler-Lagrangeexpanded}, we arrive at \\eqref{E:dMis0Expansion} - \\eqref{E:NtriangleErrorProperties}.\n\t\nThe fact that $N_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda}$ possesses the symmetry properties \\eqref{E:Nminussignproperty1} - \\eqref{E:Nsymmetryproperty} follows trivially from the fact that both  $N^{\\#\\mu \\nu \\kappa \\lambda}$ and the term in braces on the right-hand side of \\eqref{E:secondNdef} both satisfy these properties.\n\n\\end{proof}\n\n\n\\begin{remark} \\label{R:ReducedElectromagneticInhomogeneous}\n\t\n\tWith the help of the identity \\eqref{E:ContractedChristoffelIdendity}, the above proof shows that the\n\treduced equation \\eqref{E:dMis0Expansion} is obtained by adding the inhomogeneous term \n\t$- \\Gamma^{\\kappa} (g^{-1})^{\\nu \\lambda} \\Far_{\\kappa \\lambda}$ to the right-hand side of \n\tequation \\eqref{E:EquationSatisfiedbyMaxdualChainruleExpandedfirstNversion}:\n\t \n\t\n\t\\begin{align} \\label{E:ModifiedElectromagneticEquationWithInhomogeneousTerm}\n\t\tN^{\\#\\mu \\nu \\kappa \\lambda} \\mathscr{D}_{\\mu} \\Far_{\\kappa \\lambda} \n\t\t= - \\Gamma^{\\kappa} (g^{-1})^{\\nu \\lambda} \\Far_{\\kappa \\lambda}.\n\t\\end{align}\n\tWe will make use of this fact in Section \\ref{SS:WaveCoordinatesPreserved}.\n\t\n\\end{remark}\n\n\n\\subsection{The energy-momentum tensor} \\label{SS:EMT}\nIn this section, we discuss the energy-momentum tensor $T_{\\mu \\nu}$ appearing on the right-hand side of \\eqref{E:IntroEinsteinagain}. We recall that the energy-momentum tensor for an electromagnetic Lagrangian field theory is defined as follows:\n\n\\begin{align} \\label{E:electromagnetictensorupper}\n\tT^{\\#\\mu \\nu} & \\eqdef  2 \\frac{\\partial \\Ldual}{ \\partial g_{\\mu \\nu}} + (g^{-1})^{\\mu \\nu} \\Ldual.\n\\end{align}\t\nIt follows trivially from the definition \\eqref{E:electromagnetictensorupper} that $T_{\\mu \\nu}$ is symmetric:\n\n\\begin{align}\n\tT_{\\mu \\nu} & = T_{\\nu \\mu}, && (\\mu,\\nu = 0,1,2,3).\n\\end{align}\nFurthermore, we recall that if $\\Far_{\\mu \\nu}$ is a solution to the electromagnetic equations \\eqref{E:IntrodFaris0again} - \n\\eqref{E:IntrodMis0again}, then\n\n\\begin{align} \\label{E:TisDivergenceFree}\n\t\\mathscr{D}_{\\mu} T^{\\#\\mu \\nu} & = 0,&& (\\nu = 0,1,2,3).\n\\end{align}\n\nFor the class of electromagnetic energy-momentum tensors considered in this article, we can \nuse the chain rule and Lemma \\ref{L:electromagneticidentities} to express $T_{\\mu \\nu}$ as follows:\n\n\\begin{subequations}\n\\begin{align} \\label{E:electromagnetictensorloweroinTermsofLagrangian}\n\tT_{\\mu \\nu} & = - 2 \\frac{\\partial \\Ldual}{ \\partial \\Farinvariant_{(1)}} (g^{-1})^{\\kappa \\lambda}\n\t\t\\Far_{\\mu \\kappa} \\Far_{\\nu \\lambda} \n\t\t\\ - \\ \\Farinvariant_{(2)} \\frac{\\partial \\Ldual}{ \\partial \\Farinvariant_{(2)}} g_{\\mu \\nu} \n\t\t\\ + \\ g_{\\mu \\nu} \\Ldual \\\\\n\t\t& = - 2 \\frac{\\partial \\Ldual}{ \\partial \\Farinvariant_{(1)}} T_{\\mu \\nu}^{(Maxwell)}  \n\t\t\\ + \\ \\frac{1}{4} T g_{\\mu \\nu}, \\label{E:AlternateelectromagnetictensorloweroinTermsofLagrangian}\n\\end{align}\n\\end{subequations}\nwhere\n\n\\begin{align}\n\tT_{\\mu \\nu}^{(Maxwell)} \\eqdef (g^{-1})^{\\kappa \\lambda} \\Far_{\\mu \\kappa} \\Far_{\\nu \\lambda}\n\t \\ - \\ \\frac{1}{2} \\Farinvariant_{(1)} g_{\\mu \\nu}\n\\end{align}\nis the energy-momentum tensor corresponding to the linear Maxwell-Maxwell equations, and\n\n\\begin{align}\n\t(g^{-1})^{\\kappa \\lambda} T_{\\kappa \\lambda} = 4 \\Big(\\Ldual - \\Farinvariant_{(1)}\\frac{\\partial \\Ldual}{\\partial \n\t\t\\Farinvariant_{(1)}} - \\Farinvariant_{(2)}\\frac{\\partial \\Ldual}{\\partial \\Farinvariant_{(2)}} \\Big)\n\\end{align}\nis the trace of $T_{\\mu \\nu}$ with respect to $g_{\\mu \\nu}.$ Furthermore, from \\eqref{E:electromagnetictensorloweroinTermsofLagrangian} and the expansions of Lemma \\ref{L:gmhexpansions}, it follows that\n\n\\begin{align} \\label{E:TLowerExpansion}\n\tT_{\\mu \\nu} \n\t& = (m^{-1})^{\\kappa \\lambda}\\Far_{\\mu \\kappa} \\Far_{\\nu \\lambda} \n\t\t\\ - \\ \\frac{1}{4}m_{\\mu \\nu} (m^{-1})^{\\kappa \\eta} (m^{-1})^{\\lambda \\zeta} \\Far_{\\kappa \\lambda} \\Far_{\\eta \\zeta} \\\\\n\t& \\ \\ + \\ O^{\\dParameter+1}(|h||\\Far|^2) \\ + \\ O^{\\dParameter+1}(|\\Far|^3;h). \\notag\n\\end{align}\n\n\nWe now compute the right-hand side of (\\ref{E:IntroEinsteinagain}'). First, taking the trace of \\eqref{E:TLowerExpansion} with respect to $g,$ we compute that\n\n\\begin{align} \\label{E:TtraceExpansion}\n\t(g^{-1})^{\\kappa \\lambda} T_{\\kappa \\lambda} = O^{\\dParameter+1}(|h||\\Far|^2) \\ + \\ O^{\\dParameter+1}(|\\Far|^3;h).\n\\end{align}\nCombining \\eqref{E:TLowerExpansion} and \\eqref{E:TtraceExpansion}, and using the expansion \\eqref{E:gmhexpansion},\nwe have that the right-hand side of (\\ref{E:IntroEinsteinagain}') can be expressed as follows:\n\n\\begin{align} \\label{E:MBIrighthandsidefieldequations}\n\tT_{\\mu \\nu} \\ - \\ \\frac{1}{2} g_{\\mu \\nu} (g^{-1})^{\\kappa \\lambda} T_{\\kappa \\lambda} & = (m^{-1})^{\\kappa \\lambda} \n\t\t\\Far_{\\mu \\kappa} \\Far_{\\nu \\lambda} \n\t\t\\ - \\ \\frac{1}{4}m_{\\mu \\nu} (m^{-1})^{\\kappa \\eta} (m^{-1})^{\\lambda \\zeta} \\Far_{\\kappa \\lambda} \\Far_{\\eta \\zeta} \n\t\t\\ + \\ O^{\\dParameter+1}(|h||\\Far|^2) + O^{\\dParameter+1}(|\\Far|^3;h). \t\n\\end{align}\n\nTo conclude this section, we note for future use that if $\\Far_{\\mu \\nu}$ is a solution to the inhomogeneous system\n\n\\begin{subequations}\n\t\\begin{align}\n\t\t\\nabla_{\\lambda} \\Far_{\\mu \\nu} + \\nabla_{\\mu} \\Far_{\\nu \\lambda} + \\nabla_{\\nu} \\Far_{\\lambda \\mu} & = 0, \n\t\t\t&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dFis0DivergenceofT} \\\\\n\t\tN^{\\#\\mu \\nu \\kappa \\lambda} \\mathscr{D}_{\\mu} \\Far_{\\kappa \\lambda} & = \\mathfrak{I}^{\\nu}, && (\\nu = 0,1,2,3), \n\t\t\t\\label{E:dMisnot0DivertenceofT}\n\t\\end{align}\n\t\\end{subequations}\nthen with the help of Lemma \\ref{L:electromagneticidentities}, it can be shown that the following identity holds:\n\n\\begin{align} \\label{E:DivergenceofTidentity}\n\t(g^{-1})^{\\kappa \\lambda} \\mathscr{D}_{\\kappa} T_{\\lambda \\nu} & = \\mathfrak{I}^{\\kappa} \\Far_{\\nu \\kappa},&& (\\nu = 0,1,2,3).\n\\end{align}\nEquation \\eqref{E:TisDivergenceFree} corresponds to the special case $\\mathfrak{I}^{\\nu} = 0,$ \n$(\\nu = 0,1,2,3).$\n\n\n\\subsection{The modified Ricci tensor} \\label{SS:Rmunuinwave}\n\nThroughout the remainder of this article, we perform the standard wave coordinate system procedure (see e.g. \\cite{rW1984}) of replacing the Ricci tensor $R_{\\mu \\nu}$ in the Einstein's field equation \\eqref{E:IntroEinsteinagain} with a modified Ricci tensor $\\widetilde{R}_{\\mu \\nu}.$ As we will soon see, this replacement transforms equations \\eqref{E:IntroEinsteinagain} into a system of quasilinear wave equations.\n\n\\begin{definition}\nWe define the \\emph{modified Ricci tensor} $\\widetilde{R}_{\\mu \\nu}$ of the metric $g_{\\mu \\nu}$ as follows:\n\n\\begin{align} \\label{E:ModifiedRicci}\n\t\\widetilde{R}_{\\mu \\nu} \\eqdef R_{\\mu \\nu} \n\t \\ - \\ \\frac{1}{2} \\big\\lbrace  g_{\\kappa \\nu} \\mathscr{D}_{\\mu} \\Gamma^{\\kappa}\n\t \t+ g_{\\kappa \\mu} \\mathscr{D}_{\\nu} \\Gamma^{\\kappa} \\big\\rbrace \n\t\t\\ + \\ u_{\\mu \\nu \\kappa}(g,g^{-1},\\partial g) \\Gamma^{\\kappa},\n\\end{align}\nwhere the Ricci tensor $R_{\\mu \\nu}$ is defined in \\eqref{E:Riccidef}, and the ``gauge term''\n$u_{\\mu \\nu \\kappa}(g,g^{-1},\\partial g)\\Gamma^{\\kappa}$ is a smooth function of $g,$ $g^{-1},$ and $\\partial g$ \nthat will be discussed in Lemma \\ref{L:RicciInWave}. We remark that for purposes of covariant differentiation by $\\mathscr{D}$ in equation \\eqref{E:ModifiedRicci}, the $\\Gamma^{\\mu}$ are treated as the components of a vectorfield.\n\\end{definition}\n\nIn the next lemma, we perform an algebraic decomposition of the modified Ricci tensor.\n\n\\begin{lemma} \\cite[Lemmas 3.1 and 3.2]{hLiR2005} \\label{L:RicciInWave} \\textbf{(Decomposition of the modified Ricci tensor)}\n\tFor a suitable choice of the gauge term $u_{\\mu \\nu \\kappa}(g,g^{-1},\\partial g)\\Gamma^{\\kappa},$\n\tthe modified Ricci tensor $\\widetilde{R}_{\\mu \\nu}$ of the metric $g_{\\mu \\nu} = m_{\\mu \\nu} + h_{\\mu \\nu}$ can be decomposed \n\tas follows:\n\t\n\\begin{align}  \\label{E:RicciInWave}\n\t\\widetilde{R}_{\\mu \\nu} & = - \\frac{1}{2} \\Big\\lbrace \\widetilde{\\Square}_g g_{\\mu \\nu} - \n\t\t\\mathscr{P}(\\nabla_{\\mu}h, \\nabla_{\\nu}h) \n\t\t- \\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h) \\Big\\rbrace \\ + \\ O^{\\infty}(|h||\\nabla h|^2),\n\\end{align}\nwhere\n\n\\begin{align}\n\t\\widetilde{\\Square}_g \\eqdef (g^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}\n\\end{align}\nis the \\textbf{reduced wave operator} corresponding to $g_{\\mu \\nu},$ and the quadratic terms $\\mathscr{P}(\\nabla_{\\mu} \\cdot, \\nabla_{\\nu} \\cdot),$ \n$\\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\cdot,\\cdot)$ are defined by their action on tensorfields $\\Pi_{\\mu \\nu},$ $\\Theta_{\\mu \\nu},$ and $h_{\\mu \\nu}$ \nas follows:\n\n\\begin{align}\n\t\\mathscr{P}(\\nabla_{\\mu} \\Pi, \\nabla_{\\nu} \\Theta) \n\t& \\eqdef \\frac{1}{4}(\\nabla_{\\mu} \\Pi_{\\kappa}^{\\ \\kappa})(\\nabla_{\\nu} \\Theta_{\\lambda}^{\\ \\lambda})\n\t\t\\ - \\ \\frac{1}{2}(\\nabla_{\\mu} \\Pi^{\\kappa \\lambda})(\\nabla_{\\nu} \\Theta_{\\kappa \\lambda}), \n\t\t\\label{E:PNullform} \\\\\n\t\\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h) \\label{E:hAddedUpNullForms}\n\t\t& \\eqdef (m^{-1})^{\\lambda \\lambda'} \\mathscr{Q}_0(\\nabla h_{\\lambda \\mu}, \\nabla h_{\\lambda' \\nu}) \n\t\t\t\\ - \\ (m^{-1})^{\\kappa \\kappa'}(m^{-1})^{\\lambda \\lambda'} \n\t\t\t\\mathscr{Q}_{\\kappa \\lambda'}(\\nabla h_{\\lambda \\mu}, \\nabla \th_{\\kappa' \\nu}) \\\\ \n\t\t& \\ \\ +  \\ (m^{-1})^{\\kappa \\kappa'}(m^{-1})^{\\lambda \\lambda'} \n\t\t\t\\mathscr{Q}_{\\mu \\kappa}(\\nabla h_{\\kappa' \\lambda'}, \\nabla h_{\\lambda \\nu}) \n\t\t\t\\ + \\ (m^{-1})^{\\kappa \\kappa'}(m^{-1})^{\\lambda \\lambda'} \n\t\t\t\\mathscr{Q}_{\\nu \\kappa}(\\nabla h_{\\kappa' \\lambda'}, \\nabla h_{\\lambda \\mu}) \\notag \\\\\n\t\t& \\ \\ + \\ \\frac{1}{2} (m^{-1})^{\\kappa \\kappa'}(m^{-1})^{\\lambda \\lambda'}\n\t\t\t\\mathscr{Q}_{\\lambda' \\mu}(\\nabla h_{\\kappa \\kappa'},\\nabla h_{\\lambda \\nu}) \n\t\t\t\\ + \\ \\frac{1}{2} (m^{-1})^{\\kappa \\kappa'}(m^{-1})^{\\lambda \\lambda'}\n\t\t\t\\mathscr{Q}_{\\lambda' \\nu}(\\nabla h_{\\kappa \\kappa'},\\nabla h_{\\lambda \\mu}). \\notag \n\\end{align}\n\n\nThe bilinear forms $\\mathscr{Q}_0(\\cdot,\\cdot)$ and $\\mathscr{Q}_{\\mu \\nu}(\\cdot,\\cdot),$ which appear on the right-hand side of \\eqref{E:hAddedUpNullForms}, are known as the \\textbf{standard null forms}. They are defined through their action on the derivatives of scalar-valued functions $\\psi,$ $\\chi$ by\n\n\\begin{subequations}\n\\begin{align}\n\t\\mathscr{Q}_0(\\nabla \\psi,\\nabla \\chi) & \\eqdef (m^{-1})^{\\kappa \\lambda} (\\nabla_{\\kappa} \\psi)(\\nabla_{\\lambda} \\chi)\n\t\t\\label{E:StandardNullForm0}, \\\\\n\t\\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi, \\nabla \\chi) & \\eqdef (\\nabla_{\\mu} \\psi)(\\nabla_{\\nu} \\chi) \n\t\t- (\\nabla_{\\nu} \\psi)(\\nabla_{\\mu} \\chi). \\label{E:StandardNullFormmunu}\n\\end{align}\n\\end{subequations}\n\n\\end{lemma}\n\n\n\\begin{proof}\n\tThis decomposition is carried out in Lemmas 3.1 and 3.2 of \\cite{hLiR2005}.\n\\end{proof}\n\n\nWe conclude this section by observing that (\\ref{E:IntroEinsteinagain}'), \\eqref{E:MBIrighthandsidefieldequations}, and \\eqref{E:RicciInWave}\ntogether imply that under the wave coordinate condition \\eqref{E:wavecoordinategauge1}, and under the assumption \\eqref{E:Ldualassumptions} on the Lagrangian, the Einstein field equation \\eqref{E:IntroEinsteinagain} is equivalent to the following equation:\n\n\\begin{align} \\label{E:WaveCoordinateVersionofEinsteinFieldEquation}\n\t\\widetilde{\\Square}_g g_{\\mu \\nu} & = \\mathscr{P}(\\nabla_{\\mu}h, \\nabla_{\\nu}h) \n\t\t\\ + \\ \\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h) \n\t\t\\ - \\ 2 (m^{-1})^{\\kappa \\lambda} \\Far_{\\mu \\kappa} \\Far_{\\nu \\lambda} \n\t\t\\ + \\ \\frac{1}{2}m_{\\mu \\nu} (m^{-1})^{\\kappa \\eta} (m^{-1})^{\\lambda \\zeta} \\Far_{\\kappa \\lambda} \\Far_{\\eta \\zeta} \\\\\n\t& \\ \\ + \\ O^{\\infty}(|h||\\nabla h|^2) \\ + \\ O^{\\dParameter+1}(|h||\\Far|^2) \\ + \\ O^{\\dParameter+1}(|\\Far|^3;h). \\notag \n\\end{align}\n\n\n\n\\subsection{Summary of the reduced system} \\label{SS:ReducedEquations}\nIn this section, we summarize the above results by stating the form of the reduced Einstein nonlinear-electromagnetic system system that we work with for most of the remainder of the article, namely equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}; the derivation of this version of the reduced equations follows easily from the previous results of this section Section \\ref{S:ENESinWaveCoordinates}. We remind the reader that the reduced equations are obtained by \nby adding the inhomogeneous term $-\\Gamma^{\\kappa} (g^{-1})^{\\nu \\lambda} \\Far_{\\kappa \\lambda}$ to the right-hand side of \nequation \\eqref{E:EquationSatisfiedbyMaxdualChainruleExpandedfirstNversion}\nand by substituting the modified Ricci tensor in place of the Ricci tensor in equation \\eqref{E:IntroEinsteinagain}, and that in a wave coordinate system, the reduced system is equivalent to the system \\eqref{E:IntroEinsteinagain} - \\eqref{E:IntrodMis0again}.\n\n\\begin{center}\n\t{\\LARGE \\textbf{The Reduced System}}\n\\end{center}\n\nThe reduced system (where $g_{\\mu \\nu} = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)}$ and the unknowns are viewed to be $(h_{\\mu \\nu}^{(1)}, \\Far_{\\mu \\nu})$) can be expressed as\n\n\\begin{subequations}\n\\begin{align}\n\t\\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)} & = \\mathfrak{H}_{\\mu \\nu} - \\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(0)},&&\n\t(\\mu, \\nu = 0,1,2,3),  \\label{E:Reducedh1Summary} \\\\\n\t\\nabla_{\\lambda} \\Far_{\\mu \\nu} + \\nabla_{\\mu} \\Far_{\\nu \\lambda} + \\nabla_{\\nu} \\Far_{\\lambda \\mu} & = 0,&& (\\lambda, \\mu, \\nu = 0,1,2,3), \n\t\t\\label{E:ReduceddFis0Summary} \\\\\n\tN^{\\#\\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda} & = \\mathfrak{F}^{\\nu},&& (\\nu = 0,1,2,3),\n\t\t\\label{E:ReduceddMis0Summary} \n\\end{align}\n\\end{subequations}\nwhere $\\widetilde{\\Square}_g \\eqdef (g^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}$ is the reduced wave operator corresponding to \n$g_{\\mu \\nu}.$ \n\nThe quantities $\\mathfrak{H}_{\\mu \\nu}, N^{\\#\\mu \\nu \\kappa \\lambda},$ and $\\mathfrak{F}^{\\nu}$ can be decomposed \ninto principal terms and error terms (which are denoted with a ``$\\triangle$'') as follows:\n\n\\begin{subequations}\n\\begin{align}\n\t\\mathfrak{H}_{\\mu \\nu} & = \\mathscr{P}(\\nabla_{\\mu} h, \\nabla_{\\nu} h) \\ + \\ \\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h)\n\t\t \\ + \\ \\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\Far, \\Far) \\ + \\ \\mathfrak{H}_{\\mu \\nu}^{\\triangle},  \n\t\t \\label{E:ReducedhInhomogeneous} \\\\\n\t\\mathfrak{F}^{\\nu} & =  \\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla h, \\Far) \\ + \\ \\mathfrak{F}_{\\triangle}^{\\nu}, \n\t\t\\label{E:EMBIFarInhomogeneous} \\\\\n\tN^{\\#\\mu \\nu \\kappa \\lambda} & = \\frac{1}{2} \\Big\\lbrace (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \n\t\t\\lambda} (m^{-1})^{\\nu \\kappa}\n\t\t\t\\ - \\ h^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} + h^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\n\t\t\t\\ - \\ (m^{-1})^{\\mu \\kappa} h^{\\nu \\lambda} + (m^{-1})^{\\mu \\lambda} h^{\\nu \\kappa} \\Big\\rbrace\n\t\t\t\\ + \\ N_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda}, \\label{E:NSummarydef}\n\\end{align}\nwhere $\\mathscr{P}(\\nabla_{\\mu} h, \\nabla_{\\nu} h)$ is defined in \\eqref{E:PNullform}, \n$\\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h)$ is defined in \\eqref{E:hAddedUpNullForms}, and\n\n\\begin{align}\n\t\\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\Far, \\Gar) \n\t\t& = -2(m^{-1})^{\\kappa \\lambda} \\Far_{\\mu \\kappa} \\Gar_{\\nu \\lambda} \\ + \\ \\frac{1}{2}m_{\\mu \\nu}(m^{-1})^{\\kappa \n\t\t\\lambda}(m^{-1})^{\\lambda \\kappa}\\Far_{\\kappa \\lambda} \\Gar_{\\kappa \\lambda}, \\label{E:Q2h} \\\\\n\t\\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla h, \\Far) & = (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\lambda \\lambda'} (m^{-1})^{\\nu \\nu'} \n\t\t(\\nabla_{\\mu} h_{\\nu' \\lambda'}) \\Far_{\\kappa \\lambda}, \\label{E:Q2Far} \\\\\n\t\\mathfrak{H}_{\\mu \\nu}^{\\triangle} & = O^{\\infty}(|h||\\nabla h|^2) \\ + \\ O^{\\dParameter+1}(|h||\\Far|^2) \\ + \\ O^{\\dParameter+1}(|\\Far|^3;h), \n\t\t\\label{E:HtriangleSmallAlgebraic} \\\\\n\t\\mathfrak{F}_{\\triangle}^{\\nu} & =  O^{\\dParameter}(|h||\\nabla h||\\Far|) \\ + \\ O^{\\dParameter}(|\\nabla h||\\Far|^2;h),\n\t\t\\label{E:FtriangleSmallAlgebraic} \\\\\n\tN_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda} & = O^{\\dParameter}\\big(|(h,\\Far)|^2\\big). \n\t\t\\label{E:NtriangleSmallAlgebraic} \n\\end{align}\n\\end{subequations}\n\n\nFurthermore, the left-hand side of \\eqref{E:ReduceddMis0Summary} can be expressed as\n\n\\begin{subequations}\n\\begin{align}  \\label{E:NNullFormDecomposition}\n\tN^{\\#\\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \n\t\t& = \\frac{1}{2} \\big[ (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa} \n\t\t\t\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \n\t\t\\ - \\ \\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far) \n\t\t\\ - \\ \\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\Far)\n\t\t\\ + \\ N_{\\triangle}^{\\#\\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda}, \n\\end{align}\nwhere\t\n\t\t\n\\begin{align}\n\t\\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far) \n\t& = (m^{-1})^{\\mu \\mu'}(m^{-1})^{\\kappa \\kappa'} (m^{-1})^{\\nu \\lambda} h_{\\mu' \\kappa'}\\nabla_{\\mu} \\Far_{\\kappa \\lambda},\n\t\t\\label{E:PFar} \\\\\n\t\\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\Far) \n\t& = (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\nu'} (m^{-1})^{\\lambda \\lambda'} h_{\\nu' \\lambda'}\\nabla_{\\mu} \\Far_{\\kappa \\lambda}. \\label{E:Q1Far}\n\\end{align}\n\\end{subequations}\n\n\n\\section{The Initial Value Problem} \\label{S:IVP}\nIn this section, we discuss the abstract initial data and the constraint equations for the Einstein-nonlinear electromagnetic system. We then use the abstract initial data to construct initial data for the reduced equations that satisfy the wave coordinate condition at $t=0.$ Finally, we sketch a proof of the well-known fact that the wave coordinate condition is preserved by the solutions to the reduced equations that are launched by this data; this result shows that the wave coordinate gauge is a viable gauge for studying the Einstein-nonlinear electromagnetic system.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{The abstract initial data} \\label{SS:AbstractData}\n\nThe initial value problem formulation of the Einstein equations goes back to the seminal work \\cite{CB1952} by Choquet-Bruhat. In this article, initial data for the Einstein-nonlinear electromagnetic system consist of the $3-$dimensional manifold $\\Sigma_0 = \\mathbb{R}^3$ together with the following fields on $\\Sigma_0:$ \na Riemannian metric $\\mathring{\\underline{g}}_{jk},$ a covariant two-tensor $\\mathring{K}_{jk},$ and a pair of one-forms $\\mathring{\\mathfrak{\\Displacement}}_{j}, \\mathring{\\mathfrak{\\Magneticinduction}}_{j}.$ After we construct the ambient Lorentzian spacetime $(\\mathfrak{M},g_{\\mu \\nu})$, $\\mathring{\\underline{g}}_{jk}$ and $\\mathring{K}_{jk}$ will respectively be the first and second fundamental forms of $\\Sigma_0,$ while $\\mathring{\\mathfrak{\\Displacement}}_{j}, \\mathring{\\mathfrak{\\Magneticinduction}}_{j},$ which are defined below in Section \\ref{SS:EBDH}, will be an electromagnetic decomposition of $\\Far_{\\mu \\nu}|_{\\Sigma_0}$ into a pair of one-forms that are both $m-$tangent and $g-$tangent to $\\Sigma_0.$ \n\nIt is well-known that one cannot consider arbitrary data for the Einstein-nonlinear electromagnetic system. The data are subject to the following constraints: \n\n\\begin{subequations}\n\\begin{align} \n\t\\mathring{\\underline{R}} - \\mathring{K}_{ab} \\mathring{K}^{ab} + \n\t\t\\big[(\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{K}_{ab} \\big]^2 & = \n\t\t2T(\\hat{N},\\hat{N})|_{\\Sigma_0},&& \\label{E:Gauss} \\\\\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{\\underline{\\mathscr{D}}}_a \\mathring{K}_{bj} - \n\t\t(\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{\\underline{\\mathscr{D}}}_j \\mathring{K}_{ab} & = \n\t\tT(\\hat{N},\\frac{\\partial}{\\partial x^j})|_{\\Sigma_0},&&  (j=1,2,3), \\label{E:Codazzi} \n\\end{align}\n\\end{subequations}\n\n\\begin{subequations}\n\\begin{align}\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\underline{\\mathring{\\mathscr{D}}}_a \\mathring{\\mathfrak{\\Displacement}}_b & = 0, \n\t\t\\label{E:DivergenceD0} \\\\\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\underline{\\mathring{\\mathscr{D}}}_a \\mathring{\\mathfrak{\\Magneticinduction}}_b & = 0, \n\t\t\\label{E:DivergenceB0}\n\\end{align}\n\\end{subequations}\nwhere $\\mathring{\\underline{\\mathscr{D}}}$ is the Levi-Civita connection corresponding to \n$\\mathring{\\underline{g}}_{jk},$ $\\mathring{\\underline{R}}$ is the scalar curvature of $\\mathring{\\underline{g}}_{jk},$  $T_{\\mu \\nu}$ is defined in \\eqref{E:electromagnetictensorloweroinTermsofLagrangian}, and $\\hat{N}^{\\mu}$ is the future-directed unit $g-$normal to $\\Sigma_0.$ The right-hand sides of \\eqref{E:Gauss} - \\eqref{E:Codazzi} can (in principle) be computed in terms of and $\\mathring{\\underline{g}}_{jk},$ $\\mathring{\\mathfrak{\\Displacement}}_j,$ and $\\mathring{\\mathfrak{\\Magneticinduction}}_j$ with the help of the relations \\eqref{E:AbstractEBDHinertialcomponents}, which connect these quantities to $\\Far_{\\mu \\nu}|_{\\Sigma_0}.$\nIn equations \\eqref{E:Gauss} - \\eqref{E:Codazzi}, indices are lowered and raised with the Riemannian metric $\\mathring{\\underline{g}}_{jk}$ and its inverse $(\\mathring{\\underline{g}}^{-1})^{jk}.$ The constraints \\eqref{E:Gauss} - \\eqref{E:Codazzi} are manifestations of the \\emph{Gauss} and \\emph{Codazzi} equations respectively. These equations relate the geometry of the ambient spacetime $(\\mathfrak{M},g_{\\mu \\nu})$ (which has to be constructed) to the geometry inherited by an embedded Riemannian hypersurface (which will be $(\\Sigma_0,\\mathring{\\underline{g}}_{jk})$ after construction). Without providing the rather standard details (see e.g. \\cite{dC2008}), we remark that they are consequences of the following assumptions:\n\n\\begin{itemize}\n\t\\item $\\Sigma_0$ is a submanifold of the spacetime manifold $\\mathfrak{M}$\n\t\\item $\\mathring{\\underline{g}}_{jk}$ is the first fundamental form of $\\Sigma_0,$ and\n\t\t$\\mathring{K}_{jk}$ is the second fundamental form of $\\Sigma_0$\n\t\\item The Einstein-nonlinear electromagnetic system is satisfied along $\\Sigma_0$\n\n\t\n\t\n\t\n\t\n\t\\item Along $\\Sigma_0$ (viewed as a subset of $\\mathfrak{M}$),\n\t$\\mathfrak{\\Magneticinduction}_{\\mu} = - \\Fardual_{\\mu \\kappa}\\hat{N}^{\\kappa}$ \n\tand $\\mathfrak{\\Displacement}_{\\mu} = - \\Maxdual_{\\mu \\kappa} \\hat{N}^{\\kappa},$\n\twhere $\\hat{N}^{\\mu}$ is the future-directed unit $g-$normal to $\\Sigma_0.$\n\t\n\n\t\n\t\n\t\n\t\n\\end{itemize}\n\n\nWe recall that under the above assumptions, $\\mathring{\\underline{g}}$ and $\\mathring{K}$ are defined by\n\n\\begin{align}\n\t\\mathring{\\underline{g}}|_p(X,Y) & = g|_p(X,Y), && \\forall X,Y \\in T_p \\Sigma_0, \\\\\n\t\\mathring{K}|_p(X,Y) & = g|_p(\\mathscr{D}_X \\hat{N},Y), && \\forall X,Y \\in T_p \\Sigma_0, \n\\end{align}\nwhere $\\hat{N}$ is the future-directed unit $g-$normal\\footnote{Under the above assumptions, it follows that at every point $p \\in \\Sigma_0,$ $\\hat{N}^{\\mu} = (A^{-1},0,0,0).$} to $\\Sigma_0$ at $p,$ and $\\mathscr{D}$ is the Levi-Civita connection corresponding to $g.$ Furthermore, if $X,Y$ are vectorfields tangent to $\\Sigma_0,$ then\n\n\\begin{align}\n\t\\mathscr{D}_X Y = \\mathring{\\underline{\\mathscr{D}}}_X Y + \\mathring{K}(X,Y) \\hat{N}. \n\\end{align}\n\nWe also remind the reader that our stability theorem requires that the abstract initial data decay according to the rates\n\\eqref{E:metricdataexpansion} - \\eqref{E:BdecayAssumption}.\n\n\\subsection{The initial data for the reduced equations} \\label{SS:ReducedData}\n\nWe assume that we are given ``abstract'' initial data $(\\mathring{\\underline{g}}_{jk}, \\mathring{K}_{jk},\n\\mathring{\\mathfrak{\\Displacement}}_j, \\mathring{\\mathfrak{\\Magneticinduction}}_j),$ $(j,k=1,2,3),$ on the manifold $\\mathbb{R}^3$\nfor the Einstein equations as discussed in the previous section. In this section, we will use this data to construct data $(g_{\\mu \\nu}|_{t=0},$ $\\partial_t g_{\\mu \\nu}|_{t=0},$ $\\Far_{\\mu \\nu}|_{t=0}),$ $(\\mu, \\nu = 0,1,2,3)$ for the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} that satisfy the wave coordinate condition $\\Gamma^{\\mu}|_{t=0} = 0.$ We begin by recalling that $\\chi(z)$ is a fixed cut-off function with the following properties:\n\n\\begin{align} \\label{E:chidef}\n\t\\chi \\in C^{\\infty}, \\qquad \\chi \\equiv 1 \\ \\mbox{for} \\ z \\geq 3\/4, \\qquad \\chi \\equiv 0 \\ \\mbox{for} \\ z \\leq 1\/2.\n\\end{align}\nWe then define the function $A(x^1,x^2,x^3) \\geq 0$ by\n\n\\begin{align} \\label{E:aSquareddef}\n\tA^2 & \\eqdef 1 - \\frac{2M}{r} \\chi(r), && r \\eqdef |x|.\n\\end{align}\n\n\n\nWe define the data for the spacetime metric $g_{\\mu \\nu}$ by\n\n\\begin{subequations}\n\\begin{align}\n\tg_{00}|_{t=0} & = -A^2, &&  g_{0j}|_{t=0} = 0, && g_{jk}|_{t=0} = \\mathring{\\underline{g}}_{jk}, \\label{E:ReducedMetricData} \\\\\n\t\\partial_t g_{00}|_{t=0} & = 2A^3 (\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{K}_{ab},\n\t\t&& \\partial_t g_{0j}|_{t=0} = A^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{bj}\n\t\t- \\frac{1}{2} A^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_j \\mathring{\\underline{g}}_{ab} - A \\partial_j A, \n\t\t&& \\partial_t g_{jk}|_{t=0} = 2A \\mathring{K}_{jk}, \\label{E:ReducedMetricTimeDerivativeData}\n\\end{align}\n\\end{subequations}\nand the data for the Faraday tensor $\\Far_{\\mu \\nu}$ by\n\n\\begin{subequations}\n\\begin{align}\n\t\\Far_{j0}|_{t=0} & = \\mathring{\\Electricfield}_j, \\\\\n \t\t\\Far_{jk}|_{t=0} & = [ijk] \\mathring{\\Magneticinduction}_i. \n\\end{align}\n\\end{subequations}\nThe one-forms $\\mathring{\\Electricfield}_j$ and $\\mathring{\\Magneticinduction}_j$\ncan be expressed in terms of $\\mathring{\\underline{h}}_{jk}$ and the one-forms\n$\\mathring{\\mathfrak{\\Displacement}}_j$ and $\\mathfrak{\\mathring{\\Magneticinduction}}_j$ appearing in the constraint equations \\eqref{E:DivergenceD0} -  \\eqref{E:DivergenceB0} by using the relations \\eqref{E:AbstractEBDHinertialcomponents} and \\eqref{E:EBDHinertialcomponents} below. The precise form of this relations depends on the choice of Lagrangian $\\Ldual,$ but in the small-data regime, the estimates \\eqref{E:ElectricfieldDataInTermsofIntrinsic} \\eqref{E:IntialInductionintermsofInitialQuantities}, and \\eqref{E:IntialDisplacementintermsofInitialQuantities} hold.\n\nWe now state the main result of this section, which is a lemma showing that the wave coordinate condition is satisfied at $t=0.$\n\n\\begin{lemma} \\label{L:Gammamuare0initially} \\textbf{(Wave coordinate condition holds at $t=0$)}\n\tSuppose that the initial data $(g_{\\mu \\nu}|_{t=0}, \\partial_t g_{\\mu \\nu}|_{t=0}),$ $(\\mu, \\nu = \n\t0,1,2,3),$ for the reduced equations are constructed from abstract initial data $(\\mathring{\\underline{g}}_{jk}, \n\t\\mathring{K}_{jk}),$ $(j,k=1,2,3)$ as described above. Then the \n\twave coordinate condition holds initially:\n\n\\begin{align} \\label{E:Gammamuare0initially}\n\t& \\Gamma^{\\mu}|_{t=0},&& (\\mu= 0,1,2,3).\n\\end{align}\n\\end{lemma}\n\\begin{proof}\n\tLemma \\ref{L:Gammamuare0initially} follows from the expression \\eqref{E:wavecoordinategauge3}, \n\tthe definitions \\eqref{E:ReducedMetricData} - \\eqref{E:ReducedMetricTimeDerivativeData}, and simple calculations. \n\\end{proof}\n\n\nNote that the above definitions induce the following data for the spacetime metric ``remainder'' piece\n$h_{\\mu \\nu}^{(1)},$ which is defined by \\eqref{E:gmhexpansion} - \\eqref{E:h0defIntro}:\n\n\\begin{subequations}\n\\begin{align}\n\th_{00}^{(1)}|_{t=0} & = 0, && h_{0j}^{(1)}|_{t=0} = 0,\n\t\t&& h_{jk}^{(1)}|_{t=0} = \\mathring{\\underline{h}}_{jk}^{(1)}, \\\\\n\t\\partial_t h_{00}^{(1)}|_{t=0} & = 2A^3 (\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{K}_{ab},\n\t\t&& \\partial_t h_{0j}^{(1)}|_{t=0} = A^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{bj}\n\t\t- \\frac{1}{2} A^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_j \\mathring{\\underline{g}}_{ab} - A \\partial_j A,\n\t\t&& \\partial_t h_{jk}^{(1)}|_{t=0} = 2A \\mathring{K}_{jk}.\n\\end{align}\n\\end{subequations}\nSimilarly, the following data are induced in $h_{\\mu \\nu} = h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)},$ which is defined in \\eqref{E:hdefIntro}:\n\n\\begin{subequations}\n\\begin{align}\n\th_{00}|_{t=0} & = \\chi(r) \\frac{2M}{r}, && h_{0j}|_{t=0} = 0,\n\t\t&& h_{jk}|_{t=0} = \\mathring{\\underline{h}}_{jk}^{(1)} + \\chi(r) \\frac{2M}{r}, \\label{E:InducedhData} \\\\\n\t\\partial_t h_{00}|_{t=0} & = 2A^3 (\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{K}_{ab},\n\t\t&& \\partial_t h_{0j}|_{t=0} = A^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{bj}\n\t\t- \\frac{1}{2} A^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_j \\mathring{\\underline{g}}_{ab} - A \\partial_j A,\n\t\t&& \\partial_t h_{jk}|_{t=0} = 2A \\mathring{K}_{jk}.\n\\end{align}\n\\end{subequations}\nWe will make use of these facts in our proof of Proposition \\ref{P:SmallNormImpliesSmallEnergy} below.\n\n\n\\subsection{Preservation of the wave coordinate gauge} \\label{SS:WaveCoordinatesPreserved}\nIn this section, we sketch a proof of the fact that if the reduced data are constructed from abstract data \nas described in Section \\ref{SS:ReducedData}, then the wave coordinate condition $\\Gamma^{\\mu} = 0$ is preserved by the flow of the reduced equations. \nThis result requires the assumption that the abstract data satisfy the constraints \\eqref{E:Gauss} - \\eqref{E:DivergenceB0}. To simplify the discussion, we assume in this section that the data are smooth. However, the result also holds in the regularity class we use during our global existence proof. We remark that this result is quite standard, and that we have included it only for convenience.\n\n\\begin{proposition} \\label{P:PreservationofWaveCoordianteGauge}\n\t\\textbf{(Preservation of the wave coordinate gauge)}\n\tSuppose that $(g_{\\mu \\nu}|_{t=0}, \\partial_t g_{\\mu \\nu}|_{t=0}, \\Far_{\\mu \\nu}|_{t=0}),$ \n\t$(\\mu, \\nu = 0,1,2,3),$ are smooth initial data for the \n\treduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} constructed from abstract initial data \n\tsatisfying the constraints \\eqref{E:Gauss} - \\eqref{E:DivergenceB0} as described in Section \\ref{SS:ReducedData}. In \n\tparticular, by Lemma \\ref{L:Gammamuare0initially}, the wave coordinate condition \n\t$\\Gamma^{\\mu}|_{t=0}$ holds. Assume further that the reduced data are small enough so that they lie within the regime of \n\thyperbolicity\\footnote{Since our electromagnetic equations are perturbations of the linear Maxwell-Maxwell equations, there \n\twill always be such a regime.} of \n\tthe reduced equations. Let $(g_{\\mu \\nu},\\Far_{\\mu \\nu})$ be the corresponding solution to the reduced equations\n\tthat is launched by the data. Then $\\Gamma^{\\mu} \\equiv 0$ holds in the entire maximal globally hyperbolic development of the\n\tdata\\footnote{Roughly speaking, this is the largest possible solution that is uniquely determined by the data.}.\n\\end{proposition}\n\n\\noindent{\\textit{Sketch of proof}:} Our goal is to show that whenever we have a smooth solution to the \nreduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}, the corresponding $\\Gamma^{\\mu}$ satisfy a system of wave equations with principal part equal to $(g^{-1})^{\\kappa \\lambda} \\partial_{\\kappa} \\partial_{\\lambda}$ and with trivial initial data $\\Gamma^{\\mu}|_{t=0} = \\partial_t \\Gamma^{\\mu}|_{t=0} = 0.$ The conclusion that $\\Gamma^{\\mu} \\equiv 0$ in the maximal globally hyperbolic development of the data then follows from a standard uniqueness theorem based on energy estimates (see e.g. \\cite{lH1997}, \\cite{cS2008} for the ideas on how to prove such a theorem). To derive the equations satisfied by the $\\Gamma^{\\mu},$ we will view $\\Gamma^{\\mu}$ as a vectorfield for purposes of covariant differentiation. We apply $(g^{-1})^{\\nu \\lambda} \\mathscr{D}_{\\lambda}$ to each side of equation \\eqref{E:Modified}, use the Bianchi identity $(g^{-1})^{\\nu \\lambda} \\mathscr{D}_{\\lambda} \\big(R_{\\mu \\nu} - \\frac{1}{2}R g_{\\mu \\nu}\\big) = 0,$ the fact that \n$(g^{-1})^{\\nu \\lambda} \\mathscr{D}_{\\lambda} T_{\\mu \\nu} = - \\Gamma^{\\kappa} (g^{-1})^{\\beta \\lambda} \\Far_{\\kappa \\lambda} \\Far_{\\mu \\beta}$ (see Remark \\ref{R:ReducedElectromagneticInhomogeneous} and \\eqref{E:DivergenceofTidentity}), the curvature relation $\\mathscr{D}_{\\mu} \\mathscr{D}_{\\kappa} \\Gamma^{\\kappa} = \\mathscr{D}_{\\kappa} \\mathscr{D}_{\\mu} \\Gamma^{\\kappa}\n- R_{\\mu \\kappa} \\Gamma^{\\kappa},$ and expand the covariant derivatives in terms of coordinate derivatives and Christoffel symbols to deduce that the $\\Gamma^{\\mu}$ are solutions to the following \\emph{hyperbolic} system of wave equations:\n\n\\begin{align} \\label{E:Gammawaveequationcoordinates}\n\t(g^{-1})^{\\kappa \\lambda} \\partial_{\\kappa} \\partial_{\\lambda} \\Gamma^{\\mu} & =\n\t\tA_{\\ \\ \\lambda}^{\\mu \\kappa}(g,g^{-1},\\partial g) \\partial_{\\kappa} \\Gamma^{\\lambda}\n\t\t+ B_{\\ \\kappa}^{\\mu} (g,g^{-1},\\partial g,\\Far)\\Gamma^{\\kappa}, && (\\mu=0,1,2,3),\n\\end{align}\nwhere the $A_{\\ \\ \\lambda}^{\\mu \\kappa}\\big(g(t,x),g^{-1}(t,x),\\partial g(t,x)\\big)$ and \n$B_{\\ \\kappa}^{\\mu}\\big(g(t,x),g^{-1}(t,x), \\partial g(t,x),\\Far(t,x)\\big)$ are smooth functions of $(t,x).$ \n\nTo complete our sketch of the proof, it remains to show that $\\partial_t \\Gamma^{\\mu}|_{t=0} = 0.$ We first recall \n(see Remark \\ref{R:ReducedElectromagneticInhomogeneous}) that equation \\eqref{E:RicciInWave} is obtained by adding the gauge term $- \\frac{1}{2} \\big\\lbrace  g_{\\kappa \\nu} \\mathscr{D}_{\\mu} \\Gamma^{\\kappa} + g_{\\kappa \\mu} \\mathscr{D}_{\\nu} \\Gamma^{\\kappa} \\big\\rbrace \\ + \\ u_{\\mu \\nu \\kappa}(g,g^{-1},\\partial g) \\Gamma^{\\kappa}$ to the expression \\eqref{E:Riccidef} for $R_{\\mu \\nu}.$ Consequently, it follows that for a solution to the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}, we have that\n\n\\begin{align} \\label{E:Modified}\n\tR_{\\mu \\nu} \\ - \\ \\frac{1}{2}R g_{\\mu \\nu}  \n\t\t\\ - \\ T_{\\mu \\nu} \n\t\t& = \\frac{1}{2} \\big\\lbrace  g_{\\kappa \\nu} \\mathscr{D}_{\\mu} \\Gamma^{\\kappa}\n\t \t\t+ g_{\\kappa \\mu} \\mathscr{D}_{\\nu} \\Gamma^{\\kappa} \\big\\rbrace\n\t\t\\ - \\ u_{\\mu \\nu \\kappa}(g,g^{-1},\\partial g)\\Gamma^{\\kappa}&& \\\\\n\t\t& \\ \\ - \\ g_{\\mu \\nu} \\mathscr{D}_{\\lambda} \\Gamma^{\\lambda}\n\t \t\t\\ + \\ \\frac{1}{2} g_{\\mu \\nu} (g^{-1})^{\\kappa \\lambda} u_{\\kappa \\lambda \\delta}(g,g^{-1},\\partial g)\\Gamma^{\\delta}, \n\t\t\t&&(\\mu,\\nu=0,1,2,3). \\notag\n\\end{align}\t\nThe left-hand side of \\eqref{E:Modified} is simply the difference of the left and right sides of the Einstein equations \\eqref{E:IntroEinstein}. Since the abstract initial data $(\\mathring{\\underline{g}}_{jk}, \\mathring{K}_{jk},\n\\mathring{\\mathfrak{\\Displacement}}_j, \\mathring{\\mathfrak{\\Magneticinduction}}_j),$ $(j,k=1,2,3),$ are assumed to satisfy the constraint equations \\eqref{E:Gauss} - \\eqref{E:Codazzi}, it follows that the left-hand side of \\eqref{E:Modified} is equal to $0$ at $t=0$ after contracting\\footnote{In fact, one derives the constraint equations by assuming that these contractions are $0$ at $t=0.$} against $\\hat{N}^{\\mu} \\hat{N}^{\\nu}$ or $\\hat{N}^{\\mu} X^{\\nu},$ where $\\hat{N}^{\\mu}$ is the future-directed unit $g-$normal to $\\Sigma_0$ and $X^{\\mu}$ is any vector tangent to $\\Sigma_0.$\n\nRecalling that $\\hat{N}^{\\mu}|_{t=0} = A^{-1}\\delta_0^{\\mu},$ and choosing $X^{\\nu} = \\delta_j^{\\nu},$\nit therefore follows that the right-hand side must also be equal to $0$ at $t=0$ upon contraction:\n\n\\begin{subequations}\n\\begin{align} \\label{E:InitialGammaupderivativeconditionNormalNormal}\n\t\\bigg\\lbrace g_{\\kappa 0} \\mathscr{D}_t \\Gamma^{\\kappa}\n\t \t\t\\ - \\ u_{0 0 \\kappa}(g,g^{-1},\\partial g)\\Gamma^{\\kappa}\n\t\t\\ - \\ g_{0 0} \\mathscr{D}_{\\lambda} \\Gamma^{\\lambda}\n\t \t\t\\ + \\ \\frac{1}{2} g_{0 0} (g^{-1})^{\\kappa \\lambda} u_{\\kappa \\lambda \\delta}(g,g^{-1},\\partial g) \\Gamma^{\\delta} \n\t \t\t\\bigg\\rbrace\\Big|_{t = 0} = 0, &&  \\\\\n\t\t\\bigg\\lbrace \\frac{1}{2} \\big[  g_{\\kappa j} \\mathscr{D}_t \\Gamma^{\\kappa}\n\t \t\t+ g_{\\kappa 0} \\mathscr{D}_j \\Gamma^{\\kappa} \\big]\n\t\t\\ - \\ u_{0 j \\kappa}(g,g^{-1},\\partial g)\\Gamma^{\\kappa}\n\t\t\\ - \\ g_{0j} \\mathscr{D}_{\\lambda} \\Gamma^{\\lambda}\n\t \t\t\\ + \\ \\frac{1}{2} g_{0 j} (g^{-1})^{\\kappa \\lambda} u_{\\kappa \\lambda \\delta}(g,g^{-1},\\partial g)\\Gamma^{\\delta} \n\t \t\t\\bigg\\rbrace\\Big|_{t = 0} = 0, && (j=1,2,3). \t\t\n\t\t\t\\label{E:InitialGammaupderivativeconditionNormalTangential}\n\\end{align}\n\\end{subequations}\nExpanding the covariant differentiation in \\eqref{E:InitialGammaupderivativeconditionNormalNormal} - \\eqref{E:InitialGammaupderivativeconditionNormalTangential} in terms of coordinate derivatives and Christoffel symbols, and\nusing \\eqref{E:ReducedMetricData} - \\eqref{E:ReducedMetricTimeDerivativeData} plus the fact that the initial data were constructed so as to satisfy $\\Gamma^{\\mu}|_{t=0} = 0,$ it is easy to check that $\\partial_t \\Gamma^{\\mu}$ must \\emph{also necessarily} be trivial at $t=0:$\n\n\\begin{align}\n\t\\partial_t \\Gamma^{\\mu}|_{t=0} & = 0, && (\\mu = 0,1,2,3).\n\\end{align}\nThis completes our sketch of the proposition.\n\n\\hfill $\\qed$\n\n\n\n\\section{Geometry and the Minkowskian Null Frame} \\label{S:NullFrame}\n\nIn this section, we introduce the families of ingoing Minkowskian null cones $C_{s}^-,$  \noutgoing Minkowskian light cones $C_{q}^+,$ constant Minkowskian time slices $\\Sigma_t,$ and Euclidean spheres $S_{r,t}.$\nWe then discuss the well-known notion of a Minkowskian null frame, which allows us to geometrically decompose the tangent space \nas a direct sum $T|_p \\mathbb{R}^{1+3} =  \\mbox{span} \\lbrace \\uL|_p \\rbrace \\, \\oplus \\, \\mbox{span} \\lbrace L|_p \\rbrace \\, \\oplus \\, T|_p S_{r,t}.$ These decompositions allow us to geometrically decompose tensorfields. In Section \\ref{SS:NullComponents}, we provide a full description of the null decomposition of a two-form $\\Far$ into its \\emph{Minkowskian null components}. This decomposition will be essential to our subsequent analysis of the decay properties of the Faraday tensor. In Section \\ref{SS:NullDecompElectromagnetic}, we will derive equations for these null components under the assumption that $\\Far$ is a solution to the reduced electromagnetic equations \\eqref{E:ReduceddFis0Summary} - \n\\eqref{E:ReduceddMis0Summary}. In Section \\ref{S:DecayFortheReducedEquations}, we will use the equations for the null components to deduce ``upgraded'' pointwise decay estimates for the lower-order Lie derivatives of $\\Far;$ these estimates are essential for closing our global existence bootstrap argument in Section \\ref{S:GlobalExistence}.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{The Minkowskian null frame} \\label{SS:NullFrame}\n\nBefore proceeding, we introduce the subsets $C_{q}^+,$ $C_{s}^-,$ $\\Sigma_t,$ $S_{r,t}.$\n\n\\begin{definition}\n\tIn our wave coordinate system $(t,x),$ we define the \\emph{outgoing Minkowski \n\tnull cones} $C_{q}^+,$ \\emph{ingoing Minkowski null cones} $C_{s}^-,$ \n\tthe \\emph{constant Minkowskian time slices} $\\Sigma_t$ and the Euclidean spheres \n\t$S_{r,t}$ as follows:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\tC_{q}^+ & \\eqdef \\lbrace (\\tau,y) \\ | \\ |y| - \\tau = q \\rbrace, \\\\\n\t\tC_{s}^- & \\eqdef \\lbrace (\\tau,y) \\ | \\ |y| + \\tau = s \\rbrace, \\\\\n\t\t\\Sigma_t & \\eqdef \\lbrace (\\tau,y) \\ | \\ \\tau = t \\rbrace, \\\\ \n\t\tS_{r,t} & \\eqdef \\lbrace (\\tau,y) \\ | \\ \\tau = t, |y| = r \\rbrace, \n\t\\end{align}\n\t\\end{subequations}\n\tIn the above formulas, $y \\eqdef (y^1,y^2,y^3),$ and \n\t$|y| \\eqdef \\big[(y^1)^2 + (y^2)^2 + (y^2)^2 \\big]^{1\/2}.$\n\\end{definition}\n\nWe also introduce the following vectorfields, which play a fundamental role throughout this article.\n\n\\begin{definition} \\label{D:uLLdef}\n\tWe define the \\emph{ingoing Minkowski-null geodesic vectorfield} $\\uL$ and the \n\t\\emph{outgoing Minkowski-null geodesic vectorfield} $L$ by\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\uL^{\\mu} & = (1,-\\omega^1,-\\omega^2,-\\omega^3), \\label{E:uLdef} \\\\\n\t\tL^{\\mu} & = (1,\\omega^1,\\omega^2,\\omega^3), \\label{E:Ldef}\n\t\\end{align}\n\t\\end{subequations}\n\twhere $\\omega^j \\eqdef x^j\/r.$ By ``Minkowski-null,'' we mean that $m(\\uL,\\uL) = m(L,L) = 0.$\n\tNote that $\\uL$ is tangent to the ingoing cones $C_{s}^-,$ that $L$ is tangent to the outgoing cones $C_{q}^+,$\n\tand that $\\uL,$ $L$ are both $m-$orthogonal to the $S_{r,t}.$ By ``geodesic,'' we mean that\n\t$\\nabla_{\\uL} \\uL = \\nabla_L L = 0.$\n\\end{definition}\n\nNote that\n\n\\begin{subequations}\n\\begin{align}\n\t\\uL & = \\partial_t - \\partial_r, \\\\\n\tL & = \\partial_t + \\partial_r.\n\\end{align}\n\\end{subequations}\n\nWe now recall the definitions of the Minkowskian first fundamental forms of the surfaces $\\Sigma_t$ and $S_{r,t}.$\n\n\\begin{definition} \\label{D:FirstFundamental}\n\tThe \\emph{Minkowskian first fundamental} forms of the surfaces $\\Sigma_t$ and $S_{r,t}$ are respectively defined to be the \n\tfollowing intrinsic metrics:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\um_{\\mu \\nu} & \\eqdef \\mbox{diag}(0,1,1,1), \\label{E:FirstFundSigmatDef} \\\\\n\t\t\\angm_{\\mu \\nu} & \\eqdef m_{\\mu \\nu} + \\frac{1}{2}(\\uL_{\\mu} L_{\\nu} + L_{\\mu} \\uL_{\\nu}). \\label{E:angmdef}\n\t\\end{align}\n\t\\end{subequations}\n\\end{definition}\n\\noindent Recall that $\\um|_p(X,Y) = m|_p(X,Y)$ for $X,Y \\in T|_p \\Sigma_t,$ and $\\angm(X,Y) = m(X,Y)$ for $X,Y \\in T|_p S_{r,t}.$\nNote also that the tensorfields $\\um_{\\mu}^{\\ \\nu}$ and $\\angm_{\\mu}^{\\ \\nu}$ respectively $m-$orthogonally project onto the $\\Sigma_t$ and the $S_{r,t}.$ \n\nWe now defined a related tensorfield corresponding to the outgoing Minkowski null cones $C_{q}^+.$\n\n\\begin{definition} \\label{D:ConeProjection}\n\tThe tensorfield $\\coneproject_{\\mu}^{\\ \\nu},$ which $m-$orthogonally projects vectors $X^{\\mu}$ onto \n\tthe outgoing cones $C_{q}^+,$ can be expressed as follows:\n\t\n\t\\begin{align} \\label{E:ConeProjection}\n\t\t\\coneproject_{\\mu}^{\\ \\nu} & \\eqdef \\delta_{\\mu}^{\\nu} + \\frac{1}{2} L_{\\mu} \\uL^{\\nu}. \n\t\\end{align}\n\\end{definition}\n\nFurthermore, we recall the definitions of the Minkowskian volume forms of Minkowski space and of the surfaces $\\Sigma_t$ and $S_{r,t}.$\n\n\\begin{definition}\n\tThe \\emph{Minkowskian volume forms} of Minkowski spacetime, the surfaces $\\Sigma_t,$ and the Euclidean spheres $S_{r,t}$\n\tare respectively defined relative to our wave coordinate system as follows:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\Minkvolume_{\\mu \\nu \\kappa \\lambda} & \\eqdef [\\mu \\nu \\kappa \\lambda], \\label{E:Minkvolumedef} \\\\\n\t\t\\uvolume_{\\nu \\kappa \\lambda} & \\eqdef \\Minkvolume_{0 \\nu \\kappa \\lambda}, \\label{E:Sigmatvolumedef} \\\\\n\t\t\\angupsilon_{\\mu \\nu} & \\eqdef \\Minkvolume_{\\mu \\nu \\kappa \\lambda} \\uL^{\\kappa} L^{\\lambda},\t\\label{E:Spheresvolumedef}\n\t\\end{align}\n\t\\end{subequations}\n\twhere $[\\mu \\nu \\kappa \\lambda]$ is totally anti-symmetric with normalization $[0123] = 1.$\n\\end{definition}\n\nWe also recall what it means for a spacetime tensorfield to be $m-$tangent to the surfaces $\\Sigma_t$ or $S_{r,t}.$\n\n\\begin{definition} \\label{D:Tangency}\n\tLet $U$ be a type $\\binom{n}{m}$ spacetime tensorfield. We say that $U$ is $m-$tangent to the time slices $\\Sigma_t$ if \n\t\n\t\\begin{align}\n\t\tU_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} \n\t\t= \\um_{\\mu_1}^{\\ \\mu'_1} \\cdots \\um_{\\mu_m}^{\\ \\mu'_m} \n\t\t\\um_{\\nu'_1}^{\\ \\nu_1}\n\t\t\\cdots \\um_{\\nu'_n}^{\\ \\nu_1} U_{\\mu'_1 \\cdots \\mu'_m}^{\\ \\ \\ \\ \\ \\ \\ \n\t\t\\nu'_1 \\cdots \\nu'_n}.\n\t\\end{align}\n\t\tEquivalently, $U$ is $m-$tangent to the $\\Sigma_t$ if and only if every wave coordinate component of $U$ containing\n\t\ta $0$ index vanishes.\n\t\t\n\tSimilarly, we say that $U$ is $m-$tangent to the spheres $S_{r,t}$ if\n\t\n\t\\begin{align}\n\t\tU_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} \n\t\t= \\angm_{\\mu_1}^{\\ \\mu'_1} \\cdots \\angm_{\\mu_m}^{\\ \\mu'_m} \\angm_{\\nu'_1}^{\\ \\nu_1}\n\t\t\\cdots \\angm_{\\nu'_n}^{\\ \\nu_1} U_{\\mu'_1 \\cdots \\mu'_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu'_1 \n\t\t\\cdots \\nu'_n}.\n\t\\end{align}\n\tEquivalently, $U$ is $m-$tangent to the spheres $S_{r,t}$ if and only if any contraction of any index of $U$ with either $\\uL$ or $L$ vanishes.\n\t\n\\end{definition}\n\n\n\nWe are now ready to introduce the notion of a \\emph{Minkowskian null frame}. We complement the vectorfields $\\uL,$ $L$ with a locally-defined pair of $m-$orthogonal vectorfields $e_1,$ $e_2$ that are  tangent to the spheres $S_{r,t},$ and therefore $m-$orthogonal to $\\uL,$ $L.$ The resulting collection of vectorfields $\\mathcal{N} \\eqdef \\lbrace \\uL,L,e_1,e_2 \\rbrace$ is known as \\emph{Minkowskian null frame}. It spans the tangent space $T|_p \\mathbb{R}^{1+3}$ at each point $p$ where it is defined.\n\nWe leave the proof of the following lemma, which summarizes some of the important properties of the geometric quantities introduced in this section, as an exercise for the reader.\n\n\\begin{lemma} \\label{L:PropertiesofuLandL} \\textbf{(Null frame field properties)}\nThe following identities hold:\n\n\\begin{subequations}\n\\begin{align}\n\t\\nabla_L L & = \\nabla_{\\uL} \\uL = 0,&& \\label{E:LanduLaregeodesic} \\\\\n\t\\nabla_L \\uL & = \\nabla_{\\uL} L = 0,&& \\label{E:nablaLuLis0} \\\\\n\tL^{\\kappa} \\uL_{\\kappa} & = - 2,&& \\label{E:LunderlineLcontracted} \\\\\n\te_A^{\\kappa} L_{\\kappa} & = e_A^{\\kappa} \\uL_{\\kappa} = 0,&& (A = 1,2),  \\label{E:LunderlineLnormaltospheres} \\\\\n\tm_{\\kappa \\lambda} e_A^{\\kappa} e_B^{\\lambda} & = \\delta_{AB},&& (A,B = 1,2),\n\\end{align}\n\\end{subequations}\n\n\\begin{align} \\label{E:NablaLangmandNablauLangmVanish}\n\t\\nabla_{\\uL} \\angm_{\\mu \\nu} = \\nabla_L \\angm_{\\mu \\nu} & = 0,&& (\\mu,\\nu =0,1,2,3),\n\\end{align}\n\n\\begin{align}\n\t\\nabla_{\\uL} \\angupsilon_{\\mu \\nu} = \\nabla_L \\angupsilon_{\\mu \\nu} & = 0,&& (\\mu,\\nu =0,1,2,3). \n\\end{align}\nSee Definition \\ref{D:NablaXDef} concerning our use of notation in these formulas.\n\\end{lemma}\n\n\\hfill $\\qed$\n\nLater in the article, we will see that the decay rates of the null components (see Section \\ref{SS:NullComponents}) \nof $\\Far$ will be distinguished according to the kinds of contractions of $\\Far$ taken against $\\uL, L, e_1,$ and $e_2.$ With these ideas in mind, we introduce the following sets of vectorfields:\n\n\\begin{subequations}\n\\begin{align} \\label{E:Framefieldsubsets}\n\t\\mathcal{L} \\eqdef \\lbrace L \\rbrace, & & \\mathcal{T} \\eqdef \\lbrace L, e_1, e_2 \\rbrace,\n\t& & \\mathcal{N} \\eqdef \\lbrace \\uL, L, e_1, e_2 \\rbrace.\n\\end{align}\n\\end{subequations}\nIn order to measure the size of the contractions of various tensors and their covariant derivatives against vectors\nbelonging to the sets $\\mathcal{L}, \\mathcal{T}, \\mathcal{N},$ we introduce the following definitions. \n\n\\begin{definition} \\label{D:contractionnomrs}\n\nIf $\\mathcal{V}, \\mathcal{W}$ denote any two of the above sets, and $P$ is a type $\\binom{0}{2}$\ntensor, then we define the following pointwise seminorms:\n\n\\begin{subequations}\n\n\\begin{align}\n\t|P|_{\\mathcal{V} \\mathcal{W}} & \\eqdef \\sum_{V \\in \\mathcal{V}, W \\in \\mathcal{W}} |V^{\\kappa} W^{\\lambda} P_{\\kappa \n\t\t\\lambda}|, \\label{E:contractionnorm} \\\\\n\t|\\nabla P|_{\\mathcal{V} \\mathcal{W}} & \\eqdef \\sum_{N \\in \\mathcal{N}, V \\in \\mathcal{V}, W \\in \\mathcal{W}} \n\t\t|V^{\\kappa} W^{\\lambda} N^{\\gamma} \\nabla_{\\gamma} P_{\\kappa \\lambda}|, \\\\\n\t|\\conenabla P|_{\\mathcal{V} \\mathcal{W}} & \\eqdef \\sum_{T \\in \\mathcal{T}, V \\in \\mathcal{V}, W \\in \\mathcal{W}} \n\t\t|V^{\\kappa} W^{\\lambda} T^{\\gamma} \\nabla_{\\gamma} P_{\\kappa \\lambda}|.\n\\end{align}\n\\end{subequations}\n\nWe often use the abbreviations $|P| \\eqdef |P|_{\\mathcal{N} \\mathcal{N}},$ \n$|\\nabla P| \\eqdef |\\nabla P|_{\\mathcal{N} \\mathcal{N}},$\nand $|\\conenabla P| \\eqdef |\\conenabla P|_{\\mathcal{N} \\mathcal{N}}.$\n\n\\end{definition}\n\nThe above definition generalizes in an obvious way to arbitrary type $\\binom{n}{m}$ tensorfields\n$U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n}.$ Observe that for any such tensorfield, the following\ninequalities hold \\emph{in our wave coordinate system}:\n\n\\begin{align} \\label{E:AlternateNablaNorm}\n\t|U| & \\approx \\sum_{\\mu_1,\\cdots,\\mu_m,\\nu_1,\\cdots,\\nu_n = 0}^3\n\t|U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n}|.\n\\end{align}\n\n\n\n\\subsection{Minkowskian null frame decomposition of a tensorfield}\n\nFor an arbitrary vectorfield $X$ and frame vector $N \\in \\mathcal{N},$ we define \n\n\\begin{align} \\label{E:XlowerUdef}\n\tX_N & \\eqdef X_{\\kappa} N^{\\kappa}, \\ \\mbox{where} \\ X_{\\mu} \\eqdef m_{\\mu \\kappa} X^{\\kappa}. \n\\end{align}\nThe components $X_N$ are known as the \\emph{Minkowskian null components} of $X.$ In the sequel, we will abbreviate\n\n\\begin{align}\n\tX_A \\eqdef X_{e_A}, && \\nabla_A \\eqdef \\nabla_{e_A}, \\ \\mbox{etc}.\n\\end{align}\nIt follows from \\eqref{E:XlowerUdef} that\n\\begin{align}\n\tX & = X^{\\kappa} \\partial_{\\kappa} = X^{L} L + X^{\\uL} \\uL\n\t\t+ X^{A} e_A,  \\label{X:AbstractLuLAdecomp} \\\\\n\tX^{L} & = - \\frac{1}{2}X_{\\uL}, && X^{\\uL} = - \\frac{1}{2}X_L, && X^A = X_A.\n\t\t\\label{E:XupperUdef}\n\\end{align}\nFurthermore, it is easy to check that\n\\begin{align} \\label{E:XYnullframeinnerproduct}\n\tm(X,Y) \\eqdef m_{\\kappa \\lambda}X^{\\kappa}X^{\\lambda} = X^{\\kappa}Y_{\\kappa} = -\\frac{1}{2}X_{L}Y_{\\uL} - \n\t\\frac{1}{2}X_{\\uL}Y_{L} + \\delta^{AB} X_A Y_B. \n\\end{align}\n\nThe above null decomposition of a vectorfield generalizes in the obvious way to higher order tensorfields. In the next section,\nwe provide a detailed version of the null decomposition of two-forms $\\Far,$ since this decomposition is needed\nfor our derivation of decay estimates later in the article; see e.g.\nProposition \\ref{P:EOVNullDecomposition} and Proposition \\ref{P:EnergyInhomogeneousTermAlgebraicEstimate}.\n\n\n\n\n\n\\subsection{The detailed Minkowskian null decomposition of a two-form} \\label{SS:NullComponents}\n\n\n\\begin{definition} \\label{D:null}\nGiven any two-form $\\Far,$ we define its \\emph{Minkowskian null components} to be the following pair of one-forms\n$\\ualpha_{\\mu},$ $\\alpha_{\\mu},$ and the following pair of scalar $\\rho,$ $\\sigma:$\n\n\\begin{subequations}\n\\begin{align}\n\t\\ualpha_{\\mu} & \\eqdef \\angm_{\\mu}^{\\ \\nu} \\Far_{\\nu \\lambda} \\uL^{\\lambda},&& (\\mu = 0,1,2,3),\n\t\t\\label{E:ualphadef} \\\\\n\t\\alpha_{\\mu} & \\eqdef \\angm_{\\mu}^{\\ \\nu} \\Far_{\\nu \\lambda} L^{\\lambda},&& (\\mu = 0,1,2,3), \\\\\n\t\\rho & \\eqdef \\frac{1}{2} \\Far_{\\kappa \\lambda}\\uL^{\\kappa} L^{\\lambda},&& \\\\\n\t\\sigma & \\eqdef \\frac{1}{2} \\angupsilon^{\\kappa \\lambda} \\Far_{\\kappa \\lambda}.&& \\label{E:sigmadef}\n\\end{align}\n\\end{subequations}\n\n\\end{definition}\n\nIt is a simple exercise to check that $\\ualpha_{\\mu}$ and $\\alpha_{\\mu}$ are $m-$tangent to the spheres $S_{r,t}:$\n\n\\begin{subequations}\n\\begin{align}\n\t\\ualpha_{\\kappa}\\uL^{\\kappa} & = 0, & \\ualpha_{\\kappa}L^{\\kappa} & = 0, \\\\\n\t\\alpha_{\\kappa}\\uL^{\\kappa} & = 0, & \\alpha_{\\kappa}L^{\\kappa} & = 0.\n\\end{align}\n\\end{subequations}\nFurthermore, relative to the null frame $\\mathcal{N} \\eqdef \\lbrace \\uL, L, e_1, e_2 \\rbrace,$ we have that\n\n\\begin{subequations}\n\\begin{align}\n\t\\underline{\\alpha}_A & = \\Far_{A \\uL},&& (A = 1,2), \\\\\n\t\\alpha_A & = \\Far_{AL},&& (A = 1,2), \\\\\n\t\\rho & = \\frac{1}{2} \\Far_{\\uL L},&& \\\\\n\t\\sigma & = \\Far_{12}.&&\n\\end{align}\n\\end{subequations}\n\nIn terms of the seminorms introduced in Definition \\ref{D:contractionnomrs}, it follows that\n\n\\begin{subequations}\n\\begin{align}\n\t|\\Far| & \\approx |\\Far|_{\\mathcal{N}\\mathcal{N}} \\approx |\\ualpha| + |\\alpha| + |\\rho| + |\\sigma|, \\\\\n\t|\\Far|_{\\mathcal{L}\\mathcal{N}} & \\approx |\\alpha| + |\\rho|, \\\\\n\t|\\Far|_{\\mathcal{T} \\mathcal{T}} & \\approx |\\alpha| + |\\sigma|. \n\\end{align}\n\\end{subequations}\n\nThe null components of $\\FarMinkdual$ (the Minkowskian Hodge duality operator $\\ostar$ is defined in Section \\ref{SS:Hodge}) can be expressed in terms of the above null components of $\\Far.$ Denoting the null components\\footnote{We use the symbol $\\odot$ in order to avoid confusion with the Minkowskian Hodge duality operator $\\ostar;$ i.e., it is not true that ${^{\\ostar \\hspace{-.03in}}(\\ualpha[\\Far])} = \\ualpha[\\FarMinkdual].$} of $\\FarMinkdual$ by $\\ualphadot, \\alphadot, \\rhodot, \\sigmadot,$ we leave it as a simple exercise to the reader to check that  \n\n\\begin{subequations}\n\\begin{align}\n\t\\ualphadot_A & = - \\underline{\\alpha}^B \\angupsilon_{BA},&& (A = 1,2), \\label{E:Fardualalpha} \\\\\n\t\\alphadot_A & = \\alpha^B \\angupsilon_{BA},&& (A = 1,2), \\\\\n\t\\rhodot & = \\sigma,&& \\\\  \n\t\\sigmadot & = - \\rho.&& \\label{E:Fardualsigma}\n\\end{align}\n\\end{subequations}\n\n\n\n\n\\section{Differential Operators} \\label{S:DifferentialOperators}\nIn this section, we introduce a collection of differential operators that will be used throughout the remainder of the article.\nIn order to define these operators, we also introduce subsets $\\mathcal{O}$ and $\\mathcal{Z}$ of Minkowskian conformal Killing fields. Finally, we prove a collection of lemmas that expose useful properties of these operators, and that illustrate various relationships between them.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{Covariant derivatives}\n\nAs previously mentioned, throughout the article, $\\nabla$ denotes the Levi-Civita connection of the Minkowski metric $m.$\nLet $\\um$ and $\\angm$ be the first fundamental forms of the $\\Sigma_t$ and $S_{r,t}$ as defined in\nDefinition \\ref{D:FirstFundamental}, and let $\\unabla,$ $\\angn$ be their corresponding Levi-Civita connections. We state as a lemma the following well-known identities, which relates the connections $\\unabla,$ $\\angn$ to $\\nabla$ through the first fundamental forms.\n\n\\begin{lemma} \\textbf{(Relationships between connections)}\n\tIf $U$ is any type $\\binom{n}{m}$ tensorfield $m-$tangent to the $\\Sigma_t,$ then\n\n\\begin{align} \n\t\t\\unabla_{\\lambda} \n\t\t\tU_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} & = \\um_{\\lambda}^{\\ \\lambda'}\n\t\t\t\\um_{\\mu_1}^{\\ \\mu'_1} \\cdots \n\t\t\t\\um_{\\mu_m}^{\\ \\mu'_m} \\um_{\\nu'_1}^{\\ \\nu_1} \\cdots \\um_{\\nu'_n}^{\\ \\nu_n} \n\t\t\t\\nabla_{\\lambda'} U_{\\mu'_1 \\cdots \\mu'_m}^{\\ \\ \\ \\ \n\t\t\t\\ \\ \\ \\nu'_1 \\cdots \\nu'_n}, \n\t\t\t\\label{E:SigmatIntrinsicintermsofExtrinsic} \n\\end{align}\n\nSimilarly, if If $U$ is any type $\\binom{n}{m}$ tensorfield $m-$tangent to $S_{r,t}$ then\n\n\\begin{align}\n\t\t\\angn_{\\lambda}\n\t\t\tU_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} & = \\angm_{\\lambda}^{\\ \\lambda'}\n\t\t\t\\angm_{\\mu_1}^{\\ \\mu'_1} \\cdots \n\t\t\t\\angm_{\\mu_m}^{\\ \\mu'_m} \\angm_{\\nu'_1}^{\\ \\nu_1} \\cdots \\angm_{\\nu'_n}^{\\ \\nu_n} \n\t\t\t\\nabla_{\\lambda'} U_{\\mu'_1 \\cdots \\mu'_m}^{\\ \\ \\ \\ \n\t\t\t\\ \\ \\ \\nu'_1 \\cdots \\nu'_n}. \\label{E:SphereIntrinsicintermsofExtrinsic}\n\\end{align}\n\n\\end{lemma}\n\n\nWe recall the following fundamental properties of the connections $\\nabla,$ $\\unabla,$ and $\\angn:$\n\n\\begin{subequations}\n\\begin{align}\n\t\\nabla_{\\lambda} m_{\\mu \\nu} & = 0 = \\nabla_{\\lambda} (m^{-1})^{\\mu \\nu},&& (\\lambda, \\mu, \\nu = 0,1,2,3), \n\t\t\\label{E:nablamis0} \\\\\n\t\\unabla_{\\lambda} \\um_{\\mu \\nu} & = 0,&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\\\\n\t\\angn_{\\lambda} \\angm_{\\mu \\nu} & = 0,&& (\\lambda, \\mu, \\nu = 0,1,2,3). \\label{E:angnablaangmis0}\n\\end{align}\n\\end{subequations}\n\n\nWe will also make use of the projection of the operator $\\nabla$ onto the favorable directions, i.e., the directions tangent to the outgoing Minkowski cones $C_q^+.$ \n\n\\begin{definition} \\label{D:ConeProjectedDerivative}\n\tIf $U$ is any type $\\binom{n}{m}$ spacetime tensorfield, then we define the projected Minkowskian \n\tcovariant derivative $\\conenabla U$ by\n\t\n\t\\begin{align} \\label{E:ConeProjectedDerivative}\n\t\t\\conenabla_{\\lambda} U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} \n\t\t& = \\coneproject_{\\lambda}^{\\ \\lambda'} \\nabla_{\\lambda'}U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \n\t\t\t\\ \\ \\ \\nu_1 \\cdots \\nu_n},\n\\end{align}\nwhere the projection $\\coneproject_{\\mu}^{\\ \\nu}$ is defined in \\eqref{E:ConeProjection}.\n\n\\end{definition}\n\n\\begin{remark}\n\tNote that only the $\\lambda$ component is projected onto the outgoing cones, so that the tensorfield\n\t$\\conenabla_{\\lambda} U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n}$ need not be $m-$tangent to \n\tthe outgoing Minkowski cones.\n\\end{remark}\n\n\n\\begin{definition} \\label{D:NablaXDef}\n\tIf $X$ is any vectorfield, then we define the covariant derivative operators $\\nabla_X$ \n\tand $\\angn_X$ by\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\nabla_X & \\eqdef X^{\\kappa} \\nabla_{\\kappa}, \\\\\n\t\t\\angn_{X} & \\eqdef X^{\\kappa} \\angn_{\\kappa}.\n\t\\end{align}\n\t\\end{subequations}\n\t\n\\end{definition}\n\t\n\n\n\\subsection{Minkowskian conformal Killing fields} \\label{SS:ConformalKillingFields}\nIn this section, we introduce the special set of vectorfields $\\mathcal{Z}$ that appears in the definition \\eqref{E:EnergyIntro} of our energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t),$ and in the weighted Klainerman-Sobolev inequality\n\\eqref{E:KSIntro}. We begin by recalling that a \\emph{Minkowskian conformal Killing field} is a vectorfield $Z$ such that\n\n\\begin{align} \\label{E:ConformalKillingDef}\n\t\\nabla_{\\mu} Z_{\\nu} + \\nabla_{\\nu} Z_{\\mu} = {^{(Z)}\\phi} m_{\\mu \\nu}\n\\end{align}\nfor some function ${^{(Z)}\\phi}(t,x).$ The tensorfield \n\n\\begin{align} \\label{E:MinkowskianDeformationTensordef}\n\t{^{(Z)}\\pi_{\\mu \\nu}} \\eqdef \\nabla_{\\mu} Z_{\\nu} + \\nabla_{\\nu} Z_{\\mu}\n\\end{align}\nis known as the \\emph{Minkowskian deformation tensor} of $Z.$ If ${^{(Z)}\\pi_{\\mu \\nu}} = 0,$ then $Z$ is known as a \\emph{Minkowskian Killing field}.\nWe also recall that the conformal Killing fields of the Minkowski metric $m_{\\mu \\nu}$ form a Lie Algebra generated under the Lie bracket $[\\cdot,\\cdot]$ (see \\eqref{E:bracket}) that is generated by the following $15$ vectorfields (see e.g. \\cite{dC2008}): \n\n\\begin{enumerate}\n\t\\item the four \\emph{translations} $\\partial_{\\mu} = \\frac{\\partial}{\\partial x^{\\mu}}, \\qquad (\\mu =0,1,2,3),$ \n\t\\item the three \\emph{rotations} $\\Omega_{jk} \\eqdef x_j \\frac{\\partial}{\\partial x^k} - x_k \\frac{\\partial}{\\partial x^j}, \n\t\t\\qquad (1 \\leq j < k \\leq 3),$\n\t\\item the three \\emph{Lorentz boosts} $\\Omega_{jk} \\eqdef -t \\frac{\\partial}{\\partial x^j} - x_j \\frac{\\partial}{\\partial t}, \n\t\t\\qquad (j = 1,2,3),$\n\t\\item the \\emph{scaling} vectorfield $S \\eqdef x^{\\kappa} \\frac{\\partial}{\\partial x^{\\kappa}},$\t\n\t\\item the four \\emph{acceleration} vectorfields $K_{(\\mu)} \\eqdef - 2 x_{\\mu} S \n\t+ g_{\\kappa \\lambda}x^{\\kappa} x^{\\lambda} \\frac{\\partial}{\\partial x^{\\mu}}, \\qquad \\mu = (0,1,2,3).$ \n\\end{enumerate}\nIt can be checked that the translations, rotations, and Lorentz boosts are in fact Killing fields of $m_{\\mu \\nu}.$\n\nTwo subsets of the above conformal Killing fields will play a prominent role in the remainder of the article, namely the rotations $\\mathcal{O}$ and a larger set $\\mathcal{Z},$ which are defined by\n\n\\begin{subequations}\n\\begin{align}\n\t\\mathcal{O} & \\eqdef \\big\\lbrace \\Omega_{jk} \\big\\rbrace_{1 \\leq j < k \\leq 3}, \\label{E:Rotationsetdef} \\\\\n\t\\mathcal{Z} & \\eqdef \\big\\lbrace \\frac{\\partial}{\\partial x^{\\mu}}, \\Omega_{\\mu \\nu}, S \\big\\rbrace_{0 \\leq \\mu < \\nu \\leq 3}.  \\label{E:Zsetdef} \n\\end{align}\n\\end{subequations}\nThe vectorfields in $\\mathcal{Z}$ satisfy a strong version of the relation \\eqref{E:ConformalKillingDef}. That is,\nif $Z \\in \\mathcal{Z},$ then\n\n\\begin{align} \\label{E:CovariantDerivativesofZareConstant}\n\t\\nabla_{\\mu} Z_{\\nu} = {^{(Z)}c}_{\\mu \\nu},\n\\end{align}\nwhere the components ${^{(Z)}c}_{\\mu \\nu}$ are \\emph{constants} in our wave coordinate system. In particular, we compute for future use that\n\n\\begin{subequations}\n\\begin{align}\n\t\\nabla_{\\mu} S_{\\nu} & = m_{\\mu \\nu}, \\label{E:ScalingCovariantDerivative} \\\\\n\t\\nabla_{\\mu} (\\Omega_{\\kappa \\lambda})_{\\nu} & = m_{\\mu \\kappa} m_{\\nu \\lambda} - m_{\\mu \\lambda} m_{\\nu \\kappa}.\n\t\\label{E:SpacetimeRotationsCovariantDerivative}\n\\end{align}\n\\end{subequations}\nWe note in addition that if $Z \\in \\mathcal{Z},$ then there exists a \\emph{constant} $c_Z$ such that\n\n\\begin{align} \\label{E:ZDeformationTensorinTermsofcZm}\n\t\\nabla_{\\mu} Z_{\\nu} + \\nabla_{\\nu} Z_{\\mu} = c_Z m_{\\mu \\nu}.\n\\end{align}\nFurthermore, by contracting each side of \\eqref{E:ZDeformationTensorinTermsofcZm} against $(m^{-1})^{\\mu \\nu},$ it follows that\n\n\\begin{align}\n\tc_Z = \\frac{1}{4} {^{(Z)}\\pi_{\\kappa}^{\\ \\kappa}} = \\frac{1}{2} {^{(Z)}c}_{\\kappa}^{\\ \\kappa}.\n\\end{align}\n\n\n\n\\subsection{Lie derivatives}\nAs mentioned in Section \\ref{SS:DiscussionofProof}, it is convenient to use Lie derivatives to differentiate \nthe electromagnetic equations \\eqref{E:ReduceddFis0Summary} - \\eqref{E:ReduceddMis0Summary}. In this section,\nwe recall some basic facts concerning Lie derivatives.\n\nWe recall that if $X,Y,$ are any pair of vectorfields, then relative to an arbitrary coordinate system,\ntheir \\emph{Lie bracket} $[X,Y]$ can be expressed as\n\n\\begin{align} \\label{E:bracket}\n\t[X,Y]^{\\mu} & = X^{\\kappa} \\partial_{\\kappa} Y^{\\mu} - Y^{\\kappa} \\partial_{\\kappa} X^{\\mu}.\n\\end{align}\nFurthermore, we have that \n\n\\begin{align} \\label{E:LieXY}\n\t\\Lie_X Y = [X,Y], \n\\end{align}\t\nwhere $\\Lie$ denotes the \\emph{Lie derivative operator}. Given a type $\\binom{0}{m}$ tensorfield $U,$ and vectorfields $Y_{(1)}, \\cdots Y_{(m)},$ the Leibniz rule for $\\Lie$ implies that \\eqref{E:LieXY} generalizes as follows:\n\n\\begin{align} \\label{E:Liederivativebracketexpression}\n\t(\\Lie_X U)(Y_{(1)}, \\cdots, Y_{(m)}) & = X \\lbrace U(Y_{(1)}, \\cdots, Y_{(m)}) \\rbrace\n\t\t- \\sum_{i=1}^n U(Y_{(1)}, \\cdots, Y_{(i-1)}, [X,Y_{(i)}], Y_{(i+1)}, \\cdots, Y_{(m)}).\n\\end{align} \n\nUsing Lemma \\ref{L:Liederivativeintermsofnabla} below, we see that\nthe left-hand side of \\eqref{E:ZDeformationTensorinTermsofcZm} is equal\nthe Lie derivative of the Minkowski metric. It therefore follows that if $Z \\in \\mathcal{Z},$ then\n\n\\begin{subequations}\n\\begin{align}\n\t\\Lie_Z m_{\\mu \\nu} & = c_Z m_{\\mu \\nu}, \\label{E:LieZonmlower} \\\\\n\t(\\Lie_Z m^{-1})^{\\mu \\nu} & = - c_Z (m^{-1})^{\\mu \\nu}, \\label{E:LieZonmupper}\n\\end{align}\n\\end{subequations}\nwhere the constant $c_Z$ is defined in \\eqref{E:ZDeformationTensorinTermsofcZm}. \n\n\\subsection{Modified covariant and modified Lie derivatives}\n\nIt will be convenient for us to work with \\emph{modified Minkowski covariant derivatives} $\\nablamod_{Z}$ and\n\\emph{modified Lie derivatives}\\footnote{Note that these are not the same\nmodified Lie derivatives that appear in \\cite{lBnZ2009}, \\cite{dCsK1993}, and \\cite{nZ2000}.} $\\Liemod_Z.$\n\n\\begin{definition} \\label{D:ModifiedDerivatives}\n\tFor $Z \\in \\mathcal{Z},$ we define the modified Minkowski covariant derivative $\\nablamod_Z$ by\n\t\n\t\\begin{align} \\label{E:Covariantmoddef}\n\t\t\\nablamod_Z \\eqdef \\nabla_Z + c_Z,\n\t\\end{align}\n\twhere $c_Z$ denotes the constant from \\eqref{E:ZDeformationTensorinTermsofcZm}.\n\t\n\tFor each vectorfield $Z \\in \\mathcal{Z},$ we define the modified Lie derivative $\\Liemod_Z$ by\n\t\\begin{align} \\label{E:Liemoddef}\n\t\t\\Liemod_Z \\eqdef \\Lie_Z + 2c_Z,\n\t\\end{align}\n\twhere $c_Z$ denotes the constant from \\eqref{E:ZDeformationTensorinTermsofcZm}.\n\\end{definition}\nThe crucial features of the above definitions are captured by Lemmas \\ref{L:NablaModZLiemodMinkowskiWaveOperatorCommutator} and\n\\ref{L:LiemodZLiemodMaxwellCommutator} below. The first shows that for each $Z \\in \\mathcal{Z},$\n$\\nablamod_Z \\Square_m \\phi = \\Square_m \\nabla_Z \\phi,$ where $\\Square_m = (m^{-1})^{\\kappa \\lambda}\\nabla_{\\kappa} \\nabla_{\\lambda}$ is the \\emph{Minkowski} wave operator. The second shows that\n $\\Liemod_Z \\Big\\lbrace \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda}(m^{-1})^{\\nu \\kappa} \\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda}\\Big\\rbrace$ $= \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda}(m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \n\\Lie_Z \\Far_{\\kappa \\lambda}.$ Furthermore, Lemma \\ref{L:Liecommuteswithcoordinatederivatives} shows that $\\Lie_Z \\nabla_{[\\lambda}\\Far_{\\mu \\nu]}$ = $\\nabla_{[\\lambda}\\Lie_Z \\Far_{\\mu \\nu]},$ where $[\\cdots]$ denotes anti-symmetrization.\nThese commutation identities suggest that the operators $\\Liemod_Z$ and $\\nablamod_Z$ are\npotentially useful operators for differentiating equations \\eqref{E:Reducedh1Summary} and \n\\eqref{E:ReduceddFis0Summary} - \\eqref{E:ReduceddMis0Summary} respectively. This suggestion is borne out in\nPropositions \\ref{P:EnergyInhomogeneousTermAlgebraicEstimate} and \\ref{P:DIPointwise}, which show that the\ninhomogeneous terms generated by differentiating the nonlinear equations have a special algebraic structure, a structure that will be exploited during our global existence bootstrap argument.\n\n\n\n\n\n\n\\subsection{Vectorfield algebra}\n\n\nWe introduce here some notation that will allow us to compactly express iterated derivatives. If $\\mathcal{A}$\nis one of the sets from \\eqref{E:Rotationsetdef} - \\eqref{E:Zsetdef}, then we label the vectorfields in $\\mathcal{A}$ as $Z^{\\iota_{1}}, \\cdots, Z^{\\iota_{d}},$ where $d$ is the cardinality of $\\mathcal{A}.$ Then for any multi-index \n$I = (\\iota_1, \\cdots, \\iota_k)$ of length $k,$ where each $\\iota_i \\in \\lbrace 1,2,\\cdots, d \\rbrace,$ we make the\nfollowing definition.\n\n\\begin{definition} \\label{D:iterated}\nThe iterated derivative operators are defined by\n\n\\begin{subequations}\n\\begin{align} \n\t\\nabla_{\\mathcal{A}}^I & \\eqdef \\nabla_{Z^{\\iota_1}} \\circ \\cdots \\circ \\nabla_{Z^{\\iota_k}},  \\label{E:iteratedCovariant} \\\\\n\t\\nablamod_{\\mathcal{A}}^I & \\eqdef \\nablamod_{Z^{\\iota_1}} \\circ \\cdots \\circ \\nablamod_{Z^{\\iota_k}}, \\\\\n\t\\Lie_{\\mathcal{A}}^I & \\eqdef \\Lie_{Z^{\\iota_1}} \\circ \\cdots \\circ \\Lie_{Z^{\\iota_k}}, \\label{E:iteratedLie} \\\\\n\t\\Liemod_{\\mathcal{A}}^I & \\eqdef \\Liemod_{Z^{\\iota_1}} \\circ \\cdots \\circ \\Liemod_{Z^{\\iota_k}}, \n\\end{align}\n\\end{subequations}\netc. \n\\end{definition}\n\nSimilarly, if $I = (\\mu_1, \\cdots, \\mu_k)$ is a coordinate multi-index of length $k,$ where \n$\\mu_1, \\cdots, \\mu_k \\in \\lbrace 0,1,2,3 \\rbrace,$ and $U$ is a tensorfield, then\nwe use shorthand notation such as\n\n\\begin{align}\n\t\\nabla^I U \\eqdef \\nabla_{\\mu_1} \\cdots \\nabla_{\\mu_k} U,\n\\end{align}\netc.\n\nUnder the above conventions, the Leibniz rule can be written as e.g.\n\n\\begin{align}\n\t\\Lie_{\\mathcal{Z}}^I (UV) = \\sum_{I_1 + I_2 = I} (\\Lie_{\\mathcal{Z}}^I U)(\\Lie_{\\mathcal{Z}}^I V),\n\\end{align}\netc., where by a sum over $I_1 + I_2 = I,$ we mean a sum over all order preserving partitions of the index $I$ into two \nmulti-indices; i.e., if $I = (\\iota_1, \\cdots, \\iota_k),$ then $I_1 = (\\iota_{i_1}, \\cdots, \\iota_{i_a}), \nI_2 = (\\iota_{i_{a+1}}, \\cdots, \\iota_{i_k}),$ where $i_1, \\cdots, i_k$ is any re-ordering of the integers\n$1,\\cdots,k$ such that $i_1 < \\cdots < i_a,$ and $i_{a+1} < \\cdots < i_k.$ \n\n\n\nThe next standard lemma provides a useful expression relating Lie derivatives to covariant derivatives.\n\n\\begin{lemma} \\cite{rW1984} \\label{L:Liederivativeintermsofnabla} \\textbf{(Lie derivatives in terms of covariant derivatives)}\nLet $X$ be a vectorfield, and let $U$ be a tensorfield of type $\\binom{n}{m}.$ Then $\\Lie_X U$ can be expressed in terms of covariant derivatives of $U$ and $X$ as follows:\n\n\\begin{align} \\label{E:Liederivativeintermsofnabla} \n\t\\Lie_X U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} =\n\t\t\\nabla_X U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} \n\t& + U_{\\kappa \\mu_2 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n}\\nabla_{\\mu_1}X^{\\kappa} \n\t\t+ \\cdots + U_{\\mu_1 \\cdots \\mu_{m-1} \\kappa}^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n}\\nabla_{\\mu_m}X^{\\kappa} \\\\\n\t& - U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\kappa \\cdots \\nu_n} \\nabla_{\\kappa}X^{\\nu_1}\n\t \t- \\cdots - U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_{n-1} \\kappa} \\nabla_{\\kappa}X^{\\nu_n}. \\notag\n\\end{align}\n\n\\end{lemma}\n\n\\hfill $\\qed$\n\n\nThe next lemma shows that the operators $\\Lie_Z$ and $\\Liemod_Z$ commute with $\\nabla$ if $Z \\in \\mathcal{Z}.$\n\n\\begin{lemma} \\label{L:Liecommuteswithcoordinatederivatives} \\textbf{($\\Lie_Z$ and $\\nabla$ commute)}\nLet $\\nabla$ denote the Levi-Civita connection corresponding to the Minkowski metric $m,$ and let $I$ be a \n$\\mathcal{Z}-$multi-index. Let $\\Liemod_{\\mathcal{Z}}^I$ be the iterated modified Lie derivative from Definitions \\ref{D:ModifiedDerivatives} and \\ref{D:iterated}. Then\n\n\\begin{align} \\label{E:Liecommuteswithcoordinatederivatives}\n\t[\\nabla, \\Lie_{\\mathcal{Z}}^I] & = 0, && [\\nabla, \\Liemod_{\\mathcal{Z}}^I] = 0.\n\\end{align}\n\nIn an arbitrary coordinate system, equations \\eqref{E:Liecommuteswithcoordinatederivatives} are equivalent to the following relations, which hold for all type $\\binom{n}{m}$ tensorfields $U:$\n\n\\begin{align}\n\t\\nabla_{\\mu}\\big\\lbrace \\Lie_{\\mathcal{Z}}^I U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} \\big\\rbrace \n\t& = \\Lie_{\\mathcal{Z}}^I \\big\\lbrace \\nabla_{\\mu}U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} \\big\\rbrace, \\\\\n\t\\nabla_{\\mu}\\big\\lbrace \\Liemod_{\\mathcal{Z}}^I U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} \\big\\rbrace \n\t& = \\Liemod_{\\mathcal{Z}}^I \\big\\lbrace \\nabla_{\\mu} U_{\\mu_1 \\cdots \\mu_m}^{\\ \\ \\ \\ \\ \\ \\ \\ \\nu_1 \\cdots \\nu_n} \\big\\rbrace. \\notag\n\\end{align}\n \n\n\\end{lemma}\n\n\\begin{proof}\n\tThe relation \\eqref{E:Liecommuteswithcoordinatederivatives} can be shown via induction in $|I|$ using\n\t\\eqref{E:Liederivativeintermsofnabla} and the fact that $\\nabla\\nabla Z = 0.$ \n\\end{proof}\n\n\nThe next lemma captures the commutation properties of vectorfields $Z \\in \\mathcal{Z}.$\n\n\\begin{lemma} \\label{L:ConformalKillingFieldCommuatators} \\cite[pg. 139]{dCsK1990} \n\\textbf{(Lie bracket relations)}\n\tRelative to the wave coordinate system\n\t$\\lbrace x^{\\mu} \\rbrace_{\\mu = 0,1,2,3},$ the vectorfields belonging to the subset $\\mathcal{Z} \\eqdef \\big\\lbrace \n\t\\frac{\\partial}{\\partial x^{\\mu}}, \\Omega_{\\mu \\nu}, S \\big\\rbrace_{0 \\leq \\mu < \\nu \\leq 3}$ of the Minkowskian conformal \n\tKilling fields satisfy the following commutation relations, where ${^{(Z)} c_{\\mu}^{\\ \\kappa}}$ is defined in \n\t\\eqref{E:CovariantDerivativesofZareConstant}:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\left[\\frac{\\partial}{\\partial x^{\\mu}}, \\frac{\\partial}{\\partial x^{\\nu}}\\right] & = 0\n\t\t\t= {^{(\\frac{\\partial}{\\partial x^{\\nu}})} c_{\\mu}^{\\ \\kappa}} \\frac{\\partial}{\\partial x^{\\kappa}}, \n\t\t\t&& (\\mu, \\nu = 0,1,2,3), \\label{E:translationscommute} \\\\\n\t\t\\left[\\frac{\\partial}{\\partial x^{\\lambda}}, \\Omega_{\\mu \\nu}\\right] & = m_{\\lambda \\mu}\\frac{\\partial}{\\partial x^{\\nu}} \n\t\t\t- m_{\\lambda \\nu} \\frac{\\partial}{\\partial x^{\\mu}}\n\t\t\t= {^{(\\Omega_{\\mu \\nu})} c_{\\lambda}^{\\ \\kappa}} \\frac{\\partial}{\\partial x^{\\kappa}}, && (\\lambda, \\mu, \\nu = 0,1,2,3), \n\t\t\t\t\\\\\n\t\t\\left[\\frac{\\partial}{\\partial x^{\\mu}}, S \\right] & = \\frac{\\partial}{\\partial x^{\\mu}}\n\t\t\t= {^{(S)} c_{\\mu}^{\\ \\kappa}} \\frac{\\partial}{\\partial x^{\\kappa}}, && (\\mu = 0,1,2,3), \\\\\n\t\t\\left[\\Omega_{\\kappa \\lambda}, \\Omega_{\\mu \\nu} \\right] & = m_{\\kappa \\mu} \\Omega_{\\nu \\lambda} \n\t\t\t- m_{\\kappa \\nu} \\Omega_{\\mu \\lambda} + m_{\\lambda \\mu} \\Omega_{\\kappa \\nu}\n\t\t\t- m_{\\lambda \\nu} \\Omega_{\\kappa \\mu}, && (\\kappa, \\lambda, \\mu, \\nu = 0,1,2,3), \\\\\n\t\t\\left[\\Omega_{\\mu \\nu}, S\\right] & = 0, && (\\mu, \\nu = 0,1,2,3).\n\t\\end{align}\t\n\t\\end{subequations}\n\\end{lemma}\n\n\\hfill $\\qed$\n\nWe now state the following simple commutation lemma.\n\n\n\\begin{lemma} \\label{L:NablapartialmuNablaZCommutatorExpression} \n\\textbf{($\\nabla_Z,$ $\\nabla_{\\frac{\\partial}{\\partial x^{\\mu}}}$ commutation relations)}\nLet $Z \\in \\mathcal{Z}.$ Then relative to the wave coordinate system\n$\\lbrace x^{\\mu} \\rbrace_{\\mu = 0,1,2,3},$ the differential operators $\\nabla_{\\frac{\\partial}{\\partial x^{\\mu}}}$ and $\\nabla_{Z}$ satisfy the following commutation relations:\n\n\\begin{align} \\label{E:NablapartialmuNablaZCommutatorExpression}\n\t[\\nabla_{\\frac{\\partial}{\\partial x^{\\mu}}}, \\nabla_Z] = {^{(Z)} c_{\\mu}^{\\ \\kappa}} \\frac{\\partial}{\\partial x^{\\kappa}},\n\\end{align}\nwhere ${^{(Z)} c_{\\mu}^{\\ \\kappa}}$ is defined in \\eqref{E:CovariantDerivativesofZareConstant}.\n\n\\end{lemma}\n\n\n\\begin{proof}\nThe relation \\eqref{E:NablapartialmuNablaZCommutatorExpression} follows from Lemma \\ref{L:ConformalKillingFieldCommuatators} \nand the identity $[\\nabla_X, \\nabla_Y] = \\nabla_{[X,Y]},$ which holds for all pairs of vectorfields $X,Y;$ this identity holds\nbecause of the torsion-free property of the connection $\\nabla$ and because the Riemann curvature tensor of the Minkowski metric $m_{\\mu \\nu}$ completely vanishes.\n\\end{proof}\n\n\nThe next lemma shows that the operators $\\nabla$ and $\\nabla_{\\mathcal{Z}}^I$ commute up to lower-order terms.\n\n\\begin{lemma} \\label{L:NablaZICommutesWithCovariantDerivativePlusErrorTerms} \\textbf{($\\nabla$ and \n$\\nabla_{\\mathcal{Z}}^I$ commutation inequalities)}\nLet $U$ be a type $\\binom{n}{m}$ tensorfield, and let $I$ be a $\\mathcal{Z}-$multi-index. Then the following inequality holds:\n\n\\begin{align} \\label{E:NablaZICommutesWithCovariantDerivativePlusErrorTerms}\n\t|\\nabla_{\\mathcal{Z}}^I \\nabla U| & \\lesssim |\\nabla\\nabla_{\\mathcal{Z}}^I U|\n\t\\ + \\ \\sum_{|J| \\leq |I| - 1} |\\nabla\\nabla_{\\mathcal{Z}}^J U|.\n\\end{align}\n\n\\end{lemma}\n\n\\begin{proof}\n\tUsing \\eqref{E:AlternateNablaNorm}, we have that\n\t\n\t\\begin{align} \\label{E:NablaZINablaPNormapproximatelyNablaZISumOverNablamuPNorm}\n\t\t|\\nabla_{\\mathcal{Z}}^I \\nabla U| & \\approx \\sum_{\\mu = 0}^3 \n\t\t\t|\\nabla_{\\mathcal{Z}}^I \\nabla_{\\frac{\\partial}{\\partial x^{\\mu}}}U|.\n\t\\end{align}\n\tWe therefore repeatedly apply Lemma \\ref{L:NablapartialmuNablaZCommutatorExpression} to deduce that\n\tthere exist constants $C_{I;J}^{\\nu}$ such that\n\t\n\t\\begin{align} \\label{E:NablapartialmuNablaZICommutatorExpression}\n\t\t\t\\nabla_{\\mathcal{Z}}^I \\nabla_{\\frac{\\partial}{\\partial x^{\\mu}}}U\n\t\t\t& = \\nabla_{\\frac{\\partial}{\\partial x^{\\mu}}} \\nabla_{\\mathcal{Z}}^I U\n\t\t\t\t\\ + \\ \\sum_{|J| \\leq |I| - 1} \\sum_{\\nu = 0}^3 C_{I;J}^{\\nu}\n\t\t\t\t\\nabla_{\\frac{\\partial}{\\partial x^{\\nu}}} \\nabla_{\\mathcal{Z}}^J U.\n\t\\end{align}\n\tInequality \\eqref{E:NablaZICommutesWithCovariantDerivativePlusErrorTerms}\n\tnow follows from applying \\eqref{E:AlternateNablaNorm} to each side of \\eqref{E:NablapartialmuNablaZICommutatorExpression}.\n\t\n\t\n\\end{proof}\n\n\nThe next lemma captures some important differential identities.\n\n\\begin{lemma} \\label{L:LieDerivativeCommuatorsVanish} \\textbf{(Geometric differential identities)}\n\t Let $\\uL,$ $L$ be the Minkowski-null geodesic vectorfields defined in \\eqref{E:uLdef} - \\eqref{E:Ldef}, and let\n\t $O \\in \\mathcal{O}.$ Then the vectorfields, $\\uL,$ $L,$ $O$ mutually commute:\n\n\\begin{align}\n\t[\\uL, L] & = 0, && [\\uL,O] = 0, && [L, O] = 0. \\label{E:RotationLuLBracketis0} \n\\end{align}\n\t\n\tFurthermore, let $\\Minkvolume_{\\kappa \\lambda \\mu \\nu},$ $\\angm_{\\mu \\nu}$ denote the volume forms defined in\n\t\\eqref{E:Minkvolumedef} and \\eqref{E:Spheresvolumedef}. Then\n\t\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\Lie_O \\Minkvolume_{\\kappa \\lambda \\mu \\nu} & = 0, \\label{E:LieO4epsilonis0} \\\\\n\t\t\\Lie_O \\angm_{\\mu \\nu} & = 0, \\label{E:LieOanggis0} \\\\\n\t\t\\Lie_O \\angupsilon_{\\mu \\nu} & = 0. \\label{E:LieO2epsilonis0}\n\t\\end{align}\n\t\\end{subequations}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\\eqref{E:RotationLuLBracketis0} can be checked via simple calculations using the definitions \\eqref{E:uLdef} - \\eqref{E:Ldef} of $\\uL$ and $L,$ the \n\tdefinitions of the rotations $O \\in \\mathcal{O}$ given at the beginning of Section \\ref{SS:ConformalKillingFields}, and the \t  Lie bracket formula \\eqref{E:bracket}.\n\t\\eqref{E:LieO4epsilonis0} follows from the well-known identity $\\Lie_X \\Minkvolume_{\\kappa \\lambda \\mu \\nu} = \\frac{1}{2} {^{(X)}\\pi_{\\ \\beta}^{\\beta}}\n\t\\Minkvolume_{\\kappa \\lambda \\mu \\nu},$ where ${^{(X)}\\pi_{\\mu \\nu}}$ is defined in \\eqref{E:MinkowskianDeformationTensordef}, together with\n\tthe fact that $\\Lie_O m_{\\mu \\nu} = {^{(O)}\\pi_{\\mu \\nu}} = 0$ (i.e., that $O$ is a Killing field of $m_{\\mu \\nu}$). \n\t\\eqref{E:LieOanggis0} and \\eqref{E:LieO2epsilonis0} then follow from definitions \\eqref{E:angmdef}, \n\t\\eqref{E:Spheresvolumedef}, and \\eqref{E:RotationLuLBracketis0} - \\eqref{E:LieO4epsilonis0}.\n\\end{proof}\n\n\nThe next lemma shows that the modified covariant derivatives $\\nablamod_{\\mathcal{Z}}^I$ have favorable commutation properties\nwith the Minkowski wave operator.\n\n\\begin{lemma} \\label{L:NablaModZLiemodMinkowskiWaveOperatorCommutator} \\textbf{($\\nablamod_{\\mathcal{Z}}^I$ and\n\t$\\Square_m$ commutation properties)}\n\tLet $I$ be a $\\mathcal{Z}-$multi-index, and let \n\t$\\phi$ be any function. Let $\\nablamod_{\\mathcal{Z}}^I$ be the iterated modified Minkowski covariant derivative operator from \n\tDefinitions \\ref{D:ModifiedDerivatives} and \\ref{D:iterated}, and let \n\t$\\Square_m \\eqdef (m^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}$ denote the Minkowski wave operator.\n\tThen\n\t\n\t\\begin{align} \\label{E:NablaModZLiemodMinkowskiWaveOperatorCommutator}\n\t\t\\nablamod_{\\mathcal{Z}}^I \\Square_m \\phi & = \\Square_m \\nabla_{\\mathcal{Z}}^I \\phi.\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tUsing the symmetry of the tensorfield $\\nabla_{\\kappa} \\nabla_{\\lambda} \\phi,$ together with \\eqref{E:nablamis0}, \n\t\\eqref{E:ZDeformationTensorinTermsofcZm}, and definition \\eqref{E:Covariantmoddef}, we compute that\n\t\n\t\\begin{align}\n\t\t\\Square_m \\nabla_Z \\phi = (m^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda} \n\t\t\t\\big(Z^{\\zeta} \\nabla_{\\zeta} \\phi \\big)\n\t\t& = \\nabla_Z \\Square_m \\phi  \n\t\t\t\\ + \\ 2(\\nabla^{\\kappa} Z^{\\lambda}) \\nabla_{\\lambda} \\nabla_{\\kappa} \\phi \\\\\n\t\t& = \\nabla_Z \\Square_m \\phi \n\t\t\t\\ + \\ (\\nabla^{\\kappa} Z^{\\lambda} + \\nabla^{\\lambda} Z^{\\kappa}) \\nabla_{\\kappa} \\nabla_{\\lambda} \\phi \n\t\t\t \\notag \\\\\n\t\t& = \\nabla_Z \\Square_m \\phi \\ + \\ c_Z \\Square_m \\phi \\notag \\\\\n\t\t& \\eqdef \\nablamod_Z \\Square_m \\phi. \\notag \n\t\\end{align}\n\tThis proves \\eqref{E:NablaModZLiemodMinkowskiWaveOperatorCommutator} in the case $|I| = 1.$ The general case\n\tnow follows inductively.\n\t\n\\end{proof}\n\nThe next lemma shows that the modified Lie derivative $\\Lie_{\\mathcal{Z}}^I$ operator has favorable commutation properties with the linear Maxwell term $\\nabla_{\\mu} \\Far^{\\mu \\nu} = \\frac{1}{2}\\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda}.$\n\n\\begin{lemma} \\label{L:LiemodZLiemodMaxwellCommutator} \\textbf{(Commutation properties of $\\Liemod_{\\mathcal{Z}}^I$ with linear \n\tMaxwell term)}\n\tLet $I$ be a $\\mathcal{Z}-$multi-index, and let $\\Far$ be a\n\ttwo-form. Let $\\Liemod_{\\mathcal{Z}}^I$ be the iterated modified Lie derivative from Definitions \n\t\\ref{D:ModifiedDerivatives} and \\ref{D:iterated}. Then\n\t\n\t\\begin{align} \\label{E:LiemodZLiemodMaxwellCommutator}\n\t\t\\Liemod_{\\mathcal{Z}}^I \\Big\\lbrace & \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - \n\t\t\t(m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \\Big\\rbrace \\\\\n\t\t& = \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} \n\t\t\t- (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \n\t\t\t\\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}. \\notag\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tLet $Z \\in \\mathcal{Z}.$ By the Leibniz rule, \\eqref{E:LieZonmupper}, and Lemma \\ref{L:Liecommuteswithcoordinatederivatives}, \n\twe have that \n\t\n\t\\begin{align}\n\t\t\\Lie_Z \\Big\\lbrace & \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} \n\t\t\t(m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \\Big\\rbrace \\\\\n\t\t& = -2c_Z \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} \n\t\t\t(m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \\notag \\\\\n\t\t& \\ \\ + \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} \n\t\t\t(m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Lie_Z \\Far_{\\kappa \\lambda}. \\notag\n\t\\end{align}\n\tIt thus follows from Definition \\ref{D:ModifiedDerivatives} that \n\t\n\t\\begin{align}\n\t\t\\Liemod_{Z}\\Big\\lbrace & \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} \t\n\t\t\t(m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \\Big\\rbrace \\\\\n\t\t& = \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} \n\t\t\t(m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu} \\Lie_Z \\Far_{\\kappa \\lambda}. \\notag\n\t\\end{align}\n\tThis implies \\eqref{E:LiemodZLiemodMaxwellCommutator} in the case $|I| = 1.$ The general case now follows inductively.\n\\end{proof}\n\n\nThe next lemma shows that some of the differential operators we have introduced commute with the null decomposition of a two-form.\n\n\\begin{lemma} \\label{L:LieRotationcommuteswithnulldecomp} \\textbf{(Differential operators that commute with the null \n\tdecomposition)}\n\tLet $\\Far$ be a two-form and let $\\ualpha,$ $\\alpha,$ $\\rho,$ and $\\sigma$ be its null components.\n\tLet $O \\in \\mathcal{O}$ be any of the rotational Minkowskian Killing fields $\\Omega_{jk},$ \n\t$(1 \\leq j < k \\leq 3).$ Then $\\Lie_O \\ualpha[\\Far] = \\ualpha[\\Lie_O \\Far],$\n\t$\\Lie_O \\alpha[\\Far] = \\alpha[\\Lie_O \\Far],$ $\\Lie_O \\rho[\\Far] = \\rho[\\Lie_O \\Far],$ \n\tand $\\Lie_O \\sigma[\\Far] = \\sigma[\\Lie_O \\Far].$ An analogous result\n\tholds the operators $\\nabla_{\\uL}$ and $\\nabla_L;$ i.e.,\n\t$\\Lie_O, \\nabla_{\\uL},$ and $\\nabla_L$ commute with the null decomposition of $\\Far.$\n\\end{lemma}\n\n\\begin{proof}\n\tLemma \\ref{L:LieRotationcommuteswithnulldecomp} follows from \n\tLemma \\ref{L:PropertiesofuLandL}, \\eqref{E:nablamis0}, and Lemma \\ref{L:LieDerivativeCommuatorsVanish}.\n\\end{proof}\n\nThe next lemma shows that \\emph{weighted} covariant derivatives can be estimated by covariant derivatives with respect to\nvectorfields $Z \\in \\mathcal{Z}.$\n\n\\begin{lemma} \\label{L:PointwisetandqWeightedNablainTermsofZestiamtes} \\cite[Lemma 5.1]{hLiR2010}\n\t\\textbf{(Weighted pointwise differential operator inequalities)}\n\tFor any tensorfield $U$ and any two-tensor $\\Pi,$ we have the following pointwise estimates:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t(1 + t + |q|) |\\conenabla U| \\ + \\ (1 + |q|)|\\nabla U| & \\lesssim \\sum_{|I| \\leq 1} |\\nabla_{\\mathcal{Z}}^I U|, && \n\t\t\t\\label{E:WeightedDerivativesinTermsofNablaZI} \\\\\n\t\t|\\conenabla^2 U| \\ + \\ r^{-1} |\\conenabla U| & \\lesssim r^{-1}(1 + t + |q|)^{-1}\n\t\t\t\\sum_{|I| \\leq 2} |\\nabla_{\\mathcal{Z}}^I U|, && |\\conenabla^2 U| \n\t\t\t\\eqdef |\\conenabla \\conenabla U|, \\\\\n\t\t|\\Pi^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda} U| & \\lesssim \\Big\\lbrace(1 + t + |q|)^{-1} |\\Pi| \n\t\t\t\\ + \\ (1 + |q|)^{-1} |\\Pi|_{\\mathcal{L} \\mathcal{L}}\\Big\\rbrace \\sum_{|I| \\leq 1} |\\nabla\\nabla_{\\mathcal{Z}}^I U|.\n\t\t\t&&\n\t\\end{align}\n\t\\end{subequations}\n\\end{lemma}\n\n\\hfill $\\qed$\n\nThe next lemma shows that rotational Lie derivatives can be used to approximate weighted $S_{r,t}-$intrinsic\ncovariant derivatives. \n\n\\begin{lemma} \\label{L:RotationalLieDerivativesinTermsofrWeightedAngularDerivatives} \\cite[Lemma 7.0.17]{jS2010a} \n\t\t\\textbf{(Weighted covariant derivatives approximated by rotational Lie derivatives)}\n\t\tLet $U$ be any tensorfield $m-$tangent to the spheres $S_{r,t}$ and $k \\geq 0$ be any integer.\n\t\tThen with $r \\eqdef |x|,$ we have that\n\t\t\n\t\t\\begin{align} \\label{E:RotationalLieDerivativesinTermsofrWeightedAngularDerivatives}\n\t\t\t\\sum_{|I| \\leq k} r^{|I|} |\\angn^I U| \\approx \\sum_{|I| \\leq k} |\\Lie_{\\mathcal{O}}^I U|.\n\t\t\\end{align}\n\\end{lemma}\n\n\n\\begin{corollary} \\label{C:rWeightedAngularDerivativesinTermsofLieDerivatives}\n\tLet $\\Far$ be a two-form, and let $\\ualpha[\\Far],$ $\\alpha[\\Far],$ $\\rho[\\Far],$ $\\sigma[\\Far]$\n\tdenote its null components. Then with $r = |x|,$ we have that\n\t\n\t\\begin{align} \\label{E:rWeightedAngularDerivativesinTermsofLieDerivatives}\n\t\tr |\\angn \\ualpha[\\Far]| & \\lesssim \\sum_{|I| \\leq 1} |\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far]|.\n\t\\end{align}\n\tFurthermore, analogous inequalities hold for $\\alpha[\\Far],$ $\\rho[\\Far],$ and $\\sigma[\\Far].$\n\n\\end{corollary}\n\n\\begin{proof}\n\tInequality \\eqref{E:rWeightedAngularDerivativesinTermsofLieDerivatives} follows from \n\tLemma \\ref{L:LieRotationcommuteswithnulldecomp} and \n\tLemma \\ref{L:RotationalLieDerivativesinTermsofrWeightedAngularDerivatives}.\n\\end{proof}\n\n\nFinally, the following proposition provides pointwise inequalities relating various \nLie and covariant derivative operators under various contraction seminorms.\n \n\n\\begin{proposition} \\label{P:LievsCovariantLContractionRelation}\n\t\\textbf{(Lie derivative and covariant derivative inequalities)}\n\tLet $U$ be a tensorfield. Then\n\t\n\t\\begin{align} \\label{E:LieZIinTermsofNablaZI}\n\t\t\\sum_{|I| \\leq k} |\\Lie_{\\mathcal{Z}}^I U| \\approx \\sum_{|I| \\leq k} |\\nabla_{\\mathcal{Z}}^I U|.\n\t\\end{align}\n\t\n\tFurthermore, let $P$ be a symmetric or an anti-symmetric type $\\binom{0}{2}$ tensorfield. Then the following inequalities \n\thold:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\sum_{|I| \\leq k} |\\nabla\\Lie_{\\mathcal{Z}}^I P| \n\t\t& \\lesssim \\sum_{|I| \\leq k} |\\nabla\\nabla_{\\mathcal{Z}}^I P|,\n\t\t\t\\label{E:NablaLieZIinTermsofNablaNablaZI} \\\\\n\t\t\\sum_{|I| \\leq k} |\\conenabla \\Lie_{\\mathcal{Z}}^I P| \n\t\t& \\lesssim \\sum_{|I| \\leq k} |\\conenabla \\nabla_{\\mathcal{Z}}^I P|,\n\t\t\t\\label{E:TangentialDerivativesLieZIvsTangentialDerivativesNablaZI}\n\t\\end{align}\n\t\t\n\t\\begin{align} \n\t\t|\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{L}\\mathcal{L}}\n\t\t& \\lesssim |\\nabla_{\\mathcal{Z}}^I P|_{\\mathcal{L}\\mathcal{L}}\n\t\t\t\\ + \\ \\underbrace{\\sum_{|J| \\leq |I|-1} |\\nabla_{\\mathcal{Z}}^J P|_{\\mathcal{L} \\mathcal{T}}}_{\\mbox{absent if $|I| =\n\t\t\t0$}}\n\t\t\t\\ + \\ \\underbrace{\\sum_{|J'| \\leq |I|-2} |\\nabla_{\\mathcal{Z}}^{J'} P|}_{\\mbox{absent if $|I| \\leq 1$}}, \n\t\t\t\\label{E:LieZILLinTermsofNablaZILLLieZJLTPlusJunk} \\\\\n\t\t|\\nabla \\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{L}\\mathcal{L}}\n\t\t& \\lesssim |\\nabla \\nabla_{\\mathcal{Z}}^I P|_{\\mathcal{L}\\mathcal{L}}\n\t\t\t\\ + \\ \\underbrace{\\sum_{|J| \\leq |I|-1} |\\nabla_{\\mathcal{Z}}^J P]|_{\\mathcal{L} \\mathcal{T}}}_{\\mbox{absent if \n\t\t\t$|I| = 0$}}\n\t\t\t\\ + \\ \\underbrace{\\sum_{|J'| \\leq |I|-2} |\\nabla\\nabla_{\\mathcal{Z}}^{J'} P|}_{\\mbox{absent if $|I| \\leq 1$}},\n\t\t\t\\label{E:NablaLieZILLinTermsofNablaNablaZILLNablaLieZJLTPlusJunk}\n\t\\end{align}\n\t\n\t\n\t\n\t\n\t\n\t\\begin{align} \\label{E:NablaFarGoodInTermsofqWeightedLieZIFarGood}\n\t\t|\\nabla P|_{\\mathcal{L} \\mathcal{N}} \n\t\t+ |\\nabla P|_{\\mathcal{T} \\mathcal{T}}\n\t\t& \\lesssim (1 + |q|)^{-1} \\sum_{|I| \\leq 1} \\big(|\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{L} \\mathcal{N}} \n\t\t\t+ |\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{T} \\mathcal{T}} \\big)\n\t\t\t\\ + \\ (1 + t + |q|)^{-1} \\sum_{|I| \\leq 1} |\\Lie_{\\mathcal{Z}}^I P|.\n\t\\end{align}\n\t\\end{subequations}\n\t\n\\end{proposition}\n\n\n\\begin{proof}\n\tInequality \\eqref{E:LieZIinTermsofNablaZI} follows inductively using \n\t\\eqref{E:CovariantDerivativesofZareConstant} and \\eqref{E:Liederivativeintermsofnabla}. \n\t\n\tTo prove the remaining inequalities, for each $Z \\in \\mathcal{Z},$ we define the contraction operator $\\mathcal{C}_{Z}$ by\n\t\n\t\\begin{align} \\label{E:ContractionOperatorDef}\n\t\t(\\mathcal{C}_{Z} P)_{\\mu \\nu} \\eqdef P_{\\kappa \\nu}{^{(Z)}c_{\\mu}^{\\ \\kappa}} \\ + \\ P_{\\mu \\kappa}{^{(Z)}c_{\\nu}^{\\ \n\t\t\\kappa}},\n\t\\end{align}\n\twhere the covariantly constant tensorfield ${^{(Z)}c_{\\mu}^{\\ \\kappa}}$ is defined in \n\t\\eqref{E:CovariantDerivativesofZareConstant}. It follows from definition \\eqref{E:ContractionOperatorDef} and \n\tLemma \\ref{L:Liederivativeintermsofnabla} that\n\t\n\t\\begin{align}\n\t\t\\Lie_{\\mathcal{Z}} P = \\nabla_Z P \\ + \\ \\mathcal{C}_{Z} P.\n\t\\end{align}\n\tSince each $Z \\in \\mathcal{Z}$ is a conformal Killing field, and since \n\t$L^{\\mu}L^{\\nu} m_{\\mu \\nu} = 0,$ it follows that\n\t$L^{\\mu}L_{\\nu}{^{(Z)} c_{\\mu}^{\\ \\nu}} = 0.$ Also using the fact that each ${^{(Z)}c_{\\mu}^{\\ \\nu}}$ is a constant, we have \n\tthat\n\t\n\t\\begin{align}\n\t\t|\\mathcal{C}_{Z} P|_{\\mathcal{L} \\mathcal{L}} & \\lesssim |P|_{\\mathcal{L} \\mathcal{T}}\n\t\t\t\\label{E:LowerOrderLieTermLLContractioninTermsofLTContraction}, \\\\\n\t\t|\\mathcal{C}_{Z} P| & \\lesssim |P|. \\label{E:LowerOrderLieTermBounded}\n\t\\end{align}\n\nIf $I = (\\iota_1, \\cdots, \\iota_k)$ is a $\\mathcal{Z}-$multi-index with $1 \\leq |I| = k,$ then using the fact that the components ${^{(Z)}c_{\\mu}^{\\ \\kappa}}$ are constants, we have that\n\n\\begin{align} \\label{E:LieZIConformalKillingFieldNablaZIRelation}\n\t\\Lie_{\\mathcal{Z}}^I P \\eqdef \\Lie_{Z^{\\iota_1}} \\circ \\cdots \\circ \\Lie_{Z^{\\iota_k}} P \n\t& = (\\nabla_{Z^{\\iota_1}} \\ + \\ \\mathcal{C}_{Z^{\\iota_1}}) \\circ \\cdots \n\t\t\\circ (\\nabla_{Z^{\\iota_k}} \\ + \\ \\mathcal{C}_{Z^{\\iota_k}}) P \\\\\n\t& = \\nabla_{\\mathcal{Z}}^I P + \n\t\t\\sum_{i=1}^{k} \\mathcal{C}_{Z^{\\iota_i}} \\circ \\nabla_{Z^{\\iota_1}} \\circ \\cdots  \t\n\t\t\\circ \\nabla_{Z^{\\iota_{i-1}}} \\circ \\nabla_{Z^{\\iota_{i+1}}} \\circ \\cdots \\circ \n\t\t\\nabla_{Z^{\\iota_k}} P \\notag \\\\\n\t& \\ \\ + \\ \\underbrace{\\mathop{\\sum_{I_1 + I_2 = I}}_{|I|_2 \\leq k - 2}\n\t\t\\mathcal{C}_{\\mathcal{Z}}^{I_1} \\nabla_{\\mathcal{Z}}^{I_2} P}_{\\mbox{absent if $k = 1$}}. \\notag\n\\end{align}\nInequality \\eqref{E:NablaLieZIinTermsofNablaNablaZI} now follows from applying $\\nabla$ to each side of \\eqref{E:LieZIConformalKillingFieldNablaZIRelation}, from using the fact that the operator $\\nabla$ commutes through the operators $\\mathcal{C}_Z,$ and from \\eqref{E:LowerOrderLieTermBounded}. Inequality \\eqref{E:TangentialDerivativesLieZIvsTangentialDerivativesNablaZI} follows from similar reasoning.\nInequalities \\eqref{E:LieZILLinTermsofNablaZILLLieZJLTPlusJunk} and \\eqref{E:NablaLieZILLinTermsofNablaNablaZILLNablaLieZJLTPlusJunk} also follow from similar reasoning, together with \\eqref{E:LowerOrderLieTermLLContractioninTermsofLTContraction}. \n\n\nTo prove \\eqref{E:NablaFarGoodInTermsofqWeightedLieZIFarGood}, we first observe that\nby \\eqref{E:WeightedDerivativesinTermsofNablaZI}, \\eqref{E:LieZIinTermsofNablaZI}, \nand \\eqref{E:TangentialDerivativesLieZIvsTangentialDerivativesNablaZI}, we have that\n\n\\begin{align} \\label{E:PrelminaryEstimateNablaFarGoodInTermsofqWeightedLieZIFarGood}\n\t|\\nabla P|_{\\mathcal{L} \\mathcal{N}} \n\t\t\\ + \\ |\\nabla P|_{\\mathcal{T} \\mathcal{T}}\n\t& \\lesssim |\\nabla_{\\uL} P|_{\\mathcal{L} \\mathcal{N}} \n\t\t\\ + \\ |\\nabla_{\\uL} P|_{\\mathcal{T} \\mathcal{T}}\n\t\t\\ + \\ |\\conenabla P| \\\\\n\t& \\lesssim |\\nabla_{\\uL} P|_{\\mathcal{L} \\mathcal{N}} \n\t\t\\ + \\ |\\nabla_{\\uL} P|_{\\mathcal{T} \\mathcal{T}} \n\t\t\\ + \\ (1 + t + |q|)^{-1} \\sum_{|I| \\leq 1} |\\Lie_{\\mathcal{Z}}^I P|. \\notag\n\\end{align}\nTherefore, from \\eqref{E:PrelminaryEstimateNablaFarGoodInTermsofqWeightedLieZIFarGood}, we see that to prove \\eqref{E:NablaFarGoodInTermsofqWeightedLieZIFarGood}, it suffices to prove that the following inequality holds for any\nsymmetric or anti-symmetric type $\\binom{0}{2}$ tensorfield $P:$\n\n\\begin{align} \\label{E:uLDerivativeofFarGoodInTermsofqWeightedLieZIFarGood}\n\t\t|\\nabla_{\\uL} P|_{\\mathcal{L} \\mathcal{N}} + |\\nabla_{\\uL} P|_{\\mathcal{T} \\mathcal{T}}\n\t\t\t& \\lesssim (1 + |q|)^{-1} \\sum_{|I| \\leq 1} \\big(|\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{L} \\mathcal{N}} \n\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{T} \\mathcal{T}} \\big).\n\\end{align}\nTo this end, we use the vectorfields $S = x^{\\kappa} \\partial_{\\kappa},$ $\\Omega_{0j} = - t \\partial_j - x_j \\partial_t$ to decompose\n\n\\begin{align}\n\t\\uL & = - q^{-1}(S + \\omega^a \\Omega_{0a}), \\ \\omega^a \\eqdef x^a\/r,\n\\end{align}\nwhich implies that\n\n\\begin{align} \\label{E:qNablauLFarinTermsofNablaZFar}\n\t- q \\nabla_{\\uL} P_{\\mu \\nu} = \\nabla_S P_{\\mu \\nu} \\ + \\ \\omega^a \\nabla_{\\Omega_{0a}} P_{\\mu \\nu}.\n\\end{align}\n\nUsing \\eqref{E:ScalingCovariantDerivative}, \\eqref{E:SpacetimeRotationsCovariantDerivative}, \nand \\eqref{E:Liederivativeintermsofnabla}, we compute that\n\n\\begin{align}\n\t\\nabla_S P_{\\mu \\nu} & = \\Lie_{S} P_{\\mu \\nu} \\ - \\ 2 P_{\\mu \\nu}, \\\\\n\t\\omega^a \\nabla_{\\Omega_{0a}} P_{\\mu \\nu}  \n\t& = \\omega^a \\Lie_{\\Omega_{0a}} P_{\\mu \\nu}\n\t\t\\ - \\ \\frac{1}{2} \\Big\\lbrace \\uL_{\\mu} L^{\\kappa} P_{\\kappa \\nu} \n\t\t- L_{\\mu} \\uL^{\\kappa} P_{\\kappa \\nu} \n\t\t+ \\uL_{\\nu} L^{\\kappa} P_{\\mu \\kappa} \n\t\t- L_{\\nu} \\uL^{\\kappa} P_{\\mu \\kappa}\\Big\\rbrace.\n\\end{align}\nCombining these two identities with \\eqref{E:qNablauLFarinTermsofNablaZFar}, we conclude that\n\n\\begin{align} \\label{E:qWeigthedNablauLFarinTermsofLieDerivatives}\n\t- q \\nabla_{\\uL} P_{\\mu \\nu} = \\Lie_{S} P_{\\mu \\nu} \\ + \\ \\omega^a \\Lie_{\\Omega_{0a}} P_{\\mu \\nu}\n\t- 2 P_{\\mu \\nu} \\ - \\ \\frac{1}{2} \\Big\\lbrace \\uL_{\\mu} L^{\\kappa} P_{\\kappa \\nu} \n\t\t- L_{\\mu} \\uL^{\\kappa} P_{\\kappa \\nu} \n\t\t+ \\uL_{\\nu} L^{\\kappa} P_{\\mu \\kappa} \n\t\t- L_{\\nu} \\uL^{\\kappa} P_{\\mu \\kappa}\\Big\\rbrace.\n\\end{align}\nContracting \\eqref{E:qWeigthedNablauLFarinTermsofLieDerivatives} against the sets\n$\\mathcal{L}\\mathcal{N}$ and $\\mathcal{T}\\mathcal{T},$ it follows that\n\n\\begin{align} \\label{E:NablauLFarGoodComponentsinTermsofLieZIFarGoodComponentsLargeq}\n\t|q||\\nabla_{\\uL} P|_{\\mathcal{L} \\mathcal{N}} \\ + \\ |q||\\nabla_{\\uL} P|_{\\mathcal{T} \\mathcal{T}}\n\t& \\lesssim \\sum_{|I| \\leq 1}\n\t\t\\big(|\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{L} \\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{T} \\mathcal{T}}\\big).\n\\end{align}\nFurthermore, by decomposing\n\n\\begin{align}\n\t\\uL = \\partial_t - \\partial_r = \\partial_t - \\omega^a \\partial_a,\n\\end{align}\nand using the fact that ${^{(\\frac{\\partial}{\\partial t})}c_{\\mu}^{\\ \\nu}} = \n{^{(\\frac{\\partial}{\\partial x^j})}c_{\\mu}^{\\ \\nu}} = 0$ \n(where ${^{(Z)}c}_{\\mu \\nu}$ is defined in \\eqref{E:CovariantDerivativesofZareConstant}), it follows that\n\n\\begin{align} \\label{E:NablauLFarinTermsofLieDerivativesSmallq}\n\t\\nabla_{\\uL} P_{\\mu \\nu} = \\Lie_{\\frac{\\partial}{\\partial t}} P_{\\mu \\nu} \n\t\\ - \\ \\omega^a \\Lie_{\\frac{\\partial}{\\partial x^a}}P_{\\mu \\nu}.\n\\end{align}\nContracting \\eqref{E:NablauLFarinTermsofLieDerivativesSmallq} against the sets\n$\\mathcal{L}\\mathcal{N}$ and $\\mathcal{T}\\mathcal{T},$ we have that\n\n\\begin{align} \\label{E:NablauLFarGoodComponentsinTermsofLieZIFarGoodComponentsSmallq}\n\t|\\nabla_{\\uL} P|_{\\mathcal{L} \\mathcal{N}} + |\\nabla_{\\uL} P|_{\\mathcal{T} \\mathcal{T}}\n\t& \\lesssim \t\\sum_{|I| = 1}\n\t\t\\big(|\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{L} \\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^I P|_{\\mathcal{T} \\mathcal{T}} \\big).\n\\end{align}\nAdding \\eqref{E:NablauLFarGoodComponentsinTermsofLieZIFarGoodComponentsLargeq} and \\eqref{E:NablauLFarGoodComponentsinTermsofLieZIFarGoodComponentsSmallq}, we arrive at inequality\n\\eqref{E:uLDerivativeofFarGoodInTermsofqWeightedLieZIFarGood}. This completes our proof of \\eqref{E:NablaFarGoodInTermsofqWeightedLieZIFarGood}.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Reduced Equation Satisfied by \\texorpdfstring{$\\nabla_{\\mathcal{Z}}^I h^{(1)}$}{the Derivatives of the Metric Remainder Piece}} \\label{S:EquationSatisfiedbyNablaZIh1}\nIn this short section, we assume that $h_{\\mu \\nu}^{(1)}$ is a solution to the reduced equation\n\\eqref{E:Reducedh1Summary}. We provide a proposition that gives a preliminary description of the inhomogeneities\nin the equation satisfied by $\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)}.$\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\n\\begin{proposition} \\label{P:InhomogeneousTermsNablaZIh1}\n\\textbf{(Inhomogeneities for $\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)}$)}\nSuppose that $h_{\\mu \\nu}^{(1)}$ is a solution to the reduced equation \\eqref{E:Reducedh1Summary}, and \nlet $I$ be any $\\mathcal{Z}-$multi-index. Then $\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)}$ is a solution to the inhomogeneous system\n\n\\begin{align}\n\t\\widetilde{\\Square}_{g} \\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)} & = \\mathfrak{H}_{\\mu \\nu}^{(1;I)}, \n\t\t\\label{E:InhomogeneousTermsNablaZIh1} \\\\\n\t\\mathfrak{H}_{\\mu \\nu}^{(1;I)} & = \\nablamod_{\\mathcal{Z}}^I \\mathfrak{H}_{\\mu \\nu} \n\t\t- \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square} h_{\\mu \\nu}^{(0)} \t\n\t\t- \\Big\\lbrace \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)} - \\widetilde{\\Square}_{g} \n\t\t\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)} \\Big\\rbrace  \\label{E:mathfrakH1Idef} \\\\\n\t& = \\nablamod_{\\mathcal{Z}}^I \\mathfrak{H}_{\\mu \\nu} \n\t\t- \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square} h_{\\mu \\nu}^{(0)} \t\n\t\t- \\Big\\lbrace \\nablamod_{\\mathcal{Z}}^I \\big(H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda} h_{\\mu \\nu}^{(1)}\\big) \n\t\t- H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}\n\t\t\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)} \\Big\\rbrace. \\notag\t\n\\end{align}\n\n\\end{proposition}\n\n\\begin{proof}\n\tProposition \\ref{P:InhomogeneousTermsNablaZIh1} follows from differentiating each side of \\eqref{E:Reducedh1Summary}\n\twith modified covariant derivatives $\\nablamod_{\\mathcal{Z}}^I$ and applying Lemma \n\t\\ref{L:NablaModZLiemodMinkowskiWaveOperatorCommutator}.\n\\end{proof}\n\n\n\n\n\n\n\\section{The Equations of Variation, the Canonical Stress, and Electromagnetic Energy Currents} \\label{E:EOVandStress}\nIn this section, we introduce the electromagnetic equations of variation, which are linearized versions of the \nreduced electromagnetic equations. The significance of the equations of variation is the following: if $\\Far$ is a solution to the reduced electromagnetic equations \\eqref{E:ReduceddFis0Summary} - \\eqref{E:ReduceddMis0Summary}, then $\\Lie_{\\mathcal{Z}}^I \\Far$ is a solution to the equations of variation. We then provide a preliminary description of the structure of the inhomogeneous terms in the equations of variation satisfied by $\\Lie_{\\mathcal{Z}}^I \\Far.$ Additionally, we introduce the canonical stress tensorfield and use it to construct energy currents, which are vectorfields that will be used in Section \\ref{S:WeightedEnergy} to derive weighted energy estimates for solutions to the equations of variation.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{Equations of variation}\n\nThe equations of variation in the unknowns $\\dot{\\Far}_{\\mu \\nu}$ are the linearization of \\eqref{E:ReduceddFis0Summary} - \\eqref{E:ReduceddMis0Summary} around a background $(h_{\\mu \\nu}, \\Far_{\\mu \\nu}).$ More specifically, the equations of variation are the\nsystem\n\n\\begin{subequations} \n\\begin{align}\n\t\\nabla_{\\lambda} \\dot{\\Far}_{\\mu \\nu} + \\nabla_{\\mu} \\dot{\\Far}_{\\nu \\lambda} + \\nabla_{\\nu} \\dot{\\Far}_{\\lambda \\mu}\n\t\t& = \\dot{\\mathfrak{F}}_{\\lambda \\mu \\nu},&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:EOVdFis0} \\\\\n\tN^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda} & = \\dot{\\mathfrak{F}}^{\\nu},&& (\\nu = 0,1,2,3), \n\t\t\\label{E:EOVdMis0} \n\\end{align}\n\\end{subequations}\nwhere $N^{\\# \\mu \\nu \\kappa \\lambda}$ is the $(h_{\\mu \\nu}, \\Far_{\\mu \\nu})-$dependent tensorfield defined in \\eqref{E:NSummarydef}, and $\\dot{\\mathfrak{F}}_{\\lambda \\mu \\nu},$ $\\dot{\\mathfrak{F}}^{\\nu}$ are inhomogeneous terms that need to be specified. In this article, the equations of variation will arise when we differentiate the reduced equations \\eqref{E:ReduceddFis0Summary} - \\eqref{E:ReduceddMis0Summary} with modified Lie derivatives. In this case, the quantities \n$\\Lie_{\\mathcal{Z}}^I \\Far_{\\mu \\nu}$ will play the role of $\\dot{\\Far}.$ The next proposition, which is a companion of Proposition \\ref{P:InhomogeneousTermsNablaZIh1}, provides a preliminary expression of the inhomogeneous terms that arise in the study of the equations of variation satisfied by $\\Lie_{\\mathcal{Z}}^I \\Far_{\\mu \\nu}.$\n\n\n\\begin{proposition} \\label{P:InhomogeneoustermsLieZIFar}\n\t\\textbf{(Inhomogeneities for $\\Lie_{\\mathcal{Z}}^I \\Far_{\\mu \\nu}$)}\n\tIf $\\Far_{\\mu \\nu}$ is a solution to the reduced electromagnetic equations \n\t\\eqref{E:ReduceddFis0Summary} - \\eqref{E:ReduceddMis0Summary} and $I$ is a\n\t$\\mathcal{Z}-$multi-index, then $\\dot{\\Far}_{\\mu \\nu} \\eqdef \\Lie_{\\mathcal{Z}}^I \\Far_{\\mu \\nu}$ is a solution to the \n\tequations of variation \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0} (corresponding to the background \n\t$(h_{\\mu \\nu}, \\Far_{\\mu \\nu})$) with inhomogeneous terms $\\dot{\\mathfrak{F}}_{\\lambda \\mu \\nu} \\eqdef\n\t\\mathfrak{F}_{\\lambda \\mu \\nu}^{(I)}$ and $\\dot{\\mathfrak{F}}^{\\nu} \\eqdef \\mathfrak{F}_{(I)}^{\\nu},$ where\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t\\mathfrak{F}_{\\lambda \\mu \\nu}^{(I)} & = 0, && (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:EOVInhomogeneousTermsJvanish} \\\\\n\t\t\\mathfrak{F}_{(I)}^{\\nu} & = \\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu}\n\t\t\t+ \\Big\\lbrace N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I\\Far_{\\kappa \\lambda}\n\t\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big)\\Big\\rbrace,\n\t\t\t&& (\\nu = 0,1,2,3).\n\t\t\t\\label{E:LiemodZIdifferentiatedEOVInhomogeneousterms}\n\t\\end{align}\n\t\\end{subequations}\n\t\n\tFurthermore, there exist constants $\\widetilde{C}_{1;I_1,I_2}, \\widetilde{C}_{2;I_1,I_2},\n\t\\widetilde{C}_{\\mathscr{P};I_1,I_2}, \\widetilde{C}_{\\mathfrak{F}_{\\triangle};J}, \\widetilde{C}_{\\dParameter_{\\triangle};I_1,I_2}$ such \n\tthat\n\t\n\t\\begin{subequations}\n\t\\begin{align}\t\t\n\t\t\\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu} \n\t\t& = \\sum_{|I_1| + |I_2| \\leq |I|} \\widetilde{C}_{2;I_1,I_2}\n\t\t\t\t\\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla\\Lie_{\\mathcal{Z}}^{I_1} h, \\Lie_{\\mathcal{Z}}^{I_2} \\Far) \n\t\t\t \\label{E:LiemodZIFExpanded} \\\\\n\t\t& \\ \\ + \\sum_{|J| \\leq |I|} \\widetilde{C}_{\\mathfrak{F}_{\\triangle};J} \n\t\t\t\\Lie_{\\mathcal{Z}}^J \\mathfrak{F}_{\\triangle}^{\\nu}, \\notag \\\\\n\t\tN^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}\n\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big) \n\t\t& = \\mathop{\\sum_{|I_1| + |I_2| \\leq |I|}}_{|I_2| \\leq |I| - 1} \\widetilde{C}_{\\mathscr{P};I_1,I_2}\n\t\t\t\\mathscr{P}_{(\\Far)}^{\\nu}(\\Lie_{\\mathcal{Z}}^{I_1} h, \\nabla \\Lie_{\\mathcal{Z}}^{I_2}\\Far) \n\t\t \t\\label{E:LiemodZINnablaFarCommutatorTerms} \\\\\n\t\t& \\ \\ + \\mathop{\\sum_{|I_1| + |I_2| \\leq |I|}}_{|I_2| \\leq |I| - 1} \\widetilde{C}_{1;I_1,I_2}\n\t\t\t\\mathscr{Q}_{(1;\\Far)}^{\\nu}(\\Lie_{\\mathcal{Z}}^{I_1} h, \\nabla \\Lie_{\\mathcal{Z}}^{I_2}\\Far)\n\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\mathop{\\sum_{|I_1| + |I_2| \\leq |I|}}_{|I_2| \\leq |I| - 1} \\widetilde{C}_{\\dParameter_{\\triangle};I_1,I_2}\n\t\t\t(\\Lie_{\\mathcal{Z}}^{I_1} N_{\\triangle}^{\\# \\mu \\nu \\kappa \\lambda}) \n\t\t\t\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^{I_2} \\Far_{\\kappa \\lambda}. \\notag\n\t\\end{align}\n\t\\end{subequations}\n\tIn the above formulas, $\\mathfrak{F}_{\\triangle}^{\\nu}$ and $N_{\\triangle}^{\\# \\mu \\nu \\kappa \\lambda}$ are the error terms\n\tappearing in \\eqref{E:FtriangleSmallAlgebraic} and \\eqref{E:NtriangleSmallAlgebraic} respectively, \n\twhile $\\mathscr{P}_{(\\Far)}^{\\nu}(\\cdot, \\cdot)$ and $\\mathscr{Q}_{(i;\\Far)}^{\\nu}(\\cdot, \\cdot),$ $(i=1,2),$  \n\t$(\\nu = 0,1,2,3),$ are the quadratic forms defined in \\eqref{E:PFar}, \\eqref{E:Q1Far}, and \\eqref{E:Q2Far} respectively. \n\\end{proposition}\n\n\\begin{proof}\n\t\n\tTo prove \\eqref{E:EOVInhomogeneousTermsJvanish}, we first recall equation \\eqref{E:ReduceddFis0Summary}, which states \n\tthat $\\Far_{\\mu \\nu}$ is a solution to $\\nabla_{[\\kappa} \\Far_{\\mu \\nu]} = 0,$ where $[\\cdots]$ denotes anti-symmetrization.\n\tUsing \\eqref{E:Liecommuteswithcoordinatederivatives} it therefore follows that\n\t\n\t\\begin{align}\n\t\t0 = \\Lie_{\\mathcal{Z}}^I \\nabla_{[\\lambda} \\Far_{\\mu \\nu]} = \\nabla_{[\\lambda} \\Lie_{\\mathcal{Z}}^I \\Far_{\\mu \\nu]},\n\t\\end{align}\n\twhich is the desired result. \n\t\n\tTo derive \\eqref{E:LiemodZIdifferentiatedEOVInhomogeneousterms}, we simply differentiate each side of\n\t\\eqref{E:EOVdMis0} with $\\Liemod_{\\mathcal{Z}}^I$ to conclude that \n\t$\\Liemod_{\\mathcal{Z}}^I \\big( N^{\\# \\mu \\nu \\kappa \\lambda} \n\t\\nabla_{\\mu} \\Far_{\\kappa \\lambda} \\big) = \\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu}.$ Trivial algebraic manipulation \n\tthen leads to the fact that $N^{\\# \\mu \\nu \\kappa \\lambda} \n\t\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I\\Far_{\\kappa \\lambda} = \\mathfrak{F}_{(I)}^{\\nu},$ where $\\mathfrak{F}_{(I)}^{\\nu}$ \n\tis defined by \\eqref{E:LiemodZIdifferentiatedEOVInhomogeneousterms}.\n\t\n \tEquation \\eqref{E:LiemodZIFExpanded} follows from \\eqref{E:EMBIFarInhomogeneous},\n \tthe Definition \\ref{D:ModifiedDerivatives} of $\\Liemod_Z,$ and Lemma \\ref{L:nullformvectorfieldcommutation}, which is proved\n \tin Section \\ref{S:UsefulLemmas}.\n\t\n\tTo prove \\eqref{E:LiemodZINnablaFarCommutatorTerms}, we first recall equation \\eqref{E:NNullFormDecomposition}:\n\t\n\t\\begin{align} \\label{E:NNullFormDecompositionAgain}\n\t\tN^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \n\t\t& = \\frac{1}{2} \\big[ (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa} \n\t\t\t\\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda} \n\t\t\t\\ - \\ \\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far) \n\t\t\t\\ - \\ \\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\Far)\n\t\t\t\\ + \\ N_{\\triangle}^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda}.\n\t\\end{align}\t\n\tThe commutator term arising from the $\\big[ (m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} \n\t\t- (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa} \\big] \\nabla_{\\mu} \\Far_{\\kappa \\lambda}$ term on the right-hand side of \n\t\\eqref{E:NNullFormDecompositionAgain} vanishes. More specifically, we use \\eqref{E:LiemodZLiemodMaxwellCommutator} to conclude that\n\t\n\t\\begin{align} \\label{E:LieMinkowskiCommutatorTermVanishes}\n\t\t\t\\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa} \\big]  \n\t\t\t\t\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}\n\t\t\t\\ - \\ \\Liemod_{\\mathcal{Z}}^I \\Big\\lbrace \n\t\t\t\t\\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} \n\t\t\t\t- (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\\big] \\nabla_{\\mu}\\Far_{\\kappa \\lambda} \\Big\\rbrace = 0.\n\t\\end{align}\n\tTherefore, it follows from \\eqref{E:NNullFormDecompositionAgain} and \\eqref{E:LieMinkowskiCommutatorTermVanishes} that\n\t\n\t\t\\begin{align}\t\t\n\t\t\tN^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}\n\t\t\t\\ - \\ \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big) \n\t\t\t& = \\Liemod_{\\mathcal{Z}}^I \\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far)\n\t\t\t\\ - \\ \\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla\\Lie_{\\mathcal{Z}}^I\\Far)\n\t\t\t\\label{E:ProofLiemodZINnablaFarCommutatorTerms} \\\\\n\t\t& \\ \\ + \\ \\Liemod_{\\mathcal{Z}}^I \\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\Far)\n\t\t\t\\ - \\ \\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla\\Lie_{\\mathcal{Z}}^I\\Far) \\notag \\\\\n\t\t& \\ \\ + \\ N_{\\triangle}^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \n\t\t\t\\Far_{\\kappa \\lambda} \\ - \\ \\Liemod_{\\mathcal{Z}}^I (N_{\\triangle}^{\\# \\mu \\nu \\kappa \\lambda} \n\t\t\t\\nabla_{\\mu} \\Far_{\\kappa \\lambda}). \\notag\n\t\t\\end{align}\n\t\tThe expression \\eqref{E:LiemodZINnablaFarCommutatorTerms} now follows from \\eqref{E:ProofLiemodZINnablaFarCommutatorTerms},\n\t\tthe Leibniz rule, the Definition \\ref{D:ModifiedDerivatives} of $\\Liemod_Z,$ \n\t\tLemma \\ref{L:Liecommuteswithcoordinatederivatives}, and Lemma \\ref{L:nullformvectorfieldcommutation}.\n\n\n\\end{proof}\n\n\n\n\n\n\\subsection{The canonical stress} \\label{SS:CanonicalStress}\n\nThe notion of the \\emph{canonical stress tensorfield} $\\Stress_{\\ \\nu}^{\\mu}$ in the context of PDE energy estimates\nwas introduced by Christodoulou in \\cite{dC2000}. As explained in Section \\ref{SSS:EnergyandStress}, from the point of view of energy estimates, it plays the role of an energy-momentum-type tensor for the equations of variation. Its two key properties are i) its divergence is lower-order (in the sense of the number of derivatives falling on the variations $\\dot{\\Far}_{\\mu \\nu}$); and ii) contraction against a suitable (covector, vector) pair $(\\xi_{\\mu}, X^{\\nu})$ leads to a positive energy density that can be used achieve $L^2$ control of solutions $\\dot{\\Far}_{\\mu \\nu}$ to the equations of variation. \nAs we will see, property i) is captured by Lemma \\ref{L:DivergenceofStress} and \\eqref{E:currentdivergence}, while property ii)\nis captured by \\eqref{E:dotJ0estimate}, \\eqref{E:FirstweightedenergyFar}, and \\eqref{E:SecondweightedenergyFar}. In order to understand the origin of the canonical stress, we first introduce Christodoulou's \\emph{linearized Lagrangian} \\cite{dC2000}.\n\n\n\\begin{definition}\n\nGiven an electromagnetic Lagrangian $\\mathscr{L}[\\cdot]$ (as described in Section \\ref{SS:Lagrangianformluationofnonlinearelectromagnetism}) and a ``background'' $(h_{\\mu \\nu}, \\Far_{\\mu \\nu}),$ we define the linearized Lagrangian by\n\n\\begin{align} \\label{E:LinearizedLagrangian}\n\t\\dot{\\mathscr{L}} = \\dot{\\mathscr{L}}[\\dot{\\Far};h,\\Far] \n\t\t\\eqdef \\frac{1}{2} \\frac{\\partial^2 \\Ldual[h,\\Far]}{\\partial \\Far_{\\zeta \\eta} \\partial \\Far_{\\kappa \n\t\t\\lambda}} \\dot{\\Far}_{\\zeta \\eta} \\dot{\\Far}_{\\kappa \\lambda} \n\t\t= - \\frac{1}{4}N^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \\dot{\\Far}_{\\kappa \\lambda},\n\\end{align}\nwhere $N^{\\#\\zeta \\eta \\kappa \\lambda}$ is the $(h_{\\mu \\nu}, \\Far_{\\mu \\nu})-$dependent tensorfield defined in \\eqref{E:firstNdef}.\n\n\\end{definition}\n\nThe merit of the above definition is the following: the principal part (from the point of view of number of derivatives) of the Euler-Lagrange equations (assuming that we view $(h,\\Far)$ as a background, $\\dot{\\Far}$ to be the unknowns, and that \nan appropriately defined action\\footnote{A suitable action $\\mathcal{A}_{\\mathfrak{C}}[\\dot{\\Far}]$ is e.g. of the form  $\\mathcal{A}_{\\mathfrak{C}}[\\dot{\\Far}] \\eqdef \\int_{\\mathfrak{C} \\Subset \\mathfrak{M}} \\dot{\\mathscr{L}}[\\dot{\\Far};h,\\Far] \\, d^4x,$ where $\\mathfrak{C}$ is a compact subset of spacetime.} is stationary with respect to closed variations of $\\dot{\\Far}$) corresponding to $\\dot{\\mathscr{L}}[\\dot{\\Far};h,\\Far]$ is identical to the principal part of the electromagnetic equations of variation \\eqref{E:EOVdMis0}; i.e., $\\dot{\\mathscr{L}}[\\dot{\\Far};h,\\Far]$ generates the linearized equations.\n\n\n\\begin{definition}\nGiven a linearized Lagrangian $\\dot{\\mathscr{L}}[\\dot{\\Far};h,\\Far],$ the canonical stress tensorfield $\\Stress_{\\ \\nu}^{\\mu}$ is defined as follows:\n\n\\begin{align} \\label{E:Stressdef}\n\t\\Stress_{\\ \\nu}^{\\mu} \\eqdef - 2\\frac{\\partial \\dot{\\mathscr{L}}}{\\partial \\dot{\\Far}_{\\mu \\zeta}}\\dot{\\Far}_{\\nu \\zeta}\n\t\t\\ + \\ \\delta_{\\nu}^{\\mu} \\dot{\\mathscr{L}} \n\t\t=\tN^{\\# \\mu \\zeta \\kappa \\lambda} \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{\\nu \\zeta}\n\t\t\t\\ - \\ \\frac{1}{4} \\delta_{\\nu}^{\\mu} N^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \\dot{\\Far}_{\\kappa \\lambda},\n\\end{align}\nwhere $N^{\\# \\mu \\nu \\kappa \\lambda}$ is defined in \\eqref{E:firstNdef}.\n\\end{definition}\n\\noindent Note that in contrast to the energy-momentum tensor $T_{\\mu \\nu},$ \n$\\Stress_{\\mu \\nu} \\eqdef m_{\\mu \\kappa} \\Stress_{\\ \\nu}^{\\kappa}$ is in general not symmetric.\n\n\nBecause of our assumption \\eqref{E:Ldualassumptions} concerning the Lagrangian, $\\Stress_{\\ \\nu}^{\\mu}$ is equal to the \nenergy-momentum tensor (in $\\dot{\\Far}$) for the linear Maxwell-Maxwell equations in Minkowski space, plus small corrections. More specifically, it follows from definition \\ref{E:Stressdef} and the decomposition \\eqref{E:NSummarydef} that\n\n\\begin{align} \\label{E:StressExpansion}\n\t\\Stress_{\\ \\nu}^{\\mu} & = \\overbrace{\\dot{\\Far}^{\\mu \\zeta} \\dot{\\Far}_{\\nu \\zeta} \n\t\t\\ - \\ \\frac{1}{4} \\delta_{\\nu}^{\\mu} \n\t\t\\dot{\\Far}_{\\zeta \\eta}\\dot{\\Far}^{\\zeta \\eta}}^{\\mbox{terms from linear Maxwell-Maxwell equations \n\t\tin Minkowski spacetime}} \\\\\n\t\t& \\ \\ \\overbrace{- \\ h^{\\mu \\kappa} \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{\\nu}^{\\ \\zeta}\n\t\t\t\\ - \\ h^{\\kappa \\lambda} \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{\\nu \\lambda}\n\t\t\t\\ + \\ \\frac{1}{2} \\delta_{\\nu}^{\\mu} h^{\\kappa \\lambda} \\dot{\\Far}_{\\kappa \\eta}\\dot{\\Far}_{\\lambda}^{\\ \\eta}}^{\\mbox{corrections to Minkowskian \n\t\t\tlinear Maxwell-Maxwell equations arising from $h$}}\n\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\ \\underbrace{N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda} \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{\\nu \\zeta}\n\t\t\t\\ - \\ \\frac{1}{4} \\delta_{\\nu}^{\\mu} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \n\t\t\t\\dot{\\Far}_{\\kappa \\lambda}.}_{\\mbox{error terms}} \\notag\n\\end{align}\n\nThe next lemma captures the lower-order divergence property enjoyed by $\\Stress_{\\ \\nu}^{\\mu}.$ \n\n\\begin{lemma} \\label{L:DivergenceofStress} \\textbf{(Divergence of the canonical stress)}\nLet $\\dot{\\Far}_{\\mu \\nu}$ be a solution to the equations of variation \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0}\ncorresponding to the background $(h_{\\mu \\nu}, \\Far_{\\mu \\nu}),$ and let $\\dot{\\mathfrak{F}}_{\\lambda \\mu \\nu},$ $\\dot{\\mathfrak{F}}^{\\nu}$ be the inhomogeneous terms from the right-hand sides of \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0}. Let $\\Stress_{\\ \\nu}^{\\mu}$\nbe the canonical stress tensorfield defined in \\eqref{E:Stressdef}. Then\n\n\\begin{align} \\label{E:divergenceofStress}\n\t\\nabla_{\\mu} \\Stress_{\\ \\nu}^{\\mu} & = - \\ \\frac{1}{2} N^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \n\t\t\\dot{\\mathfrak{F}}_{\\nu \\kappa \\lambda}\n\t\t\\ + \\ \\dot{\\Far}_{\\nu \\eta} \\dot{\\mathfrak{F}}^{\\eta}\n\t\t\\ + \\ (\\nabla_{\\mu}N^{\\# \\mu \\zeta \\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{\\nu \\zeta}\n\t\t\\ - \\ \\frac{1}{4} (\\nabla_{\\nu}N^{\\#\\zeta \\eta \\kappa \\lambda}) \\dot{\\Far}_{\\zeta \\eta} \\dot{\\Far}_{\\kappa \\lambda}, \\\\\n\t& = \\ - \\frac{1}{2} N^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \\dot{\\mathfrak{F}}_{\\nu \\kappa \\lambda}\n\t\t\\ + \\ \\dot{\\Far}_{\\nu \\eta} \\dot{\\mathfrak{F}}^{\\eta} \\notag \\\\\n\t& \\ \\ - \\ (\\nabla_{\\mu} h^{\\mu \\kappa}) \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{\\nu}^{\\ \\zeta}\n\t\t\t\\ - \\ (\\nabla_{\\mu} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{\\nu \\lambda}\n\t\t\t\\ + \\ \\frac{1}{2} (\\nabla_{\\nu} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\eta}\\dot{\\Far}_{\\lambda}^{\\ \\eta}\n\t\t\\notag \\\\\n\t& \\ \\ + \\ (\\nabla_{\\mu}N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{\\nu \\zeta}\n\t\t\\ - \\ \\frac{1}{4} (\\nabla_{\\nu}N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda}) \\dot{\\Far}_{\\zeta \\eta} \\dot{\\Far}_{\\kappa \n\t\t\\lambda}. \\notag\n\\end{align}\n\n\\end{lemma}\n\n\\begin{proof}\n\tTo obtain \\eqref{E:divergenceofStress}, we use use the equations \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0}, \n\ttogether with the properties \\eqref{E:Nminussignproperty1} - \\eqref{E:Nsymmetryproperty}, which are also satisfied by\n\tthe tensorfield $N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda}.$\n\\end{proof}\n\n\n\n\n\\subsection{Electromagnetic energy currents}\n\nIn this section, we introduce the energy current that will be used to derive the \nweighted energy estimate \\eqref{E:FirstweightedenergyFar} for a solution $\\dot{\\Far}$ to the equations\nof variation \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0}.\n\n\\begin{definition} \\label{D:Jdotdef}\n\tLet $h_{\\mu \\nu}$ be a symmetric type $\\binom{0}{2}$ tensorfield, and let\n\t$\\Far_{\\mu \\nu},$ $\\dot{\\Far}_{\\mu \\nu}$ be a pair of two-forms. Let $w(q)$ be the weight defined in \\eqref{E:weight},\n\tand let $X^{\\nu} \\eqdef w(q)\\delta_0^{\\nu}$ be the ``multiplier'' vectorfield.\n\tWe define the \\emph{energy current} $\\dot{J}_{(h,\\Far)}^{\\mu}[\\dot{\\Far}]$ corresponding to the variation \n\t$\\dot{\\Far}_{\\mu \\nu}$ and the background $(h_{\\mu \\nu}, \\Far_{\\mu \\nu})$ to be the vectorfield\n\t\n\t\\begin{align} \\label{E:Jdotdef}\n\t\t\\dot{J}_{(h,\\Far)}^{\\mu}[\\dot{\\Far}] & \\eqdef - \\Stress_{\\ \\nu}^{\\mu} X^{\\nu} = - w(q) \\Stress_{\\ 0}^{\\mu},\n\t\\end{align}\n\twhere $\\Stress_{\\ \\nu}^{\\mu}$ is the canonical stress tensorfield from \\eqref{E:Stressdef}.\n\\end{definition}\n\n\n\n\\begin{lemma} \\label{L:currentproperties} \\textbf{(Positivity of $\\dot{J}_{(h,\\Far)}^0$)}\n\tLet $\\dot{J}_{(h,\\Far)}^{\\mu}[\\dot{\\Far}]$ be the energy current defined in \\eqref{E:Jdotdef}. Then\n\t\n\t\\begin{align} \\label{E:dotJ0estimate}\n\t\t\\dot{J}_{(h,\\Far)}^0 & = \\frac{1}{2} |\\dot{\\Far}|^2 w(q) \n\t\t\\ + \\ \\Big\\lbrace O^{\\infty}(|h|;\\Far) + O^{\\dParameter}\\big(|(h,\\Far)|^2 \\big) \\Big\\rbrace|\\dot{\\Far}|^2 w(q).\n\t\\end{align}\n\t\n\tFurthermore, if $\\dot{\\Far}_{\\mu \\nu}$ is a solution to the equations of variation \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0}\n\twith inhomogeneous terms $\\dot{\\mathfrak{F}}_{\\lambda \\mu \\nu} \\equiv 0,$\n\tthen the Minkowskian divergence of $\\dot{J}_{(h,\\Far)}$ can be expressed as follows:\n\t\n\t\\begin{align} \\label{E:currentdivergence}\n\t\t\\nabla_{\\mu} \\dot{J}_{(h,\\Far)}^{\\mu} & = - \\ \\frac{1}{2} w'(q) (\\dot{\\alpha}^2 + \\dot{\\rho}^2 + \\dot{\\sigma}^2) \\\\\n\t\t& \\ \\ - \\ w(q) \\Big \\lbrace \\dot{\\Far}_{0 \\eta} \\dot{\\mathfrak{F}}^{\\eta} \n\t\t\t- (\\nabla_{\\mu} h^{\\mu \\kappa}) \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta}\n\t\t\t- (\\nabla_{\\mu} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda}\n\t\t\t+ \\frac{1}{2} (\\nabla_{t} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\eta}\\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big \\rbrace \\notag  \\\\\n\t\t& \\ \\ - \\ w'(q) \\Big\\lbrace - L_{\\mu}h^{\\mu \\kappa} \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta} \n\t\t\t- L_{\\mu} h^{\\kappa \\lambda} \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda} \n\t\t\t- \\frac{1}{2} h^{\\kappa \\lambda} \\dot{\\Far}_{\\kappa \\eta} \\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big\\rbrace \\notag \\\\\n\t\t& \\ \\ - \\ w(q) \\Big \\lbrace (\\nabla_{\\mu}N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\lambda} \n\t\t\t\\dot{\\Far}_{0 \\zeta} - \\frac{1}{4} (\\nabla_{t} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda}) \\dot{\\Far}_{\\zeta \\eta} \n\t\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big \\rbrace \\notag \\\\ \n\t\t& \\ \\ - \\ w'(q) \\Big\\lbrace L_{\\mu} N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda} \n\t\t\t\\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{0 \\zeta} + \\frac{1}{4} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda} \n\t\t\t\\dot{\\Far}_{\\zeta \\eta} \\dot{\\Far}_{\\kappa \\lambda} \\Big\\rbrace,  \\notag\n\t\\end{align}\n\twhere $\\dot{\\alpha} \\eqdef \\alpha[\\dot{\\Far}],$ $\\dot{\\rho} \\eqdef \\rho[\\dot{\\Far}],$ \n\tand $\\dot{\\sigma} \\eqdef \\sigma[\\dot{\\Far}]$ are the ``favorable'' Minkowskian null components of $\\dot{\\Far}$\n\tdefined in Section \\ref{SS:NullComponents}.\n\t\n\\end{lemma}\n\n\\begin{remark}\n\tThe term $\\frac{1}{2} w'(q) (\\dot{\\alpha}^2 + \\dot{\\rho}^2 + \\dot{\\sigma}^2)$ appearing on the right-hand side of\n\tof \\eqref{E:currentdivergence} is of central importance for closing the bootstrap argument during\n\tour global existence proof. It manifests itself as the additional positive space-time integral\n\t$\\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}}^2 \n\t+ |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}}^2 \\big) w'(q) \\,d^3x \\, d \\tau$ on the left-hand side of \n\t\\eqref{E:FirstweightedenergyFar} below, and provides a means\n\tfor controlling some of the spacetime integrals that emerge in Section \\ref{SS:MainTheoremFarInhomogeneities}. \n\\end{remark}\n\n\n\\begin{proof}\n\t\\eqref{E:dotJ0estimate} follows from \\eqref{E:StressExpansion}, simple calculations, and \\eqref{E:NtriangleSmallAlgebraic}.\n\t\n\tTo prove \\eqref{E:currentdivergence}, we first recall that since $q = r - t,$ it follows that $\\nabla_{\\mu} q = L_{\\mu},$\n\twhere $L$ is defined in \\eqref{E:Ldef}. Hence, we have that $\\nabla_{\\mu} w(q) = w'(q) L_{\\mu}.$ \n\tUsing this fact, \\eqref{E:StressExpansion}, and \\eqref{E:divergenceofStress}, we calculate that\n\t\n\t\\begin{align}\n\t\t\\nabla_{\\mu} \\dot{J}_{(h,\\Far)}^{\\mu} & = - \\ w(q) \\dot{\\Far}_{0 \\eta} \\dot{\\mathfrak{F}}^{\\eta} \\\\\n\t\t& \\ \\ - \\ w(q) \\Big \\lbrace - (\\nabla_{\\mu} h^{\\mu \\kappa}) \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta}\n\t\t\t- (\\nabla_{\\mu} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda}\n\t\t\t+ \\frac{1}{2} (\\nabla_{t} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\eta}\\dot{\\Far}_{\\lambda}^{\\ \\eta}\n\t\t\t\\Big \\rbrace \\notag \\\\\n\t\t& \\ \\ - \\ w(q) \\Big \\lbrace (\\nabla_{\\mu}N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\lambda} \n\t\t\t\\dot{\\Far}_{0 \\zeta} - \\frac{1}{4} (\\nabla_{t} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda}) \\dot{\\Far}_{\\zeta \\eta} \n\t\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big \\rbrace \\notag \\\\\n\t\t& \\ \\ - \\ w'(q) \\Big\\lbrace \\underbrace{L_{\\mu} \\dot{\\Far}^{\\mu \\zeta} \\dot{\\Far}_{0 \\zeta} \n\t\t\t+ \\frac{1}{4} \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}^{\\kappa \\lambda}}_{\\frac{1}{2}(\\dot{\\alpha}^2 + \\dot{\\rho}^2 + \n\t\t\t\t\\dot{\\sigma}^2)} \\Big \\rbrace \\notag \\\\\n\t\t& \\ \\ - \\ w'(q) \\Big\\lbrace - L_{\\mu}h^{\\mu \\kappa} \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta} \n\t\t\t- L_{\\mu} h^{\\kappa \\lambda} \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda} \n\t\t\t- \\frac{1}{2} h^{\\kappa \\lambda} \\dot{\\Far}_{\\kappa \\eta} \\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big\\rbrace \\notag \\\\\n\t\t& \\ \\ - \\ w'(q) \\Big\\lbrace L_{\\mu} N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda} \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{0 \\zeta}\n\t\t\t+ \\frac{1}{4} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \n\t\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big\\rbrace.  \\notag \\\\\n\t\\end{align}\n\tThe expression \\eqref{E:currentdivergence} thus follows.\n\\end{proof}\n\n\n\n\\section{Decompositions of the Electromagnetic Equations} \\label{S:DecompositionsofElectromagneticEquations}\n\nIn this section we perform two decompositions of the electromagnetic equations. The first is a null decomposition of the equations of variation, which will be used in Section \\ref{S:DecayFortheReducedEquations} to derive pointwise decay estimates for the lower-order Lie derivatives of $\\Far_{\\mu \\nu}.$ The second is a decomposition of the electromagnetic equations into constraint and evolution equations for the Minkowskian one-forms $\\Electricfield_{\\mu},$ $\\Magneticinduction_{\\mu},$ which are \nrespectively known as the electric field and magnetic induction. This decomposition will be used in Section \\ref{S:SmallDataAssumptions} to prove that our smallness condition on the abstract data necessarily implies a smallness condition on the initial energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0)$ of the corresponding solution to the reduced equations. We remark that the Minkowskian one-forms $\\Displacement_{\\mu},$ $\\Magneticfield_{\\mu},$ which are respectively known as the electric displacement and the magnetic field, and also the geometric electromagnetic one-forms $\\mathfrak{\\Electricfield}_{\\mu},$ $\\mathfrak{\\Magneticinduction}_{\\mu},$ $\\mathfrak{\\Displacement}_{\\mu},$ $\\mathfrak{\\Magneticfield}_{\\mu}$ will play a role in the discussion.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{The Minkowskian null decomposition of the electromagnetic equations of variation} \\label{SS:NullDecompElectromagnetic} \nIn this section, we decompose the equations of variation into equations for the null components of $\\dot{\\Far}.$ The main advantage of our decomposition, which is given in Proposition \\ref{P:EOVNullDecomposition}, is the following: the terms in each equation can be separated into two classes: i) a derivative of a null component in a ``nearly-Minkowski-null'' direction\\footnote{By ``nearly-Minkowski-null,'' we mean vectors that are nearly parallel to $\\uL$ or $L,$ with some corrections coming from the presence of a non-zero $h$ in the case of the null component $\\dot{\\ualpha}.$} (which appears on the left-hand side of the inequality); and ii) some error terms (which appear on the right-hand side of the inequality). Although from the point of view of differentiability the error terms are not lower-order, it will turn out that they are lower-order in terms of decay rates. In this way, the equations can be viewed as \\emph{ordinary differential equations} for the null components of $\\dot{\\Far}$ with inhomogeneous terms; this point of view is fully realized in Proposition \\ref{P:ODEsNullComponentsLieZIFar}. The key point is that the ODEs we derive will be amenable to Gronwall estimates: in Section \\ref{S:DecayFortheReducedEquations}, we will use this line of argument to derive pointwise decay estimates for the null components of the lower-order Lie derivatives of a solution $\\Far$ to the electromagnetic equations \\eqref{E:ReduceddFis0Summary} - \\eqref{E:ReduceddMis0Summary}. These estimates will be an improvement over what can be deduced from the weighted Klainerman-Sobolev inequality \\eqref{E:PhiKlainermanSobolev} alone.\n\nWe begin the analysis by using \\eqref{E:NSummarydef} to write the equations of variation \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0}\nin the following form:\n\n\\begin{subequations} \n\\begin{align}\n\t\\nabla_{\\lambda} \\dot{\\Far}_{\\mu \\nu} + \\nabla_{\\mu} \\dot{\\Far}_{\\nu \\lambda} + \\nabla_{\\nu} \\dot{\\Far}_{\\lambda \\mu}\n\t\t= 0, \\label{E:EOVdFis0nullsection} \n\\end{align}\n\\begin{align}\n\t\\bigg\\lbrace \\frac{1}{2} \\big[(m^{-1})^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} - (m^{-1})^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa}\n\t\t& - h^{\\mu \\kappa} (m^{-1})^{\\nu \\lambda} + h^{\\mu \\lambda} (m^{-1})^{\\nu \\kappa} \n\t\t\\label{E:EOVdMis0nullsection} \\\\\n\t& - (m^{-1})^{\\mu \\kappa} h^{\\nu \\lambda} + (m^{-1})^{\\mu \\lambda} h^{\\nu \\kappa} \\big]\n\t\t+ N_{\\triangle}^{\\# \\mu \\nu \\kappa \\lambda} \\bigg\\rbrace \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda} \n\t\t= \\dot{\\mathfrak{F}}^{\\nu}. \\notag  \n\\end{align}\n\\end{subequations}\n\n\n\nIn our calculations below, we will make use of the identities\n\n\\begin{align} \\label{E:nablaALnablaAuL}\n\t\\nabla_A \\uL = -r^{-1} e_A, && \\nabla_A L = r^{-1} e_A, \n\\end{align}\nwhich can be directly calculated in our wave coordinate system using \\eqref{E:uLdef} - \\eqref{E:Ldef}. We will also make use\nof the identity \n\n\\begin{align} \\label{E:nablaAeBcovriantintrintermsofinsicnablaAeBcovraint}\n\t\\angn_A e_B & = \\nabla_A e_B \\ + \\ \\frac{1}{2} m(\\nabla_A e_B, \\uL) L \\ + \\ \\frac{1}{2} m(\\nabla_A e_B, L) \\uL \\\\\n\t& =  \\nabla_A e_B \\ - \\ \\frac{1}{2} m(e_B, \\nabla_A \\uL) L \\ - \\ \\frac{1}{2} m(e_B, \\nabla_A L)\\uL \\notag \\\\\n\t& = \\nabla_A e_B \\ + \\ \\frac{1}{2}r^{-1} \\delta_{AB} (L - \\uL), \\notag\n\\end{align}\nwhich follows from \\eqref{E:SphereIntrinsicintermsofExtrinsic} and \\eqref{E:nablaALnablaAuL}.\n\n\nFurthermore, if $U$ is a type $\\binom{0}{m}$ tensorfield, and $X_{(i)},$ $(1 \\leq i \\leq m),$ and $Y$ are vectorfields, then \nby the Leibniz rule, we have that\n\n\\begin{align} \\label{E:nablaLeibnizrule}\n\t\\nabla_Y \\big\\lbrace U(X_{(1)}, \\cdots, X_{(m)}) \\big\\rbrace \n\t& = (\\nabla_Y U)(X_{(1)} \\cdots, X_{(m)}) \n\t\t\\ + \\ U(\\nabla_Y X_{(1)}, X_{(2)}, \\cdots, X_{(m)}) \\ + \\ \\cdots \\ + \\ U(X_{(1)}, X_{(2)}, \\cdots, \\nabla_Y X_{(m)}).\n\\end{align}\nSimilarly, if $U$ is $m-$tangent to the spheres $S_{r,t},$ then\n\n\\begin{align} \\label{E:instrinsicnablaLiebnizrule}\n\t\\angn_A (U_{B_{(1)}, \\cdots, B_{(m)}}) & \\eqdef \\angn_{e_A} \\big\\lbrace U(e_{B_{(1)}}, \\cdots, e_{B_{(m)}}) \\big\\rbrace \\\\\n\t& = (\\angn_A U)(e_{B_{(1)}}, \\cdots, e_{B_{(m)}}) \\ + \\ U(\\angn_A e_{B_{(1)}}, e_{B_{(2)}}, \\cdots, e_{B_{(m)}}) \\notag \\\\\n\t& \\ \\ + \\ \\cdots \\ + \\ U(e_{B_{(1)}}, e_{B_{(2)}}, \\cdots, \\angn_A e_{B_{(m)}}). \\notag\n\\end{align}\n\nApplying \\eqref{E:nablaLeibnizrule} and \\eqref{E:instrinsicnablaLiebnizrule} to $\\Far,$ and using \\eqref{E:nablaALnablaAuL}, \\eqref{E:nablaAeBcovriantintrintermsofinsicnablaAeBcovraint}, and \\eqref{E:Fardualalpha} - \\eqref{E:Fardualsigma},\nwe compute (as in \\cite[pg. 161]{dCsK1990}) the following identities, which we state as a lemma.\n\n\\begin{lemma} \\label{L:FarDerivativeNullComponentRelations} \\textbf{(Contracted derivatives expressed in terms of the null components)}\nLet $\\Far$ be a two-form, and let $\\ualpha,$ $\\alpha,$ $\\rho,$ and $\\sigma$ be its null components. Then the following identities hold:\n\n\\begin{subequations}\n\\begin{align}\n\t\\nabla_A \\Far_{B \\uL} & = \\angn_A \\ualpha_B \\ - \\ r^{-1}(\\rho \\delta_{AB} + \\sigma \\angupsilon_{AB}), \n\t\t\\label{E:nablaAFarBuLintermsofintrinsic} \\\\\n\t\\nabla_A \\Far_{BL} & = \\angn_A \\alpha_B \\ - \\ r^{-1}(\\rho \\delta_{AB} - \\sigma \\angupsilon_{AB}), \\\\\n\t\\nabla_A \\FarMinkdual_{B \\uL} & = - \\ \\angupsilon_{CB}\\angn_A \\ualpha_C \n\t\t\\ - \\ r^{-1}(\\sigma \\delta_{AB} - \\rho \\angupsilon_{AB}), \\\\\n\t\\nabla_A \\FarMinkdual_{BL} & = \\angupsilon_{CB}\\angn_A \\alpha_C \\ - \\ r^{-1}(\\sigma \\delta_{AB} \n\t\t\\ + \\ \\rho \\angupsilon_{AB}), \\\\\n\t\\frac{1}{2} \\nabla_A \\Far_{\\uL L} & = \\angn_A \\rho \\ + \\ \\frac{1}{2} r^{-1}(\\ualpha_A + \\alpha_A), \\\\\n\t\\frac{1}{2} \\nabla_A \\FarMinkdual_{\\uL L} & = \\angn_A \\sigma \\ + \\ \\frac{1}{2} r^{-1}(-\\angupsilon_{BA}\\ualpha_B \n\t\t\\ + \\ \\angupsilon_{BA} \\alpha_B), \\\\\n\t\\nabla_A \\Far_{BC} & = \\angupsilon_{BC} \\Big\\lbrace \\angn_A \\sigma + \\frac{1}{2}r^{-1}\n\t\t(-\\angupsilon_{DA}\\ualpha_D + \\angupsilon_{DA} \\alpha_D) \\Big\\rbrace. \\label{E:nablaAFarBCintermsofintrinsic}\n\\end{align}\n\\end{subequations}\nNote that in all of our expressions, contractions are taken after differentiating; e.g., \n$\\nabla_A \\Far_{BL} \\eqdef e_A^{\\mu} e_B^{\\kappa} \\uL^{\\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda}.$\n\n\\end{lemma}\n\n\\begin{remark} \\label{R:FarDerivativeNullComponentRelations}\n\t\n\tThe identities in Lemma \\ref{L:FarDerivativeNullComponentRelations} can be \n\treinterpreted as identities for spacetime tensors that are $m-$tangent to the spheres $S_{r,t}.$ That is,\n\tthey can be rephrased in terms of our wave coordinate frame with the help of the projection $\\angm_{\\mu}^{\\ \\nu}$ \n\tand the spherical volume form $\\angupsilon_{\\mu}^{\\ \\nu}$ defined in \\eqref{E:angmdef} and \\eqref{E:Spheresvolumedef} \n\trespectively. For example, equation \\eqref{E:nablaAFarBuLintermsofintrinsic} is equivalent \n\tto the following equation:\n\t\n\t\\begin{align}\n\t\t\\angm_{\\mu}^{\\ \\mu'} \\angm_{\\nu}^{\\ \\nu'} \\uL^{\\kappa} \\nabla_{\\mu}\\Far_{\\nu' \\kappa}\n\t\t& = \\angm_{\\mu}^{\\ \\mu'} \\angm_{\\nu}^{\\ \\nu'} \\nabla_{\\mu'} \\ualpha_{\\nu'} \n\t\t\t\\ - \\ r^{-1}(\\rho \\angm_{\\mu \\nu} \\ + \\ \\sigma \\angupsilon_{\\mu \\nu}).\n\t\\end{align}\n\tWe will use the spacetime coordinate frame version of the identities in our proof of Proposition \\ref{P:EOVNullDecomposition}.\n\\end{remark}\n\n\n\nWe now derive equations for the null components of a solution $\\dot{\\Far}$ to \\eqref{E:EOVdFis0nullsection} - \\eqref{E:EOVdMis0nullsection}.\n\n\n\\begin{proposition} \\label{P:EOVNullDecomposition}\n\\textbf{(Minkowskian null decomposition of the equations of variation)}\nLet $\\dot{\\Far}$ be a solution to the equations of variation \\eqref{E:EOVdFis0nullsection} - \\eqref{E:EOVdMis0nullsection}, and let $\\dot{\\ualpha} \\eqdef \\ualpha[\\dot{\\Far}],$ $\\dot{\\alpha} \\eqdef \\alpha[\\dot{\\Far}],$ $\\dot{\\rho} \\eqdef \\rho[\\dot{\\Far}],$ $\\dot{\\sigma} \\eqdef \\sigma[\\dot{\\Far}]$ denote its Minkowskian null components.\nAssume that the source term $\\dot{\\mathfrak{F}}_{\\lambda \\mu \\nu}$ on the right-hand side of \\eqref{E:EOVdFis0nullsection} vanishes.\\footnote{By Proposition \\ref{P:InhomogeneoustermsLieZIFar}, this assumption holds for the variations of interest in this article.} Then the following equations are verified by the null components:\n\n\\begin{subequations}\n\\begin{align}\n\t& \\nabla_L \\dot{\\ualpha}_{\\nu} \\ + \\ r^{-1}\\dot{\\ualpha}_{\\nu} \\ + \\ \\angm_{\\nu}^{\\ \\kappa} \\nabla_{\\kappa} \\dot{\\rho} \n\t\t\\ - \\ \\angupsilon_{\\nu}^{\\ \\kappa} \\nabla_{\\kappa} \\dot{\\sigma} && \\label{E:dotualphaEOVnulldecomp} \\\\\n\t& \\ \\ - \\ \\overbrace{\\angm_{\\nu}^{\\ \\lambda} h^{\\mu \\kappa} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}}^{\\eqdef \\angm_{\\nu \n\t\t\t\\lambda} \\mathscr{P}_{(\\Far)}^{\\lambda}(h, \\nabla \\dot{\\Far})}\n\t\t\\ - \\ \\overbrace{\\angm_{\\nu \\nu'} (m^{-1})^{\\mu \\kappa} h^{\\nu' \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \n\t\t\t\\lambda}}^{\\eqdef \\angm_{\\nu \\lambda} \\mathscr{Q}_{(1;\\Far)}^{\\lambda}(h, \\nabla \\dot{\\Far})} \n\t\t\\ + \\ \\angm_{\\nu \\nu'} N_{\\triangle}^{\\# \\mu \\nu' \\kappa \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}\n\t\t&& = \\ \\angm_{\\nu \\nu'} \\dot{\\mathfrak{F}}^{\\nu'}, \\notag \\\\\n\t& \\nabla_{\\uL} \\dot{\\alpha}_{\\nu} \\ - \\ r^{-1}\\dot{\\alpha}_{\\nu} \\ - \\ \\angm_{\\nu}^{\\ \\kappa} \\nabla_{\\kappa} \\dot{\\rho} \n\t\t\\ - \\ \\angupsilon_{\\nu}^{\\ \\kappa} \\nabla_{\\kappa} \\dot{\\sigma}&& \\label{E:uLdotalphaEOVnulldecomp} \\\\\n\t& \\ \\ - \\ \\overbrace{\\angm_{\\nu}^{\\ \\lambda} h^{\\mu \\kappa} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}}^{\\eqdef \\angm_{\\nu \n\t\t\\lambda} \\mathscr{P}_{(\\Far)}^{\\lambda}(h, \\nabla \\dot{\\Far})}\n\t\t\\ - \\ \\overbrace{\\angm_{\\nu \\nu'} (m^{-1})^{\\mu \\kappa} h^{\\nu' \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}}^{\\eqdef \n\t\t\t\\angm_{\\nu \\lambda} \\mathscr{Q}_{(1;\\Far)}^{\\lambda}(h, \\nabla \\dot{\\Far})} \n\t\t\\ + \\ \\angm_{\\nu \\nu'} N_{\\triangle}^{\\# \\mu \\nu' \\kappa \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}\n\t\t&& = \\ \\angm_{\\nu \\nu'} \\dot{\\mathfrak{F}}^{\\nu'}, \\notag  \\\\\n\t& \\nabla_{\\uL} \\dot{\\rho} \\ + \\ \\angm^{\\mu \\nu} \\nabla_{\\mu} \\dot{\\ualpha}_{\\nu} \\ - \\ 2 r^{-1} \\dot{\\rho}\n\t\t\\ - \\ \\overbrace{\\uL^{\\lambda} h^{\\mu \\kappa} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}}^{\n\t\t\\uL_{\\lambda} \\mathscr{P}_{(\\Far)}^{\\lambda}(h, \\nabla \\dot{\\Far})} && \\label{E:uLdotnablarhoEOVnulldecomp} \\\\\n\t& \\ \\ - \\ \\overbrace{\\uL_{\\nu} (m^{-1})^{\\mu \\kappa} h^{\\nu \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}}^{\n\t\t\\uL_{\\nu} \\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\dot{\\Far})} \n\t\t\\ + \\ \\uL_{\\nu} N_{\\triangle}^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}\n\t\t&& = \\ \\uL_{\\lambda} \\dot{\\mathfrak{F}}^{\\lambda}, \\notag \\\\\n\t& \\nabla_{\\uL} \\dot{\\sigma} \\ - \\ 2 r^{-1} \\dot{\\sigma} \n\t\t\\ + \\ \\angupsilon^{\\mu \\nu} \\nabla_{\\mu} \\dot{\\ualpha}_{\\nu} && = \\ 0, \\label{E:uLdotnablasigmaEOVnulldecomp} \\\\\n\t& \\nabla_L \\dot{\\rho} \\ - \\ \\angm^{\\mu \\nu} \\nabla_{\\mu} \\dot{\\alpha}_{\\nu} \\ + \\ 2 r^{-1} \\dot{\\rho}&&\n\t\t\\label{E:nablaLdotrhoEOVnulldecomp} \\\\\n\t& \\ \\ + \\ \\overbrace{L^{\\lambda} h^{\\mu \\kappa} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}}^{L_{\\lambda} \t\n\t\t\\mathscr{P}_{(\\Far)}^{\\lambda}(h, \\nabla \\dot{\\Far})}\n\t\t\\ + \\ \\overbrace{L_{\\nu} (m^{-1})^{\\mu \\kappa} h^{\\nu \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}}^{\n\t\t\tL_{\\nu} \\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\dot{\\Far})}  \n\t\t\\ - \\ L_{\\nu} N_{\\triangle}^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}\n\t\t&& = \\ - L_{\\lambda} \\dot{\\mathfrak{F}}^{\\lambda}, \\notag \\\\\n\t& \\nabla_L \\dot{\\sigma} \\ + \\ 2 r^{-1} \\dot{\\sigma} \\ + \\ \\angupsilon^{\\mu \\nu} \\nabla_{\\mu} \\dot{\\ualpha}_{\\nu} && = \\ 0. \n\t\t\\label{E:nablaLdotsigmaEOVnulldecomp}\n\\end{align}\nIn the above expressions, the quadratic terms $\\mathscr{P}_{(\\Far)}^{\\lambda}(h, \\nabla \\dot{\\Far})$ \nand $\\mathscr{Q}_{(1;\\Far)}^{\\lambda}(h, \\nabla \\dot{\\Far})$ are as defined in Section \\ref{SS:ReducedEquations}.\n\\end{subequations}\n\n\\begin{remark} \\label{R:FavorableAngular}\nNote that in the above equations, we have that e.g. $\\angm_{\\nu}^{\\ \\kappa} \\nabla_{\\kappa} = \\angm_{\\nu}^{\\ \\kappa} \\angn_{\\kappa}$ and $\\angupsilon_{\\nu}^{\\ \\kappa} \\nabla_{\\kappa} = \\angupsilon_{\\nu}^{\\ \\kappa} \\angn_{\\kappa},$ so that these operators only involve favorable angular derivatives.\n\\end{remark}\n\n\\end{proposition}\n\n\\begin{proof}\n\tTo obtain \\eqref{E:dotualphaEOVnulldecomp} and \\eqref{E:uLdotalphaEOVnulldecomp}, we contract \\eqref{E:EOVdFis0nullsection} \n\tagainst $\\uL^{\\lambda} L^{\\mu} e_A^{\\nu},$ \\eqref{E:EOVdMis0nullsection} against $(e_A)_{\\nu},$\n\tand use Lemma \\ref{L:FarDerivativeNullComponentRelations} plus Remark \\ref{R:FarDerivativeNullComponentRelations}\n\tto deduce that\n\t\n\t\\begin{align}\n\t\t& \\nabla_L \\ualpha_{\\nu} \\ - \\ \\nabla_{\\uL} \\alpha_{\\nu} \\ + \\ 2 \\angm_{\\nu}^{\\ \\nu'} \\nabla_{\\nu'} \\rho \n\t\t\t\\ + \\ r^{-1}(\\ualpha_{\\nu} + \\alpha_{\\nu}) && = \\ 0, \\\\\n\t\t& \\nabla_L \\ualpha_{\\nu} \\ + \\ \\nabla_{\\uL} \\alpha_{\\nu} \\ - \\ 2 \\angupsilon_{\\nu}^{\\ \\kappa} \\nabla_{\\kappa} \\sigma\n\t\t\t\\ + \\ r^{-1}(\\ualpha_{\\nu} - \\alpha_{\\nu}) &&  \\\\\n\t\t& \\ \\ - \\ 2 \\angm_{\\nu}^{\\ \\lambda} h^{\\mu \\kappa} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda} \n\t\t\t\\ - \\ 2 \\angm_{\\nu \\nu'} (m^{-1})^{\\mu \\kappa} h^{\\nu' \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}\n\t\t\t\\ + \\ \\angm_{\\nu \\nu'} N_{\\triangle}^{\\# \\mu \\nu' \\kappa \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda}\n\t\t\t&& = \\ 2 \\angm_{\\nu \\nu'} \\dot{\\mathfrak{F}}^{\\nu'}. \\notag\n\t\\end{align}\n\tAdding the two above equations gives \\eqref{E:dotualphaEOVnulldecomp}, while subtracting the first from the second\n\tgives \\eqref{E:uLdotalphaEOVnulldecomp}.\n\t\n\tSimilarly, to deduce \\eqref{E:uLdotnablasigmaEOVnulldecomp},\n\twe contract \\eqref{E:EOVdFis0nullsection} \n\tagainst $\\uL^{\\lambda} e_A^{\\ \\mu} e_B^{\\ \\nu},$\n\tand then contract against $\\angupsilon_{AB};$ to deduce \\eqref{E:nablaLdotsigmaEOVnulldecomp}, we contract\n\t\\eqref{E:EOVdFis0nullsection} against $L^{\\lambda} e_A^{\\ \\mu} e_B^{\\ \\nu},$\n\tand then against $\\angupsilon_{AB};$ to deduce \\eqref{E:uLdotnablarhoEOVnulldecomp}, we contract \n\t\\eqref{E:EOVdMis0nullsection} against $\\uL_{\\nu};$\n\tand to deduce \\eqref{E:nablaLdotrhoEOVnulldecomp}, we contract\n\t\\eqref{E:EOVdMis0nullsection} against $-L_{\\nu}.$\n\t\n\\end{proof}\t\n\n\n\n\\subsection{Electromagnetic one-forms} \\label{SS:EBDH}\nIn this section, we introduce the one-forms $\\mathfrak{\\Electricfield},$ $\\mathfrak{\\Magneticinduction},$ \n$\\mathfrak{\\Displacement},$ and $\\mathfrak{\\Magneticfield},$ which are derived from a geometric decomposition of\n$\\Far$ with the help of the spacetime metric $g_{\\mu \\nu}.$ We also introduce the one-forms $\\Electricfield,$ $\\Magneticinduction,$ $\\Displacement,$ and $\\Magneticfield,$ which are derived from a Minkowskian decomposition of $\\Far.$ We then derive an equivalent version of the electromagnetic equations, namely constraint and electromagnetic evolution equations for the Minkowskian one-forms. These quantities play a role only in Section \\ref{S:SmallDataAssumptions}, where they are used to connect the smallness of the abstract initial data to the smallness of the energy of the corresponding reduced solution at $t=0.$ Furthermore, we show that the abstract one-forms $\\mathring{\\mathfrak{\\Displacement}}, \\mathring{\\mathfrak{\\Magneticinduction}},$ satisfy the constraints \\eqref{E:DivergenceD0Intro} - \\eqref{E:DivergenceB0Intro}\nif and only if the corresponding Minkowskian one-forms $\\mathring{\\Displacement}, \\mathring{\\Magneticinduction},$ satisfy a Minkowskian version of the constraints.\n\n\nWe will perform our electromagnetic decompositions of the equations with the help of two versions of the electromagnetic equations, namely \\eqref{E:dFis0Dversion}, \\eqref{E:dMis0Dversion} and \\eqref{E:dFis0nablaversion}, \\eqref{E:dMis0nablaversion}. We restate them here for convenience:\n\n\\begin{subequations}\n\t\\begin{align}\n\t\t\\mathscr{D}_{\\lambda} \\Far_{\\mu \\nu} + \\mathscr{D}_{\\mu} \\Far_{\\nu \\lambda} + \\mathscr{D}_{\\nu} \\Far_{\\lambda \\mu} & = 0,&&\n\t\t\t(\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dFis0ElectrodecompMathscrD} \\\\\n\t\t\\mathscr{D}_{\\lambda} \\Max_{\\mu \\nu} + \\mathscr{D}_{\\mu} \\Max_{\\nu \\lambda} + \\mathscr{D}_{\\nu} \\Max_{\\lambda \\mu} & = 0,&&\n\t\t\t(\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dMis0ElectrodecompMathscrD}\n\t\\end{align}\n\\end{subequations}\nand\n\n\\begin{subequations}\n\t\\begin{align}\n\t\t\\nabla_{\\lambda} \\Far_{\\mu \\nu} + \\nabla_{\\mu} \\Far_{\\nu \\lambda} + \\nabla_{\\nu} \\Far_{\\lambda \\mu} & = 0,&&\n\t\t\t(\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:dFis0Electrodecomp} \\\\\n\t\t\\nabla_{\\lambda} \\Max_{\\mu \\nu} + \\nabla_{\\mu} \\Max_{\\nu \\lambda} + \\nabla_{\\nu} \\Max_{\\lambda \\mu} & = 0,&&\n\t\t\t(\\lambda, \\mu, \\nu = 0,1,2,3). \\label{E:dMis0Electrodecomp}\n\t\\end{align}\n\\end{subequations}\n\n\n\n\n\n\n\n\n\\begin{comment}\n\n\n\\begin{subequations}\n\t\\begin{align}\n\t\t\\mathscr{D}_{\\mu} \\Fardual^{\\# \\mu \\nu} & = 0, && (\\nu = 0,1,2,3), \\label{E:DivFardual0Electrodecomp} \\\\\n\t\t\\mathscr{D}_{\\mu} \\Maxdual^{\\# \\mu \\nu} & = 0, && (\\nu = 0,1,2,3), \\label{E:DivMaxdual0Electrodecomp}\n\t\\end{align}\n\\end{subequations}\n\n\\begin{subequations}\n\t\\begin{align}\n\t\t\\nabla_{\\mu} \\FarMinkdual^{\\mu \\nu} & = 0, && (\\nu = 0,1,2,3), \\label{E:DivFarMinkdual0Electrodecomp} \\\\\n\t\t\\nabla_{\\mu} \\MaxMinkdual^{\\mu \\nu} & = 0, && (\\nu = 0,1,2,3). \\label{E:DivMaxMinkdual0Electrodecomp}\n\t\\end{align}\n\\end{subequations}\n\n\\end{comment}\nBefore decomposing the equations, we first define the aforementioned geometric electromagnetic one-forms.\n\n\\begin{definition} \\label{D:IntrinsicOneForms}\n\tLet $\\hat{N}^{\\mu} = \\hat{N}^{\\mu}(t,x)$ denote the future-directed unit $g-$normal to the hypersurface $\\Sigma_t.$ \n\tThen in components relative to an arbitrary coordinate system, we define the following one-forms:\n\n\t\\begin{align} \\label{E:AbstractEBDHinertialcomponents}\n\t\\mathfrak{\\Electricfield}_{\\mu} & = \\Far_{\\mu \\kappa}\\hat{N}^{\\kappa}, & \\mathfrak{\\Magneticinduction}_{\\mu} & = \n\t\t- \\Fardual_{\\mu \\kappa}\\hat{N}^{\\kappa}, & \\mathfrak{\\Displacement}_{\\mu} & = - \t\n\t\\Maxdual_{\\mu \\kappa} \\hat{N}^{\\kappa},  \t& \\mathfrak{\\Magneticfield}_{\\mu} & = - \\Max_{\\mu \\kappa}\\hat{N}^{\\kappa}.\n\t\t\\end{align}\n\t\\textbf{Note that in the above expressions, $\\star$ denotes the Hodge duality operator corresponding to the spacetime metric \n\t$g.$}\n\\end{definition}\n\nWe now define the Minkowskian electromagnetic one-forms.\n\n\\begin{definition} \\label{D:EBDHinertialcomponents}\nIn components relative to the wave coordinate coordinate system $\\lbrace x^{\\mu} \\rbrace_{\\mu = 0,1,2,3},$ we define the \\emph{electric field} $\\Electricfield,$ the \\emph{magnetic induction} $\\Magneticinduction,$ the \\emph{electric displacement} $\\Displacement,$ and the \\emph{magnetic field} $\\Magneticfield$ by\n\n\\begin{align} \\label{E:EBDHinertialcomponents}\n\t\\Electricfield_{\\mu} & = \\Far_{\\mu 0}, & \\Magneticinduction_{\\mu} & = - \\FarMinkdual_{\\mu 0}, & \\Displacement_{\\mu} & = - \t\n\t\\MaxMinkdual_{\\mu 0},  \t& \\Magneticfield_{\\mu} & = - \\Max_{\\mu 0}.\n\\end{align}\n\t\\textbf{Note that in the above expressions, $\\ostar$ denotes the Hodge duality operator corresponding to the \n\tMinkowski metric $m.$}\n\\end{definition}\n\nObserve that \\eqref{E:EBDHinertialcomponents} implies that\n\n\\begin{align} \\label{E:FarspatialintermsofB}\n\t\\Far_{jk} & = [ijk]\\Magneticinduction_i,&& \\Magneticinduction_j = \\frac{1}{2} [jab] \\Far_{ab},&& \n\t\\Displacement_j = \\frac{1}{2} [jab] \\Max_{ab}, && (j,k = 1,2,3).\n\\end{align}\n\n\\begin{remark}\n\tOur definition of $\\Magneticinduction$ coincides with the one commonly found in the physics literature, but it has the \n\topposite sign convention of the definition given in \\cite{dCsK1990}.\n\\end{remark}\n\nIt follows from the anti-symmetry of $\\Far_{\\mu \\nu}$ and $\\Max_{\\mu \\nu}$ that $\\Electricfield_{\\mu},$ $\\Magneticinduction_{\\mu},$ $\\Displacement_{\\mu},$ and $\\Magneticfield_{\\mu}$ are $m-$tangent to the hyperplanes $\\Sigma_t;$ i.e., we have that $\\Electricfield_0 = \\Magneticinduction_0 = \\Displacement_0 = \\Magneticfield_0 = 0.$ We may therefore view these four quantities as one-forms that are intrinsic to $\\Sigma_t.$ Similarly, we have that\n$\\mathfrak{\\Electricfield}_{\\mu}\\hat{N}^{\\mu} = \\mathfrak{\\Magneticinduction}_{\\mu}\\hat{N}^{\\mu}=\n\\mathfrak{\\Displacement}_{\\mu}\\hat{N}^{\\mu}  = \\mathfrak{\\Magneticfield}_{\\mu}\\hat{N}^{\\mu} = 0.$\n\n\nUsing Definition \\ref{D:EBDHinertialcomponents}, the assumption \\eqref{E:Ldualassumptions} on the Lagrangian, \n\\eqref{E:MaxintermsofFarMinkDualPlusError}, \\eqref{E:FarspatialintermsofB}, \nand \\eqref{E:DintermsofEBh} - \\eqref{E:HintermsofEBh} it follows that\n\n\\begin{subequations}\n\\begin{align}\n\t\\Displacement & = \\Electricfield \\ + \\ O^{\\dParameter+1}\\big(|h||(\\Electricfield, \\Magneticinduction)| \\big)\n\t\t\\ + \\ O^{\\dParameter+1}\\big(|(\\Electricfield, \\Magneticinduction)|^3;h \\big), \\label{E:DintermsofEBh} \\\\\n\t\\Magneticfield  & = \\Magneticinduction \\ + \\ O^{\\dParameter+1}\\big(|h||(\\Electricfield, \\Magneticinduction)| \\big)\n\t\t\\ + \\ O^{\\dParameter+1}\\big(|(\\Electricfield, \\Magneticinduction)|^3;h \\big), \\label{E:HintermsofEBh} \\\\\n\t\\Electricfield  & = \\Displacement \\ + \\ O^{\\dParameter+1}\\big(|h||(\\Displacement, \\Magneticinduction)| \\big)\n\t\t\\ + \\ O^{\\dParameter+1}\\big(|(\\Displacement, \\Magneticinduction)|^3;h \\big), \\label{E:EintermsofDBh} \\\\\n\t\\Magneticfield & = \\Magneticinduction \\ + \\ O^{\\dParameter+1}\\big(|h||(\\Displacement, \\Magneticinduction)| \\big)\n\t\t\\ + \\ O^{\\dParameter+1}\\big(|(\\Displacement, \\Magneticinduction)|^3;h \\big).\n\\end{align}\n\\end{subequations}\n\nFurthermore, recalling that $\\hat{N}^{\\nu}|_{\\Sigma_0} = A \\delta_0^{\\nu},$ where $A \\eqdef \\sqrt{1 - \\frac{2M}{r}\\chi(r)},$ \nand using \\eqref{E:FargDualIntermsofFarmDual}, and \\eqref{E:InducedhData}, it follows that\n\n\t\\begin{align} \\label{E:ElectricfieldDataInTermsofIntrinsic}\n\t\t\\mathring{\\Electricfield} & = \\mathring{\\Displacement} \\ + \\ \n\t\tO^{\\dParameter+1}\\big(|\\mathring{h}^{(1)}||(\\mathring{\\Displacement}, \n\t\t\t\\mathring{\\Magneticinduction})|;\\chi(r)M\/r\\big) \n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|\\chi(r)M\/r||(\\mathring{\\Displacement}, \\mathring{\\Magneticinduction})|;\\mathring{h}^{(1)} \\big)\n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|(\\mathring{\\Displacement}, \\mathring{\\Magneticinduction})|^3;\\mathring{h}^{(1)};\\chi(r)M\/r\\big)\n\t\\end{align}\n\tand\n\n\t\\begin{subequations}\n\t\t\\begin{align}\n\t\t\\mathring{\\Magneticinduction} & = \\mathring{\\mathfrak{\\Magneticinduction}} \n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|\\chi(r)M\/r||\\mathring{\\mathfrak{\\Magneticinduction}}|\\big)\n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|\\mathring{h}^{(1)}||(\\mathring{\\mathfrak{\\Magneticinduction}},  \n\t\t\t\\mathring{\\mathfrak{\\Displacement}})|;|\\chi(r)M\/r|\\big), \n\t\t\t\\label{E:IntialInductionintermsofInitialQuantities} \\\\\n\t\t\\mathring{\\Displacement} & = \\mathring{\\mathfrak{\\Displacement}} \n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|\\mathring{h}^{(1)}||\\chi(r)M\/r||\\mathring{\\mathfrak{\\Displacement}}|\\big)\n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|(\\mathring{\\mathfrak{\\Magneticinduction}},  \n\t\t\t\\mathring{\\mathfrak{\\Displacement}})|;|\\chi(r)M\/r|\\big), \n\t\t\t\\label{E:IntialDisplacementintermsofInitialQuantities} \\\\\n\t\t\\mathring{\\mathfrak{\\Magneticinduction}} & = \\mathring{\\Magneticinduction}\n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|\\chi(r)M\/r||\\mathring{\\Magneticinduction}|\\big)\n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|\\mathring{h}^{(1)}||(\\mathring{\\Magneticinduction}, \\mathring{\\Displacement})|;|\\chi(r)M\/r|\\big), \\\\\n\t\t\\mathring{\\mathfrak{\\Displacement}} & = \\mathring{\\Displacement} \n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|\\chi(r)M\/r||\\mathring{\\Displacement}|\\big)\n\t\t\t\\ + \\ O^{\\dParameter+1}\\big(|\\mathring{h}^{(1)}||(\\mathring{\\Magneticinduction},  \\mathring{\\Displacement})|;|\\chi(r)M\/r|\\big).\n\t\t\t\\label{E:InitialIntrinsicDisplacementIntermsofInitialQuantities}\n\t\\end{align}\n\t\\end{subequations}\n\n\n\\begin{remark}\n\tLogically speaking, the ADM mass $M$ (and hence also the coordinates of the unit normal vector $\\hat{N}|_{\\Sigma_0}$)\n\tis only well-defined \\emph{after} one has solved the abstract Einstein constraint equations \\eqref{E:GaussIntro} - \n\t\\eqref{E:DivergenceB0Intro}. Thus, the\n\trelations \\eqref{E:ElectricfieldDataInTermsofIntrinsic} - \\eqref{E:IntialDisplacementintermsofInitialQuantities}\n\tshould be thought of as defining $\\mathring{\\Electricfield}_j, \\mathring{\\Magneticinduction}_j, \n\t\\mathring{\\Displacement}_j$ in terms of $\\mathring{\\mathfrak{\\Electricfield}}_j, \\mathring{\\mathfrak{\\Magneticinduction}}_j$ \n\tand \\emph{not} the other way around.\n\\end{remark}\n\nThe main goal of this section is to deduce the following proposition, which is a decomposition of the electromagnetic equations into \\emph{constraint} equations and \\emph{evolution} equations.\n\n\n\\begin{proposition} \\label{P:ElectromagneticDecomp}\n\\textbf{(Electromagnetic constraint and evolution equations)}\nUnder the assumption \\eqref{E:Ldualassumptions} on $\\Ldual,$ the electromagnetic equations \\eqref{E:dFis0Electrodecomp} - \\eqref{E:dMis0Electrodecomp} are equivalent to the following pairs of constraint equations and evolution equations:\n\n\\begin{center} \n\t\\textbf{Constraint Equations}\n\\end{center}\n\\begin{subequations}\n\\begin{align}\n\t(\\um^{-1})^{ab} \\unabla_a \\Displacement_b & = 0, \\label{E:Dconstraint} \\\\ \n\t(\\um^{-1})^{ab} \\unabla_a \\Magneticinduction_b & = 0, \\label{E:Bconstraint}\n\\end{align}\n\\end{subequations}\n\n\\vspace{.5in}\n\n\\begin{center} \n\t\\textbf{Evolution Equations}\n\\end{center}\n\\begin{subequations}\n\\begin{align}\n\t\\partial_t \\Magneticinduction_j & = - [jab] \\unabla_a \\Electricfield_b, \n\t\t\\label{E:partialtBisolated} \\\\\n\t\\partial_t \\Electricfield_j & = [jab] \\unabla_a \\Magneticinduction_b \n\t\t\\ + \\ O^{\\dParameter}\\big(|h||\\unabla(\\Electricfield,\\Magneticinduction)|;(\\Electricfield,\\Magneticinduction) \\big)\n\t\t\\ + \\ O^{\\dParameter}\\big(|(\\Electricfield,\\Magneticinduction)|^2|\\unabla(\\Electricfield,\\Magneticinduction)|;h \\big)\n\t\t\\ + \\ O^{\\dParameter}\\big(|\\nabla h| |(\\Electricfield, \\Magneticinduction)|; h \\big). \n\t\t\t\\label{E:partialtEisolated} \n\\end{align}\n\\end{subequations}\n\nFurthermore, if the one-forms $\\mathring{\\Displacement}, \\mathring{\\Magneticinduction}$ are related to the one-forms\n$\\mathring{\\mathfrak{\\Displacement}}, \\mathring{\\mathfrak{\\Magneticinduction}}$ as implicitly determined by the \nrelations \\eqref{E:AbstractEBDHinertialcomponents} - \\eqref{E:EBDHinertialcomponents} (together with the fact that\n$\\hat{N}^{\\mu} = A^{-1} \\delta_0^{\\mu}$), then equations \\eqref{E:Dconstraint} - \\eqref{E:Bconstraint} hold \nfor $\\mathring{\\Displacement}, \\mathring{\\Magneticinduction}$ (i.e., along $\\Sigma_0$) if and only if the following equations hold:\n\n\\begin{center} \n\t\\textbf{Abstract Constraint Equations}\n\\end{center}\n\n\\begin{subequations}\n\\begin{align}\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{\\underline{\\mathscr{D}}}_a \\mathring{\\mathfrak{\\Displacement}}_b & = 0, \n\t\t\\label{E:AbstractDconstraint} \\\\\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{\\underline{\\mathscr{D}}}_a \\mathring{\\mathfrak{\\Magneticinduction}}_b & = 0.\n\t\t\\label{E:AbstractBconstraint} \n\\end{align}\n\\end{subequations}\nIn the above expressions, $\\mathring{\\underline{g}}_{jk}$ is the first-fundamental form of $\\Sigma_0,$ and $\\mathring{\\underline{\\mathscr{D}}}$ is the Levi-Civita connection corresponding to $\\mathring{\\underline{g}}_{jk}.$ \n\n\n\\end{proposition}\n\n\\begin{remark}\nIn equations \\eqref{E:Dconstraint} - \\eqref{E:Bconstraint}, $(\\um^{-1})^{ab} \\unabla_a$ is the standard Euclidean divergence operator, while in equations \\eqref{E:partialtBisolated} - \\eqref{E:partialtEisolated}, $[jab] \\unabla_a$ is the standard Euclidean curl operator.\n\\end{remark}\n\n\\begin{remark}\n\tUsing equations \\eqref{E:partialtBisminuscurlE} - \\eqref{E:partialtDiscurlH}, it is easy to check that if a classical \n\tsolution to the evolution equations satisfies the constraints at $t=0,$  then it necessarily satisfies the constraints \n\t\\eqref{E:Dconstraint} - \\eqref{E:Bconstraint} at all later times (as long as it persists).\n\\end{remark}\n\n\\begin{proof}\nWe first show that \\eqref{E:Dconstraint} follows from either \\eqref{E:dFis0ElectrodecompMathscrD} or \\eqref{E:dFis0Electrodecomp}, and that \\eqref{E:Dconstraint} holds if and only if \\eqref{E:AbstractDconstraint} holds. To this end, we first note that since $\\hat{N}^{\\mu}$ is the future-directed unit $g-$normal to $\\Sigma_t$ and \n$g_{\\mu \\nu} = \\mathring{\\underline{g}}_{\\mu \\nu} - \\hat{N}_{\\mu} \\hat{N}_{\\nu}$ along $\\Sigma_0,$ \nthe following identities hold for any one-form $X_{\\mu}$ $g-$tangent to $\\Sigma_0$ and any two-form $P_{\\mu \\nu}:$\n\n\\begin{align}\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{\\underline{\\mathscr{D}}}_a X_b \n\t& = (g^{-1})^{\\kappa \\lambda} \\mathscr{D}_{\\kappa} X_{\\lambda} - X_{\\lambda} \\hat{N}^{\\kappa} \n\t\t\\mathscr{D}_{\\kappa} \\hat{N}^{\\lambda}, \\label{E:IntrinsicExtrinsicDivergenceIdentity} \\\\\n\t(g^{-1})^{\\kappa \\lambda} P_{\\lambda \\nu} \\mathscr{D}_{\\kappa} \\hat{N}^{\\nu} \n\t& = P_{\\lambda \\nu} \\hat{N}^{\\nu} \\hat{N}^{\\kappa} \\mathscr{D}_{\\kappa} \\hat{N}^{\\lambda}. \n\t\t\\label{E:TwoFormhatNContractionIdentity}\n\\end{align}\n\nUsing \\eqref{E:IntrinsicExtrinsicDivergenceIdentity} and \\eqref{E:TwoFormhatNContractionIdentity}\nwith $X_{\\mu} \\eqdef \\mathfrak{\\Magneticinduction}_{\\mu}$ and $P_{\\mu \\nu} \\eqdef \\Fardual_{\\mu \\nu}$\nwe compute that the following identities hold along $\\Sigma_0:$\n\n\\begin{align} \\label{E:DivergenceIntrinsicBintermsofFarandhatN}\n\t(\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{\\underline{\\mathscr{D}}}_{a} \\mathfrak{\\Magneticinduction}_b \n\t& = (g^{-1})^{\\kappa \\lambda} \\mathscr{D}_{\\kappa} \\mathfrak{\\Magneticinduction}_{\\lambda}\t\n\t\t\t- \\mathfrak{\\Magneticinduction}_{\\lambda} \\hat{N}^{\\kappa} \\mathscr{D}_{\\kappa} \\hat{N}^{\\lambda} \n\t\t\t \\\\\n\t& = - (g^{-1})^{\\kappa \\lambda} \\mathscr{D}_{\\kappa} (\\Fardual_{\\lambda \\nu} \\hat{N}^{\\nu})\t\n\t\t\t+ \\Fardual_{\\lambda \\nu} \\hat{N}^{\\nu} \\hat{N}^{\\kappa} \\mathscr{D}_{\\kappa} \\hat{N}^{\\lambda}\n\t\t\t\\notag \\\\\n\t& = - \\frac{1}{2}g_{\\nu \\nu'} \\hat{N}^{\\nu'} \\epsilon^{\\# \\mu \\nu \\kappa \\lambda} \\mathscr{D}_{\\mu} \\Far_{\\kappa \\lambda}. \n\t\t\\notag \n\\end{align}\nIdentities analogous to \\eqref{E:DivergenceIntrinsicBintermsofFarandhatN} hold if we make the replacements \n$\\Big(\\mathring{\\underline{g}}^{-1}, g, \\mathring{\\underline{\\mathscr{D}}},\n\\mathscr{D}, \\star, \\hat{N}^{\\mu}, \\epsilon^{\\# \\mu \\nu \\kappa \\lambda}, \\mathfrak{\\Magneticinduction} \\Big) \\rightarrow \n\\Big(\\um^{-1}, m, \\unabla, \\nabla, \\ostar, \\hat{n}^{\\mu}, \\Minkvolume^{\\mu \\nu \\kappa \\lambda}, \\Magneticinduction \\Big),$\nwhere $\\hat{n}^{\\mu}(t,x)$ is the future-directed unit Minkowskian unit normal to $\\Sigma_t.$ Now by \\eqref{E:DivergenceIntrinsicBintermsofFarandhatN} and the Minkowskian analogy of \\eqref{E:DivergenceIntrinsicBintermsofFarandhatN}, equations \\eqref{E:Dconstraint} and\n\\eqref{E:AbstractDconstraint} follow from either \\eqref{E:dFis0ElectrodecompMathscrD} or \\eqref{E:dFis0Electrodecomp},\nsince either \\eqref{E:dFis0ElectrodecompMathscrD} or \\eqref{E:dFis0Electrodecomp} are sufficient to guarantee that the right-hand side of \\eqref{E:DivergenceIntrinsicBintermsofFarandhatN} is $0.$ Furthermore, since\n$g_{\\nu \\nu'} \\hat{N}^{\\nu'}$ and $m_{\\nu \\nu'} \\hat{n}^{\\nu'}$ are proportional along $\\Sigma_0,$ since $\\epsilon^{\\# \\mu \\nu \\kappa \\lambda}$ and $\\upsilon^{\\mu \\nu \\kappa \\lambda}$ are proportional, and since the Christoffel symbols of $\\mathscr{D}$ and $\\nabla$ are symmetric in their two lower indices, it follows that\n\n\\begin{align} \\label{E:IntrinsicConstraintEquivalenttoMinkowskiConstraint}\n\tg_{\\nu \\nu'} \\hat{N}^{\\nu'} \\epsilon^{\\# \\mu \\nu \\kappa \\lambda} \\mathscr{D}_{\\mu} \\Far_{\\kappa \\lambda}|_{\\Sigma_0} & = 0 \\\\\n\t\t& \\iff \\notag \\\\\n\tm_{\\nu \\nu'} \\hat{n}^{\\nu'} \\upsilon^{\\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Far_{\\kappa \\lambda}|_{\\Sigma_0} & = 0. \\notag\n\\end{align}\nHence, \\eqref{E:Bconstraint} holds along $\\Sigma_0$ if and only if \\eqref{E:AbstractBconstraint} holds along $\\Sigma_0.$ \nThe derivation of \\eqref{E:Dconstraint} and \\eqref{E:AbstractDconstraint} along $\\Sigma_0$ from \\eqref{E:dMis0ElectrodecompMathscrD} or \\eqref{E:dMis0Electrodecomp} and the proof of the equivalence of \\eqref{E:Dconstraint} and \\eqref{E:AbstractDconstraint} along $\\Sigma_0$ are similar.\n\nWe now set $\\lambda = 0, \\mu = a, \\nu = b$ in \\eqref{E:dFis0Electrodecomp}, then contract against the Euclidean volume form $[jab]$ use \\eqref{E:EBDHinertialcomponents} - \\eqref{E:FarspatialintermsofB} to deduce that\n\n\\begin{align} \\label{E:partialtBisminuscurlE}\n\t\\partial_t \\Magneticinduction_j = - [jab] \\unabla_a \\Electricfield_b.\n\\end{align}\nSimilarly, we set $\\lambda = 0, \\mu = a, \\nu = b$ in \\eqref{E:dMis0Electrodecomp}, contract against $[jab],$\nand use \\eqref{E:EBDHinertialcomponents} - \\eqref{E:FarspatialintermsofB} to deduce that\n\n\\begin{align} \\label{E:partialtDiscurlH}\n\t\\partial_t \\Displacement_j = [jab] \\unabla_a \\Magneticfield_b.\n\\end{align}\nFinally, we use \\eqref{E:partialtBisminuscurlE}, \\eqref{E:partialtDiscurlH},\nand \\eqref{E:DintermsofEBh} - \\eqref{E:HintermsofEBh}, to deduce \\eqref{E:partialtBisolated} - \\eqref{E:partialtEisolated}. \n\n\\end{proof}\n\n\n\n\n\n\n\\section{The Smallness Condition on the Abstract Data} \\label{S:SmallDataAssumptions}\n\\setcounter{equation}{0}\n\nIn this section, we assume that we are given abstract initial data $(\\mathring{\\underline{g}}_{jk} = \\delta_{jk} + \\mathring{\\underline{h}}_{jk}^{(0)} + \\mathring{\\underline{h}}_{jk}^{(1)} ,\\mathring{K}_{jk},\\mathring{\\mathfrak{\\Displacement}}_j,\\mathring{\\mathfrak{\\Magneticinduction}}_j),$ $(j,k=1,2,3),$\non the manifold $\\mathbb{R}^3$ satisfying the constraint equations \\eqref{E:Gauss} - \\eqref{E:DivergenceB0}. Our goal is to describe in detail the smallness condition on $(\\mathring{\\underline{h}}_{jk}^{(0)}, \\mathring{\\underline{h}}_{jk}^{(1)} ,\\mathring{K}_{jk},\\mathring{\\mathfrak{\\Displacement}}_j,\\mathring{\\mathfrak{\\Magneticinduction}}_j)$\nthat will lead to global existence for the reduced system \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}, under the assumption that its initial data $(g_{\\mu \\nu}|_{t=0},\\partial_t g_{\\mu \\nu}|_{t=0},\\Far_{\\mu \\nu}|_{t=0}),$ $(\\mu, \\nu = 0,1,2,3),$\nare constructed from the abstract initial data as described in Section \\ref{SS:ReducedData}. Recall that our global existence argument is heavily based on the analysis of $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t),$ which is the energy defined in \\eqref{E:EnergyIntro}. In particular, $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0)$ must be sufficiently small in order for us to close the argument. The energy depends on \\emph{both normal and tangential} Minkowskian covariant derivatives of the quantities $(\\nabla_{\\lambda} h_{\\mu \\nu}^{(1)},\\Far_{\\mu \\nu})$ at $t=0.$ \nOn the other hand, our smallness condition will be expressed in terms of the ADM mass $M$ and $E_{\\dParameter;\\upgamma}(0),$ which is a weighted Sobolev norm of $(\\unabla_i \\mathring{\\underline{h}}_{jk}^{(1)} ,\\mathring{K}_{jk},\\mathring{\\mathfrak{\\Displacement}}_j,\\mathring{\\mathfrak{\\Magneticinduction}}_j)$\ndepending only on \\emph{tangential} derivatives of the abstract data. More specifically, our smallness condition is expressed in terms of the weighted Sobolev norms $\\| \\cdot \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}$ introduced in Definition \\ref{D:HNdeltanorm}. The main result of this section is contained in Proposition \\ref{P:SmallNormImpliesSmallEnergy}, which shows that if $E_{\\dParameter;\\upgamma}(0) + M$ is sufficiently small and $(h_{\\mu \\nu}^{(1)},\\Far_{\\mu \\nu})$ is the corresponding solution to the reduced equations, then $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0) \\lesssim E_{\\dParameter;\\upgamma}(0) + M.$ Thus, Proposition \\ref{P:SmallNormImpliesSmallEnergy} allows us to deduce the smallness of $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0)$ from the smallness of quantities that depend exclusively on the abstract initial data.\n\n\n\\noindent \\hrulefill\n\\ \\\\\n\nWe begin by introducing the weighted Sobolev norm discussed in the above paragraph.\n\n\\begin{definition} \\label{D:HNdeltanorm}\n\tLet $U(x)$ be a tensorfield defined along the Euclidean space $\\mathbb{R}^3.$\n\tThen for any integer $\\dParameter \\geq 0,$ and any real number $\\eta,$ we define the $H_{\\eta}^{\\dParameter}$ norm of $U$ by\n\t\n\t\\begin{align} \\label{E:HNdeltanorm}\n\t\t\\| U \\|_{H_{\\eta}^{\\dParameter}}^2 \\eqdef \\sum_{|I| \\leq \\dParameter } \\int_{x \\in \\mathbb{R}^3} (1 + \n\t\t\t|x|^2)^{(\\eta + |I|)} |\\underline{\\nabla}^I U(x)|^2 \\, d^3 x.\n\t\\end{align}\n\\end{definition}\n\nWe also introduce the following norm, which can be controlled in terms of a suitable\n$H_{\\eta}^{\\dParameter}$ norm via a Sobolev embedding result; see Proposition \\ref{P:SobolevEmbeddingHNdeltaCNprimedeltamprime}.\n \n\n\\begin{definition} \\label{D:CNdeltanorm}\n\tLet $U(x)$ be a tensorfield defined along the Euclidean space $\\mathbb{R}^3.$ \n\tThen for any integer $\\dParameter \\geq 0,$ and any real number $\\eta,$ we define the $C_{\\eta}^{\\dParameter}$ norm of $U$ \n\tby\n\t\n\t\\begin{align} \\label{E:CNdeltanorm}\n\t\t\\| U \\|_{C_{\\eta}^{\\dParameter}}^2 \\eqdef \t\\| U \\|_{C_{\\eta}^{\\dParameter}}^2 \n\t\t\\eqdef \\sum_{|I| \\leq \\dParameter } \\mbox{ess} \\sup_{x \\in \\mathbb{R}^3} (1 + |x|^2)^{(\\eta + |I|)} \n\t\t|\\underline{\\nabla}^I U(x)|^2.\n\t\\end{align}\n\\end{definition}\n\nWe are now ready to introduce our norm $E_{\\dParameter;\\upgamma}(0) \\geq 0$ on the abstract initial data. Recall that as discussed in Section \\ref{SS:AbstractData}, the data are the following four fields on $\\mathbb{R}^3:$ \n$(\\mathring{\\underline{g}}_{jk} = \\delta_{jk} + \\underbrace{\\mathring{\\underline{h}}_{jk}^{(0)} + \\mathring{\\underline{h}}_{jk}^{(1)}}_{\\mathring{\\underline{h}}_{jk}}, \\mathring{K}_{jk}, \\mathring{\\mathfrak{\\Displacement}}_j, \\mathring{\\mathfrak{\\Magneticinduction}}_j),$ $(j,k = 1,2,3).$\n\n\\begin{definition} \\label{D:DataNorm}\nThe norm $E_{\\dParameter;\\upgamma}(0) \\geq 0$ of the abstract initial data is defined by\n\n\\begin{align}   \\label{E:DataNorm}\n\tE_{\\dParameter;\\upgamma}^2(0) \n\t& \\eqdef \\| \\underline{\\nabla} \\mathring{\\underline{h}}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{K} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{\\mathfrak{\\Displacement}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{\\mathfrak{\\Magneticinduction}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2.\n\\end{align}\n\n\n\\end{definition}\n\n\\vspace{.5in}\n\\begin{center}\n\t\\textbf{\\LARGE The Smallness Condition}\n\\end{center}\n\n\\bigskip\n\nOur smallness condition for global existence is\n\n\\begin{align} \\label{E:NormSmallnessCondition}\n\tE_{\\dParameter;\\upgamma}(0) + M \\leq \\varepsilon_{\\dParameter},\n\\end{align}\nwhere $\\varepsilon_{\\dParameter}$ is a sufficiently small positive number.\n\\vspace{.5in}\n\nRecall that the energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\geq 0$ is defined by\n\n\\begin{align} \\label{E:EnergydefSmallnessSection}\n\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}^2(t) & \\eqdef \\underset{0 \\leq \\tau \\leq t}{\\mbox{sup}} \n\t\t\\sum_{|I| \\leq \\dParameter } \\int_{\\Sigma_{\\tau}} \n\t\t\\Big\\lbrace |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|^2 \\Big\\rbrace w(q) \\, d^3 x,\n\\end{align}\nwhere $\\nabla$ denotes the \\emph{full Minkowski spacetime} covariant derivative operator, and the weight $w(q)$ is defined in \\eqref{E:weight}. The next proposition, which is the main result of this section, shows that the smallness of $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0)$ follows from the smallness of $E_{\\dParameter;\\upgamma}(0) + M.$\n\n\n\\begin{proposition} \\label{P:SmallNormImpliesSmallEnergy}\n\t\\textbf{(The smallness of the initial energy)}\n\tLet $(\\mathring{\\underline{g}}_{jk}  \\delta_{jk} + \\mathring{\\underline{h}}_{jk}^{(0)} + \\mathring{\\underline{h}}_{jk}^{(1)}, \n\t\\mathring{K}_{jk}, \\mathring{\\mathfrak{\\Displacement}_j}, \\mathring{\\mathfrak{\\Magneticinduction}}_j),$ be abstract initial \n\tdata on the manifold $\\mathbb{R}^3$ for the Einstein-nonlinear electromagnetic system \\eqref{E:IntroEinstein} - \n\t\\eqref{E:IntrodMis0} and\tassume that the abstract initial data \n\tare asymptotically flat in the sense that \\eqref{E:metricdataexpansion} - \\eqref{E:BdecayAssumption} \n\thold. Let $(g_{\\mu \\nu}|_{t=0} = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)}|_{t=0} + h_{\\mu \\nu}^{(1)}|_{t=0}, \n\t\\partial_t g_{\\mu \\nu}|_{t=0} = \\partial_t h_{\\mu \\nu}^{(0)}|_{t=0} + \\partial_th_{\\mu \\nu}^{(1)}|_{t=0}, \n\t\\Far_{\\mu \\nu}|_{t=0}),$ $(\\mu, \\nu = 0,1,2,3),$ be the corresponding initial data \n\tfor the reduced system \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} as defined in Section \\ref{SS:ReducedData},\n\tand let $(g_{\\mu \\nu} = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)}, \n\t\\Far_{\\mu \\nu})$ be the solution to the reduced system launched by this data. Let $\\dParameter \\geq 8$ be an integer.\n\tThen there exists a constant $\\varepsilon_0 > 0$ independent of $\\dParameter $ and constants $C_{\\dParameter} > 0,$ $\\widetilde{C}_{\\dParameter} > 0$\n\tsuch that if $E_{\\dParameter;\\upgamma}(0) + M \\leq \\varepsilon \\leq \\varepsilon_0,$ \n\tthen \n\t\n\t\\begin{align} \\label{E:SmallNormImpliesSmallEnergy}\n\t\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0) \n\t\t& \\leq C_{\\dParameter} \\big\\lbrace E_{\\dParameter;\\upgamma}(0) + M \\big\\rbrace\n\t\t\t\\leq \\widetilde{C}_{\\dParameter} \\varepsilon.\n\t\\end{align}\n\t\n\\end{proposition}\n\n\\begin{remark} \\label{R:Nomu}\n\tNote that $q \\geq 0$ holds at $t=0.$ Therefore, $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0)$ does not depend on the constant\n\t$\\upmu.$\n\\end{remark}\n\nThe proof of Proposition \\ref{P:SmallNormImpliesSmallEnergy} will be given at the end of this section. \nWe first establish some technical lemmas.\n\n\n\\begin{lemma} \\label{L:EnergyWithFarReplacedbyElectricandMagnetic} \\textbf{(Energy in terms of the \n\tspacetime metric remainder piece, the electric field, and the magnetic induction)}\n\tLet $\\Far_{\\mu \\nu}$ be a two-form, let the pair of one-forms $(\\Electricfield_{\\mu},\\Magneticinduction_{\\mu})$ be its\n\tMinkowskian electromagnetic decomposition as defined in Section \\ref{SS:EBDH},\n\tand let $h_{\\mu \\nu}^{(1)}$ be an arbitrary type $\\binom{0}{2}$ tensorfield. Let\n\t$\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t)$ be the energy defined in \\eqref{E:EnergydefSmallnessSection}.\n\tThen\n\t\n\t\\begin{align} \\label{E:EnergyWithFarReplacedbyElectricandMagnetic}\n\t\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}^2(t) & \\approx \\underset{0 \\leq \\tau \\leq t}{\\sup} \n\t\t\t\\sum_{|I| \\leq \\dParameter } \\int_{\\Sigma_{\\tau}} \n\t\t\t\\Big\\lbrace |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 \n\t\t\t+ |\\nabla_{\\mathcal{Z}}^I \\Electricfield|^2 + |\\nabla_{\\mathcal{Z}}^I \\Magneticinduction|^2 \\Big\\rbrace w(q) \\, d^3 x.\n\t\\end{align}\n\n\\end{lemma}\n\n\n\\begin{proof}\n\t\\eqref{E:EnergyWithFarReplacedbyElectricandMagnetic} easily follows from the identity\n\t$|\\nabla_{\\mathcal{Z}}^I \\Far|^2 = 2 |\\nabla_{\\mathcal{Z}}^I \\Electricfield|^2 + 2 |\\nabla_{\\mathcal{Z}}^I \n\t\\Magneticinduction|^2,$ the verification of which we leave to the reader.\n\\end{proof}\n\n\n\\begin{lemma} \\label{L:NablaZIUL2InTermsofrWeightedPartialtkUnderlineNablaJNorms}\n\tThe following estimates hold for any sufficiently differentiable spacetime tensorfield $U(t,x)$\n\tdefined in a neighborhood of $\\Sigma_0 \\eqdef \\lbrace(t,x) \\mid t=0 \\rbrace,$ where $w(q)$ is the weight\n\tdefined in \\eqref{E:weight}:\n\t\n\t\\begin{align} \\label{E:NablaZIUL2InTermsofrWeightedPartialtkUnderlineNablaJNorms}\n\t\t\\Big(\\sum_{|I| \\leq \\dParameter } w^{1\/2}(q)|\\nabla_{\\mathcal{Z}}^I U| \\Big)|_{\\Sigma_0} \n\t\t& \\approx \\Big(\\sum_{|I| \\leq \\dParameter } (1 + r)^{1\/2 + \\upgamma + |I|} |\\nabla^I U| \\Big)\\big|_{\\Sigma_0} \\\\\n\t\t& \\approx \\Big( \\sum_{|J| + k \\leq \\dParameter } (1 + r)^{1\/2 + \\upgamma + |J| + k} \n\t\t|\\underline{\\nabla}^J \\partial_t^k U|\\Big)\\big|_{\\Sigma_0}. \\notag \n\t\\end{align}\n\t\n\tThe same estimates hold if $\\nabla_{\\mathcal{Z}}^I$ is replaced with $\\Lie_{\\mathcal{Z}}^I.$ The notation\n\t$\\big|_{\\Sigma_0}$ is meant to indicate that the estimates only hold along $\\Sigma_0.$\n\n\\end{lemma}\n\n\\begin{proof}\n\tBy iterating the identity $\\frac{\\partial}{\\partial x^{\\mu}} = \\frac{x^{\\kappa} \\Omega_{\\kappa \\mu} + x_{\\mu}S}{qs},$ \n\tand noting that $q = r = s$ along $\\Sigma_0,$ it follows that\n\t\n\t\\begin{align} \\label{E:TranslationaldervativeslessthanqweightedLieZKnorm}\n\t\t(1 + r)^{|I|} |\\nabla^I U| \\lesssim \\sum_{|J| \\leq |I|} |\\nabla_{\\mathcal{Z}}^J U|.\n\t\\end{align}\n\tIt thus follows that\n\t\n\t\\begin{align}\n\t\t\\Big(\\sum_{|I| \\leq \\dParameter } (1 + r)^{1\/2 + \\upgamma + |I|} |\\nabla^I U|\\Big)\\big|_{\\Sigma_0}\n\t\t& \\lesssim \\Big(\\sum_{|I| \\leq \\dParameter } w^{1\/2}(q)|\\nabla_{\\mathcal{Z}}^I U| \\Big)\\big|_{\\Sigma_0}.\n\t\\end{align}\n\tOn the other hand, the opposite inequality follows easily from expanding the operator $\\nabla_{\\mathcal{Z}}^I$ \n\tand using the Leibniz rule plus \\eqref{E:CovariantDerivativesofZareConstant}. This proves the first $\\approx$ in \n\t\\eqref{E:NablaZIUL2InTermsofrWeightedPartialtkUnderlineNablaJNorms}. The second $\\approx$ is trivial. We have\n\tthus established \\eqref{E:NablaZIUL2InTermsofrWeightedPartialtkUnderlineNablaJNorms}.\n\tTo establish the same estimates with the operator $\\Lie_{\\mathcal{Z}}^I$ in place of $\\nabla_{\\mathcal{Z}}^I,$\n\twe simply use \\eqref{E:LieZIinTermsofNablaZI}.\n\n\\end{proof}\n\n\n\n\\begin{corollary} \\label{C:InitialEnergyInTermsofTangentialandTimeDerivatives}\n\tUnder the assumptions of Lemma \\ref{L:EnergyWithFarReplacedbyElectricandMagnetic}, we have that\n\t\n\t\\begin{align}   \\label{E:InitialEnergyInTermsofTangentialandTimeDerivatives}\n\t\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}^2(0) \n\t\t& \\approx \\sum_{k + |I| \\leq \\dParameter } \n\t\t\t\\int_{\\mathbb{R}^3} (1 + |x|)^{1 + 2\\upgamma + 2|I|} \\Big\\lbrace |\\partial_t^k \\underline{\\nabla}^I \n\t\t\t\\partial_t h^{(1)}|^2(0,x) + |\\underline{\\nabla}^I \\underline{\\nabla} h^{(1)}|^2(0,x) \\Big\\rbrace \\, d^3 x \\\\\n\t\t& \\ \\ +  \\int_{\\mathbb{R}^3} (1 + |x|)^{1 + 2\\upgamma + 2|I|} \\Big\\lbrace|\\partial_t^k \\underline{\\nabla}^I \n\t\t\t\\Electricfield|^2(0,x) \n\t\t\t+ |\\partial_t^k \\underline{\\nabla}^I \\Magneticinduction|^2(0,x) \\Big\\rbrace \\, d^3 x. \\notag\n\t\\end{align}\n\t\n\\end{corollary}\n\n\n\\begin{proof}\n\tCorollary \\ref{C:InitialEnergyInTermsofTangentialandTimeDerivatives}\n\tfollows easily from Lemmas \\ref{L:EnergyWithFarReplacedbyElectricandMagnetic}\n\tand \\ref{L:NablaZIUL2InTermsofrWeightedPartialtkUnderlineNablaJNorms}.\n\\end{proof}\n\t\n\n\n\\begin{lemma} \\label{L:SolveforTimeDerivativesinTermsofInherentDerivatives}\n\tLet $k \\geq 1$ and $\\dParameter \\geq 8$ be integers, and let $J$ be a $\\underline{\\nabla}-$multi-index. Assume that $|J| +|K| \n\t\\leq \\dParameter.$ Assume that $(h_{\\mu \\nu}^{(1)}, \\Far_{\\mu \\nu})$ is a solution to the reduced equations \n\t\\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}, and define the arrays $V,$ $V^{(0)},$ $V^{(1)},$ $W,$ $W^{(0)},$ $W^{(1)}$ by\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\tV & \\eqdef (h, \\underline{\\nabla} h, \\partial_t h, \\Electricfield, \\Magneticinduction) = V^{(0)} + V^{(1)}, \n\t\t\t\\label{E:Vdef} \\\\\n\t\tV^{(0)} & \\eqdef (h^{(0)}, \\underline{\\nabla} h^{(0)}, \\partial_t h^{(0)}, 0, 0), \\\\\n\t\tV^{(1)} & \\eqdef (h^{(1)}, \\underline{\\nabla} h^{(1)}, \\partial_t h^{(1)}, \\Electricfield, \\Magneticinduction), \\\\\n\t\tW & \\eqdef (0,\\underline{\\nabla} h, \\partial_t h, \\Electricfield, \\Magneticinduction) = W^{(0)} + W^{(1)}, \\\\\n\t\tW^{(0)} & \\eqdef (0,\\underline{\\nabla} h^{(0)}, \\partial_t h^{(0)}, 0, 0), \\\\\n\t\tW^{(1)} & \\eqdef (0,\\underline{\\nabla} h^{(1)}, \\partial_t h^{(1)}, \\Electricfield, \\Magneticinduction).\n\t\t\\label{E:W1def}\n\t\\end{align}\n\t\\end{subequations}\n\tIn the above expressions, the tensorfields $h_{\\mu \\nu}^{(0)},$ $h_{\\mu \\nu}^{(1)}$ are defined by \\eqref{E:gmhexpansion} - \t\n\t\\eqref{E:h0defIntro}, while the electromagnetic one-forms $\\Electricfield_{\\mu},$ $\\Magneticinduction_{\\mu}$ are defined in \t\n\t\\eqref{E:EBDHinertialcomponents}. Assume further that $|V^{(1)}| + M \\leq \\varepsilon.$ Then if $\\varepsilon$ is sufficiently \n\tsmall, $\\partial_t^k \\underline{\\nabla}^J W^{(1)}$ can be written as the following finite linear combination:\n\t\n\t\\begin{align} \n\t\t\\underline{\\nabla}^J \\partial_t^k W^{(1)} & = \\sum terms,\n\t\\end{align}\n\twhere each $term$ can be written as\n\t\n\t\\begin{align}\t \\label{E:PartialtkunderlinenablaJW1inTermsofInstrinsic}\n\t\t\tterm & = \\sum_{s=0}^{|J| + k + 1} \\sum_{|I_1| + \\cdots + |I_s| \\leq |J| + k} \n\t\t\t\tF_{(I_1, \\cdots, I_s;J;k;s)}(t,x)\\mathscr{M}_{(I_1, \\cdots, I_s;J;k;s)}(V)[\\underline{\\nabla}^{I_1} W^{(1)}, \\cdots, \n\t\t\t\t\\underline{\\nabla}^{I_s} W^{(1)}], \n\t\\end{align}\n\twhere \n\t\n\t\\begin{enumerate}\n\t\t\\item The array-valued functions $\\mathscr{M}_{(I_1, \\cdots, I_s;J;k;s)}(V)[\\underline{\\nabla}^{I_1} W^{(1)}, \\cdots, \n\t\t\t\\underline{\\nabla}^{I_s} W^{(1)}]$ are continuous in a neighborhood of $V = 0$ and\n\t\t\tare multi-linear in the arguments $[\\underline{\\nabla}^{I_1} W^{(1)}, \\cdots, \n\t\t\t\\underline{\\nabla}^{I_s} W^{(1)}].$\n\t\t\\item The array-valued functions $F_{(I_1, \\cdots, I_s;J;k;s)}(t,x)$\n\t\t\tare smooth and satisfy \\\\\n\t\t\t$|F_{(I_1, \\cdots, I_s;J;k;s)}(t,x)| \\lesssim  M (1 + t + |x|)^{-(3 + |J| + k )}$ if $s = 0$ \n\t\t\t(i.e., if there are no multi-linear arguments $[\\cdots]$), where $M$ is the ADM mass.\n\t\t\\item In the case $s \\geq 1,$ $|F_{(I_1, \\cdots, I_s;J;k;s)}(t,x)| \\lesssim (1 + t + |x|)^{-d},$ \n\t\t\twhere $d \\geq |J| + k - (|I_1| + \\cdots + |I|_s) - (s-1).$\n\t\t\\end{enumerate}\n\t\t\n\t\n\\end{lemma}\n\n\\begin{proof}\n\n\tWe first claim that we can write the reduced system \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} \t\t\n\tas a finite linear combination\n\n\t\\begin{subequations}\t\n\t\\begin{align} \\label{E:partialtW1isaSumofTerms}\n\t\t\\partial_t W^{(1)} = \\sum terms,\n\t\\end{align}\n\twhere each term can be written in the form\n\t\n\t\\begin{align} \\label{E:SolveforPartialtV}\n\t\tterm & = \\sum_{|I| = 1} \\mathscr{M}_{(I;0;1;1)}(V)[\\unabla^I W^{(1)}]\n\t\t\t+ \\mathscr{M}_{(0;0;1;2)}(V)[W^{(1)}, W^{(1)}] \\\\ \n\t\t& \\ \\ + f_{(0;0;1;1)}(t,x) \\mathscr{M}_{(0;0;1;1)}(V)[W^{(1)}]\n\t\t\t+ f_{(0;0;1;0)}(t,x)\\mathscr{M}_{(0;0;1;0)}(V), \\notag \n\t\\end{align}\n\t\\end{subequations}\n\twhere the functions $\\mathscr{M}_{(\\cdots)}(V)[\\cdots],$ which depend on the \n\t$\\dParameter +1-$times continuously differentiable Lagrangian $\\Ldual$\n\tfor the electromagnetic equations, have the properties stated in the conclusions of the theorem; and\n\t$f_{(0;0;1;1)}(t,x),$ $f_{(0;0;1;0)}(t,x)$ are smooth functions satisfying\n\t$|\\nabla^{I}f_{(0;0;1;1)}(t,x)| \\lesssim (1 + t + |x|)^{-2 + |I|},$ \n\t$|\\nabla^{I}f_{(0;0;1;0)}(t,x)| \\lesssim M(1 + t + |x|)^{-3 + |I|}$ for any $\\nabla-$multi-index $I.$\n\tLet us accept the claim \\eqref{E:SolveforPartialtV} for now; we will briefly discuss the derivation of \n\t\\eqref{E:SolveforPartialtV} at the end of the proof. We also note that\n\t\n\t\\begin{align} \n\t\t\\partial_t V & = \\partial_t W^{(1)} + \\Pi_1 W^{(1)} + \\partial_t V^{(0)},  \n\t\t\t\\label{E:PartialtVintermsofPartialtWandW} \\\\\n\t\t\\unabla V & = \\unabla W^{(1)} + \\Pi_2 W^{(1)} + \\unabla V^{(0)}, \n\t\t\t\\label{E:UnderlineNablaVintermsofUnderlineNablaWandW}\t\n\t\\end{align}\n\twhere $\\Pi_1 W^{(1)} \\eqdef (\\partial_t h^{(1)}, 0, 0, 0, 0),$\n\t$\\Pi_2 W^{(1)} \\eqdef (\\unabla h^{(1)}, 0, 0, 0, 0),$ and $V^{(0)}(t,x)$ satisfies \\\\\n\t$|\\nabla^{I} \\partial_t V^{(0)}(t,x)|$ $+ |\\nabla^{I} \\unabla V^{(0)}(t,x)|$ $\\lesssim  \n\t(1 + t + |x|)^{-2 + |I|}$ for any $\\nabla-$multi-index $I$ (see Lemma \\ref{L:h0decayestimates}).\n\tNow with the help of \\eqref{E:PartialtVintermsofPartialtWandW} - \\eqref{E:UnderlineNablaVintermsofUnderlineNablaWandW}, the\n\tchain rule, and the Leibniz rule, we repeatedly partially differentiate \\eqref{E:SolveforPartialtV} with\n\trespect to time and spatial derivatives, using the resulting equations to replace time derivatives with spatial derivatives,\n\tthereby inductively arriving at an expression of the form \\eqref{E:PartialtkunderlinenablaJW1inTermsofInstrinsic} featuring\n\tthe properties (i) - (iii). The properties (ii) - (iii) capture the fact that each additional differentiation of \n\t$\\partial_t W^{(1)}$ either a) creates an additional decay factor of $(1 + t+ |x|)^{-1}$ \\big(when the derivative falls on \n\tone of the $f_{\\cdots}(t,x)$\\big); b) increases one of the powers $|I_j|$ (when the derivative is spatial and falls on one of \n\tthe multilinear factors $[\\cdots, \\unabla^{I_j} W^{(1)}, \\cdots]);$ or c) increases $s$ by one (when the derivative falls on\n\t$\\mathscr{M}(V),$ thereby creating an additional multi-linear factor of $\\nabla W^{(1)}$ via the chain rule).\n\t\n\nWe now return to the issue of expressing $\\partial_t W^{(1)}$ in the form \\eqref{E:partialtW1isaSumofTerms} - \\eqref{E:SolveforPartialtV}. We will make repeated use of the splitting $h = h^{(0)} + h^{(1)},$ where $h^{(0)}$ is the smooth function of $(t,x)$ with the decay properties \\eqref{E:nablaIh0Linfinity}, which are proved in Section \\ref{SS:PreliminaryLinfinityEstimates}. We first note that $\\partial_t \\Electricfield$ and $\\partial_t \\Magneticinduction$ can be expressed in the desired form using \\eqref{E:partialtBisolated} - \\eqref{E:partialtEisolated}, together with the splitting of $h$ and the properties \\eqref{E:nablaIh0Linfinity}. Next, the quantities $\\partial_t \\unabla h_{\\mu \\nu}^{(1)}$ can be expressed in the desired form through the trivial identity $\\partial_t \\unabla h_{\\mu \\nu}^{(1)} = \\unabla \\partial_t h_{\\mu \\nu}^{(1)}.$ The quantities $\\partial_t^2 h_{\\mu \\nu}^{(1)}$ can be expressed in the desired form by using equation \\eqref{E:Reducedh1Summary} to isolate them. We remark that the $\\mathscr{M}_{I;0;1;1}(V)[\\unabla^I W^{(1)}]$ term on the right-hand side of \\eqref{E:SolveforPartialtV} arises from the spatial derivatives and mixed space-time derivatives of $h^{(1)}$ contained in the term $\\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)}$ on the left-hand side of \\eqref{E:Reducedh1Summary}. Furthermore, the $\\mathscr{M}_{0;0;1;2}(V)[W^{(1)}, W^{(1)}]$ term on the right-hand side of \\eqref{E:SolveforPartialtV} arise from the quadratic and higher-order-in $W^{(1)}$ terms on the right-hand sides of \\eqref{E:Reducedh1Summary} and \\eqref{E:partialtEisolated}, while the $f_{0;0;1;1}(t,x) \\mathscr{M}_{0;0;1;1}(V)[W^{(1)}]$ term on the right-hand side of \\eqref{E:SolveforPartialtV} arises from the $h^{(0)}$ and $\\nabla h^{(0)}-$containing factors that arise from the terms on the right-hand sides of \\eqref{E:Reducedh1Summary} and \\eqref{E:partialtEisolated} that contain a linear factor of $h$ or $\\nabla h.$ Finally, the $f_{0;0;1;0}(t,x)\\mathscr{M}_{0;0;1;0}(V)$ term on the right-hand side of \\eqref{E:SolveforPartialtV} arises from the $\\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(0)}$ term on the right-hand side of \\eqref{E:Reducedh1Summary}, and from the $O(|\\nabla h^{(0)}|^2)$ terms arising from splitting the $O(|\\nabla h|^2)$ terms on the right-hand side of \\eqref{E:Reducedh1Summary}. \n\n\\end{proof}\n\n\n\n\n\\begin{corollary} \\label{C:WeightedL2PartialtkunderlinenablaJW1inTermsofInstrinsic}\n\tAssume the hypotheses of Proposition \\ref{P:SmallNormImpliesSmallEnergy}, \n\tincluding the smallness condition $E_{\\dParameter;\\upgamma}(0) + M \\leq \\varepsilon.$\n\tLet $k\\geq 0$ be an integer, let $J$ be a $\\unabla$ multi-index, and \n\tassume that $|J| + k \\leq \\dParameter .$ Let $V(t,x),$ $\\cdots, W^{(1)}(t,x)$ be the array-valued functions defined in \\eqref{E:Vdef} - \n\t\\eqref{E:W1def}, let $\\mathring{V}(x) = V(0,x),$ $\\cdots, \\mathring{W}^{(1)}(x) = W^{(1)}(0,x),$\n\tand assume that $\\| \\mathring{V}^{(1)} \\|_{L^{\\infty}} + \\| \\mathring{W}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}} \\leq \\varepsilon.$\n\tThen if $\\varepsilon$ is sufficiently small, the following inequality holds:\n\t\n\t\n\t\\begin{align} \\label{E:WeightedL2PartialtkunderlinenablaJW1inTermsofInstrinsic}\n\t\t\\big\\| (1 + |x|)^{(1\/2 + \\upgamma + |J| + k)} \\unabla^J \\partial_t^k W^{(1)}(0,x) \\big\\|_{L^2}\n\t\t& \\lesssim \\| \\mathring{W}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}} + M.\n\t\\end{align}\n\t\n\\end{corollary}\n\n\\begin{proof}\n\t\n\tLet us first consider the case $s=0$ in \\eqref{E:PartialtkunderlinenablaJW1inTermsofInstrinsic}.\n\tThen using that $|F_{(0;J;k;0)}(t,x)| \\lesssim M (1 + |x|)^{-(3 + |J| + k)}$ (i.e., property (ii) from Lemma \n\tthe conclusions of \\ref{L:SolveforTimeDerivativesinTermsofInherentDerivatives}) and recalling that $0 < \\upgamma < 1\/2,$ it \n\tfollows that \n\t\n\t\\begin{align} \\label{E:W0WeightedL2}\n\t\t\\big\\| & (1 + |x|)^{1\/2 + \\upgamma} F_{(0;J;k;0)}(0,x)\\mathscr{M}_{(0;J;k;0)}\\big(\\mathring{V}(x)\\big) \\big\\|_{L^2}^2 \\\\\n\t\t& = \\int_{x \\in \\mathbb{R}^3} \n\t\t(1 + |x|)^{1 + 2 \\upgamma} |F_{(0;J;k;0)}(0,x) \\mathscr{M}_{(0;J;k;0)}\\big(\\mathring{V}(x)\\big)|^2 \\, d^3 x \\notag \\\\\n\t\t& \\lesssim M^2 \\int_{x \\in \\mathbb{R}^3} (1 + |x|)^{-4} \\, d^3 x \\lesssim M^2. \\notag\n\t\\end{align}\n\t\n\t\n\tFor the case $s \\geq 1,$ we first use Proposition \\ref{P:SobolevEmbeddingHNdeltaCNprimedeltamprime} to deduce that\n\tfor all $\\unabla-$indices $K$ with $|K| \\leq \\dParameter -2,$ we have\n\t\n\t\\begin{align} \n\t\t|\\unabla^K \\mathring{W}^{(1)}(x)| & \\lesssim (1 + |x|)^{-(|K| + 1)} \n\t\t\t\\| \\mathring{W}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{|K| + 2}}.  \n\t\t\t\\label{E:NablaIW1WeightedSobolevEmbeddingPointwiseBound}\n\t\\end{align}\n\tThen (without loss of generality assuming $|I_1| \\leq |I_2| \\leq \\cdots \\leq |I_s$) we use \\\\\n\t$|F_{(I_1, \\cdots, I_s;J;k;s)}(t,x)| \\lesssim (1 + t + |x|)^{-\\big(|J| + k - (|I_1| + \\cdots + |I|_s) - \n\t(s-1)\\big)}$ (i.e., property (iii)), together with \\eqref{E:NablaIW1WeightedSobolevEmbeddingPointwiseBound}, to deduce\n\t\n\t\\begin{align} \\label{E:CrucialWeightedL2EstimateforProductofDifferentiatedW1Terms}\n\t\t\\big\\| & (1 + |x|)^{1\/2 + \\upgamma + |J| + k} F_{(I_1, \\cdots, I_s;J;k;s)}(0,x)\n\t\t\t\\mathscr{M}_{(I_1, \\cdots, I_s;J;k;s)}\\big(\\mathring{V}(x)\\big)[\\unabla^{I_1}\\mathring{W}^{(1)}(x), \n\t\t\t\\cdots, \\unabla^{I_s} \\mathring{W}^{(1)}(x)] \\big\\|_{L^2} \\\\\n\t\t& \\lesssim \\Big\\lbrace \n\t\t\t\\prod_{i=1}^{s-1} \\big\\| (1 + |x|)^{|I_1| + \\cdots + |I_{s-1}| + (s-1)} \\unabla^{I_i} \n\t\t\t\\mathring{W}^{(1)}(x) \\big\\|_{L^{\\infty}} \n\t\t\t\\Big\\rbrace \\big\\| (1 + |x|)^{1\/2 + \\upgamma + |I_s|} \\unabla^{I_s} \\mathring{W}^{(1)}(x) \\big\\|_{L^2} \\notag \\\\\n\t\t& \\lesssim  \\big\\| (1 + |x|)^{1\/2 + \\upgamma + |I_s|} \\unabla^{I_s} \\mathring{W}^{(1)}(x) \\big\\|_{L^2}\n\t\t\t\\lesssim \\| \\mathring{W}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}. \\notag \n\t\\end{align}\n\tCombining \\eqref{E:W0WeightedL2} and \\eqref{E:CrucialWeightedL2EstimateforProductofDifferentiatedW1Terms},\n\twe arrive at \\eqref{E:WeightedL2PartialtkunderlinenablaJW1inTermsofInstrinsic}.\n\t\n\\end{proof}\n\n\n\t\nWe are now ready for the proof of the proposition.\n\t\n\\textbf{Proof of Proposition \\ref{P:SmallNormImpliesSmallEnergy}:}\t\n\tWe first remark that \\emph{the estimates derived in this proof are valid under the assumption that $\\varepsilon$ is \n\tsufficiently small}. Recall that $g_{\\mu \\nu}(t,x) = m_{\\mu \\nu} + \\chi\\big(\\frac{r}{t}\\big) \\chi(r) \\frac{2M}{r} \\delta_{\\mu \\nu} \n\t+ h_{\\mu \\nu}^{(1)}(t,x).$ Also recall that according to the assumptions of the proposition, \n\t\n\t\\begin{subequations}\n\t\\begin{align} \\label{E:InitialSpacetimeh1inTermsofInstrinsich1}\n\t\th^{(1)}(0,x) & = \\begin{pmatrix}\n                        0 & 0 & 0 & 0 \\\\\n                        0 & \\mathring{\\underline{h}}_{11}^{(1)} & \\mathring{\\underline{h}}_{12}^{(1)}\n                        \t& \\mathring{\\underline{h}}_{13}^{(1)} \\\\\n                        0 & \\mathring{\\underline{h}}_{21}^{(1)} & \\mathring{\\underline{h}}_{22}^{(1)}\n                        \t& \\mathring{\\underline{h}}_{23}^{(1)} \\\\\n                        0 & \\mathring{\\underline{h}}_{31}^{(1)} & \\mathring{\\underline{h}}_{32}^{(1)}\n                        \t& \\mathring{\\underline{h}}_{33}^{(1)} \n                    \\end{pmatrix}, \\\\\n \t\t\\partial_t h^{(1)}(0,x) \n \t\t\t& = \\begin{pmatrix}\n     \t\t2A^3 (\\mathring{\\underline{g}}^{-1})^{ab} \\mathring{K}_{ab} & A^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{b1} \n       \t\t& A^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{b2} & A^2 \n       \t\t(\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{b3} \\\\\n      \tA^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{b1} & 2A \\mathring{K}_{11} \n      \t\t& 2A \\mathring{K}_{12} & 2A \\mathring{K}_{13} \\\\\n      \tA^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{b2} & 2A \\mathring{K}_{21} \n      \t\t& 2A \\mathring{K}_{22} & 2A \\mathring{K}_{23} \\\\\n      \tA^2 (\\mathring{\\underline{g}}^{-1})^{ab} \\partial_a \\mathring{\\underline{g}}_{b3} & 2A \\mathring{K}_{31} \n      \t\t& 2A \\mathring{K}_{32} & 2A \\mathring{K}_{33}\n   \\end{pmatrix},  \\label{E:InitialpartialtSpacetimeh1inTermsofInstrinsic}\n\t\\end{align}\n\t\\end{subequations}\n\twhere $A(x) = \\sqrt{1- 2M \\chi(r)\/r},$ and \n\t$\\mathring{\\underline{g}}_{jk}(x) = \\delta_{jk} + 2M \\chi(r)\/r \\delta_{jk} + \\mathring{\\underline{h}}_{jk}^{(1)}(x).$\n\tNote that $(\\mathring{\\underline{g}}^{-1})^{jk} = \\delta^{jk} + O^{\\infty}(|M \\chi(r)\/r|;\\mathring{\\underline{h}}^{(1)}) \n\t+ O^{\\infty}(|\\mathring{\\underline{h}}^{(1)}|;M \\chi(r)\/r).$ Our immediate objectives are to relate\n\t$\\| \\partial_t h^{(1)}(0,\\cdot) \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}$ and $\\| \\mathring{\\Electricfield} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}$\n\tto the inherent quantities $\\| \\mathring{\\underline{h}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}},$ \n\t$\\| \\mathring{K} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}},$ \n\t$\\| \\mathring{\\mathfrak{\\Displacement}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}},$ \n\t$\\| \\mathring{\\mathfrak{\\Magneticinduction}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}},$ and $M.$\n\tTo this end, we first observe that the following estimates hold for sufficiently small $M:$ \n\t\n\t\\begin{align}\n\t\t|\\underline{\\nabla}^I \\big(M \\frac{\\chi(r)}{r}\\big)| & \\lesssim M(1 + r)^{-(1 + |I|)}, \\label{E:unablaASchwarzschildTailDecayEstimates} && \\\\\n\t\t|A(x)| & \\lesssim 1, && \\\\\n\t\t|\\underline{\\nabla}^I A(x)| & \\lesssim M(1 + r)^{-(1 + |I|)}, && |I| \\geq 1. \\label{E:unablaADecayEstimates}\n\t\\end{align}\n\tUsing \\eqref{E:InitialSpacetimeh1inTermsofInstrinsich1} - \\eqref{E:InitialpartialtSpacetimeh1inTermsofInstrinsic}, the decay \n\testimates \\eqref{E:unablaASchwarzschildTailDecayEstimates} - \\eqref{E:unablaADecayEstimates},\n\tthe Leibniz rule, Corollary \\ref{C:CompositionProductHNdelta},\n\tthe definition of $\\| \\cdot \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}},$ the fact that \n\t$0 < \\upgamma < 1\/2,$ and elementary calculations,\n\tit is easy to check that\n\t\n\t\\begin{align} \\label{E:PartialtSpacetimeh1inTermsofInstrinsicK}\n\t\t\\| \\partial_t h(0,\\cdot) \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}} \\lesssim \\| \\mathring{\\underline{h}}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}\n\t\t+ \\| \\mathring{K} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}} + M.\n\t\\end{align}\n\tFurthermore, by  \\eqref{E:DintermsofEBh}, \\eqref{E:EintermsofDBh}, and Corollary \\ref{C:CompositionProductHNdelta}, \n\twe have that\n\t\n\t\\begin{align} \\label{E:InitialElectricMagneticWeightedSobolevinTermsofInitialDisplacementMagnetic}\n\t\t\\| \\mathring{\\Electricfield} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}} + \\| \\mathring{\\Magneticinduction} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}\n\t\t& \\approx \\| \\mathring{\\Displacement} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}} \n\t\t+ \\| \\mathring{\\Magneticinduction} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}.\n\t\\end{align}\n\tSimilarly, by we have that\n\t\\begin{align} \\label{E:InitialIntrinsicDisplacementMagneticintermsofInitialDisplacementMagnetic}\n\t\t\\| \\mathring{\\Displacement} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}} + \\| \\mathring{\\Magneticinduction} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}\n\t\t& \\approx \\| \\mathring{\\mathfrak{\\Displacement}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}} \n\t\t+ \\| \\mathring{\\mathfrak{\\Magneticinduction}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}.\n\t\\end{align}\n\tBy \\eqref{E:PartialtSpacetimeh1inTermsofInstrinsicK}, \n\t\\eqref{E:InitialElectricMagneticWeightedSobolevinTermsofInitialDisplacementMagnetic},\n\t\\eqref{E:InitialIntrinsicDisplacementMagneticintermsofInitialDisplacementMagnetic},\n\tand Proposition \\ref{P:SobolevEmbeddingHNdeltaCNprimedeltamprime},\n\tit follows that if $E_{\\dParameter;\\upgamma}(0) + M$ is sufficiently small, then the smallness conditions\\footnote{As in the\n\tLindblad-Rodnianski proof of Corollary \\ref{C:WeakDecay} below, the smallness condition \n\t$|h^{(1)}(0,x)| \\lesssim \\varepsilon (1 + r)^{-1 - \\upgamma}$ \n\tfollows from integrating the smallness condition $|\\partial_r h^{(1)}(0,x)| \\lesssim \\varepsilon (1 + r)^{-2 - \\upgamma},$\n\twhich is a consequence of Proposition \\ref{P:SobolevEmbeddingHNdeltaCNprimedeltamprime},\n\tfrom spatial infinity and using the decay assumption \\eqref{E:h1AbstractDataAsymptotics} for \n\t$|\\mathring{h}^{(1)}(x)|$ at spatial infinity.}\n\tfor $\\|\\mathring{V}^{(1)} \\|_{L^{\\infty}}$ and $\\| \\mathring{W}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}$ \n\tin the hypotheses of Lemma \\ref{L:SolveforTimeDerivativesinTermsofInherentDerivatives} \n\tand Corollary \\ref{C:WeightedL2PartialtkunderlinenablaJW1inTermsofInstrinsic} hold. \n\tTherefore, combining Corollaries \\ref{C:InitialEnergyInTermsofTangentialandTimeDerivatives} and \n\t\\ref{C:WeightedL2PartialtkunderlinenablaJW1inTermsofInstrinsic}, \\eqref{E:PartialtSpacetimeh1inTermsofInstrinsicK},\n\t\\eqref{E:InitialElectricMagneticWeightedSobolevinTermsofInitialDisplacementMagnetic}, and\n\t\\eqref{E:InitialIntrinsicDisplacementMagneticintermsofInitialDisplacementMagnetic},\n\twe deduce that if $\\varepsilon$ is sufficiently small, then\n\t\n\t\\begin{align}\n\t\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}^2(0) \n\t\t& \\lesssim \\| \\underline{\\nabla} \\mathring{\\underline{h}}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\t+ \\| \\partial_t h^{(1)}(0,\\cdot) \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\t+ \\| \\mathring{\\Electricfield} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\t+ \\| \\mathring{\\Magneticinduction} \t\\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 + M^2 \\\\\n\t\t& \\lesssim \\| \\underline{\\nabla} \\mathring{\\underline{h}}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\t+ \\| \\mathring{K} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\t+ \\| \\mathring{\\mathfrak{\\Displacement}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\t+ \\| \\mathring{\\mathfrak{\\Magneticinduction}} \t\\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 + M^2 \\notag \\\\\n\t\t& \\eqdef E_{\\dParameter;\\upgamma}^2(0) + M^2. \\notag\n\t\\end{align}\n\tThis concludes our proof of Proposition \\ref{P:SmallNormImpliesSmallEnergy}. \\hfill $\\qed$\n\t\n\t\n\n\n\n\n\\section{Algebraic Estimates of the Nonlinearities} \\label{S:AlgebraicEstimates}\n\nIn this section, we provide algebraic estimates for the inhomogeneous terms that arise from\ndifferentiating the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}. We also \nuse the equations of Proposition \\ref{P:EOVNullDecomposition} to derive ordinary differential inequalities\nfor the null components of $\\dot{\\Far} = \\Lie_{\\mathcal{Z}}^I \\Far.$ Furthermore, we provide algebraic estimates\nfor the inhomogeneous terms appearing on the right-hand sides of these inequalities. Many of the estimates derived in this section rely on the wave coordinate condition.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{Statement and proofs of the propositions}\n\nThe proofs of the propositions given in this section use the results of a collection of technical lemmas, which \nwe relegate to the end of the section. We begin by quoting the following proposition proved in \\cite{hLiR2010},\nwhich is central to many of the estimates. The basic idea is the following: many of our estimates for coupled quantities would break down if we could not achieve good control of the components $h_{LL}$ and $h_{LT}.$ Amazingly, as shown in \\cite{hLiR2005} and \\cite{hLiR2010}, the wave coordinate condition allows for \\emph{independent, improved} estimates of exactly these components.\n\n\n\\begin{proposition} \\cite[Proposition 8.2]{hLiR2010} \\label{P:harmonicgauge} \n\t\\textbf{(Algebraic consequences of the wave coordinate condition)}\n\tLet $g$ be a Lorentzian metric satisfying the wave coordinate condition \\eqref{E:wavecoordinategauge1} relative to the \n\tcoordinate system $\\lbrace x^{\\mu} \\rbrace_{\\mu=0,1,2,3}.$ Let $I$ be a $\\mathcal{Z}-$multi-index, \n\tassume that $|\\nabla_{\\mathcal{Z}}^J h| \\leq \\varepsilon$ holds for all $\\mathcal{Z}-$multi-indices $J$ satisfying $|J| \\leq \n\t\\lfloor |I|\/2 \\rfloor,$ where $h_{\\mu \\nu} \\eqdef g_{\\mu \\nu} - m_{\\mu \\nu}.$ Then if $\\varepsilon$ is sufficiently small,\n\tthe following pointwise estimates hold for the tensor $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu}:$\n\t\n\t\\begin{subequations}\n\t\\begin{align} \\label{E:nablaZIHLTpointwiseEstimate}\n\t\t|\\nabla\\nabla_{\\mathcal{Z}}^I H|_{\\mathcal{L} \\mathcal{T}} & \\lesssim \\sum_{|J| \\leq |I|} \n\t\t |\\conenabla \\nabla_{\\mathcal{Z}}^J H|\n\t\t \\ + \\ \\underbrace{\\sum_{|J| \\leq |I| - 1} |\\nabla\\nabla_{\\mathcal{Z}}^J H|}_{\\mbox{Absent if $|I|=0.$}}\n\t\t\t\\ + \\ \\sum_{|I_1| + |I_2| \\leq |I|} |\\nabla_{\\mathcal{Z}}^{I_1} H||\\nabla\\nabla_{\\mathcal{Z}}^{I_2}H|, \\\\\n\t|\\nabla\\nabla_{\\mathcal{Z}}^I H|_{\\mathcal{L} \\mathcal{L}} & \\lesssim \\sum_{|J| \\leq |I|} \n\t\t|\\conenabla \\nabla_{\\mathcal{Z}}^J H|\n\t\t\t\\ + \\ \\underbrace{\\sum_{|J| \\leq |I| - 2} |\\nabla\\nabla_{\\mathcal{Z}}^{J}H|}_{\\mbox{Absent if $|I| \\leq 1.$}}\n\t\t\t\\ + \\ \\sum_{|I_1| + |I_2| \\leq |I|} |\\nabla_{\\mathcal{Z}}^{I_1} H||\\nabla\\nabla_{\\mathcal{Z}}^{I_2}H|. \n\t\t\t\\label{E:nablaZIHLLpointwiseEstimate}\n\t\\end{align}\n\t\\end{subequations}\n\t\n\tFurthermore, analogous estimates hold for the tensor $h_{\\mu \\nu}.$ \n\t\n\\end{proposition}\n\n\\hfill $\\qed$\n\n\nThe next lemma provides an analogous version of the proposition for the ``remainder'' pieces of $(g^{-1})^{\\mu \\nu}$\nand $g_{\\mu \\nu}.$\n\n\\begin{lemma} \\label{L:NablaZIh1LLh1LTwaveCoordinateAlgebraicEstimate} \\cite[Slight extension of Lemma 15.4]{hLiR2010}\n\t\\textbf{(Algebraic\/analytic consequences of the wave coordinate condition)}\n\tLet $g$ be a Lorentzian metric satisfying the wave coordinate condition \\eqref{E:wavecoordinategauge1} relative to the \n\tcoordinate system $\\lbrace x^{\\mu} \\rbrace_{\\mu = 0,1,2,3},$ and let $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \n\t\\nu}.$ Let $k \\geq 0$ be an integer, and assume that there is a constant $\\varepsilon$ such that $|\\nabla_{\\mathcal{Z}}^J h| \n\t\\leq \\varepsilon$ holds for all $\\mathcal{Z}-$multi-indices $J$ satisfying $|J| \\leq k\/2,$ where \n\t$h_{\\mu \\nu} \\eqdef g_{\\mu \\nu} - m_{\\mu \\nu}.$ Let \n\t\n\t\\begin{align} \\label{E:NablaZIh1LLh1LTwaveCoordinateAlgebraicEstimate}\n\t\tH_{(1)}^{\\mu \\nu} \\eqdef H^{\\mu \\nu} - H_{(0)}^{\\mu \\nu}, \\qquad H_{(0)}^{\\mu \\nu} \\eqdef \n\t\t\t- \\chi\\big(\\frac{r}{t}\\big) \\chi(r)\n\t\t\t\\frac{2M}{r} \\delta^{\\mu \\nu},\n\t\\end{align}\n\twhere $H_{(1)}^{\\mu \\nu}$ is the tensor obtained by subtracting the Schwarzschild part \n\t$H_{(0)}^{\\mu \\nu}$ from $H^{\\mu \\nu},$ and let $\\chi_0(1\/2 < z < 3\/4)$ denote the characteristic function of the interval \n\t$[1\/2,3\/4].$ Assume further that $M \\leq \\varepsilon.$ Then if $\\varepsilon$ is sufficiently small, the following\n\tpointwise estimates hold\n\t\n\t\\begin{align}\n\t\t\\sum_{|I| \\leq k} |\\nabla\\nabla_{\\mathcal{Z}}^I H_{(1)}|_{\\mathcal{L} \\mathcal{L}}\n\t\t\\ + \\ \\sum_{|J| \\leq k - 1} |\\nabla\\nabla_{\\mathcal{Z}}^J H_{(1)}|_{\\mathcal{L} \\mathcal{T}} \n\t\t& \\lesssim \\sum_{|I| \\leq k} |\\conenabla \\nabla_{\\mathcal{Z}}^I H_{(1)}| \n\t\t\t \\\\\n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^I H_{(1)}| \n\t\t\t\\ + \\ \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-2} |\\nabla_{\\mathcal{Z}}^I H_{(1)}| \\notag \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| \\leq k} |\\nabla_{\\mathcal{Z}}^{I_1} H_{(1)}| |\\nabla\\nabla_{\\mathcal{Z}}^{I_2} H_{(1)}|  \n\t\t \t\\ + \\ \\underbrace{\\sum_{|J'| \\leq k - 2} |\\nabla\\nabla_{\\mathcal{Z}}^{J'} H_{(1)}|}_{\\mbox{Absent if $k \\leq 1$}}   \n\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon (1 + t + |q|)^{-2} \\chi_0(1\/2 < r\/t < 3\/4) \n\t\t\t\\ + \\ \\varepsilon^2 (1 + t + |q|)^{-3}. \\notag\n\t\\end{align}\n\t\n\tAdditionally, let\n\t\\begin{align}\n\t\th_{\\mu \\nu}^{(1)} \\eqdef h_{\\mu \\nu} - h_{\\mu \\nu}^{(0)}, \\qquad h_{\\mu \\nu}^{(0)} \n\t\t\t\\eqdef \\chi\\big(\\frac{r}{t}\\big) \\chi(r) \\frac{2M}{r} \\delta_{\\mu \\nu},\n\t\\end{align}\n\twhere $h_{\\mu \\nu}^{(1)}$ is the tensorfield obtained by subtracting the Schwarzschild part $h_{\\mu \\nu}^{(0)}$ \n\tfrom $h_{\\mu \\nu}.$ Then an estimate analogous to \\eqref{E:NablaZIh1LLh1LTwaveCoordinateAlgebraicEstimate} holds if we \n\treplace the tensorfield $H_{(1)}$ with the tensorfield $h^{(1)}.$ \n\t\n\\end{lemma}\n\n\\begin{proof}\n\tThe estimates for the tensorfield $H_{(1)}^{\\mu \\nu}$ were proved as \\cite[Lemma 15.4]{hLiR2010}. The analogous estimates for \n\tthe tensorfield $h_{\\mu \\nu}^{(1)}$ follow from those for $H_{(1)}^{\\mu \\nu},$ together with the fact that\n\t$H_{(1); \\mu \\nu} = - h_{\\mu \\nu}^{(1)} + O^{\\infty}(|h^{(0)} + h^{(1)}|^2)$ and the decay estimates for $h^{(0)}$ stated in \n\tLemma \\ref{L:h0decayestimates}.\n\\end{proof}\n\n\n\nWe now turn to the following proposition, which captures the algebraic structure of the inhomogeneous term $\\mathfrak{H}_{\\mu \\nu}$ appearing on the right-hand side of the reduced equation \\eqref{E:Reducedh1Summary}.\n\n\n\\begin{proposition} \\label{P:AlgebraicInhomogeneous} \n\\cite[Extension of Proposition 9.8]{hLiR2010}\n\t\\textbf{(Algebraic estimates of $\\mathfrak{H}_{\\mu \\nu}$ and $\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}_{\\mu \\nu}$)}\n\tLet $\\mathfrak{H}_{\\mu \\nu}$ be the inhomogeneous term on the right-hand side of the reduced equation \n\t\\eqref{E:Reducedh1Summary}, and assume that the wave coordinate condition \\eqref{E:wavecoordinategauge1} holds. Then\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t|\\mathfrak{H}|_{\\mathcal{T} \\mathcal{N}} & \\lesssim |\\conenabla h||\\nabla h| \n\t\t\t\\ + \\ \\big(|\\Far|_{\\mathcal{L}\\mathcal{N}} + |\\Far|_{\\mathcal{T}\\mathcal{T}} \\big)|\\Far|\n\t\t\t \\ + \\ O^{\\infty}(|h||\\nabla h|^2) \\ + \\ O^{\\dParameter+1}(|h||\\Far|^2) \\ + \\ O^{\\dParameter+1}|\\Far|^3;h), \\label{E:InhomogeneousHTUAlgebraic} \\\\\n\t\t|\\mathfrak{H}| & \\lesssim |\\nabla h|_{\\mathcal{T} \\mathcal{N}}^2 \\ + \\ |\\conenabla h||\\nabla h|\n\t\t\t\\ + \\ |\\Far|^2 \\ + \\ O^{\\infty}(|h||\\nabla h|^2) \\ + \\ O^{\\dParameter+1}(|h||\\Far|^2) \\ + \\ O^{\\dParameter+1}(|\\Far|^3;h). \\label{E:InhomogeneousHAlgebraic} \n\t\\end{align}\n\t\n\t\n\tIn addition, assume that there exists an $\\varepsilon > 0$ \n\tsuch that $|\\nabla_{\\mathcal{Z}}^J h| + |\\Lie_{\\mathcal{Z}}^J \\Far| \\leq \\varepsilon$ \n\tholds for all $\\mathcal{Z}-$multi-indices $|J| \\leq \\lfloor |I|\/2 \\rfloor.$ \n\tThen if $\\varepsilon$ is sufficiently small, the following pointwise estimates hold:\n\t\n\t\n\t\\begin{align} \\label{E:ZIinhomogeneoushpointwise}\n\t\t|\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}| \n\t\t& \\lesssim \\sum_{|I_1| + |I_2| \\leq |I|}\n\t\t\t\\Big\\lbrace |\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h|_{\\mathcal{T} \\mathcal{N}} \n\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h |_{\\mathcal{T} \\mathcal{N}} \n\t\t\t\\ + \\ |\\conenabla \\nabla_{\\mathcal{Z}}^{I_1}h | |\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h| \\Big\\rbrace \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| \\leq |I|} |\\Lie_{\\mathcal{Z}}^{I_1} \\Far| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far|\n\t\t\t\\ + \\ \\underbrace{\\sum_{|I_1| + |I_2| \\leq |I| - 2} |\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h|\n\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h|}_{\\mbox{Absent if $|I| \\leq 1$}} \t\n\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| + |I_3| \\leq |I|} \n\t\t\t\\Big\\lbrace|\\nabla_{\\mathcal{Z}}^{I_1} h| |\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h| |\\nabla\\nabla_{\\mathcal{Z}}^{I_3}h| \n\t\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{I_1} h||\\Lie_{\\mathcal{Z}}^{I_2}\\Far| |\\Lie_{\\mathcal{Z}}^{I_3} \\Far| \n\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^{I_1} \\Far||\\Lie_{\\mathcal{Z}}^{I_2}\\Far| |\\Lie_{\\mathcal{Z}}^{I_3} \\Far| \\Big\\rbrace. \\notag \n\\end{align}\n\\end{subequations}\n\\end{proposition}\n\n\\begin{proof}\n\tUsing \\eqref{E:ReducedhInhomogeneous}, we can decompose $\\mathfrak{H}_{\\mu \\nu}$ \n\tinto\n\t\n\t\\begin{align}\n\t\t\\mathfrak{H}_{\\mu \\nu} & = (i)_{\\mu \\nu} + (ii)_{\\mu \\nu} + (iii)_{\\mu \\nu} + (iv)_{\\mu \\nu}, \n\t\\end{align}\n\twhere\n\t\n\t\\begin{align}\n\t\t(i)_{\\mu \\nu} & \\eqdef \\mathscr{P}(\\nabla_{\\mu} h, \\nabla_{\\nu} h),   \n\t\t\t\\label{E:PieceiReducedhInhomogeneous} \\\\\n\t\t(ii)_{\\mu \\nu} & \\eqdef \\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h), \\\\\n\t\t(iii)_{\\mu \\nu} & \\eqdef \\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\Far, \\Far), \n\t\t\t\\label{E:PieceiiiReducedhInhomogeneous} \\\\\n\t\t(iv)_{\\mu \\nu} & \\eqdef O^{\\infty}(|h||\\nabla h|^2) \\ + \\ O^{\\dParameter+1}(|h||\\Far|^2) \\ + \\ O^{\\dParameter+1}(|\\Far|^3;h).\n\t\t\t\\label{E:PieceivReducedhInhomogeneous}\n\t\\end{align}\n\tWe will analyze each of the four pieces separately.\n\t\n\tThe facts that $|(i)|_{\\mathcal{T} \\mathcal{N}} \\lesssim$ the \n\tright-hand side of \\eqref{E:InhomogeneousHTUAlgebraic} and that $|(i)| \\lesssim$ the right-hand side of \n\t\\eqref{E:InhomogeneousHAlgebraic} follow from \n\tProposition \\ref{P:harmonicgauge}, \\eqref{E:PSpecialNullStructure}, and\n\t\\eqref{E:PTUSpecialNullStructure}. The fact that $|\\nabla_{\\mathcal{Z}}^I(i)| \\lesssim$ \n\tthe right-hand side of \\eqref{E:ZIinhomogeneoushpointwise} follows from Proposition \\ref{P:harmonicgauge},\n\t\\eqref{E:SpecialPLeibnizRule}, and \\eqref{E:PSpecialNullStructure}.\n\t\n\tThe facts that $|(ii)|_{\\mathcal{T} \\mathcal{N}} \\lesssim$ the right-hand side of \n\t\\eqref{E:InhomogeneousHTUAlgebraic}, and that $|(ii)| \\lesssim$ the right-hand side of \n\t\\eqref{E:InhomogeneousHAlgebraic} both follow from \\eqref{E:Q1hNullFormEstimate}.\n\tThat $|\\nabla_{\\mathcal{Z}}^I(ii)| \\lesssim$\n\tthe right-hand side of \\eqref{E:ZIinhomogeneoushpointwise} follows from \n\t\\eqref{E:Q1hLeibnizRule} and \\eqref{E:Q1hNullFormEstimate}.\n\t\n\tThe fact that $|(iii)|_{\\mathcal{T} \\mathcal{N}} \\lesssim$ the right-hand side of \n\t\\eqref{E:InhomogeneousHTUAlgebraic} follows from \\eqref{E:Q2TUhNullFormEstimate}, while the fact that \n\t$|(iii)| \\lesssim$ the right-hand side of \\eqref{E:InhomogeneousHAlgebraic} follows from \n\t\\eqref{E:Q2hNullFormEstimate}. The fact that $|\\nabla_{\\mathcal{Z}}^I(iii)| \\lesssim$ the right-hand side of \n\t\\eqref{E:ZIinhomogeneoushpointwise} follows from \\eqref{E:LieZIinTermsofNablaZI},\n\t\\eqref{E:Q2hLeibnizRule}, and \\eqref{E:Q2hNullFormEstimate}.\n\t\n\t\n\tThe desired estimates for term $(iv)$ follow easily with the help of the Leibniz rule and \\eqref{E:LieZIinTermsofNablaZI}.\n\n\\end{proof}\n\n\n\nThe next proposition captures the special algebraic structure of the reduced inhomogeneous term $\\mathfrak{F}_{(I)}^{\\nu}$ defined in \\eqref{E:LiemodZIdifferentiatedEOVInhomogeneousterms}.\n\n\n\\begin{proposition} \\label{P:EnergyInhomogeneousTermAlgebraicEstimate}\n\t\\textbf{(Algebraic estimates of $\\mathfrak{F}_{(I)}^{\\nu}$)}\n\tLet $\\mathfrak{F}^{\\nu}$ be the inhomogeneous term \\eqref{E:EMBIFarInhomogeneous} in the reduced electromagnetic\n\tequations, let $I$ be a $\\mathcal{Z}-$multi-index with $|I|=k,$\n\tand let $X_{\\nu}$ be any covector. In addition, assume that there exists an $\\varepsilon > 0$ \n\tsuch that $|\\nabla_{\\mathcal{Z}}^J h| + |\\Lie_{\\mathcal{Z}}^J \\Far| \\leq \\varepsilon$ \n\tholds for all $\\mathcal{Z}-$multi-indices $|J| \\leq \\lfloor k\/2 \\rfloor.$ \n\tThen if $\\varepsilon$ is sufficiently small, the following pointwise estimates hold:\n\t\n\t\\begin{subequations}\n\t\\begin{align} \\label{E:LieZIFarNullFormInhomogeneousTermAlgebraicEstimate}\n\t\t|X_{\\nu} \\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu}|\n\t\t& \\lesssim \\sum_{|I_1| + |I_2| \\leq k} |X| \n\t\t\t|\\conenabla \\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far|\n\t\t\t\\ + \\ \\sum_{|I_1| + |I_2| \\leq k} |X||\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h| \n\t\t\t\\big(|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{T} \n\t\t\t\\mathcal{T}}\\big) \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| + |I_3| \\leq k} \n\t\t\t|X||\\nabla_{\\mathcal{Z}}^{I_1} h| |\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h| \n\t\t\t|\\Lie_{\\mathcal{Z}}^{I_3} \\Far|\n\t\t\t\\ + \\ \\sum_{|I_1| + |I_2| + |I_3| \\leq k} \n\t\t\t|X||\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \n\t\t\t|\\Lie_{\\mathcal{Z}}^{I_3} \\Far| \\notag \\\\\n\t\t& \\lesssim \n\t\t\t(1 + t + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_2| \\leq k} |X| \n\t\t\t|\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far|\n\t\t\t\\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_2| \\leq k} |X| |\\nabla_{\\mathcal{Z}}^{I_1} h| \n\t\t\t\\big(|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{T} \n\t\t\t\\mathcal{T}}\\big)  \\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| + |I_3| \\leq k + 1}}_{|I_2|, |I_3| \\leq k} \n\t\t\t|X| \\Big\\lbrace |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} h| |\\Lie_{\\mathcal{Z}}^{I_3} \\Far|\n\t\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \n\t\t\t|\\Lie_{\\mathcal{Z}}^{I_3} \\Far| \\Big\\rbrace. \\notag\n\t\\end{align}\n\tIn addition, the same estimates hold for $|X_{\\nu} \\Lie_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu}|.$\n\t\n\tFurthermore, let $N^{\\# \\mu \\nu \\kappa \\lambda}$ be the tensorfield from \n\tthe reduced electromagnetic equation \\eqref{E:ReduceddMis0Summary}. \n\tThen if $\\varepsilon$ is sufficiently small and $k \\geq 1,$ the following pointwise commutator estimate holds:\n\t\n\t\\begin{align} \\label{E:EnergyInhomogeneousTermAlgebraicEstimate}\n\t\t\\Big|X_{\\nu} \\Big\\lbrace & N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I\\Far_{\\kappa \\lambda}\n\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big) \\Big\\rbrace \\Big|\t\\\\\n\t\t& \\lesssim (1 + |q|)^{-1} \\mathop{\\sum_{|I'| = k}}_{|J| \\leq 1} \n\t\t\t\t|X| |\\nabla_{\\mathcal{Z}}^{I'} h|_{\\mathcal{L}\\mathcal{L}} |\\Lie_{\\mathcal{Z}}^J \\Far| \n\t\t\t\\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|J| \\leq 1}}_{|I'| = k} \n\t\t\t\t|X| |\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L}\\mathcal{L}} |\\Lie_{\\mathcal{Z}}^{I'} \\Far| \\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\sum_{|I'| = k} |X||h|_{\\mathcal{L}\\mathcal{T}} |\\Lie_{\\mathcal{Z}}^{I'} \\Far| \n\t\t\t\\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_1|, |I_2| \\leq k} |X| |\\nabla_{\\mathcal{Z}}^{I_1} h|\n\t\t \t\t\\big(|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{L} \\mathcal{N}} +  \n\t\t \t\t|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{T} \\mathcal{T}}\\big) \\notag \\\\\n\t\t& \\ \\ + \\ (1 + t + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_1|, |I_2| \\leq k}\n\t\t\t|X||\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far|\n\t\t\t\\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_1|, |I_2| \\leq k}\n\t\t\t\t|X|_{\\mathcal{L}}|\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_1| \\leq k - 1, |I_2| \\leq k - 1} \n\t\t\t\t|X||\\nabla_{\\mathcal{Z}}^{I_1} h|_{\\mathcal{L}\\mathcal{L}} |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \n\t\t\t\\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| \\leq k}}_{|I_1| \\leq k - 1, |I_2| \\leq k - 1} \n\t\t\t\t|X||\\nabla_{\\mathcal{Z}}^{I_1} h|_{\\mathcal{L}\\mathcal{T}} |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\underbrace{\\mathop{\\sum_{|I_1| + |I_2| \\leq k - 1}}_{|I_1| \\leq k - 2, |I_2| \\leq k - 1} \n\t\t\t|X||\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far|}_{\\mbox{absent if $k = 1$}} \\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|I_1| + |I_2| + |I_3| \\leq k + 1}}_{|I_1|, |I_2|, |I_3| \\leq k} |X| \n\t\t\t\\Big\\lbrace |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\nabla_{\\mathcal{Z}}^{I_2} h| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|\n\t\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|\n\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^{I_1} \\Far| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|\n\t\t\t\\Big\\rbrace. \\notag\n\t\\end{align}\n\t\\end{subequations}\n\t\n\\end{proposition}\n\n\n\\begin{proof}\n\tInequality \\eqref{E:LieZIFarNullFormInhomogeneousTermAlgebraicEstimate} follows from\n\n\t\\eqref{E:LieZIinTermsofNablaZI},\n\t\\eqref{E:NablaLieZIinTermsofNablaNablaZI}, \\eqref{E:TangentialDerivativesLieZIvsTangentialDerivativesNablaZI}\n\t(which allow us to estimate Lie derivatives of $h$ in terms of covariant derivatives of $h$), \\eqref{E:LiemodZIFExpanded}, and \n\t\\eqref{E:Q2FarNullFormEstimate}. \n\t\n\tInequality \\eqref{E:EnergyInhomogeneousTermAlgebraicEstimate} \n\tfollows from \\eqref{E:NtriangleSmallAlgebraic}, \n\t\\eqref{E:LieZIinTermsofNablaZI} and \\eqref{E:LieZILLinTermsofNablaZILLLieZJLTPlusJunk}\n\t(which allow us to estimate Lie derivatives of $h$ in terms of covariant derivatives of $h$), \n\t\\eqref{E:LiemodZINnablaFarCommutatorTerms},\n\t\\eqref{E:XPFarhNablaFarNullFormEstimate}, and \\eqref{E:XQ1FarhNablaFarNullFormEstimate}.\n\t\n\t\n\\end{proof}\n\n\nAs discussed at the beginning of Section \\ref{SS:NullDecompElectromagnetic}, the null components of the lower-order\nLie derivatives of $\\Far$ satisfy ordinary differential equations with controllable inhomogeneous terms. \nThe next proposition provides convenient algebraic expressions for the inhomogeneities. In Section\n\\ref{S:DecayFortheReducedEquations}, these algebraic expressions will be combined with decay estimates\nto deduce upgraded decay estimates for the null components of $\\Far$ and its lower-order Lie derivatives.\n\n\n\\begin{proposition} \\label{P:ODEsNullComponentsLieZIFar}\n\\textbf{(Ordinary differential inequalities for $\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far],$ $\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far],$ \n \t$\\rho[\\Lie_{\\mathcal{Z}}^I \\Far],$ and $\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]$)}\n\tLet $\\Far$ be a solution to the reduced electromagnetic equations \\eqref{E:ReduceddFis0Summary} - \n\t\\eqref{E:ReduceddMis0Summary}, and let $\\ualpha,$ $\\alpha,$ $\\rho,$ $\\sigma$ denote its null components. Let $\\Lambda \\eqdef \n\tL + \\frac{1}{4} h_{LL} \\uL,$ and assume that $|h| + |\\Far| \\leq \\varepsilon$ holds. Then if $\\varepsilon$ is sufficiently \n\tsmall, the following pointwise estimate holds:\n\t\n\t\\begin{align} \\label{E:ODErualpha}\n\t\tr^{-1} \\big|\\nabla_{\\Lambda} (r \\ualpha) \\big| \n\t\t& \\lesssim r^{-1} |h|_{\\mathcal{L}\\mathcal{L}} |\\ualpha|\n\t\t\t\\ + \\ \\sum_{|I| \\leq 1} r^{-1} \\big(|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L} \\mathcal{N}} \n\t\t\t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}} \\big)\n\t\t\t\\ + \\ \\sum_{|I_1| + |I_2| \\leq 1}r^{-1} |\\nabla_{\\mathcal{Z}}^{I_1}h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \\\\\n\t\t& \\ \\ + \\ \\sum_{|I| \\leq 1}(1 + |q|)^{-1} |h| \\big(|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L} \\mathcal{N}} \n\t\t  \t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}} \\big)  \\notag \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| + |I_3| \\leq 1} (1 + |q|)^{-1} \n\t\t\t\\Big\\lbrace |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\nabla_{\\mathcal{Z}}^{I_2} h| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|\n\t\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|  \n\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^{I_1} \\Far| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far| \\Big\\rbrace. \\notag\n\t\\end{align}\n\t\n\tSimilarly, for each $\\mathcal{Z}-$multi-index $I,$ let\n \t$\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far],$ $\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far],$ \n \t$\\rho[\\Lie_{\\mathcal{Z}}^I \\Far],$ and $\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]$ denote the null components of \n\t$\\Lie_{\\mathcal{Z}}^I \\Far.$ Furthermore, let $\\varpi(q)$ be any \n\tdifferentiable function of $q.$ Assume that $|\\nabla_{\\mathcal{Z}}^I h| + |\\Lie_{\\mathcal{Z}}^I \\Far| \n\t\\leq \\varepsilon$ holds for $|I| \\leq \\lfloor k\/2 \\rfloor.$ Then if $\\varepsilon$ is sufficiently small, \n\tthe following pointwise estimates also hold:\n\t\n\t\\begin{subequations}\n\t\\begin{align} \\label{E:LambdaLieZIualphaEquationGoodqWeights}\n\t\t\\sum_{|I| \\leq k} r^{-1} \\big|\\nabla_{\\Lambda} & \\big(r \\varpi(q) \\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far] \\big)\\big| \\\\\n\t\t& \\lesssim \\sum_{|I| \\leq k} r^{-1} \\varpi(q) |h|_{\\mathcal{L}\\mathcal{L}} \\big|\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far] \\big|\n\t\t\t\\ + \\ \\sum_{|I| \\leq k} \\varpi'(q) |h|_{\\mathcal{L}\\mathcal{L}} \\big|\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far]\\big| \\notag \\\\\n\t\t& \\ \\ + \\ \\underbrace{\\mathop{\\sum_{|I| \\leq k}}_{|J| \\leq 1} \\varpi(q) (1 + |q|)^{-1} \n\t\t\t\t|\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{L}} \n\t\t\t\t\\big|\\ualpha[\\Lie_{\\mathcal{Z}}^J \\Far] \\big|}_{\\mbox{absent if $k \\leq 1$}} \n\t\t\t\\ + \\ \\underbrace{\\mathop{\\sum_{|J| \\leq 1}}_{|I| \\leq k} \\varpi(q) (1 + |q|)^{-1} \n\t\t\t\t|\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L}\\mathcal{L}} \n\t\t\t\t\\big|\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far]\\big|}_{\\mbox{absent if $k = 0$}} \\notag \\\\\n\t\t& \\ \\ + \\ \\underbrace{\\sum_{|I| \\leq k} \\varpi(q) (1 + |q|)^{-1} \n\t\t\t\t|h|_{\\mathcal{L}\\mathcal{T}} \\big|\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far]\\big|}_{\\mbox{absent if $k = 0$}} \n\t\t\t\\ + \\underbrace{\\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_1| \\leq k - 1, |I_2| \\leq k - 1} \\varpi(q) (1 + |q|)^{-1} \n\t\t\t\t|\\nabla_{\\mathcal{Z}}^{I_1} h| \\big|\\ualpha[\\Lie_{\\mathcal{Z}}^{I_2} \\Far]\\big|}_{\\mbox{absent if $k = 0$}}\n\t\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\ \\sum_{|I| \\leq |k| + 1} \\varpi(q) r^{-1} \\big(|\\Lie_{\\mathcal{Z}}^{I} \\Far|_{\\mathcal{L} \\mathcal{N}} \n\t\t\t\t+ |\\Lie_{\\mathcal{Z}}^{I} \\Far|_{\\mathcal{T} \\mathcal{T}} \\big) \n\t\t\t\t\\ + \\sum_{|I_1| + |I_2| \\leq k + 1} \\varpi(q) (1 + |q|)^{-1} |\\nabla_{\\mathcal{Z}}^{I_1} h|\n\t\t \t\t\\big(|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{L} \\mathcal{N}} +  \n\t\t \t\t|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{T} \\mathcal{T}}\\big) \\notag \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| \\leq k + 1} \\varpi(q) (1 + t + |q|)^{-1}\n\t\t\t\t|\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \\notag \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| + |I_3| \\leq k + 1} \\varpi(q) (1 + |q|)^{-1} \n\t\t\t\\Big\\lbrace |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\nabla_{\\mathcal{Z}}^{I_2} h| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|\n\t\t\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|  \n\t\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^{I_1} \\Far| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far| \\Big\\rbrace, \\notag\n\t\t\\end{align}\n\t\t\n\t\\begin{align} \\label{E:alphaODE}\n\t\t\\sum_{|I| \\leq k} r \\big| \\nabla_{\\uL} \\big(r^{-1} \\alpha[\\Lie_{\\mathcal{Z}}^I \\Far] \\big) \\big| \n\t\t& \\lesssim\n\t\t\t\\sum_{|I| \\leq k + 1} r^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|\n\t\t\t\\ + \\ \\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_1| \\leq k} \n\t\t\t(1 + |q|)^{-1} |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| + |I_3| \\leq k + 1} \n\t\t\t\t(1 + |q|)^{-1} \\Big\\lbrace |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\nabla_{\\mathcal{Z}}^{I_2} h| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|\n\t\t\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|  \n\t\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^{I_1} \\Far| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far| \\Big\\rbrace, \\notag\n\t\t\\end{align}\t\n\t\n\t\\begin{align} \\label{E:rhoODE}\t\n\t\t\\sum_{|I| \\leq k} r^2 \\big| \\nabla_{\\uL} \\big(r^{-2} \\rho[\\Lie_{\\mathcal{Z}}^I \\Far] \\big) \\big| \n\t\t& \\lesssim\n\t\t\t\\sum_{|I| \\leq k + 1} r^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|\n\t\t\t\\ + \\ \\mathop{\\sum_{|I_1| + |I_2| \\leq k + 1}}_{|I_1| \\leq k} \n\t\t\t(1 + |q|)^{-1} |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \\\\\n\t\t& \\ \\ + \\ \\sum_{|I_1| + |I_2| + |I_3| \\leq k + 1} \n\t\t\t\t(1 + |q|)^{-1} \\Big\\lbrace |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\nabla_{\\mathcal{Z}}^{I_2} h| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|\n\t\t\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{I_1} h| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far|  \n\t\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^{I_1} \\Far| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| |\\Lie_{\\mathcal{Z}}^{I_3}\\Far| \\Big\\rbrace, \\notag\n\t\t\\end{align}\t\n\t\t\n\t\\begin{align} \\label{E:sigmaODE}\n\t\t\\sum_{|I| \\leq k} r^2 \\big| \\nabla_{\\uL} \\big(r^{-2} \\sigma[\\Lie_{\\mathcal{Z}}^I \\Far] \\big) \\big| \n\t\t& \\lesssim \\sum_{|I| \\leq k + 1} r^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|.\n\t\\end{align}\t\n\t\\end{subequations}\n\n\\end{proposition}\n\n\n\n\\begin{proof}\n\tOur proof of \\eqref{E:ODErualpha} is based on decomposing the terms in equation \\eqref{E:dotualphaEOVnulldecomp},\n\twhere $\\dot{\\ualpha}_{\\nu} = \\ualpha_{\\nu}[\\Far],$ \n\t$\\dot{\\mathfrak{F}}^{\\nu'} = \\mathfrak{F}^{\\nu'},$ etc. in the equation. We remind the reader\n\tthat this equation is a consequence of performing a Minkowskian null decomposition on the electromagnetic equations \n\t\\eqref{E:ReduceddFis0Summary} - \n\t\\eqref{E:ReduceddMis0Summary}. Here, $\\mathfrak{F}^{\\nu'}$ is defined in \\eqref{E:EMBIFarInhomogeneous}. We begin by noting \n\tthat the first two terms in equation \\eqref{E:dotualphaEOVnulldecomp} can be written as $r^{-1}\\nabla_L (r \\ualpha).$ We then \n\tremove the dangerous $\\frac{1}{4}h_{LL} \\nabla_{\\uL} \\ualpha_{\\nu}$ component from the quadratic term $\\angm_{\\nu \\lambda} \n\t\\mathscr{P}_{(\\Far)}^{\\lambda}(h, \\nabla \\Far) \\eqdef \\angm_{\\nu}^{\\ \\lambda} h^{\\mu \\kappa} \\nabla_{\\mu} \\Far_{\\kappa \n\t\\lambda}$ on the left-hand side of \\eqref{E:dotualphaEOVnulldecomp}, and add it to the $r^{-1}\\nabla_L (r \\ualpha_{\\nu})$ \n\tterm. Using the fact that $\\nabla_{\\Lambda} r = 1 - \\frac{1}{4}h_{LL},$ it follows that\n\tthe resulting sum can be written as $r^{-1}\\nabla_{\\Lambda} (r \\ualpha_{\\nu}) + \\frac{1}{4}r^{-1} h_{LL} \\ualpha_{\\nu}.$ We \n\tthen put the $\\frac{1}{4} r^{-1} h_{LL} \\ualpha_{\\nu}$ term on the right-hand side of \\eqref{E:ODErualpha} as the first \n\tinhomogeneous term; all the remaining terms in \\eqref{E:dotualphaEOVnulldecomp}\n\twill also be placed on the right-hand side of \\eqref{E:ODErualpha}. The left-over terms in \n\t$\\mathscr{P}_{(\\Far)}^{\\nu}(h,\\nabla \\Far)$ (after the dangerous component $\\frac{1}{4}h_{LL} \\nabla_{\\uL} \\ualpha^{\\nu}$ \n\thas been removed) are denoted by $\\widetilde{\\mathscr{P}}_{(\\Far)}^{\\nu}(h,\\nabla \\Far)$ in Lemma \n\t\\ref{L:AlgebraicTensorialEstimates} below. Now by \\eqref{E:XPNoBadComponentFarhNablaFarNullFormEstimate}, with \n\t$X_{\\nu'} \\eqdef \\angm_{\\nu \\nu'}$ (so that $|X|_{\\mathcal{L}} = 0),$ it follows that the left-over terms \n\t$X_{\\nu'} \\widetilde{\\mathscr{P}}_{(\\Far)}^{\\nu'}(h,\\nabla \\Far)$\n\tare bounded by the right-hand side of \\eqref{E:ODErualpha}. The terms $\\angn \\rho$ and $\\angn \\sigma$ appearing on the \n\tleft-hand side of \\eqref{E:dotualphaEOVnulldecomp} (see Remark \\ref{R:FavorableAngular})\n\tcan be bounded by the second term on the right-hand side of \\eqref{E:ODErualpha} via Corollary \n\t\\ref{C:rWeightedAngularDerivativesinTermsofLieDerivatives}. \n\tThe remaining terms in equation \\eqref{E:ODErualpha} that need to be bounded can be expressed as $X_{\\nu'} \n\t\\widetilde{\\mathscr{Q}}_{(1;\\Far)}^{\\nu'}(h,\\nabla \\Far),$ $X_{\\nu'} N_{\\triangle}^{\\#\\beta \\nu' \\kappa \\lambda} \n\t\\nabla_{\\beta} \\Far_{\\kappa \\lambda},$ and $X_{\\nu'} \\mathfrak{F}^{\\nu'}.$ The first of these can be bounded using \n\t\\eqref{E:XQ1FarhNablaFarNullFormEstimate}, the third with \\eqref{E:LieZIFarNullFormInhomogeneousTermAlgebraicEstimate} (in \n\tthe case $|I| = 0$), while the second (with the help of Lemma \\ref{L:PointwisetandqWeightedNablainTermsofZestiamtes}) \n\tcontributes to the cubic terms on the right-hand side of \\eqref{E:ODErualpha}.\n\t\n\tOur proof of \\eqref{E:LambdaLieZIualphaEquationGoodqWeights} is similar, but more elaborate. To begin, we\n\tdifferentiate the electromagnetic equations with the iterated modified Lie derivative $\\Liemod_{\\mathcal{Z}}^I$ to obtain the \n\tequations of variation \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0} for $\\dot{\\Far}_{\\mu \\nu} \\eqdef \\Lie_{\\mathcal{Z}}^I \n\t\\Far_{\\mu \\nu}$ with inhomogeneous terms $\\dot{\\mathfrak{F}}^{\\nu} = \\mathfrak{F}_{(I)}^{\\nu},$ where \n\t$\\mathfrak{F}_{(I)}^{\\nu}$ is defined in \\eqref{E:LiemodZIdifferentiatedEOVInhomogeneousterms}. We then perform a null \n\tdecomposition of the equations of variation, obtaining equation \\eqref{E:dotualphaEOVnulldecomp} with\n\t$\\dot{\\ualpha}_{\\nu} \\eqdef \\ualpha_{\\nu}[\\Lie_{\\mathcal{Z}}^I \\Far],$ $\\dot{\\mathfrak{F}}^{\\nu'} \\eqdef \\mathfrak{F}_{(I)}^{\\nu'},$\n\tetc. Next, we multiply equation \\eqref{E:dotualphaEOVnulldecomp}\n\tby $\\varpi(q),$ use the identities $\\nabla_{\\Lambda} r = 1 - \\frac{1}{4}h_{LL}$ and $\\nabla_{\\Lambda} q = - \n\t\\frac{1}{2}h_{LL},$\n\tand argue as above, removing the dangerous $\\frac{1}{4} h_{LL} \\nabla_{\\uL} \\ualpha_{\\nu}[\\Lie_{\\mathcal{Z}}^I \\Far]$ \n\tcomponent from the quadratic term $\\angm_{\\nu \\lambda} \\mathscr{P}_{(\\Far)}^{\\lambda}(h, \\nabla\\Lie_{\\mathcal{Z}}^I \\Far) \n\t\\eqdef \\angm_{\\nu}^{\\ \\lambda} h^{\\mu \\kappa} \\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}$\n\tand denoting the remaining terms by $\\angm_{\\nu \\lambda} \\widetilde{\\mathscr{P}}_{(\\Far)}^{\\lambda}(h,\\nabla \n\t\\Lie_{\\mathcal{Z}}^I \\Far),$ to deduce that $\\varpi(q) \\big(\\nabla_L \\ualpha_{\\nu}[\\Lie_{\\mathcal{Z}}^I \\Far] \n\t+ \\frac{1}{4}h_{LL} \\nabla_{\\uL} \\ualpha_{\\nu}[\\Lie_{\\mathcal{Z}}^I \\Far] \n\t+ r^{-1}\\ualpha_{\\nu}[\\Lie_{\\mathcal{Z}}^I \\Far] \\big)$ \n\t$= r^{-1} \\nabla_{\\Lambda} \\big(r \\varpi(q) \\ualpha_{\\nu}[\\Lie_{\\mathcal{Z}}^I \\Far]\\big) + \\frac{1}{4} r^{-1} \n\t\\varpi(q) h_{LL} \\ualpha_{\\nu}[\\Lie_{\\mathcal{Z}}^I \\Far] \n\t- \\frac{1}{2}\\varpi'(q)h_{LL}\\ualpha_{\\nu}[\\Lie_{\\mathcal{Z}}^I \\Far].$ The \n\tfirst of these three terms is the only term on the left-hand side of \\eqref{E:LambdaLieZIualphaEquationGoodqWeights}, while \n\tthe last two are brought over to the right-hand side of \\eqref{E:LambdaLieZIualphaEquationGoodqWeights}.\n\tTo bound $\\angm_{\\nu \\nu'}\\mathfrak{F}_{(I)}^{\\nu'}$ by the right-hand side of \n\t\\eqref{E:LambdaLieZIualphaEquationGoodqWeights}, we again set $X_{\\nu'} \\eqdef \\angm_{\\nu \\nu'}$ (so that \n\t$|X|_{\\mathcal{L}} = 0);$ the desired bound then follows from \\eqref{E:LieZIFarNullFormInhomogeneousTermAlgebraicEstimate}\n\tand \\eqref{E:EnergyInhomogeneousTermAlgebraicEstimate}, together with repeated use of the inequality\n\t$|\\Lie_{\\mathcal{Z}}^I \\Far| \\lesssim |\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far]| \n\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}.$\n\tThe terms $\\varpi(q) \\angn \\rho[\\Lie_{\\mathcal{Z}}^I \\Far]$ and \n\t$\\varpi(q) \\angn \\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]$ appearing on the left-hand side\n\tof \\eqref{E:dotualphaEOVnulldecomp} (see Remark \\ref{R:FavorableAngular}) can be bounded by the \n\tseventh term on the right-hand side of \\eqref{E:LambdaLieZIualphaEquationGoodqWeights} with the help of \n\tCorollary \\ref{C:rWeightedAngularDerivativesinTermsofLieDerivatives}. The remaining three terms on the left-hand side of \n\t\\eqref{E:dotualphaEOVnulldecomp} to be estimated are \n\t$X_{\\nu'} \\widetilde{\\mathscr{P}}_{(\\Far)}^{\\nu'}(h,\\nabla\\Lie_{\\mathcal{Z}}^I \n\t\\Far),$ $X_{\\nu'} \\mathscr{Q}_{(1;\\Far)}^{\\nu'}(h, \\nabla\\Lie_{\\mathcal{Z}}^I \\Far),$ \n\tand $X_{\\nu'} N_{\\triangle}^{\\#\\beta \\nu' \\kappa \\lambda} \\nabla_{\\beta} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}.$\n\tThe first of these can be bounded using \\eqref{E:XPNoBadComponentFarhNablaFarNullFormEstimate},\n\tthe second with \\eqref{E:XQ1FarhNablaFarNullFormEstimate}, while the third (with the help of Lemma \t\n\t\\ref{L:PointwisetandqWeightedNablainTermsofZestiamtes}) contributes to the cubic terms on the right-hand side of \n\t\\eqref{E:ODErualpha}.\n\t\n\tThe proofs of \\eqref{E:alphaODE} - \\eqref{E:sigmaODE}, which are based on an analysis of equations\n\t\\eqref{E:uLdotalphaEOVnulldecomp} - \\eqref{E:uLdotnablasigmaEOVnulldecomp}, are similar, but \n\tmuch simpler. We leave the details to the reader.\n\\end{proof}\n\n\n\nThe next proposition provides pointwise estimates for the challenging commutator term $\\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)} - \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(1)}$\nfrom the right-hand side of \\eqref{E:InhomogeneousTermsNablaZIh1}.\n\n\\begin{proposition} \\label{P:DIPointwise} \\cite[Proposition 5.3]{hLiR2010} \n\\textbf{(Algebraic estimates of $[\\widetilde{\\Square}_g, \\nabla_{\\mathcal{Z}}^I]$)}\n\tLet $g_{\\mu \\nu}$ be a Lorentzian metric and let $h_{\\mu \\nu} \\eqdef g_{\\mu \\nu} - m_{\\mu \\nu}$ and\n\t$H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - m^{\\mu \\nu}.$\n\tLet $\\widetilde{\\Square}_g \\eqdef \\Square_m + H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda},$ and let\n\t$I$ be a $\\mathcal{Z}-$multi-index with $1 \\leq |I|.$ Let $\\hat{\\nabla}_{\\mathcal{Z}}^I$ \n\tdenote the modified Minkowskian covariant derivative operator\n\tdefined in \\eqref{E:Covariantmoddef}. Assume that there is a constant $\\varepsilon$ such that \n\t$|\\nabla_{\\mathcal{Z}}^J h| \\leq \\varepsilon$ holds for all $\\mathcal{Z}-$multi-indices $J$ satisfying \n\t$|J| \\leq \\lfloor |I|\/2 \\rfloor.$ Then if $\\varepsilon$ is sufficiently small, the following pointwise estimate holds:\n\t\n\t\\begin{align} \\label{E:waveoperatorZIcommutaorMBIIEPointwise}\n\t\t|\\widetilde{\\Square}_g & \\nabla_{\\mathcal{Z}}^I h^{(1)}  \n\t\t\t- \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(1)}| \\\\\n\t\t& \\lesssim (1 + t + |q|)^{-1} \\mathop{\\sum_{|K| \\leq |I|}}_{|J| + (|K| - 1)_{+} \\leq |I|} \n\t\t\t|\\nabla_{\\mathcal{Z}}^J H| |\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}| \\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\sum_{|K| \\leq |I|} |\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}|\n\t\t \t\\bigg\\lbrace \\mathop{\\sum_{|J| + (|K| - 1)_{+}}}_{\\ \\leq |I|} |\\nabla_{\\mathcal{Z}}^{J} H|_{\\mathcal{L} \\mathcal{L}} \n\t\t\t\\ + \\\t\\mathop{\\sum_{|J'| + (|K| - 1)_{+}}}_{\\ \\leq |I|-1} \n\t\t\t\t|\\nabla_{\\mathcal{Z}}^{J'} H|_{\\mathcal{L} \\mathcal{T}}\n\t\t\t\\ + \\ \\underbrace{\\mathop{\\sum_{|J''| + (|K| - 1)_{+}}}_{\\ \\leq |I|-2} |\\nabla_{\\mathcal{Z}}^{J''} H|}_{\n\t\t\t\t\\mbox{Absent if $|I| \\leq 1$ or $|K| = |I|$}} \\bigg\\rbrace, \\notag\n\t\\end{align}\n\twhere $(|K|-1)_+ \\eqdef 0$ if $|K| = 0$ and $(|K|-1)_+ \\eqdef |K| - 1$ if $|K| \\geq 1.$\n\n\\end{proposition}\n\n\\hfill $\\qed$\n\n\\begin{corollary} \\label{C:boxZIh1ALinfinity}\n\t\\textbf{(Algebraic estimates of $\\big|\\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)} \\big|$)}\n\tAssume that $h_{\\mu \\nu}^{(1)},$ $(\\mu, \\nu = 0,1,2,3),$ is a solution to the reduced equation \\eqref{E:Reducedh1Summary}.\n\tThen under the assumptions of Proposition \\ref{P:DIPointwise}, we have that\n\t\n\t\\begin{align} \\label{E:boxZIh1ALinfinity}\n\t\t|\\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)}|\t\t \n\t\t& \\lesssim |\\nablamod_{\\mathcal{Z}}^I \\mathfrak{H}| \\ + \\ |\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}| \n\t\t\t\t\\ + \\ (1 + t + |q|)^{-1} \\mathop{\\sum_{|K| \\leq |I|}}_{|J| + (|K| - 1)_{+} \\leq |I|} \n\t\t\t\t\t|\\nabla_{\\mathcal{Z}}^J H| |\\nabla \\nabla_{\\mathcal{Z}}^K h^{(1)}| \\\\\n\t\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\sum_{|K| \\leq |I|}  \n\t\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}| \\Bigg\\lbrace \\mathop{\\sum_{|J| + (|K| - 1)_{+}}}_{\\ \\ \\leq |I|} \n\t\t\t\t|\\nabla_{\\mathcal{Z}}^J H|_{\\mathcal{L} \\mathcal{L}} \n\t\t\t\t\\ + \\ \\underbrace{\\mathop{\\sum_{|J'| + (|K| - 1)_{+}}}_{\\ \\leq |I|-1} |\\nabla_{\\mathcal{Z}}^{J'} H|_{\\mathcal{L} \n\t\t\t\t\\mathcal{T}}}_{\\mbox{Absent if $|I| = 0$}}\n\t\t\t\t\\ + \\ \\underbrace{\\mathop{\\sum_{|J''| + (|K| - 1)_{+}}}_{\\ \\leq |I|-2} |\\nabla_{\\mathcal{Z}}^{J''} H|}_{\\mbox{Absent if \n\t\t\t\t$|I| \\leq 1$ or $|K| = |I|$}} \\Bigg\\rbrace. \\notag\n\t\\end{align}\n\\end{corollary}\n\n\\begin{proof}\n\tSimply use Proposition \\ref{P:InhomogeneousTermsNablaZIh1} to decompose \n\t$\\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)} =  \n\t\t\\nablamod_{\\mathcal{Z}}^I {\\mathfrak{H}} - \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}\n\t\t+ \\Big\\lbrace \\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)} \n\t\t- \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(1)} \\Big\\rbrace$ \n\t\tand apply Proposition \\ref{P:DIPointwise}.\n\\end{proof}\n\n\n\n\\subsection{Useful lemmas} \\label{S:UsefulLemmas}\nIn this section, we provide the lemmas that are used in the proofs of the propositions. We will make repeated use of the following decompositions of the Minkowski metric and its inverse:\n\n\\begin{subequations}\n\\begin{align}\n\tm_{\\mu \\nu} & = - \\frac{1}{2}L_{\\mu} \\underline{L}_{\\nu} - \\frac{1}{2} \\underline{L}_{\\mu} L_{\\nu} + \\angm_{\\mu \\nu},\n\t\t\\label{E:mdecomp} \\\\\n\t(m^{-1})^{\\mu \\nu} & = - \\frac{1}{2}L^{\\mu} \\underline{L}^{\\nu} - \\frac{1}{2} \\underline{L}^{\\mu} L^{\\nu} + \\angm^{\\mu \\nu},\n\t\t\\label{E:minversedecomp}\t\n\\end{align}\n\\end{subequations}\nwhere $\\angm_{\\mu \\nu}$ is the Euclidean first fundamental form of the spheres $S_{r,t}$ defined in \\eqref{E:angmdef}.\n\n\nWe begin with a lemma that shows that the essential algebraic structure of the quadratic terms appearing on the right-hand sides of the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} is preserved under differentiation.\n\n\n\n\n\\begin{lemma} \\label{L:nullformvectorfieldcommutation} \n\\textbf{(Leibniz rules for the quadratic terms)}\nLet $\\mathscr{Q}_0(\\nabla \\psi,\\nabla \\chi),$ $\\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi, \\nabla \\chi)$ denote the standard null forms defined in \\eqref{E:StandardNullForm0} - \\eqref{E:StandardNullFormmunu}, and let \n$\\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h),$\n$\\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\Far, \\Far),$ $\\mathscr{P}(\\nabla_{\\mu} h, \\nabla_{\\nu} h),$\n$\\mathscr{P}_{(\\Far)}^{\\nu}(\\nabla h, \\Far),$ \n$\\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\Far),$ and $\\mathscr{Q}_{(2;\\Far)}^{\\nu}(h, \\nabla \\Far)$\ndenote the quadratic terms defined in \\eqref{E:hAddedUpNullForms}, \\eqref{E:Q2h}, \\eqref{E:PNullform}, \\eqref{E:PFar}, \\eqref{E:Q1Far}, and \\eqref{E:Q2Far} respectively. Let $I$ be a $\\mathcal{Z}-$multi-index. Then there exist constants \n$C_{I_1,I_2; \\mu \\nu}^{\\kappa \\lambda \\gamma \\gamma' \\delta \\delta'},$ $C_{I_1,I_2; \\mu \\nu}^{0;\\gamma \\gamma' \\delta \\delta'},$ $C_{\\mathscr{P};I_1,I_2; \\mu \\nu}^{\\kappa \\lambda},$ $C_{\\mathscr{P};I_1,I_2},$ and $C_{i;I_1,I_2}$ such that\n\n\\begin{subequations}\n\\begin{align} \\label{E:Q1hLeibnizRule}\n\t\\nabla_{\\mathcal{Z}}^I \\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h) =  \n\t\t& \\sum_{|I_1| + |I_2| \\leq |I|} C_{I_1,I_2; \\mu \\nu}^{\\kappa \\lambda \\gamma \\gamma' \\delta \\delta'}\n\t\t\t\\mathscr{Q}_{\\kappa \\lambda}(\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h_{\\gamma \\gamma'}, \n\t\t\t\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h_{\\delta \\delta'}) \\\\\n\t& + \\sum_{|I_1| + |I_2| < |I|} C_{I_1,I_2; \\mu \\nu}^{0;\\gamma \\gamma' \\delta \\delta'}\n\t\t\\mathscr{Q}_0(\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h_{\\gamma \\gamma'}, \n\t\t\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h_{\\delta \\delta'}), \\notag\n\\end{align}\n\n\\begin{align} \\label{E:Q2hLeibnizRule}\n\t\\nabla_{\\mathcal{Z}}^I \\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\Far, \\Far) =\n\t \\sum_{|I_1| + |I_2| \\leq |I|} C_{I_1,I_2} \n\t\t\\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\nabla_{\\mathcal{Z}}^{I_1} \\Far, \\nabla_{\\mathcal{Z}}^{I_2}\\Far),\n\\end{align}\n\n\\begin{align} \\label{E:SpecialPLeibnizRule}\n\t\\nabla_{\\mathcal{Z}}^I \\mathscr{P}(\\nabla_{\\mu} h, \\nabla_{\\nu} h)\n\t= \\sum_{|I_1| + |I_2| \\leq |I|} C_{\\mathscr{P};I_1,I_2; \\mu \\nu}^{\\kappa \\lambda}\n\t\\mathscr{P}(\\nabla_{\\kappa} \\nabla_{\\mathcal{Z}}^{I_1} h, \\nabla_{\\lambda} \\nabla_{\\mathcal{Z}}^{I_2} h),\n\\end{align}\n\n\\begin{align}\n\t\\Lie_{\\mathcal{Z}}^I \\mathscr{P}_{(\\Far)}^{\\nu}(\\nabla h, \\Far)\n\t\t& = \\sum_{|I_1| + |I_2| \\leq |I|} C_{\\mathscr{P};I_1,I_2} \n\t\t\\mathscr{P}_{(\\Far)}^{\\nu}(\\nabla\\Lie_{\\mathcal{Z}}^{I_1} h, \\Lie_{\\mathcal{Z}}^{I_2} \\Far), \n\t\t\\label{E:PFarLeibnizRule} \\\\\n\t\\Lie_{\\mathcal{Z}}^I \\mathscr{Q}_{(i;\\Far)}^{\\nu}(h, \\nabla \\Far) & \n\t\t= \\sum_{|I_1| + |I_2| \\leq |I|} C_{i;I_1,I_2} \n\t\t\\mathscr{Q}_{(i;\\Far)}^{\\nu}(\\Lie_{\\mathcal{Z}}^{I_1} h, \\nabla\\Lie_{\\mathcal{Z}}^{I_2}\\Far), && (i=1,2). \n\t\t\\label{E:Q1and2FarLeibnizRule} \n\\end{align}\n\\end{subequations}\n\n\\end{lemma}\n\n\\begin{proof}\n\tBy pure calculation, if $Z \\in \\mathcal{Z},$ then the following identity holds for the standard null form \n\t$\\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi, \\nabla \\chi):$\n\t\n\t\\begin{align} \\label{E:ClassicNullFormZDifferentiatedExpansion}\n\t\t\\nabla_Z \\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi, \\nabla \\chi) \n\t\t\t& = \\mathscr{Q}_{\\mu \\nu}(\\nabla\\nabla_Z \\psi, \\nabla \\chi) \n\t\t\t\t+ \\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi, \\nabla\\nabla_Z \\chi)\n\t\t\t\t- ^{(Z)} c_{\\mu}^{\\ \\kappa} \\mathscr{Q}_{\\kappa \\nu}(\\nabla \\psi, \\nabla \\chi)\n\t\t\t\t- ^{(Z)} c_{\\nu}^{\\ \\kappa} \\mathscr{Q}_{\\mu \\kappa}(\\nabla \\psi, \\nabla \\chi),\n\t\\end{align}\n\twhere $^{(Z)} c_{\\mu \\nu}$ is the covariantly constant tensorfield defined in \\eqref{E:CovariantDerivativesofZareConstant}. A \n\tsimilar identity holds for the standard null form $\\mathscr{Q}_0(\\nabla \\psi, \\nabla \\chi).$ \\eqref{E:Q1hLeibnizRule} now \n\tfollows from inductively from these facts and the Leibniz rule, since $\\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h)$ is \n\ta linear combination of standard null forms. \\eqref{E:SpecialPLeibnizRule} follows similarly.\n\t\\eqref{E:Q2hLeibnizRule} follows trivially from definition \\eqref{E:Q2h} and the Leibniz rule. \\eqref{E:PFarLeibnizRule} and \n\t\\eqref{E:Q1and2FarLeibnizRule} follow from \\eqref{E:LieZonmupper}, Lemma \\ref{L:Liecommuteswithcoordinatederivatives}, and \n\tthe Leibniz rule. \n\\end{proof}\n\n\n\nThe next lemma concerns the null structure of the standard null forms.\n\n\\begin{lemma} \\label{L:starndardnullforms} \\textbf{(Null form estimates of the standard null forms)} \n\tLet $\\mathscr{Q}_0(\\nabla \\psi,\\nabla \\chi) \\eqdef (m^{-1})^{\\kappa \\lambda} (\\nabla_{\\kappa} \\psi)(\\nabla_{\\lambda} \\chi),$ \n\t$\\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi, \\nabla \\chi) \\eqdef (\\nabla_{\\mu} \\psi)(\\nabla_{\\nu} \\chi) \n\t\t- (\\nabla_{\\nu} \\psi)(\\nabla_{\\mu} \\chi)$ denote the standard null forms defined in  \n\t\t\\eqref{E:StandardNullForm0} - \\eqref{E:StandardNullFormmunu}. Then\n\t\n\t\\begin{align} \\label{E:standardnullforms}\n\t\t|\\mathscr{Q}_0 (\\nabla \\psi, \\nabla \\chi)| + |\\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi,\\nabla \\chi)| \n\t\t\t\\lesssim |\\conenabla \\psi||\\nabla \\chi| + |\\conenabla\\chi||\\nabla \\psi|.\n\t\\end{align}\t\n\t\n\t\\begin{proof}\n\t\tThe estimate \\eqref{E:standardnullforms} for $\\mathscr{Q}_0$ easily follows from using\n\t\t\\eqref{E:mdecomp} to decompose $(m^{-1})^{\\kappa \\lambda}.$ To obtain the estimates for \n\t\t$\\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi,\\nabla \\chi),$\n\t\tfirst consider the $\\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi,\\nabla \\chi)$ to be components of a 2-covariant tensor \n\t\t$\\mathscr{Q}(\\nabla \\psi,\\nabla \\chi).$ Inequality \\eqref{E:standardnullforms} is equivalent to the following inequality:\n\t\t\n\t\t\\begin{align} \\label{E:standardnullforminequality}\n\t\t\t|\\mathscr{Q}(\\nabla \\psi, \\nabla \\chi)|_{\\mathcal{N}\\mathcal{N}}\n\t\t\t\t\\lesssim |\\conenabla \\psi||\\nabla \\chi| + |\\conenabla\\chi||\\nabla\\psi|.\n\t\t\\end{align}\n\t\tContracting $\\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi,\\nabla \\chi)$ against frame vectors $N^{\\mu},N^{\\nu} \\in \n\t\t\\mathcal{N},$ we see that the only component on the left-hand side of \\eqref{E:standardnullforminequality} that could pose \n\t\tany difficulty is $\\uL^{\\mu} \\uL^{\\nu} \\mathscr{Q}_{\\mu \\nu}(\\nabla \\psi,\\nabla \\chi).$ But the \n\t\tanti-symmetry the $\\mathscr{Q}_{\\mu \\nu}(\\cdot,\\cdot)$ implies that this component is $0.$\n\t\\end{proof}\n\\end{lemma}\n\nThe next lemma addresses the null structure of some of the terms appearing in the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}.\n\n\n\\begin{lemma} \\label{L:AlgebraicTensorialEstimates} \\textbf{(Null form estimates for the reduced equations)}\nLet $\\mathscr{P}(\\nabla_{\\mu} \\Pi, \\nabla_{\\nu} \\Theta),$ $\\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h),$ $\\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\Far, \\Gar),$ $\\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far),$ $\\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\Far),$ and $\\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla h, \\Far)$ be the quadratic forms defined in\nSection \\ref{SS:ReducedEquations}, and define the quadratic form $\\widetilde{\\mathscr{P}}_{(\\Far)}^{\\nu}(h, \\nabla \\Far)$ by\nremoving the $\\nabla_{\\uL}\\ualpha^{\\nu}[\\Far]-$containing component of $\\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far):$\n\n\\begin{align}\n\t\\widetilde{\\mathscr{P}}_{(\\Far)}^{\\nu}(h, \\nabla \\Far) \n\t& \\eqdef \\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far) - \\frac{1}{4}h_{LL} \\angm^{\\nu \\nu'} \\nabla_{\\uL} \\Far_{\\uL \\nu'} \\\\\n\t& = \\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far) + \\frac{1}{4}h_{LL} \\nabla_{\\uL} \\ualpha^{\\nu}[\\Far]. \\notag\n\\end{align}\nLet $X_{\\nu}$ be any covector, let $\\Pi_{\\mu \\nu},$ $\\Theta_{\\mu \\nu}$ be any symmetric or anti-symmetric type $\\binom{0}{2}$ tensorfields, and let $\\Far_{\\mu \\nu},$ $\\Gar_{\\mu \\nu}$ be any two-forms. Then the following pointwise inequalities hold:\n\n\\begin{subequations}\n\\begin{align}\n\t\t|\\mathscr{P}(\\nabla_{\\mu} \\Pi, \\nabla_{\\nu} \\Theta)|\n\t\t\t& \\lesssim |\\nabla \\Pi|_{\\mathcal{T} \\mathcal{N}} |\\nabla \\Theta|_{\\mathcal{T} \\mathcal{N}}\n\t\t\t\\ + \\ |\\nabla \\Pi|_{\\mathcal{L}\\mathcal{L}} |\\nabla \\Theta| \n\t\t\t\\ + \\ |\\Pi| |\\nabla \\Theta|_{\\mathcal{L}\\mathcal{L}}, \\qquad (\\mu, \\nu = 0,1,2,3),\n\t\t\t\\label{E:PSpecialNullStructure} \\\\\n\t\t\t\\sum_{T \\in \\mathcal{T}, N \\in \\mathcal{N}}|T^{\\mu} N^{\\nu}\\mathscr{P}(\\nabla_{\\mu} \\Pi, \\nabla_{\\nu} \\Theta)|\n\t\t\t& \\lesssim |\\conenabla \\Pi||\\nabla \\Theta|, \\label{E:PTUSpecialNullStructure} \\\\\n\t\t|\\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla \\Pi, \\nabla \\Theta)| & \\lesssim \n\t\t\t|\\conenabla \\Pi||\\nabla \\Theta| \\ + \\ |\\nabla \\Pi||\\conenabla \\Theta|, \\qquad (\\mu, \\nu = 0,1,2,3), \n\t\t\t\\label{E:Q1hNullFormEstimate} \\\\\n\t\t\\sum_{T \\in \\mathcal{T}, N \\in \\mathcal{N}} \n\t\t\t|T^{\\mu} N^{\\nu}\\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\Far, \\Gar)| \n\t\t\t& \\lesssim \\big(|\\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Far|_{\\mathcal{T} \\mathcal{T}}\\big)|\\Gar|\n\t\t\t\\ + \\ |\\Far| \\big(|\\Gar|_{\\mathcal{L} \\mathcal{N}} + |\\Gar|_{\\mathcal{T} \\mathcal{T}} \\big),\n\t\t\t\\label{E:Q2TUhNullFormEstimate} \\\\\n\t\t|\\mathscr{Q}_{\\mu \\nu}^{(2;h)}(\\Far, \\Gar)| & \\lesssim |\\Far||\\Gar|, \\qquad (\\mu, \\nu = 0,1,2,3), \n\t\t\t\\label{E:Q2hNullFormEstimate} \\\\\n\t\t|X_{\\nu} \\mathscr{P}_{(\\Far)}^{\\nu}(h, \\nabla \\Far)| \n\t\t& \\lesssim |X||h||\\conenabla \\Far| \n\t\t\t\\ + \\ |X||h|\\big(|\\nabla \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\nabla \\Far|_{\\mathcal{T} \\mathcal{T}}\\big)\n\t\t\t\\ + \\ |X||h|_{\\mathcal{L}\\mathcal{L}}|\\nabla \\Far|\n\t\t\t\\ + \\ |X|_{\\mathcal{L}}|h||\\nabla \\Far| \\label{E:XPFarhNablaFarNullFormEstimate} \\\\\n\t\t& \\lesssim (1 + t + |q|)^{-1} \\sum_{|I| \\leq 1} |X||h||\\Lie_{\\mathcal{Z}}^I \\Far| \n\t\t\t\\ + \\ (1 + |q|)^{-1} \\sum_{|I| \\leq 1} |X||h|\n\t\t\t\\big(|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}}\\big)\n\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\sum_{|I| \\leq 1} |X||h|_{\\mathcal{L}\\mathcal{L}}|\\Lie_{\\mathcal{Z}}^I \\Far|\n\t\t\t\\ + \\ (1 + |q|)^{-1} \\sum_{|I| \\leq 1} |X|_{\\mathcal{L}}|h||\\Lie_{\\mathcal{Z}}^I \\Far|,\n\t\t\t\\notag \\\\\n\t\t|X_{\\nu} \\widetilde{\\mathscr{P}}_{(\\Far)}^{\\nu}(h, \\nabla \\Far)| \n\t\t& \\lesssim |X||h||\\conenabla \\Far| \n\t\t\t\\ + \\ |X||h|\\big(|\\nabla \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\nabla \\Far|_{\\mathcal{T} \\mathcal{T}}\\big)\n\t\t\t\\ + \\ |X|_{\\mathcal{L}}|h||\\nabla \\Far| \\label{E:XPNoBadComponentFarhNablaFarNullFormEstimate} \\\\\n\t\t& \\lesssim (1 + t + |q|)^{-1} \\sum_{|I| \\leq 1} |X||h||\\Lie_{\\mathcal{Z}}^I \\Far| \n\t\t\t\\ + \\ (1 + |q|)^{-1} \\sum_{|I| \\leq 1} |X||h|\n\t\t\t\\big(|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}}\\big)\n\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\sum_{|I| \\leq 1} |X|_{\\mathcal{L}}|h||\\Lie_{\\mathcal{Z}}^I \\Far|,\n\t\t\t\\notag \\\\\n\t\t|X_{\\nu} \\mathscr{Q}_{(1;\\Far)}^{\\nu}(h, \\nabla \\Far)| \n\t\t& \\lesssim |X||h||\\conenabla \\Far| \n\t\t\t\\ + \\ |X||h|\\big(|\\nabla \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\nabla \\Far|_{\\mathcal{T} \\mathcal{T}}\\big) \n\t\t\t\\label{E:XQ1FarhNablaFarNullFormEstimate} \\\\\n\t\t& \\lesssim (1 + t + |q|)^{-1} \\sum_{|I| \\leq 1} |X||h||\\Lie_{\\mathcal{Z}}^I \\Far| \n\t\t\t\\ + \\ (1 + |q|)^{-1} \\sum_{|I| \\leq 1} |X||h|\n\t\t\t\\big(|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}}\\big),\n\t\t\t\\notag \\\\\n\t\t|X_{\\nu}\\mathscr{Q}_{(2;\\Far)}^{\\nu}(\\nabla h, \\Far)| \n\t\t\t& \\lesssim |X||\\conenabla h||\\Far| \\ + \\ |X||\\nabla h||\\Far|_{\\mathcal{L} \\mathcal{N}} \n\t\t\t\t\\label{E:Q2FarNullFormEstimate} \\\\\n\t\t\t& \\lesssim (1 + t + |q|)^{-1} \\sum_{|I| \\leq 1} |X||\\nabla_{\\mathcal{Z}}^I h||\\Far| \n\t\t\t\\ + \\ (1 + |q|)^{-1} \\sum_{|I| \\leq 1} |X||\\nabla_{\\mathcal{Z}}^I h|\n\t\t\t\\big(|\\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Far|_{\\mathcal{T} \\mathcal{T}}\\big).\n\t\t\t\\notag \n\t\t\\end{align}\n\\end{subequations}\n\n\\end{lemma}\n\n\n\\begin{proof}\n\tInequality \\eqref{E:Q1hNullFormEstimate} follows directly from\n\tLemma \\ref{L:starndardnullforms}, since $\\mathscr{Q}_{\\mu \\nu}^{(1;h)}(\\nabla h, \\nabla h)$\n\tis a linear combination of standard null forms. Inequality \\eqref{E:Q2hNullFormEstimate} is trivial, \n\twhile \\eqref{E:PSpecialNullStructure}, \\eqref{E:PTUSpecialNullStructure}, and the first inequalities in \n\t\\eqref{E:Q2TUhNullFormEstimate} - \\eqref{E:Q2FarNullFormEstimate} are easy to check using \\eqref{E:mdecomp} - \n\t\\eqref{E:minversedecomp}. The second inequalities in \\eqref{E:Q2TUhNullFormEstimate} - \\eqref{E:Q2FarNullFormEstimate} then \n\tfollow from the first ones, Lemma \\ref{L:PointwisetandqWeightedNablainTermsofZestiamtes}, and Proposition \n\t\\ref{P:LievsCovariantLContractionRelation}.\n\\end{proof}\n\nThe next lemma concerns the null structure of the cubic terms on the right-hand side of \\eqref{E:Firstweightedenergyscalar}.\n\n\\begin{lemma} \\cite[Lemma 4.2]{hLiR2010} \\label{L:InhomogeneousWaveEquationEnergyCurrentAlgebraicNullFormEstimates}\n\t\\textbf{(Null form estimates for quasilinear wave equations)}\n\tLet $\\Pi$ be a type $\\binom{0}{2}$ tensorfield, and let $\\phi$ be a scalar function. Then the following inequalities hold:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t|\\Pi^{\\kappa \\lambda}(\\nabla_{\\kappa} \\phi)(\\nabla_{\\lambda} \\phi)| & \\lesssim\n\t\t\t|\\Pi|_{\\mathcal{L} \\mathcal{L}}|\\nabla \\phi|^2 \\ + \\ |\\Pi||\\conenabla \\phi| |\\nabla \\phi|, \\\\\n\t\t|L_{\\kappa}\\Pi^{\\kappa \\lambda}\\nabla_{\\lambda} \\phi| & \\lesssim\n\t\t\t|\\Pi|_{\\mathcal{L} \\mathcal{L}}|\\nabla \\phi| \\ + \\ |\\Pi||\\conenabla \\phi|, \\\\\n\t\t|(\\nabla_{\\kappa} \\Pi^{\\kappa \\lambda})\\nabla_{\\lambda} \\phi| & \\lesssim\n\t\t\t|\\nabla \\Pi|_{\\mathcal{L} \\mathcal{L}} |\\nabla \\phi| \\ + \\ |\\conenabla \\Pi||\\nabla \\phi|\n\t\t\t\\ + \\ |\\nabla \\Pi| |\\conenabla \\phi|, \\\\\n\t\t|\\Pi^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda} \\phi| & \\lesssim\n\t\t\t|\\Pi|_{\\mathcal{L} \\mathcal{L}}|\\nabla\\nabla \\phi| \\ + \\ |\\conenabla \\nabla \\phi|.\n\t\\end{align}\n\t\\end{subequations}\n\t\n\\end{lemma}\n\n\\hfill $\\qed$\n\nThe following lemma addresses the null structure of the cubic terms on the right-hand \nside of \\eqref{E:currentdivergence}.\n\n\\begin{lemma} \\label{L:DivergenceofJAlgebraicNullFormEstimates}\n\t\\textbf{(Null form estimates for the electromagnetic equations of variation)}\n\tLet $h_{\\mu \\nu}$ be a type $\\binom{0}{2}$ tensorfield, and let $\\Far_{\\mu \\nu}$ be two-form.\n\tThen the following inequalities hold:\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t|(\\nabla_{\\mu} h^{\\mu \\kappa}) \\Far_{\\kappa \\zeta} \\Far_{0}^{\\ \\zeta}|\t\t\n\t\t& \\lesssim |\\nabla h|_{\\mathcal{L} \\mathcal{L}}|\\Far|^2 \\ + \\ |\\conenabla h||\\Far|^2\n\t\t\t\\ + \\ |\\nabla h||\\Far| \\big(|\\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Far|_{\\mathcal{T} \\mathcal{T}}\\big), \n\t\t\t\t\\label{E:DivergenceofJFirstNablahdotFarsquaredAlgebraicNullFormEstimate} \\\\\n\t\t|(\\nabla_{\\mu} h^{\\kappa \\lambda}) \\Far_{\\ \\kappa}^{\\mu} \\Far_{0 \\lambda}|\n\t\t& \\lesssim |\\conenabla h||\\Far|^2 \n\t\t\t\\ + \\ |\\nabla h||\\Far|\n\t\t\t\\big(|\\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Far|_{\\mathcal{T} \\mathcal{T}}\\big),\t\\\\\n\t\t|(\\nabla_{t} h^{\\kappa \\lambda}) \\Far_{\\kappa \\eta}\\Far_{\\lambda}^{\\ \\eta}|\n\t\t& \\lesssim |\\nabla h|_{\\mathcal{L} \\mathcal{L}}|\\Far|^2\n\t\t\t\\ + \\ |\\nabla h||\\Far|\n\t\t\t\\big(|\\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Far|_{\\mathcal{T} \\mathcal{T}}\\big), \\\\\n\t\t|L_{\\mu}h^{\\mu \\kappa} \\Far_{\\kappa \\zeta} \\Far_{0}^{\\ \\zeta}|\n\t\t& \\lesssim |h|_{\\mathcal{L} \\mathcal{L}}|\\Far|^2\n\t\t\t\\ + \\ |h||\\Far|\\big(|\\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Far|_{\\mathcal{T} \\mathcal{T}}\\big), \\\\\n\t\t\t|L_{\\mu} h^{\\kappa \\lambda} \\Far_{\\ \\kappa}^{\\mu} \\Far_{0 \\lambda}|\n\t\t& \\lesssim |h||\\Far||\\Far|_{\\mathcal{L} \\mathcal{N}},  \\\\\n\t\t\t|h^{\\kappa \\lambda} \\Far_{\\kappa \\eta} \\Far_{\\lambda}^{\\ \\eta}| \n\t\t& \\lesssim |h|_{\\mathcal{L} \\mathcal{L}}|\\Far|^2\n\t\t\t\\ + \\ |h||\\Far|\\big(|\\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Far|_{\\mathcal{T} \\mathcal{T}}\\big).\n\t\t\t\\label{E:DivergenceofJThirdhdotFarsquaredAlgebraicNullFormEstimate}\n\t\\end{align}\n\t\\end{subequations}\n\n\\end{lemma}\n\n\\begin{proof}\n\tInequalities \\eqref{E:DivergenceofJFirstNablahdotFarsquaredAlgebraicNullFormEstimate} -\n\t\\eqref{E:DivergenceofJThirdhdotFarsquaredAlgebraicNullFormEstimate} are easy to check\n\tusing \\eqref{E:mdecomp}.\n\\end{proof}\n\n\n\n\\section{Weighted Energy Estimates for the Electromagnetic Equations of Variation and for Systems of Nonlinear Wave Equations in a Curved Spacetime} \\label{S:WeightedEnergy}\n\nIn this section, we prove weighted energy estimates for the electromagnetic equations of variation\n\n\n\\begin{subequations} \n\t\\begin{align}\n\t\t\\nabla_{\\lambda} \\dot{\\Far}_{\\mu \\nu} + \\nabla_{\\mu} \\dot{\\Far}_{\\nu \\lambda} + \\nabla_{\\nu} \\dot{\\Far}_{\\lambda \\mu}\n\t\t\t& = \\dot{\\mathfrak{\\Far}}_{\\lambda \\mu \\nu},&& (\\lambda, \\mu, \\nu = 0,1,2,3), \\label{E:EOVdFis0again} \\\\\n\t\t\tN^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\dot{\\Far}_{\\kappa \\lambda} & = \\dot{\\mathfrak{F}}^{\\nu},\n\t\t\t&& (\\nu = 0,1,2,3). \\label{E:EOVdMis0again} \n\t\\end{align}\n\\end{subequations}\nOur estimates complement the weighted energy estimates proved in \\cite{hLiR2010} for the inhomogeneous wave equation \n\n\\begin{align}\n\t\\widetilde{\\Square}_g \\phi = \\mathfrak{I},\n\\end{align}\nand for tensorial systems of inhomogeneous wave equations with principal part $\\widetilde{\\Square}_g:$\n\n\\begin{align}\n\t\\widetilde{\\Square}_g \\phi_{\\mu \\nu} = \\mathfrak{I}_{\\mu \\nu}, && (\\mu, \\nu = 0,1,2,3).\n\\end{align}\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{The energy estimate weight function \\texorpdfstring{$w(q)$}{}}\n\nAs in \\cite{hLiR2010}, our energy estimates will involve the weight function $w(q)$ defined by \n\n\n\\begin{align} \\label{E:weight}\n\tw = w(q) = \\left \\lbrace\n\t\t\\begin{array}{lr}\n    \t1 \\ + \\ (1 + |q|)^{1 + 2 \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      1 \\ + \\ (1 + |q|)^{-2 \\upmu}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \\right.\n\\end{align}\nwhere the constants $\\upgamma$ and $\\upmu$ are subject to the restrictions stated in Section \\ref{SS:FixedConstants}.\n\nObserve that the following inequalities follow from the definition \\eqref{E:weight}:  \n  \n\\begin{align} \\label{E:weightinequality}\n  \tw' \\leq 4(1 + |q|)^{-1} w \\leq 16 \\upgamma^{-1} (1 + q_-)^{2 \\upmu} w',\n\\end{align}\nwhere $q_- = 0$ if $q \\geq 0$ and $q_- = |q|$ if $q < 0.$\n\t\n\\subsection{Weighted energy estimates}\t\n\nWe begin by deriving weighted energy estimates for the electromagnetic equations of variation.\n\n\\begin{lemma} \\label{L:weightedenergyFar} \\textbf{(Weighted energy estimates for $\\dot{\\Far}$)}\n\tAssume that $\\dot{\\Far}_{\\mu \\nu} $ is a solution to the equations of variation \\eqref{E:EOVdFis0} - \\eqref{E:EOVdMis0}\n\tcorresponding to the background $(h_{\\mu \\nu}, \\Far_{\\mu \\nu}),$ \n\twhere $h_{\\mu \\nu} \\eqdef g_{\\mu \\nu} - m_{\\mu \\nu}.$ Let \n\t$\\dot{\\alpha} \\eqdef \\alpha[\\dot{\\Far}],$ $\\dot{\\rho} \\eqdef \\rho[\\dot{\\Far}],$ and \n\t$\\dot{\\sigma} \\eqdef \\sigma[\\dot{\\Far}]$ denote the ``favorable'' null components of $\\dot{\\Far}$ as\n\tdefined in Definition \\ref{D:null}. Assume that $|h| + |\\Far| \\leq \\varepsilon.$ Then if $\\varepsilon$ is sufficiently small, \n\tand $t_1 \\leq t_2,$ the following integral inequality holds:\n\t\n\t\\begin{align}  \\label{E:FirstweightedenergyFar}\n\t\t\\int_{\\Sigma_{t_2}} |\\dot{\\Far}|^2 w(q) \\,d^3x \n\t\t\\ + \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} & \\big(\\dot{\\alpha}^2 + \\dot{\\rho}^2 + \\dot{\\sigma}^2 \\big) \n\t\t\tw'(q) \\,d^3x \\, d \\tau \\\\\n\t\t& \\lesssim \\int_{\\Sigma_{t_1}} |\\dot{\\Far}|^2 w(q) \\,d^3x \\notag  \\\\ \t\t\t\t\t\t\n\t\t& \\ \\ + \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \\Big| \\lbrace \\dot{\\Far}_{0 \\eta} \\dot{\\mathfrak{F}}^{\\eta} \n\t\t\t- (\\nabla_{\\mu} h^{\\mu \\kappa}) \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta}\n\t\t\t- (\\nabla_{\\mu} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda}\n\t\t\t+ \\frac{1}{2} (\\nabla_{t} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\eta}\\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big| w(q) \\,d^3x \n\t\t\t\\, d \\tau \\notag \\\\\n\t& \\ \\ + \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \n\t\t\t\\Big|L_{\\mu}h^{\\mu \\kappa} \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta} \n\t\t\t+ L_{\\mu} h^{\\kappa \\lambda} \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda} \n\t\t\t+ \\frac{1}{2} h^{\\kappa \\lambda} \\dot{\\Far}_{\\kappa \\eta} \\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big| w'(q) \\,d^3x \\, d \\tau \\\t\t\n\t\t\t\\notag  \\\\\n\t& \\ \\ + \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \n\t\t\\Big| (\\nabla_{\\mu}N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\lambda} \n\t\t\t\\dot{\\Far}_{0 \\zeta} - \\frac{1}{4} (\\nabla_{t} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda}) \\dot{\\Far}_{\\zeta \\eta} \n\t\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big| \tw(q) \\,d^3x \\, d \\tau \\notag \\\\\n\t& \\ \\ + \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \n\t\t\\Big|L_{\\mu} N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda} \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{0 \\zeta}\n\t\t+ \\frac{1}{4} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \n\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big| w'(q) \\,d^3x \\, d \\tau. \\notag\n\t\\end{align}\n\\end{lemma}\n\n\n\n\n\\begin{proof}\n\t\n\tIt follows from \\eqref{E:dotJ0estimate} that if $\\varepsilon$ is sufficiently small, we have that\n\t\n\t\\begin{align} \\label{E:J0positivity}\n\t\t\\frac{1}{4} |\\dot{\\Far}|^2 w(q) \\leq \\dot{J}_{(h,\\Far)}^0 \\leq |\\dot{\\Far}|^2 w(q).\n\t\\end{align}\n\tUsing \\eqref{E:currentdivergence} and the divergence theorem, it follows that\n\t\n\n\\begin{align}\n\t\\int_{\\Sigma_{t_2}} & \\dot{J}_{(h,\\Far)}^0 \\, d^3 x \\ + \\ \\frac{1}{2} \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \n\t\tw'(q) (\\dot{\\alpha}^2 + \\dot{\\rho}^2 + \\dot{\\sigma}^2) \\, d^3 x \\, d \\tau \\\\\n\t& = \\int_{\\Sigma_{t_1}} \\dot{J}_{(h,\\Far)}^0 \\, d^3 x \\notag \\\\\n\t& \\ \\ - \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} w(q) \\Big \\lbrace \\dot{\\Far}_{0 \\eta} \\dot{\\mathfrak{F}}^{\\eta} \n\t\t\t- (\\nabla_{\\mu} h^{\\mu \\kappa}) \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta}\n\t\t\t- (\\nabla_{\\mu} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda}\n\t\t\t+ \\frac{1}{2} (\\nabla_{t} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\eta}\\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big \\rbrace \n\t\t\\, d^3 x \\, d \\tau \\notag \\\\\n\t& \\ \\ - \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \n\t\tw'(q) \\Big\\lbrace - L_{\\mu}h^{\\mu \\kappa} \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta} \n\t\t\t- L_{\\mu} h^{\\kappa \\lambda} \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda} \n\t\t\t- \\frac{1}{2} h^{\\kappa \\lambda} \\dot{\\Far}_{\\kappa \\eta} \\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big\\rbrace \\, d^3 x \\, d \\tau \n\t\t\t\\notag \\\\\n\t& \\ \\ - \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \n\t\tw(q) \\Big\\lbrace (\\nabla_{\\mu}N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\lambda} \n\t\t\t\\dot{\\Far}_{0 \\zeta} - \\frac{1}{4} (\\nabla_{t} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda}) \\dot{\\Far}_{\\zeta \\eta} \n\t\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big\\rbrace \\, d^3 x \\, d \\tau \\notag \\\\\n\t& \\ \\ - \\ \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \n\t\tw'(q) \\Big\\lbrace L_{\\mu} N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda} \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{0 \\zeta}\n\t\t+ \\frac{1}{4} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \n\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big\\rbrace \\, d^3 x \\, d \\tau, \\notag\n\\end{align}\t\nwhich, with the help of \\eqref{E:J0positivity}, implies \\eqref{E:FirstweightedenergyFar}.\n\n\\end{proof}\n\nWe now recall the analogous lemma proved in \\cite{hLiR2010} for solutions to the inhomogeneous wave equation in curved spacetime.\n\n\n\n\n\n\\begin{lemma} \\cite[Lemma 6.1]{hLiR2010} \\label{L:weightedenergy} \\textbf{(Weighted energy estimates for a scalar wave \n\tequation)}\n\tAssume that the scalar-valued function $\\phi$ is a solution to the equation $\\widetilde{\\Square}_g \\phi = \\mathfrak{I},$ and \n\tlet $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu}.$ Assume that\n\tthe metric $g_{\\mu \\nu}$ is such that $|H| \\leq \\frac{1}{2}.$ Then\n\t\n\t\\begin{align}  \\label{E:Firstweightedenergyscalar} \n\t\t\\int_{\\Sigma_{t_2}} & |\\nabla \\phi|^2 w(q) \\,d^3x \\ + \\ 2 \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} |\\conenabla \\phi|^2 \n\t\t\tw'(q) \\,d^3x \\, d \\tau \\\\\n\t\t& \\leq 4 \\int_{\\Sigma_{t_1}} |\\nabla \\phi|^2 w(q) \\,d^3x \n\t\t\t\\ + \\ 4 \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \\Big| \\mathfrak{I}_{\\kappa} \\nabla_t \\phi^{\\kappa} \n\t\t\t+ (\\nabla_{\\nu} H^{\\nu \\lambda})(\\nabla_{\\lambda} \\phi)(\\nabla_t \\phi) \n\t\t\t- \\frac{1}{2} (\\nabla_t H^{\\lambda \\kappa})(\\nabla_{\\lambda} \\phi)(\\nabla_{\\kappa} \\phi)\n\t\t\t\\Big| w(q) \\,d^3x \\, d \\tau \\notag \\\\\n\t& \\ \\ + \\ 4 \\int_{t_1}^{t_2} \\int_{\\Sigma_{\\tau}} \\Big| \\underbrace{(\\omega_j H^{j \\lambda} \n\t\t-  H^{0 \\lambda})}_{L_{\\kappa} H^{\\kappa \\lambda}} (\\nabla_t \\phi)(\\nabla_{\\lambda} \\phi)  \n\t\t+ \\frac{1}{2} H^{\\lambda \\kappa} (\\nabla_{\\lambda} \\phi)(\\nabla_{\\kappa} \\phi) \\Big| w'(q) \\,d^3x \\, d \\tau. \\notag \n\t\\end{align}\n\\end{lemma}\n\n\\hfill $\\qed$\n\nWe now extend the results of the previous lemmas by estimating (under assumptions that are compatible with our global stability theorem) some of the cubic terms on the right-hand sides of \\eqref{E:FirstweightedenergyFar} and \\eqref{E:Firstweightedenergyscalar}. \n\n\\begin{proposition}  \\cite[extension of Proposition 6.2]{hLiR2010} \\label{P:weightedenergy}\n\t\\textbf{(Weighted energy estimates for the reduced equations)}\n\tLet $\\phi$ be a solution to $\\widetilde{\\Square}_g \\phi = \\mathfrak{I}$ for the metric $g_{\\mu \\nu},$ and \n\tlet $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu}.$ Let $\\upgamma$ and $\\upmu$ be positive constants\n\tsatisfying the restrictions described in Section \\ref{SS:FixedConstants}. Assume that the following pointwise estimates\n\thold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t(1 + |q|)^{-1}|H|_{LL} \\ + \\ |\\nabla H|_{LL} \\ + \\ |\\conenabla H| & \\leq C \\varepsilon(1 + t + |q|)^{-1}, \\\\\n\t\t(1 + |q|)^{-1}|H| \\ + \\ |\\nabla H| & \\leq C \\varepsilon(1 + t + |q|)^{-1\/2} (1 + |q|)^{-1\/2} (1 + q_-)^{- \\upmu},\n\t\\end{align}\n\t\\end{subequations}\n\twhere $q_{-} = 0$ if $q \\geq 0$ and $q_{-} = |q|$ if $q < 0.$ Then there exists a constant $C_1 > 0$ such that if\n\t$0 < \\varepsilon \\leq \\frac{\\upgamma}{C_1},$ then the following integral inequality holds for $t \\in [0,T):$\n\t\n\t\\begin{align} \\label{E:Secondweightedenergyscalar}\n\t\t\\int_{\\Sigma_{t}} |\\nabla \\phi|^2 w(q) \\,d^3x \n\t\t\t& \\ + \\ \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} |\\conenabla \\phi|^2 w'(q) \\,d^3x \\, d \\tau \\\\\n\t\t& \\leq 8 \\int_{\\Sigma_{0}} |\\nabla \\phi|^2 w(q) \\,d^3x \n\t\t\t\\ + \\ 16 \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \\Big(\\frac{C \\varepsilon |\\nabla \\phi|^2}{1 + \\tau} + |\\mathfrak{I}|\n\t\t\t|\\nabla \\phi| \\Big) w(q) \\,d^3x \\, d \\tau. \\notag\n\t\\end{align}\n\t\n\t\n\tFurthermore, let $\\dot{\\Far}_{\\mu \\nu}$ be a solution to the electromagnetic equations of variation \\eqref{E:EOVdFis0} - \n\t\\eqref{E:EOVdMis0} corresponding to the background $(h_{\\mu \\nu}, \\Far_{\\mu \\nu}),$ where $h_{\\mu \\nu} \\eqdef g_{\\mu \\nu} - \n\tm_{\\mu \\nu}.$ Assume that the following pointwise estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t(1 + |q|)^{-1}|h|_{\\mathcal{L} \\mathcal{L}} \n\t\t\\ + \\ |\\nabla h|_{\\mathcal{L} \\mathcal{L}} \\ + \\ |\\conenabla h| \\ + \\ |\\Far| & \\leq C \\varepsilon(1 + t + |q|)^{-1}, \\\\\n\t\t(1 + |q|)^{-1}|h| \\ + \\ |\\nabla h| \\ + \\ |\\nabla \\Far| \n\t\t& \\leq C \\varepsilon(1 + t + |q|)^{-1\/2} (1 + |q|)^{-1\/2} (1 + q_-)^{- \\upmu},\n\t\\end{align}\n\t\\end{subequations}\n\twhere $q_{-} = 0$ if $q \\geq 0$ and $q_{-} = |q|$ if $q < 0.$  Then there exists a constant $C_1 > 0$ such that if\n\t$0 < \\varepsilon \\leq \\frac{\\upgamma}{C_1},$ then the following integral inequality holds for $t \\in [0,T):$\n\t\n\t\\begin{align}  \\label{E:SecondweightedenergyFar}\n\t\\int_{\\Sigma_{t}} & |\\dot{\\Far}|^2 w(q) \\,d^3x \n\t\t\\ + \\ \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \n\t\t\\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}}^2 + |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}}^2 \\big) w'(q)\n\t\t\\,d^3x \\, d \\tau \\\\  \n\t& \\lesssim  \\int_{\\Sigma_{0}} |\\dot{\\Far}|^2 w(q) \\,d^3x \n\t\t\\ + \\ \\varepsilon \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \\frac{|\\dot{\\Far}|^2}{1 + \\tau} w(q) \\,d^3x \\, d \\tau\n\t\t\\ + \\ \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} |\\dot{\\Far}_{0 \\kappa}\\dot{\\mathfrak{F}}^{\\kappa}| w(q) \\,d^3x \\, d \\tau. \\notag \n\\end{align}\n\\end{proposition}\n\n\n\\begin{remark}\n\tProposition \\ref{P:weightedenergy} will not be used until the proof of Theorem \\ref{T:ImprovedDecay}, where it plays\n\ta key role; see Section \\ref{SS:MainArgument}. We also remark that the hypotheses of the proposition are implied by the \n\thypotheses of the theorem; see Section \\ref{SS:FixedConstants} and Remark \\ref{R:ImprovedDecay}.\n\\end{remark}\n\n\n\\begin{proof}\n\tInequality \\eqref{E:Secondweightedenergyscalar} was proved as Proposition 6.2 of \\cite{hLiR2010}.\tTheir proof was\n\tbased on using Lemma \\ref{L:InhomogeneousWaveEquationEnergyCurrentAlgebraicNullFormEstimates} to estimate the inhomogeneous \n\tterms on the right-hand side of \\eqref{E:Firstweightedenergyscalar}. Rather than reproving this inequality, we only give the \n\tproof of \\eqref{E:SecondweightedenergyFar}, which is based on \\eqref{E:FirstweightedenergyFar} and uses related ideas.\n\t\n\tWe commence with the proof of \\eqref{E:SecondweightedenergyFar}, \n\tour goal being to deduce suitable pointwise bounds for some of the terms appearing on the right-hand side of \n\t\\eqref{E:FirstweightedenergyFar}. For the cubic terms, we use Lemma \\ref{L:DivergenceofJAlgebraicNullFormEstimates}, the \n\thypotheses of the proposition, and the inequality $|ab| \\lesssim a^2 + b^2$ to conclude that\n\t\n\t\\begin{align} \\label{E:firsttensorialdotFLinfinity}\n\t\t\\Big|(\\nabla_{\\mu} h^{\\mu \\kappa}) \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta}\n\t\t\t- (\\nabla_{\\mu} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda}\n\t\t\t& + \\frac{1}{2} (\\nabla_{t} h^{\\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\eta}\\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big| \\\\\n\t\t& \\lesssim \\big(|\\nabla h|_{\\mathcal{L} \\mathcal{L}} + |\\conenabla h| \\big) |\\dot{\\Far}|^2\n\t\t\t\\ + \\ |\\nabla h| |\\dot{\\Far}| \\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}} + |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}} \\big) \n\t\t\t\\notag \\\\\n\t\t& \\lesssim \\varepsilon (1 + t + |q|)^{-1} |\\dot{\\Far}|^2 \\ + \\ \\varepsilon (1 + |q|)^{-1} (1 + q_-)^{-2 \\upmu} \n\t\t\t\\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}}^2 + |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}}^2 \\big) \\notag\n\t\\end{align}\n\tand \n\t\\begin{align} \\label{E:secondtensorialdotFLinfinity}\n\t\t\\Big|L_{\\mu}h^{\\mu \\kappa} \\dot{\\Far}_{\\kappa \\zeta} \\dot{\\Far}_{0}^{\\ \\zeta} \n\t\t\t+ L_{\\mu} h^{\\kappa \\lambda} \\dot{\\Far}_{\\ \\kappa}^{\\mu} \\dot{\\Far}_{0 \\lambda} \n\t\t\t& + \\frac{1}{2} h^{\\kappa \\lambda} \\dot{\\Far}_{\\kappa \\eta} \\dot{\\Far}_{\\lambda}^{\\ \\eta} \\Big| \\\\\n\t\t& \\lesssim |h|_{\\mathcal{L}\\mathcal{L}}|\\dot{\\Far}|^2 \n\t\t\t\\ + \\ |h||\\dot{\\Far}| \\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}} + |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}} \\big) \\notag \\\\\n\t\t& \\lesssim \\varepsilon (1+|q|)(1 + t + |q|)^{-1} |\\dot{\\Far}|^2 \n\t\t\t\\ + \\ \\varepsilon (1 + q_-)^{-2 \\mu} \n\t\t\t\\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}}^2 + |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}}^2 \\big).  \\notag\n\t\\end{align}\n\t\n\tFor the higher-order terms, we use \\eqref{E:NtriangleSmallAlgebraic},\n\tthe hypotheses of the proposition, and the inequality $|ab| \\lesssim a^2 + b^2$\n\tto deduce that\n\t\n\t\\begin{align}\n\t\t\\Big| (\\nabla_{\\mu}N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda}) \\dot{\\Far}_{\\kappa \\lambda} \n\t\t\t\\dot{\\Far}_{0 \\zeta} - \\frac{1}{4} (\\nabla_{t} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda}) \\dot{\\Far}_{\\zeta \\eta} \n\t\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big| & \\lesssim \\big(|(h,\\Far)||(\\nabla h, \\nabla \\Far)| \\big)  |\\dot{\\Far}|^2 \\\\\n\t\t& \\lesssim \\varepsilon (1 + t + |q|)^{-1} |\\dot{\\Far}|^2 \\notag\t\n\t\\end{align}\t\n\tand\n\t\t\n\t\t\\begin{align} \\label{E:fourthtensorialdotFLinfinity}\n\t\t\\Big| L_{\\mu} N_{\\triangle}^{\\# \\mu \\zeta \\kappa \\lambda} \\dot{\\Far}_{\\kappa \\lambda} \\dot{\\Far}_{0 \\zeta}\n\t\t+ \\frac{1}{4} N_{\\triangle}^{\\#\\zeta \\eta \\kappa \\lambda} \\dot{\\Far}_{\\zeta \\eta} \n\t\t\\dot{\\Far}_{\\kappa \\lambda} \\Big| & \\lesssim |(h,\\Far)|^2 |\\dot{\\Far}|^2 \\\\\n\t\t& \\lesssim \\varepsilon (1+|q|)(1 + t + |q|)^{-1} |\\dot{\\Far}|^2. \\notag\t\n\t\\end{align}\t\n\nInserting \\eqref{E:firsttensorialdotFLinfinity} - \\eqref{E:fourthtensorialdotFLinfinity} into\nthe right-hand side of \\eqref{E:FirstweightedenergyFar}, and using \\eqref{E:weightinequality}, \nwe have that\n\n\\begin{align}  \\label{E:preliminarweightedenergyFar}\n\t\\int_{\\Sigma_{t}} & |\\dot{\\Far}|^2 w(q) \\,d^3x \n\t\t\\ + \\ \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \n\t\t\\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}}^2 + |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}}^2 \\big) w'(q)\n\t\t\\,d^3x \\, d \\tau \\\\  \n\t& \\leq C \\int_{\\Sigma_{0}} |\\dot{\\Far}|^2 w(q) \\,d^3x \n\t\t\\ + \\ C_1 \\varepsilon \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \\bigg\\lbrace \\frac{|\\dot{\\Far}|^2}{1 + \\tau} w(q)  \n\t\t\\ + \\ \\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}}^2 + |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}}^2 \\big) \\frac{w'(q)}{\\upgamma} \n\t\t\\bigg \\rbrace \\,d^3x \\, d \\tau\t\t\n\t\t\\ + \\ C \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} |\\dot{\\Far}_{0 \\kappa} \\dot{\\mathfrak{F}}^{\\kappa}| w(q) \\,d^3x \\, d \\tau. \\notag \n\\end{align}\n\nNow if $C_1 \\varepsilon\/\\upgamma$ is sufficiently small, we can absorb the  \n$C_1 \\varepsilon \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \\Big\\lbrace  \\big(|\\dot{\\Far}|_{\\mathcal{L} \\mathcal{N}}^2 + |\\dot{\\Far}|_{\\mathcal{T} \\mathcal{T}}^2 \\Big) \\frac{w'(q)}{\\upgamma} \\Big \\rbrace \\,d^3x \\, d \\tau$ term on the right-hand side of \\eqref{E:preliminarweightedenergyFar} into the second term on the left-hand side at the expense of increasing the constants $C.$ Inequality \\eqref{E:SecondweightedenergyFar} thus follows.\n\n\\end{proof}\n\n\n\n\\section{Pointwise Decay Estimates for Wave Equations in a Curved Spacetime} \\label{S:WaveEquationDecay}\n\nIn this section, we state a lemma and a corollary proved in \\cite{hLiR2010}. \nThey allow one to deduce pointwise decay estimates for solutions to inhomogeneous wave equations\n(e.g., for the $h_{\\mu \\nu}$). The main advantage of these estimates is that if one has good control over the inhomogeneous terms, then the pointwise decay estimates provided by the lemma and its corollary are \\emph{improvements over what can be deduced from the weighted Klainerman-Sobolev inequalities} of Proposition \\ref{P:WeightedKlainermanSobolev}. In particular, the lemma and its corollary play a fundamental role in the proofs of Propositions \\ref{P:UpgradedDecayhA} and \\ref{P:UpgradedDecayh1A}.\n\n\\begin{remark}\n\tThe Faraday tensor analogs of Lemma \\ref{L:scalardecay} and Corollary \\ref{C:systemdecay}\n\tare contained in the estimates of Proposition \\ref{P:ODEsNullComponentsLieZIFar}. More specifically,\n\tthe analogous inequalities would arise from integrating (in the direction of the first-order vectorfield differential \n\toperators on the left-hand sides of the inequalities) the inequalities in the proposition. We will carry out these\n\tintegrations in Section \\ref{S:DecayFortheReducedEquations}, which will allow us to derive improved pointwise decay estimates \n\tfor the lower-order Lie derivatives of the Faraday tensor (improved over what can be deduced from the \n\tweighted Klainerman-Sobolev inequality \\eqref{E:KSIntro}).\n\\end{remark}\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{The decay estimate weight function \\texorpdfstring{$\\varpi(q)$}{}}\n\nAs in \\cite{hLiR2010}, our decay estimates will involve the following weight function $\\varpi(q),$\nwhich is chosen to complement the energy estimate weight function $w(q)$ defined in \\eqref{E:weight}:\n\n\n\\begin{align} \\label{E:decayweight}\n\t\\varpi = \\varpi(q) = \\left \\lbrace\n\t\t\\begin{array}{lr}\n    \t(1 + |q|)^{1 + \\upgamma'}, &  \\mbox{if} \\ q > 0, \\\\\n      (1 + |q|)^{1\/2 - \\upmu'}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \\right.\n\\end{align}\nwhere $0 < \\updelta < \\upmu' < 1\/2 - \\upmu$ and $0 < \\upgamma' < \\upgamma - \\updelta$ are fixed constants. Its\ncomplementary role will become apparent in Section \\ref{S:DecayFortheReducedEquations}.\n\n\n\\subsection{Pointwise decay estimates}\n\nWe now state the lemma concerning pointwise decay estimates for solutions to inhomogeneous quasilinear wave equations.\n\n\\begin{lemma} \\cite[Lemma 7.1]{hLiR2010} \\label{L:scalardecay} \\textbf{(Pointwise decay estimates for solutions to a scalar \n\twave equation)}\n\tLet $\\phi$ be a solution of the scalar wave equation \\eqref{E:scalar} \n\t\n\t\\begin{align} \\label{E:scalar}\n\t\t\\widetilde{\\Square}_g \\phi = \\mathfrak{I}\n\t\\end{align}\n\ton a curved background with metric $g_{\\mu \\nu}.$ Assume that the tensor $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - \n\t(m^{-1})^{\\mu \\nu}$ obeys the following pointwise estimates \n\t\n\t\\begin{align}\n\t\t|H| \\leq \\varepsilon',&& \t\t\t\n\t\t\\int_{0}^{\\infty} (1 + t)^{-1} \\big\\| H(t,\\cdot) \\big\\|_{L^{\\infty}(D_{t})} \\, d t \\leq \\frac{1}{4},\n\t\t&& |H|_{\\mathcal{L} \\mathcal{T}} \\leq \\varepsilon' (1 + t + |x|)^{-1}(1 + |q|)\n\t\\end{align}\n\tin the region\n\t\\begin{align}\n\t\tD_t \\eqdef \\lbrace x: t\/2 < r < 2 t \\rbrace\n\t\\end{align}\n\tfor $t \\in [0,T).$ Then with $\\upalpha \\eqdef \\max(1 + \\upgamma', 1\/2 - \\upmu'),$ the following pointwise estimate holds\n\tfor $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{align} \\label{E:scalardecay}\n\t\t(1 + t + |q|) \\varpi(q) |\\nabla \\phi| & \\lesssim \\sup_{0 \\leq \\tau \\leq t} \\sum_{|I| \\leq 1} \n\t\t\t\\big\\| \\varpi(q)\\nabla_{\\mathcal{Z}}^I \\phi(\\tau,\\cdot) \\big\\|_{L^{\\infty}} \\\\\n\t\t& \\ \\ + \\ \\int_{\\tau = 0}^{t} \\varepsilon' \\upalpha \\big\\| \\varpi(q) \\nabla \\phi(\\tau, \\cdot) \n\t\t\t\\big\\|_{L^{\\infty}} \\, d \\tau \n\t\t\t\\ + \\ \\int_{\\tau = 0}^{t} (1 + \\tau) \\big\\| \\varpi(q) \\mathfrak{I}(\\tau, \\cdot) \\big\\|_{L^{\\infty}(D_{\\tau})} \n\t\t\t\\, d \\tau \\notag \\\\\n\t\t& \\ \\ + \\ \\int_{\\tau = 0}^{t} \\sum_{|I| \\leq 2} (1 + \\tau)^{-1} \\big\\| \\varpi(q) \\nabla_{\\mathcal{Z}}^I \\phi(\\tau, \\cdot) \n\t\t\t\\big\\|_{L^{\\infty}(D_{\\tau})} \\, d \\tau. \\notag\n\t\\end{align}\n\t\n\\end{lemma}\n\n\\hfill $\\qed$\n\nWe now state the following corollary, which provides similar decay estimates for\nthe null components of tensorial systems of wave equations.\n\n\\begin{corollary} \\cite[Corollary 7.2]{hLiR2010}   \\label{C:systemdecay}\n\t\\textbf{(Pointwise decay estimates for solutions to a system of tensorial wave \n\tequations)}\n\tLet $\\phi_{\\mu \\nu}$ be a solution of the system \n\t\n\t\\begin{align} \\label{E:system}\n\t\t\\widetilde{\\Square}_g \\phi_{\\mu \\nu} = \\mathfrak{I}_{\\mu \\nu}\n\t\\end{align}\n\ton a curved background with a metric $g_{\\mu \\nu}.$ Assume that the tensor $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - \n\t(m^{-1})^{\\mu \\nu}$ obeys the following pointwise estimates \n\t\n\t\\begin{align}\n\t\t|H| \\leq \\frac{\\varepsilon'}{4}, &&\n\t\t\\int_{0}^{\\infty} (1 + t)^{-1} \\big \\| H(t,\\cdot) \\big \\|_{L_{\\infty}(D_t)} \\,dt \\leq \\varepsilon', &&\n\t\t|H|_{\\mathcal{L} \\mathcal{T}} \\leq \\frac{\\varepsilon'}{4} (1 + t + |q|)^{-1} (1 + |q|)\n\t\\end{align}\n\tin the region\n\t\\begin{align}\n\t\tD_{t} \\eqdef \\lbrace x: t\/2 < |x| < 2 t \\rbrace\n\t\\end{align}\n\tfor $t \\in [0,T).$ Then for any $\\mathcal{U},\\mathcal{V} \\in \\lbrace \\mathcal{L},\\mathcal{T},\\mathcal{N} \\rbrace$ and with \n\t$\\upalpha \\eqdef \\max(1 + \\upgamma', 1\/2 - \\upmu'),$ the following pointwise estimate holds for $(t,x) \\in [0,T) \\times \n\t\\mathbb{R}^3:$\n\t\n\t\\begin{align} \\label{E:systemdecay}\n\t\t(1 + t + |q|) \\varpi(q) |\\nabla \\phi|_{\\mathcal{U}\\mathcal{V}} & \\lesssim \\sup_{0 \\leq \\tau \\leq t} \\sum_{|I| \\leq 1} \n\t\t\t\\big\\| \\varpi(q)\\nabla_{\\mathcal{Z}}^I \\phi(t,x) \\big\\|_{L^{\\infty}} \\\\\n\t\t\t& \\ \\ +\t\\ \\int_{\\tau = 0}^{t} \\varepsilon' \\upalpha \\big\\| \\varpi(q) |\\nabla \\phi(\\tau, \\cdot)|_{\\mathcal{U}\\mathcal{V}} \n\t\t\t\t\\big\\|_{L^{\\infty}} \\, d \\tau \n\t\t\t\\ +\t\\ \\int_{\\tau = 0}^{t} (1 + \\tau) \\big\\| \\varpi(q) |\\mathfrak{I}(\\tau, \\cdot)|_{\\mathcal{U}\\mathcal{V}} \n\t\t\t\t\\big\\|_{L^{\\infty}(D_{\\tau})} \\, d \\tau \\notag \\\\\n\t\t\t& \\ \\ + \\ \\sum_{|I| \\leq 2} \\int_{\\tau = 0}^{t} (1 + \\tau)^{-1} \n\t\t\t\t\\big\\| \\varpi(q) \\nabla_{\\mathcal{Z}}^I \\phi(\\tau, \\cdot) \\big\\|_{L^{\\infty}(D_{\\tau})} \\, d \\tau. \\notag\n\t\\end{align}\n\t\n\\end{corollary}\n\n\\hfill $\\qed$\n\n\\section{Local Existence and the Continuation Principle for the Reduced Equations} \\label{S:LocalExistence}\n\nIn this short section, we state for convenience a standard proposition concerning local existence and a continuation principle for the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}. The continuation principle shows that an a-priori smallness condition on the energy of the solution is sufficient to deduce global existence. It therefore plays a\nfundamental role in our global stability argument of Section \\ref{S:GlobalExistence}.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\begin{proposition} \\label{P:LocalExistence}\n\t\\textbf{(Local existence and the continuation principle)}\n\tLet $(h_{\\mu \\nu}^{(1)}|_{t=0},\\partial_t h_{\\mu \\nu}^{(1)}|_{t=0}, \\Far_{\\mu \\nu}|_{t=0}),$ $(\\mu, \\nu = 0,1,2,3),$\n\tbe initial data for the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} constructed from \n\tabstract initial data \n\t$(\\mathring{\\underline{h}}_{jk}^{(1)}, \\mathring{K}_{jk}, \\mathring{\\mathfrak{\\Displacement}}_j, \t\n\t\\mathring{\\mathfrak{\\Magneticinduction}}_j),$ $(j,k = 1,2,3),$\n\ton the manifold $\\mathbb{R}^3$ satisfying the constraints \\eqref{E:Gauss} - \\eqref{E:DivergenceB0} as described in Section \n\t\\ref{SS:ReducedData}. Assume that the data are asymptotically flat in the sense of \\eqref{E:metricdataexpansion} -\n\t\\eqref{E:BdecayAssumption}. Let $\\dParameter \\geq 3$ be an integer, and let $\\upgamma > 0, \\upmu > 0$ be constants. \n\tAssume that $E_{\\dParameter;\\upgamma}(0) < \\varepsilon,$ where $E_{\\dParameter;\\upgamma}(0)$ is the norm of the abstract data \n\tdefined in \\eqref{E:DataNorm}. Then if $\\varepsilon$ is sufficiently small\\footnote{This smallness assumption \n\tensures that the reduced data lie within the regime of hyperbolicity of the reduced equations.}, these data launch a unique \n\tclassical solution to the reduced equations existing on a nontrivial maximal spacetime slab $[0,T_{max}) \\times \n\t\\mathbb{R}^3.$ The energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t)$ of the solution, which is defined in \n\t\\eqref{E:EnergyIntro}, satisfies $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0) \\lesssim \\varepsilon$ and is continuous on \n\t$[0,T_{max}).$ Furthermore, either $T_{max} = \\infty,$ or one of the following two ``breakdown'' scenarios must occur:\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\lim_{t \\uparrow T_{max}} \\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) = \\infty.$ \n\t\t\\item The solution escapes the regime of hyperbolicity of the reduced equations.\n\t\\end{enumerate}\n\t\n\\end{proposition}\n\n\\begin{remark}\n\tThe classification of the two breakdown scenarios is known as a \\emph{continuation principle}.\n\\end{remark}\n\nThe main ingredients in the proof of Proposition \\ref{P:LocalExistence} are Lemma \\ref{L:weightedenergyFar} and Lemma \\ref{L:weightedenergy}, which provide weighted energy estimates for linearized versions of the reduced equations. Based on the availability of these estimates, the proof is rather standard, and we omit the details. Readers may consult \ne.g. \\cite[Ch. VI]{lH1997}, \\cite{aM1984}, \\cite{jSmS1998}, \\cite{cS2008}, \\cite{jS2008a}, and \\cite[Ch. 16]{mT1997III} for details concerning local existence, and e.g. \\cite{jS2008b} for the ideas behind the continuation principle. \n\n\n\n\n\n\\section{The Fundamental Energy Bootstrap Assumption and Pointwise Decay Estimates for the Reduced Equations} \\label{S:DecayFortheReducedEquations}  \n\\setcounter{equation}{0}\n\nIn this section, we introduce our fundamental bootstrap assumption \\eqref{E:Bootstrap} for the energy of a solution to the reduced equations. Under this assumption, we derive a collection of pointwise decay estimates that will play a crucial role in the proof of Theorem \\ref{T:ImprovedDecay}. In particular, these decay estimates are used to deduce the factors $(1 + \\tau)^{-1}$ and $(1 + \\tau)^{- 1 + C \\varepsilon}$ in \\eqref{E:Gronwallreadyinequalityk}, which are essential for deriving the bound \\eqref{E:ImprovedEnergyInequality}. Many of the estimates we derive in this section rely upon the wave coordinate condition.\n\n\n\\noindent \\hrulefill\n\\ \\\\\nWe recall that the spacetime metric $g_{\\mu \\nu}$ is split into the pieces \n$g_{\\mu \\nu} = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)},$ and that the energy \n$\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t)$ (see \\eqref{E:EnergyIntro}) is a functional of $(h^{(1)},\\Far).$ \nOur main bootstrap assumption for the energy is \n\n\\begin{align} \\label{E:Bootstrap}\n\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\leq \\varepsilon(1 + t)^{\\updelta},\n\\end{align}\nwhere $0 < \\upgamma < 1\/2$ is a fixed constant, $\\updelta$ is a fixed constant satisfying both $0 < \\updelta < 1\/4$ and $0< \\updelta < \\upgamma,$ $0 < \\upmu < 1\/2$ is a fixed constant, (all of which will be chosen during the proof of Theorem \\ref{T:MainTheorem}), and $\\varepsilon$ is a small positive number whose required smallness is adjusted (as many times as necessary) during the derivation of our inequalities. With the help of \\eqref{E:LieZIinTermsofNablaZI}, inequality \\eqref{E:Bootstrap} implies the following more explicit consequence of the \nenergy bootstrap assumption: \n\n\n\\begin{align} \\label{E:Boostrapexplicit}\n\t\\sum_{|I| \\leq \\dParameter }  \\big\\| w^{1\/2}(q) \\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)} \\big\\|_{L^2}\n\t\t\\ + \\ \\big\\| w^{1\/2}(q) \\Lie_{\\mathcal{Z}}^I \\Far \\big\\|_{L^2} \\leq C \\varepsilon (1 + t)^{\\updelta}.\n\\end{align}\n\nIn the remaining estimates in this article, \\textbf{we will also often make the following smallness assumption on the ADM mass}:\n\n\\begin{align} \\label{E:Missmall}\n\tM \\leq \\varepsilon.\n\\end{align}\n\n\\subsection{Preliminary (weak) pointwise decay estimates} \\label{SS:PreliminaryLinfinityEstimates}\n\nIn this section, we provide some preliminary pointwise decay estimates that are essentially a consequence of the weighted Klainerman-Sobolev inequalities of Appendix \\ref{A:WeightedKS}. Unlike the upgraded pointwise decay estimates of the next section, these estimates do not take into account the special structure of the reduced equations under the wave coordinate condition.\n\nWe begin with a lemma concerning pointwise decay estimates for the Schwarzschild tail of the metric and its derivatives.\n\n\\begin{lemma} \\label{L:h0decayestimates}\n\t\t\\textbf{(Decay estimates for $h^{(0)}$)}\n\t\tLet $h^{(0)}$ be as in \\eqref{E:h0defIntro}, and let $I$ be any $\\nabla-$multi-index. Then the following\n\t\tpointwise estimate holds for $(t,x) \\in [0,\\infty) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{subequations}\n\t\\begin{align} \\label{E:nablaIh0Linfinity}\n\t\t|\\nabla^I h^{(0)}| \\leq C M (1 + t + |q|)^{-1+ |I|},\n\t\\end{align}\n\twhere $M$ is the ADM mass.\n\t\n\tFurthermore, if $I$ is any $\\nabla-$multi-index and $J$ is any $\\mathcal{Z}-$multi-index,\n\tthen the following pointwise estimate holds for $(t,x) \\in [0,\\infty) \\times \\mathbb{R}^3:$\n\t\n\t\t\\begin{align} \\label{E:decaynablaJnablaZIh0Linfinity}\n\t\t\t|\\nabla^I \\nabla_{\\mathcal{Z}}^J h^{(0)}| \\ + \\ |\\nabla_{\\mathcal{Z}}^J \\nabla^I h^{(0)}|\n\t\t\t\\leq C M (1 + t + |q|)^{- 1 + |I|}. \n\t\t\\end{align}\n  \t\\end{subequations}\n\\end{lemma}\n\n\n\\begin{remark}\n\tSince $H_{(0)\\mu \\nu} = - h_{\\mu \\nu}^{(0)}$ (where $H_{(0)}^{\\mu \\nu}$ is defined in \n\t\\eqref{E:NablaZIh1LLh1LTwaveCoordinateAlgebraicEstimate}), the above estimates also hold if we replace\n\t$h^{(0)}$ with $H_{(0)}.$\n\\end{remark}\n\n\\begin{proof}\n\tThe lemma follows from simple computations using the definition \\eqref{E:chidef} of the cut-off function $\\chi,$\n\tthe definition of $h^{(0)},$ and the definitions of the vectorfields $Z \\in \\mathcal{Z}.$\n\\end{proof}\n\n\n\n\\begin{corollary} \\label{C:WeakDecay} \\cite[Slight extension of Corollary 9.4]{hLiR2010}\n\t\\textbf{(Weak pointwise decay estimates)}\n\tLet $\\dParameter \\geq 8$ be an integer. Assume that the abstract initial data are asymptotically flat in the sense \n\tof \\eqref{E:metricdataexpansion} - \\eqref{E:BdecayAssumption}, that the ADM mass smallness condition\n\t\\eqref{E:Missmall} holds, and that the initial data for the reduced\n\tsystem are constructed from the abstract initial data as described in Section \\ref{SS:ReducedData}.\n\tLet $(g_{\\mu \\nu} \\eqdef m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \n\t\\nu}^{(1)}, \\Far_{\\mu \\nu})$ be the corresponding solution to the reduced system \\eqref{E:Reducedh1Summary} - \n\t\\eqref{E:ReduceddMis0Summary} existing on a slab $(t,x) \\in [0,T) \\times \n\t\\mathbb{R}^3,$ where $h^{(1)}$ is defined in \\eqref{E:hdefIntro}. Assume in addition that the pair \n\t$(h^{(1)}, \\Far)$ \n\tsatisfies the energy bootstrap assumption \\eqref{E:Bootstrap} on the interval $[0,T).$\n\tThen if $\\varepsilon$ is sufficiently small, the following pointwise estimates hold for $(t,x) \\in [0,T) \\times \n\t\\mathbb{R}^3:$\n\t\n\t\\begin{subequations}\n\t\\begin{align} \\label{E:weakdecaypartialLinfinity}\n\t\t|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}| \n\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far| \\leq\n\t\t\\left \\lbrace \\begin{array}{lr}\n    \tC \\varepsilon (1 + t + |q|)^{-1} (1 + t)^{\\updelta} (1 + |q|)^{-1 - \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      C \\varepsilon (1 + t + |q|)^{-1} (1 + t)^{\\updelta} (1 + |q|)^{-1\/2}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \t\\right., &&\n  \t|I| \\leq \\dParameter - 2, \n\t\\end{align}\n\n\t\\begin{align} \\label{E:weakdecayLinfinity}\n\t\t\t|\\nabla_{\\mathcal{Z}}^I h^{(1)}|  \\leq\n\t\t\t\\left \\lbrace \\begin{array}{lr}\n    \t\tC \\varepsilon (1 + t + |q|)^{-1 + \\updelta} (1 + |q|)^{- \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      \tC \\varepsilon (1 + t + |q|)^{-1 + \\updelta} (1 + |q|)^{1\/2}, & \\mbox{if} \\ q < 0,\n    \t\\end{array}\n  \t\t\\right., &&\n  \t\t|I| \\leq \\dParameter -2,  \n  \t\\end{align}\n  \t\n  \n  \t\\begin{align} \\label{E:weakdecaybarpartialLinfinity}\n\t\t|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}| \n\t\t\\ + \\ (1 + |q|)|\\conenabla \\Lie_{\\mathcal{Z}}^I \\Far| \\leq\n\t\t\\left \\lbrace \\begin{array}{lr}\n    \tC \\varepsilon (1 + t + |q|)^{-2 + \\updelta} (1 + |q|)^{- \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      C \\varepsilon (1 + t + |q|)^{-2 + \\updelta} (1 + |q|)^{1\/2}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \t\\right., &&\n  \t|I| \\leq \\dParameter - 3.  \n  \t\\end{align}\n  \t\\end{subequations}\n  \t\n  \n  \tIn addition, the tensorfield $H_{(1)}^{\\mu \\nu}$ \n  \tdefined in \\eqref{E:NablaZIh1LLh1LTwaveCoordinateAlgebraicEstimate} satisfies the same estimates as $h_{\\mu \\nu}^{(1)}.$\n  \tFurthermore, if we make the substitution $\\upgamma \\rightarrow \\updelta$ in the above inequalities, then the same estimates \n  \thold for the tensorfields $h_{\\mu \\nu}^{(0)},$ $h_{\\mu \\nu} \\eqdef h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)},$ \n  \t$H_{(0)\\mu \\nu} \\eqdef - h_{\\mu \\nu}^{(0)},$ $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu},$\n  \tand $H_{(1)}^{\\mu \\nu} \\eqdef H^{\\mu \\nu} - H_{(0)}^{\\mu \\nu}.$ \n\\end{corollary}\n\n\\begin{proof}\n\tThis Corollary is a slight extension of Corollary 9.4 of \\cite{hLiR2010}, in which estimates for $h^{(0)} = \n\t- H_{(0)},$ $h^{(1)},$ and $h$ were proved. The main idea in the proof is to use the weighted Klainerman-Sobolev \n\testimates of Proposition \\ref{P:WeightedKlainermanSobolev} under the assumption \\eqref{E:Boostrapexplicit}, together with the\n\tdecay \\eqref{E:h1AbstractDataAsymptotics} - \\eqref{E:BdecayAssumption} of the initial data at $\\infty.$ The estimates for \n\t$\\Far$ follow from the arguments of \\cite[Corollary 9.4]{hLiR2010}, while the estimates for $H_{(1)}$ and $H$ follow from \n\tthose for $h^{(1)}$ and $h$ together with \\eqref{E:Hintermsofh}.\n\\end{proof}\n\nThe next lemma uses the weak decay estimates to provide algebraic estimates for the Schwarzschild tail term\n$\\nabla_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}$ appearing on the right-hand side of \\eqref{E:Reducedh1Summary}.\n\n\\begin{lemma}\\cite[Lemma 9.9]{hLiR2010} \\label{L:weakdecayLinfinitynablaZISquaregh0} \n\\textbf{(Pointwise decay estimates for $\\nabla_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}$)}\n\tLet $h^{(0)}$ be the Schwarzschild part of $h$ as defined in \\eqref{E:h0defIntro}, and\n\tassume the hypotheses\/conclusions of Corollary \\ref{C:WeakDecay}.\n\tLet $I$ be a $\\mathcal{Z}-$multi-index subject to the restrictions stated below.\n\tThen if $\\varepsilon$ is sufficiently small, the following pointwise estimates hold for $(t,x) \\in [0,T) \\times \n\t\\mathbb{R}^3,$ where $M$ is the ADM mass: \n\t\n\t\\begin{subequations}\n\t\\begin{align} \\label{E:weakdecayLinfinitynablaZISquaregh0}\n\t\t\t|\\nabla_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}|  \\leq\n\t\t\t\\left \\lbrace \\begin{array}{lr}\n    \t\tC M \\varepsilon (1 + t + |q|)^{-4 + \\updelta} (1 + |q|)^{- \\updelta}, &  \\mbox{if} \\ q > 0,\\\\\n      \tC M (1 + t + |q|)^{-3}, & \\mbox{if} \\ q < 0,\n    \t\\end{array}\n  \t\t\\right., &&\n  \t\t|I| \\leq \\dParameter -2.  \n  \t\\end{align}\n  \t\n  \n  \tFurthermore, the following pointwise estimates also hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t  \n\t  \\begin{align} \\label{E:weakdecayLinfinitynablaZISquaregh0MoreGeneral}\n\t\t\t|\\nabla_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}|  \\leq\n\t\t\t\\left \\lbrace \\begin{array}{lr}\n    \t\tC M \\varepsilon (1 + t + |q|)^{-4}, &  \\mbox{if} \\ q > 0,\\\\\n      \tC M (1 + t + |q|)^{-3}, & \\mbox{if} \\ q < 0\n    \t\\end{array}\n  \t\t\\right. \\ + \\ C M \\sum_{|J| \\leq |I|} (1 + t + |q|)^{-3} |\\nabla_{\\mathcal{Z}}^J h^{(1)}|, &&\n  \t\t|I| \\leq \\dParameter .  \n  \t\\end{align}\n  \t\\end{subequations}\n  \t\n\\end{lemma}\n\n\\begin{proof}\n\tWe first observe that $\\widetilde{\\Square}_g h^{(0)} = \\Square_m h^{(0)} + H^{\\kappa \\lambda} \\nabla_{\\kappa} \n\t\\nabla_{\\lambda}h^{(0)},$ where $\\Square_m \\eqdef (m^{-1})^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}$\n\tis the Minkowski wave operator. Using \\eqref{E:decaynablaJnablaZIh0Linfinity}, the definition of $h^{(0)},$ \n\tthe Leibniz rule, and the fact that $\\Square_m (1\/r) = 0$ for $r > 0,$\n\tit follows that\n\t\n\t\\begin{align}\n\t\t|\\nabla_{\\mathcal{Z}}^I \\Square_m h^{(0)}| \\  \n\t\t\t& \\lesssim M (1 + t + |q|)^{-3} \\chi_0(1\/2 \\leq r\/t \\leq 3\/4),  \\\\\n  \t|\\nabla_{\\mathcal{Z}}^I \\big(H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}h^{(0)} \\big)|\t   \n  \t\t&  \\lesssim M (1 + t + |q|)^{-3} \\sum_{|J| \\leq |I|} |\\nabla_{\\mathcal{Z}}^J H|, \\label{E:BoxHappliedtoh0}\n\t\\end{align}\n\twhere $\\chi_0(1\/2 \\leq z \\leq 3\/4)$ is the characteristic function of the interval $[1\/2,3\/4].$\n\tFurthermore, using that $H = - h^{(0)} - h^{(1)} + O^{\\infty}(\\big|h^{(0)} + h^{(1)}|^2 \\big),$ it follows that\n\t\n\t\\begin{align} \\label{E:NablaZJHexpanded}\n\t\t\\sum_{|J| \\leq |I|}|\\nabla_{\\mathcal{Z}}^J H| \\lesssim \\varepsilon (1 + t + |q|)^{-1} \n\t\t\t\\ + \\ \\sum_{|J| \\leq |I|} |\\nabla_{\\mathcal{Z}}^J h^{(1)}|.\n\t\\end{align}\n\tUsing \\eqref{E:BoxHappliedtoh0}, \\eqref{E:NablaZJHexpanded}, and the estimate \\eqref{E:weakdecayLinfinity}, \n\twe have that\n\t\n\t\\begin{align} \n\t\t\t|\\nabla_{\\mathcal{Z}}^I \\big(H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}h^{(0)} \\big)|  \n\t\t\t\\lesssim \\left \\lbrace \\begin{array}{lr}\n    \t\tM \\varepsilon (1 + t + |q|)^{-4 + \\updelta} (1 + |q|)^{- \\updelta}, &  \\mbox{if} \\ q > 0,\\\\\n      \tM \\varepsilon (1 + t + |q|)^{-4 + \\updelta} (1 + |q|)^{1\/2}, & \\mbox{if} \\ q < 0,\n    \t\\end{array}\n  \t\t\\right., && |I| \\leq \\dParameter - 2, \n  \t\\end{align}\n  \tand\n  \t\n  \t\\begin{align} \n\t\t\t|\\nabla_{\\mathcal{Z}}^I \\big(H^{\\kappa \\lambda} \\nabla_{\\kappa} \\nabla_{\\lambda}h^{(0)} \\big)|  \n\t\t\t\t\\lesssim M \\varepsilon (1 + t + |q|)^{-4}  \n\t\t\t\\ + \\ M \\varepsilon (1 + t + |q|)^{-3} \\sum_{|J| \\leq |I|} |\\nabla_{\\mathcal{Z}}^J h^{(1)}|, &&\n  \t\t|I| \\leq \\dParameter. \n  \t\\end{align}\n  \tInequalities \\eqref{E:weakdecayLinfinitynablaZISquaregh0} and \\eqref{E:weakdecayLinfinitynablaZISquaregh0MoreGeneral}\n  \tnow easily follow from the above estimates.\n\t\n\\end{proof}\n\n\n\\subsection{Initial upgraded pointwise decay estimates for $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}$\nand $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}$} \\label{SS:InitialFLUTTLinfinityImprovements}\n\nIn this section, we prove some upgraded pointwise decay estimates for the ``favorable'' components of the lower-order Lie derivatives of $\\Far.$ Our estimates take into account the special structure revealed by our null decomposition of the electromagnetic equations of variations, a structure that was captured by Proposition \\ref{P:ODEsNullComponentsLieZIFar}\nand that depends in part upon the wave coordinate condition. We remark that in Section \\ref{SS:FullUpgradedLinfinityEstimates}, some of these decay estimates will be further improved (hence our use of the term ``initial upgraded'' here).\n\n\n\\begin{proposition} \\label{P:FLUTTimproveddecay}\n\t\\textbf{(Initial upgraded pointwise decay estimates for $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}$ \n\tand $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}$)}\n\tAssume the hypotheses\/conclusions of Corollary \\ref{C:WeakDecay}. Then if $\\varepsilon$ is sufficiently small, the following \n\tpointwise estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{align} \\label{E:FLUTTimproveddecay}\n\t\t|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}} \n\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}} \n\t\t\t\\leq\n\t\t\\left \\lbrace \\begin{array}{lr}\n    \tC \\varepsilon (1 + t + |q|)^{-2 + 2 \\updelta} (1 + |q|)^{- \\upgamma - \\updelta}, &  \\mbox{if} \\ q > 0, \\\\\n      C \\varepsilon (1 + t + |q|)^{-2 + 2 \\updelta} (1 + |q|)^{1\/2 - \\updelta},\t& \\mbox{if} \\ q < 0,\n    \\end{array}\n  \t\\right., &&\n  \t|I| \\leq \\dParameter - 3.  \n  \t\\end{align}\n  \t\n\\end{proposition}\n\n\n\\begin{proof}\n\tSince $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}} +  \n\t|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}| \\approx |\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far]| \n\t+ |\\rho[\\Lie_{\\mathcal{Z}}^I \\Far]| + |\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]|,$ it suffices to prove the desired decay estimates \n\tfor $|\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far]|,$ $|\\rho[\\Lie_{\\mathcal{Z}}^I \\Far]|,$ and $|\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]|$ \n\tseparately. We provide proof for the null component $\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far].$ The proofs for the components \n\t$\\rho[\\Lie_{\\mathcal{Z}}^I \\Far]$ and $\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]$ are similar, and we leave these details to the \n\treader. Let $\\mathcal{W} \\eqdef \\big\\lbrace (t,x): |x| \\geq 1 + t\/2 \\big\\rbrace \\cap \\big\\lbrace (t,x): |x| \\leq 2t - 1 \n\t\\big\\rbrace$ denote the ``wave zone'' region. Then for $(t,x) \\nin \\mathcal{W},$ we have that \n\t$1 + |q| \\approx (1 + t + |q|).$ Using this fact, for $(t,x) \\nin \\mathcal{W},$ we can bound $|\\alpha[\\Lie_{\\mathcal{Z}}^I \n\t\\Far]|$ by the right-hand side of \\eqref{E:FLUTTimproveddecay} by using the weak decay estimate\t\n\t\\eqref{E:weakdecaypartialLinfinity}.\n\t\n\tWe now consider the case $(t,x) \\in \\mathcal{W}.$ Let $f \\eqdef r^{-1} \\alpha[\\Lie_{\\mathcal{Z}}^I \\Far].$ Then using\n\t\\eqref{E:alphaODE}, the fact that $r \\approx (1 + t + |q|) \\approx (1 + s + |q|)$ on $\\mathcal{W},$\n\tand the weak decay estimates of Corollary \\ref{C:WeakDecay}, it follows that (with $\\partial_q$ defined in Section \\ref{SS:Derivatives})\n\t\n\t\\begin{align} \\label{E:partialqfbound}\n\t\t|\\partial_q f(t,x)| & \\lesssim \\left \\lbrace \\begin{array}{lr}\n    \t\\varepsilon (1 + s + |q|)^{-3 + \\updelta} (1 + |q|)^{-1 - \\upgamma} \n    \t\t\\ + \\ \\varepsilon(1 + s + |q|)^{-3 + 2\\updelta} (1 + |q|)^{-2 - 2 \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      \\varepsilon (1 + s + |q|)^{-3 + \\updelta} (1 + |q|)^{- 1\/2} \n      \t\\ + \\ \\varepsilon(1 + s + |q|)^{-3 + 2\\updelta} (1 + |q|)^{-1}, & \\mbox{if} \\ q < 0.\n    \\end{array}\n  \t\\right.&&\n\t\\end{align}\n\t\n\tLet $(\\tau(q'), y(q'))$ be the $q'-$parameterized line segment of constant $s$ and angular \n\tvalues that initiates at $(t,x)$ and terminates at the point $(t_0,x_0)$ which lies to the \\emph{past} of $(t,x)$ and on\n\tthe boundary of $\\mathcal{W}.$ Let $q,s$ be the null coordinates corresponding to $(t,x).$\n\tThen the null coordinates corresponding to $(t_0,x_0)$ are $q_0 = s\/3 - 2\/3, s_0 = s.$ Integrating\n\tthe inequality \\eqref{E:partialqfbound} along this line segment (i.e., integrating $dq'$), we have that\n\t\n\t\\begin{align} \\label{E:qIntegratedfBound}\n\t\t|f(t,x)| & \\lesssim |f(t_0,x_0)|  \\\\\n\t\t\t& \\ \\ + \\ \\int_{q'= q}^{q'= s\/3 - 2\/3} \n\t\t\t\t \\left \\lbrace \\begin{array}{lr}\n    \t\t\\varepsilon (1 + s + |q'|)^{-3 + \\updelta} (1 + |q'|)^{-1 - \\upgamma} \n    \t\t\\ + \\ \\varepsilon(1 + s + |q'|)^{-3 + 2 \\updelta} (1 + |q'|)^{-2 - 2 \\upgamma}, &  \\mbox{if} \\ q' > 0, \\\\\n       \t\\varepsilon(1 + s + |q'|)^{-3 + \\updelta} (1 + |q'|)^{- 1\/2} \n      \t\\ + \\ \\varepsilon(1 + s + |q'|)^{-3 + 2\\updelta} (1 + |q'|)^{-1}, & \\mbox{if} \\ q' < 0,\n    \\end{array}\n  \t\\right\\rbrace \\, dq' \\notag \\\\\n  \t& \\lesssim |f(t_0,x_0)| \n  \t\t\\ + \\\n  \t\t\t\\left \\lbrace \\begin{array}{lr}\n    \t\t\\varepsilon (1 + s)^{-3 + \\updelta} (1 + |q|)^{- \\upgamma} \n    \t\t\\ + \\ \\varepsilon (1 + s )^{-3 + 2 \\updelta} (1 + |q|)^{-1 - 2 \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n       \t\\varepsilon(1 + s)^{-3 + \\updelta} (1 + |q|)^{1\/2} \n      \t\\ + \\ \\varepsilon(1 + s)^{-3 + 2\\updelta} \\ln(1 + |q|), & \\mbox{if} \\ q < 0.\n    \\end{array}\n  \t\\right. \\notag\n\t\\end{align}\t\n\t\n\tUsing the facts that $r_0 \\approx 1 + |q_0| \\approx 1 + t_0 + |q_0| \\approx \n\t 1 + s_0 + |q_0| \\approx 1 + s,$ together\n\twith the weak decay estimate \\eqref{E:weakdecaypartialLinfinity}, it follows that\n\t\n\t\\begin{align} \\label{E:fTerminalPointBound}\n\t\t|f(t_0,x_0)| & \\lesssim \\varepsilon (1 + s)^{-3 - \\upgamma + \\updelta}.\n  \\end{align}\n  Combining \\eqref{E:qIntegratedfBound} and \\eqref{E:fTerminalPointBound}, and using the fact that\n  $1 + s \\approx 1 + t + |q|,$ it follows that\n  $|\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far(t,x)]|$ is bounded from above by the right-hand side of\n  \\eqref{E:FLUTTimproveddecay}. This completes our proof of \\eqref{E:FLUTTimproveddecay} for the \n  $\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far]$ component. \n  \n\\end{proof}  \n\n\n\n\n\n\\subsection{Upgraded pointwise decay estimates for $|\\nabla_{\\mathcal{Z}}^I h|,$ \n$|\\Lie_{\\mathcal{Z}}^I \\Far|,$ and fully upgraded pointwise decay estimates for $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}},$  $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}} $} \\label{SS:FullUpgradedLinfinityEstimates}\n\nIn this section, we state two propositions that strengthen some of the pointwise decay estimates proved in sections \n\\ref{SS:PreliminaryLinfinityEstimates} and \\ref{SS:InitialFLUTTLinfinityImprovements}. Their proofs, which are provided in sections \\ref{SS:ProofofUpgradedDecayhA} and \\ref{SS:Proof of PropositionUpgradedDecayh1A}, are based on a careful analysis of the special structure of the reduced equations and in particular rely upon the decompositions performed in Section \\ref{S:AlgebraicEstimates}, which rely in part upon the wave coordinate condition. These estimates play a central role in our derivation of the ``strong'' energy inequality \\eqref{E:ImprovedEnergyInequality}, which is the main step in the proof of our stability theorem.\n\n\\begin{proposition} \\cite[Extension of Proposition 10.1]{hLiR2010} \\label{P:UpgradedDecayhA}\n\t\\textbf{(Upgraded pointwise decay estimates for $\\Far$ and certain components of $h,$ $\\nabla h,$ and $\\nabla_Z h$)}\n\tAssume that the abstract initial data satisfy the constraints \\eqref{E:Gauss} - \\eqref{E:DivergenceB0}, and\n\tassume the hypotheses\/conclusions of Corollary \\ref{C:WeakDecay}. In particular, by Proposition \n\t\\ref{P:PreservationofWaveCoordianteGauge}, the wave coordinate condition \\eqref{E:wavecoordinategauge1} holds for $(t,x) \\in \n\t[0,T) \\times \\mathbb{R}^3.$ Then if $\\varepsilon$ is sufficiently small, for every vectorfield $Z \\in \\mathcal{Z},$ the \n\tfollowing pointwise estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$ \n\t\n\t\t\\begin{subequations}\n\t\t\\begin{align} \\label{E:partialhLTpartialZhLLpointwise}\n\t\t\t|\\nabla h|_{\\mathcal{L} \\mathcal{T}} \n\t\t\t\\ + \\ |\\nabla\\nabla_Z h|_{\\mathcal{L} \\mathcal{L}} \n\t\t\t\t\\leq \\left \\lbrace \\begin{array}{lr}\n\t    \tC \\varepsilon (1 + t + |q|)^{-2 + \\updelta} (1 + |q|)^{- \\updelta}, &  \\mbox{if} \\ q > 0, \\\\\n\t      C \\varepsilon (1 + t + |q|)^{-2 + \\updelta} (1 + |q|)^{1\/2}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \t\\right., \n  \t\\end{align}\n  \t\n  \t\\begin{align} \\label{E:hLTZhLLpointwise}\n\t\t\t|h|_{\\mathcal{L} \\mathcal{T}} \n\t\t\t\\ + \\ |\\nabla_Z h|_{\\mathcal{L} \\mathcal{L}} \\leq\n\t\t\t\\left \\lbrace \\begin{array}{lr}\n\t    \tC \\varepsilon (1 + t + |q|)^{-1} , &  \\mbox{if} \\ q > 0, \\\\\n\t      C \\varepsilon (1 + t + |q|)^{-1} (1 + |q|)^{1\/2 + \\updelta}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \t\\right., \n  \t\\end{align}\n  \t\\end{subequations}\n  \t\n  \t\\begin{subequations}\n  \t\\begin{align}\n  \t\t|\\nabla h|_{\\mathcal{T} \\mathcal{N}} \n  \t\t\t& \\leq C \\varepsilon (1 + t + |q|)^{-1}, \\label{E:partialhTUpointwise} \\\\\n  \t\t|\\nabla h| & \\leq C \\varepsilon (1 + t + |q|)^{-1} \\big\\lbrace 1 + \\ln (1 + t) \\big\\rbrace, \\label{E:partialhpointwise}\n  \t\\end{align}\n  \t\\end{subequations}\n  \t\n  \\begin{align} \\label{E:Farupgradedecay}\n\t\t|\\Far| & \\leq C \\varepsilon (1 + t + |q|)^{-1}.  \n\t\\end{align}\n\t\t\n\t\tFurthermore, the same estimates hold for the tensorfields $h_{\\mu \\nu}^{(0)},$ $h_{\\mu \\nu}^{(1)},$\n\t\t$H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu},$ $H_{(0)}^{\\mu \\nu},$ and $H_{(1)}^{\\mu \\nu}.$\n\\end{proposition}\n\n\n\n\\begin{proposition} \\cite[Extension of Proposition 10.2]{hLiR2010} \\label{P:UpgradedDecayh1A}\n\t\\textbf{(Upgraded pointwise decay estimates for the lower-order derivatives of $h$\n\tand $\\Far$)}\n\tUnder the assumptions of Proposition \\ref{P:UpgradedDecayhA}, let\n\t$0 < \\upgamma' < \\upgamma - \\updelta$ and $0 < \\updelta < \\upmu' < 1\/2$ be fixed constants. Let $I$ be any \n\t$\\mathcal{Z}-$multi-index subject to the restrictions stated below. Then there exist constants $M_k, C_k,$ and \n\t$\\varepsilon_k$ depending on $\\upgamma', \\upmu', \\updelta$ such that if $\\varepsilon$ is sufficiently small, then the \n\tfollowing pointwise estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{subequations}\n\t\\begin{align} \\label{E:partialZIh1Aupgraded}\n\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|\n\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far|  \\leq\n\t\t\t\\left \\lbrace \\begin{array}{lr}\n\t    \tC_k \\varepsilon (1 + t + |q|)^{-1 + M_k \\varepsilon} (1 + |q|)^{-1 - \\upgamma'}, &  \\mbox{if} \\ q > 0, \\\\\n\t      C_k \\varepsilon (1 + t + |q|)^{-1 + M_k \\varepsilon} (1 + |q|)^{-1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \t\\right., && \n\t  \t|I|= k \\leq \\dParameter - 4,\n  \\end{align} \n  \t\n  \t\n  \t\\begin{align} \\label{E:ZIh1Aupgraded}\n\t\t\t|\\nabla_{\\mathcal{Z}}^I h^{(1)}| \\leq\n\t\t\t\\left \\lbrace \\begin{array}{lr}\n\t    \tC_k \\varepsilon (1 + t + |q|)^{-1 + M_k \\varepsilon} (1 + |q|)^{- \\upgamma'}, &  \\mbox{if} \\ q > 0, \\\\\n\t      C_k \\varepsilon (1 + t + |q|)^{-1 + M_k \\varepsilon} (1 + |q|)^{1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \t\\right., &&  \n\t  \t|I|= k \\leq \\dParameter - 4,\n  \t\\end{align}\n  \t\n  \t\\begin{align} \\label{E:barpartialZIh1Aupgraded}\n\t\t\t|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}| \\ + \\ (1 + |q|)|\\conenabla \\Lie_{\\mathcal{Z}}^I \\Far| \n\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}  \n\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}} \\leq\n\t\t\t\\left \\lbrace \\begin{array}{lr}\n\t    \tC_k \\varepsilon (1 + t + |q|)^{-2 + M_k \\varepsilon} (1 + |q|)^{- \\upgamma'}, &  \\mbox{if} \\ q > 0, \\\\\n\t      C_k \\varepsilon (1 + t + |q|)^{-2 + M_k \\varepsilon} (1 + |q|)^{1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \t\\right., && \n\t  \t|I|= k \\leq \\dParameter - 5. \n  \t\\end{align}\n  \t\\end{subequations}\n  \t\n  \tFurthermore, the same estimates hold for $h_{\\mu \\nu} \\eqdef g_{\\mu \\nu} - m_{\\mu \\nu}$ and \n  \t$H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu}$ if we replace $\\upgamma'$ with $M_k \\varepsilon.$\n  \t\n\n\\end{proposition}\n\n\n\t \t\n\n \\subsection{Proof of Proposition \\ref{P:UpgradedDecayhA}} \\label{SS:ProofofUpgradedDecayhA}\n\t\tWe only prove the estimates for $h_{\\mu \\nu}$ and $\\Far_{\\mu \\nu}.$ The estimates for $h_{\\mu \\nu}^{(0)},$ $h_{\\mu \n\t\t\\nu}^{(1)},$ $H^{\\mu \\nu},$ $H_{(0)}^{\\mu \\nu},$ and $H_{(1)}^{\\mu \\nu}$ follow easily from those for \n\t\t$h_{\\mu \\nu},$ \\eqref{E:Hintermsofh}, and Lemma \\ref{L:h0decayestimates}.\n\n\\subsubsection{Proofs of \\eqref{E:partialhLTpartialZhLLpointwise} and \\eqref{E:hLTZhLLpointwise}}\n\nTo prove \\eqref{E:partialhLTpartialZhLLpointwise} and \\eqref{E:hLTZhLLpointwise}, we will argue as in Lemma 10.4 of\n\\cite{hLiR2010}; we first provide a lemma that establishes a more general version of the desired estimates.\n\n\n\n\\begin{lemma} \\label{L:UpgradedDecayhA} \\cite[Lemma 10.4]{hLiR2010} \n\t\\textbf{(Pointwise estimates for $|\\nabla\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L} \\mathcal{L}},$\n\t$|\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L} \\mathcal{L}},$ $|\\nabla\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L} \\mathcal{T}},$ \n\tand $|\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L} \\mathcal{T}}$)}\n\tUnder the hypotheses of Proposition \\ref{P:UpgradedDecayhA}, if $k \\leq \\dParameter - 3$ \n\tand $\\varepsilon$ is sufficiently small, then the following pointwise estimates hold for $(t,x) \\in [0,T) \\times \n\t\\mathbb{R}^3:$\n\t\n\t\\begin{align} \\label{E:partialZIhLLpluspartialZJhLTLinfinity}\n\t\t\\sum_{|I| \\leq k} |\\nabla\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L} \\mathcal{L}}\n\t\t\t\\ + \\ \\underbrace{\\sum_{|J| \\leq k - 1} \n\t\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L} \\mathcal{T}}}_{\\mbox{absent if $k = 0$}}\n\t\t\t\\lesssim \\underbrace{\\sum_{|K| \\leq k - 2} |\\nabla\\nabla_{\\mathcal{Z}}^K h|}_{\\mbox{absent if $k \\leq 1$}} \n\t\t\t\\ + \\ \\left\\lbrace \\begin{array}{lr}\n\t    \t\\varepsilon (1 + t + |q|)^{-2 + 2 \\updelta} (1 + |q|)^{- 2 \\updelta}, &  \\mbox{if} \\ q > 0, \\\\\n\t      \\varepsilon (1 + t + |q|)^{-2 + 2 \\updelta} (1 + |q|)^{1\/2 - \\updelta}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \t\\right., \\\\\n\t \t\\sum_{|I| \\leq k} |\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L} \\mathcal{L}}\n\t\t\t\\ + \\ \\underbrace{\\sum_{|J| \\leq k - 1} |\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L} \\mathcal{T}}}_{\\mbox{absent if $k = 0$}}\n\t\t\t\\lesssim \\underbrace{\\sum_{|K| \\leq k - 2} \\int_{\\varrho = |x|}^{\\varrho = |x| + t}\n\t\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^K h|(t + |q| - \\varrho, \\varrho x\/|x|) \\, d \\varrho }_{\\mbox{absent if $k \\leq 1$}}\n\t\t\t\\ + \\ \\left\\lbrace \\begin{array}{lr}\n\t    \t\\varepsilon (1 + t + |q|)^{-1}, &  \\mbox{if} \\ q > 0, \\\\\n\t      \\varepsilon (1 + t + |q|)^{-1} (1 + |q|)^{1\/2 + \\updelta}, & \\mbox{if} \\ q < 0.\n\t    \\end{array}\n\t  \\right. \\label{E:partialZIhLLpluspartialZJhLTLinfinityintegrated}\n\t\\end{align}\t\n\tFurthermore, the same estimates hold for the tensor $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu}.$\n\\end{lemma}\t\n\n\\begin{proof}\n\tBy Proposition \\ref{P:harmonicgauge}, we have that \n\t\n\t\\begin{align} \\label{E:harmonicgaugeagain}\n\t\t\\sum_{|I| \\leq k} |\\nabla\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L} \\mathcal{L}}\n\t\t\t\\ + \\ \\underbrace{\\sum_{|J| \\leq k - 1} |\\nabla\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L} \\mathcal{T}}}_{\\mbox{absent if \n\t\t\t$k = 0$}} \n\t\t\t\\lesssim \\underbrace{\\sum_{|K| \\leq k - 2} |\\nabla\\nabla_{\\mathcal{Z}}^{J} h|}_{\\mbox{absent if $k \\leq 1$}}\n\t\t\t\\ + \\ \\sum_{|J| \\leq k} |\\conenabla \\nabla_{\\mathcal{Z}}^J h|\n\t\t\t\\ + \\ \\sum_{|I_1| + |I_2| \\leq k} |\\nabla_{\\mathcal{Z}}^{I_1} h||\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h|.\n\t\\end{align}\n\tBy Corollary \\ref{C:WeakDecay}, we have that\n\t\n\t\\begin{align} \\label{E:Tangentialhderivatives}\n\t\t\\sum_{|J| \\leq k} |\\conenabla \\nabla_{\\mathcal{Z}}^J h| \n\t\t\\ + \\ \\sum_{|I_1| + |I_2| \\leq k} |\\nabla_{\\mathcal{Z}}^{I_1} h||\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h|  \n\t\t\\lesssim \\left \\lbrace \\begin{array}{lr}\n    \t\\varepsilon (1 + t + |q|)^{-2 + 2\\updelta} (1 + |q|)^{- 2\\updelta}, &  \\mbox{if} \\ q > 0, \\\\\n      \\varepsilon (1 + t + |q|)^{-2 + 2\\updelta} (1 + |q|)^{1\/2 - \\updelta}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \t\\right., &&\n  \tk \\leq \\dParameter - 3.\n\t\\end{align}\n\tCombining \\eqref{E:harmonicgaugeagain} and \\eqref{E:Tangentialhderivatives},\n\twe deduce \\eqref{E:partialZIhLLpluspartialZJhLTLinfinity}. Inequality \n\t\\eqref{E:partialZIhLLpluspartialZJhLTLinfinityintegrated}\n\tfollows from integrating inequality  \\eqref{E:partialZIhLLpluspartialZJhLTLinfinity}\n\tfor $|\\partial_q \\nabla_{\\mathcal{Z}}^I h| \\lesssim |\\nabla\\nabla_{\\mathcal{Z}}^I h|,$ $q \\eqdef |x| - t,$ along the \n\tlines along which the angle $\\omega \\eqdef x\/|x|$ and the null coordinate $s = |x| + t$ are constant (i.e. integrating $dq$), \n\tand using \\eqref{E:weakdecayLinfinity} at $t=0.$\n\t\n\tThe proofs of the estimates for $H^{\\mu \\nu}$ follow from the estimates for $h_{\\mu \\nu},$ \\eqref{E:Hintermsofh},\n\tand Corollary \\ref{C:WeakDecay}.\n\n\\end{proof}\n\n\nInequalities \\eqref{E:partialhLTpartialZhLLpointwise} and \\eqref{E:hLTZhLLpointwise} now follow from\ninequalities \\eqref{E:partialZIhLLpluspartialZJhLTLinfinity}, \\eqref{E:partialZIhLLpluspartialZJhLTLinfinityintegrated},\nand the weak decay estimates of Corollary \\ref{C:WeakDecay}.\n\n\\hfill $\\qed$\n\n\\subsubsection{Proof of \\eqref{E:Farupgradedecay}} \\label{SSS:ProofofFarUpgradedDecay}\n\tLet $\\mathcal{W} \\eqdef \\big\\lbrace (t,x): |x| \\geq 1 + t\/2 \\big\\rbrace \\cap \\big\\lbrace \n\t(t,x): |x| \\leq 2t - 1 \\big\\rbrace$ denote the ``wave zone'' region. Note that $r \\approx 1 + t + |q| \\approx 1 + t + s$ for \n\t$(t,x) \\in \\mathcal{W}.$ Now as in the proof of Proposition \\ref{P:FLUTTimproveddecay}, inequality \n\t\\eqref{E:Farupgradedecay}\n\tfollows from the weak decay estimates of Corollary \\ref{C:WeakDecay} if $(t,x) \\nin \\mathcal{W}.$\n\tFurthermore, we have that $|\\Far| \\approx |\\ualpha[\\Far]| + |\\alpha[\\Far]| + |\\rho[\\Far]| + |\\sigma[\\Far]|,$ and\n\tby Proposition \\ref{P:FLUTTimproveddecay}, inequality \\eqref{E:Farupgradedecay}\n\thas already been shown to hold for \n\t$|\\alpha[\\Far]| + |\\rho[\\Far]| + |\\sigma[\\Far]| \\approx |\\Far|_{\\mathcal{L}\\mathcal{N}} + |\\Far|_{\\mathcal{T}\\mathcal{T}}.$\n\t\n\t\n\tIt remains to prove the desired estimate for $|\\ualpha[\\Far(t,x)]|$ under the assumption that\n\t$(t,x) \\in \\mathcal{W}.$ To this end, we use \\eqref{E:ODErualpha}, the weak decay \n\testimates of Corollary \\ref{C:WeakDecay}, and Proposition \\ref{P:FLUTTimproveddecay} to deduce that \n\tif $(t,x) \\in \\mathcal{W},$ then\n\t\n\t\\begin{align} \\label{E:LambdarualphaBound}\n\t\t\\big|\\nabla_{\\Lambda} \\big(r\\ualpha[\\Far] \\big)\\big| \n\t\t\t& \\lesssim  \\varepsilon (1 + t + |q|)^{-3\/2 + \\updelta} \\big|r\\ualpha[\\Far] \\big| \n\t\t\t\\ + \\ \\varepsilon (1 + t + |q|)^{-2 + 3 \\updelta},\n  \\end{align}\n  where $\\Lambda \\eqdef L + \\frac{1}{4} h_{LL} \\uL.$ Let $\\big(\\tau(\\lambda),y(\\lambda) \\big) $ be the integral \n  curve\\footnote{By integral curve,\n  we mean the solution to the ODE system $\\frac{d \\tau}{d \\lambda} = \\Lambda^0(\\tau,y),$ \n  $\\frac{d y^j}{d \\lambda^j} = \\Lambda^j(\\tau,y),$ $(j=1,2,3),$ passing through the point $(t,x).$} of the \n  vectorfield $\\Lambda$ passing through the point $(t,x) = \\big(\\tau(\\lambda_1),y(\\lambda_1) \\big) \\in \\mathcal{W}.$ \n\tBy the already-proven smallness estimate \\eqref{E:hLTZhLLpointwise} for $h_{LL},$ every such integral curve must intersect \n\tthe boundary of $\\mathcal{W}$ at a point $(t_0,x_0) = \\big(\\tau(\\lambda_0),y(\\lambda_0) \\big)$ to the \\emph{past} of $(t,x).$\n\tFurthermore, by \\eqref{E:hLTZhLLpointwise} again, we have that\n\t$\\frac{d \\tau}{d\\lambda} \\approx 1$ along the integral curves, and for all $(\\tau, y) \\in \\mathcal{W},$ we have\n\tthat $|y| \\approx \\tau \\approx 1 + |\\tau| \\approx 1 + |\\tau| + \\big||y| - \\tau\\big|.$\n\tWe now set $f(\\lambda) \\eqdef \\Big||y(\\lambda)|\\ualpha\\big[\\Far\\big(\\tau(\\lambda),y(\\lambda)\\big)\\big]\\Big|,$\n\tintegrate inequality \\eqref{E:LambdarualphaBound} along the integral curve\n\t(which is contained in $\\mathcal{W}$), use the assumption $0 < \\updelta < 1\/4,$\n\tand change variables so that $\\tau$ is the integration variable to obtain \n\t\t\n\t\t\\begin{align} \\label{E:rualphaGronwallready}\n\t\t\t\\overbrace{\\big|r \\ualpha[\\Far](t,x) \\big|}^{f(\\lambda(t))}\n\t\t\t& \\leq \\overbrace{\\big|r_0 \\ualpha[\\Far(t_0,x_0)]\\big|}^{f(\\lambda_0)}\n\t\t\t\t\\ + \\ C \\varepsilon \\int_{\\lambda = \\lambda_0}^{\\lambda = \\lambda_1} [1 + \\tau(\\lambda)]^{-2 + 3 \\updelta} d \\lambda \n\t\t\t \t\\ + \\ C \\varepsilon \\int_{\\lambda = \\lambda_0}^{\\lambda = \\lambda_1} [1 + \\tau(\\lambda)]^{-3\/2 + \\updelta}  \n\t\t\t\tf(\\lambda) d \\lambda \\\\\n\t\t\t& \\leq C \\varepsilon \n\t\t\t\t\\ + \\ C \\varepsilon \\int_{\\tau = t_0}^{\\tau = t} (1 + \\tau)^{-2 + 3 \\updelta} d \\tau \n\t\t\t\t\\ + \\ C \\varepsilon \\int_{\\tau = t_0}^{\\tau = t} (1 + \\tau)^{-3\/2 + \\updelta}\n\t\t\t\tf(\\lambda \\circ \\tau) d \\tau \\notag \\\\\n\t\t\t& \\leq C \\varepsilon \\ + \\ C \\varepsilon \\int_{\\tau = t_0}^{\\tau = t} (1 + \\tau)^{-3\/2 + \n\t\t\t\t\\updelta} f(\\lambda \\circ \\tau) d \\tau, \\notag\n\t\t\\end{align}\n\t\twhere we have used \\eqref{E:weakdecaypartialLinfinity} to obtain the bound \n\t\t$\\big|r_0 \\ualpha[\\Far(t_0,x_0)] \\big| \\leq C \\varepsilon$\n\t\tfor points $(t_0,x_0)$ lying on the boundary of $\\mathcal{W}.$ Applying Gronwall's lemma to \n\t\t\\eqref{E:rualphaGronwallready}, we have that\n\t\t\n\t\t\\begin{align} \\label{E:rualphaGronwall}\n\t\t\t\\big|r \\ualpha[\\Far(t,x)] \\big| & \\leq C \\varepsilon \n\t\t\t\\exp\\bigg(C \\varepsilon \\int_{\\tau = t_0}^{\\tau = t} \n\t\t\t(1 + \\tau)^{-3\/2 + \\updelta} d \\tau \\bigg) \\\\\n\t\t\t& \\leq C \\varepsilon \\exp\\bigg(C \\varepsilon \\int_{\\tau = t_0}^{\\tau = t} (1 + \\tau)^{-3\/2 + \\updelta} d \\tau \\bigg)\n\t\t\t\\leq C \\varepsilon, \\notag\n\t\t\\end{align}\n\t\tfrom which it trivially follows that\n\t\t\n\t\t\\begin{align}\n\t\t\t\\big| \\ualpha[\\Far(t,x)] \\big| & \\leq C \\varepsilon r^{-1} \\leq C \\varepsilon (1 + t + |q|)^{-1}\n\t\t\\end{align}\n\t\tas desired.\n\t\t\\hfill $\\qed$\n\t\t\n\t\t\n\n\n\\subsubsection{Proofs of \\eqref{E:partialhTUpointwise} - \\eqref{E:partialhpointwise}}\n\nIn the next two lemmas, we will use the fact that the tensorfield $h_{\\mu \\nu} \\eqdef g_{\\mu \\nu} - m_{\\mu \\nu}$ is a solution to the\nsystem\n\n\\begin{align} \\label{E:hwavesystem}\n\t\\widetilde{\\Square}_g h_{\\mu \\nu} = \\mathfrak{H}_{\\mu \\nu},\n\\end{align}\nwhere the inhomogeneous term $\\mathfrak{H}_{\\mu \\nu}$ is defined in \\eqref{E:HtriangleSmallAlgebraic}. \n\n\\begin{lemma} \\cite[Extension of Lemma 10.5]{hLiR2010}\\label{L:Inhomogeneousdecayestimates}\n\t\\textbf{(Pointwise estimates for the $\\mathfrak{H}_{\\mu \\nu}$ inhomogeneities)}\n\tSuppose the assumptions of Proposition \\ref{P:UpgradedDecayhA} hold. Then if $\\varepsilon$ is sufficiently small, the \n\tfollowing pointwise estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t|\\mathfrak{H}|_{\\mathcal{T} \\mathcal{N}} & \\leq C \\varepsilon (1 + t + |q|)^{-3\/2 + \\updelta} |\\nabla h|\n\t\t\t\\ + \\ C \\varepsilon (1 + t + |q|)^{-5\/2 + \\updelta}, \\label{E:mathfrakHTULinfinity} \\\\\n\t\t|\\mathfrak{H}| & \\leq C \\varepsilon (1 + t + |q|)^{-3\/2 + \\updelta} |\\nabla h|\n\t\t\t\\ + \\ C |\\nabla h|_{\\mathcal{T} \\mathcal{N}}^2 \\ + \\ C \\varepsilon^2 (1 + t + |q|)^{-2}.   \\label{E:mathfrakHLinfinity} \n\t\\end{align}\n\t\\end{subequations}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\tLemma \\ref{L:Inhomogeneousdecayestimates} follows from Proposition \\ref{P:AlgebraicInhomogeneous}, Corollary \n\t\\ref{C:WeakDecay}, Proposition \\ref{P:FLUTTimproveddecay}, the already-proven estimate \\eqref{E:Farupgradedecay},\n\tand the assumption $0 < \\updelta < 1\/4.$\n\\end{proof}\n\n\n\n\\begin{lemma} \\cite[Extension of Lemma 10.6]{hLiR2010} \\label{L:partialhTUpartialhdecay} \n\t\\textbf{(Integral inequalities for $|\\nabla h|_{\\mathcal{T} \\mathcal{N}}$ and $|\\nabla h|$)}\n\tSuppose the assumptions of Proposition \\ref{P:UpgradedDecayhA} hold.\n\tThen if $\\varepsilon$ is sufficiently small, the following integral inequalities hold for $t \\in [0,T):$\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t(1 + t) \\big\\| |\\nabla h|_{\\mathcal{T} \\mathcal{N}}(t,\\cdot) \\big\\|_{L^{\\infty}}\n\t\t\t& \\leq C \\varepsilon \\ + \\ C \\varepsilon \\int_{0}^t (1 + \\tau)^{- 1\/2 + \\updelta} \n\t\t \t\\big\\| \\nabla h(\\tau, \\cdot) \\big\\|_{L^{\\infty}} \\, d\\tau,  \\label{E:partialhTUdecay} \\\\\n \t\t(1 + t) \\big\\| \\nabla h(t,\\cdot)  \\big\\|_{L^{\\infty}} \\label{E:partialhdecay}\n\t\t& \\leq C \\varepsilon \\ + \\ C \\varepsilon^2 \\ln(1 + t) \\ + \\ C \\varepsilon \\int_{0}^t (1 + \\tau)^{- 1\/2 + \\updelta} \n\t\t \t\\big\\| \\nabla h(\\tau,\\cdot) \\big\\|_{L^{\\infty}} \\, d\\tau \\\\\n\t\t & \\ \\ + \\ C \\varepsilon \\int_{0}^t (1 + \\tau) \\big\\| |\\nabla h|_{\\mathcal{T} \\mathcal{N}}^2(\\tau, \\cdot) \n\t\t \\big\\|_{L^{\\infty}} \\, d\\tau. \\notag\n\t\\end{align}\n\t\\end{subequations}\n\n\\end{lemma}\n\n\\begin{proof}\n\tFirst observe that \\eqref{E:weakdecayLinfinity} and \\eqref{E:hLTZhLLpointwise} (the version for the tensor $H$) imply that \n\tthe hypotheses of Lemma \\ref{L:scalardecay} and Corollary \\ref{C:systemdecay} hold. Therefore, using the lemma and the \n\tcorollary, with $\\varpi(q) \\eqdef 1$ and $\\upalpha \\eqdef 0,$ and noting that $h_{\\mu \\nu}$ verifies the system \n\t\\eqref{E:hwavesystem}, we have that \n\t\n\t\\begin{align} \\label{E:partialhTUfirstinequality}\n\t\t(1 + t) |\\nabla h|_{\\mathcal{T} \\mathcal{N}}\n\t\t& \\lesssim \\sup_{0 \\leq \\tau \\leq t} \n\t\t\t\\sum_{|I| \\leq 1} \n\t\t\t\\big\\| \\nabla_{\\mathcal{Z}}^I h(t,\\cdot) \\big\\|_{L^{\\infty}}  \n\t\t\t\\ + \\ \\int_{\\tau = 0}^{t} \n\t\t\t(1 + \\tau) \\big\\| |\\mathfrak{H}|_{\\mathcal{T} \\mathcal{N}} \\big\\|_{L^{\\infty}(D_{\\tau})} \\, d\\tau \n\t\t\t\\ + \\ \\sum_{|I| \\leq 2} \\int_{\\tau = 0}^{t} \n\t\t\t(1 + \\tau)^{-1} \\big\\| \\nabla_{\\mathcal{Z}}^I h \\big\\|_{L^{\\infty}(D_{\\tau})} \\, d \\tau. \n\t\\end{align}\n\tUsing \\eqref{E:weakdecayLinfinity} (the version for the tensor $h$), we estimate the \n\tthe first and third terms on the right-hand side of \\eqref{E:partialhTUfirstinequality}\n\tas follows:\n\t\n\t\\begin{align}\n\t\t\\sup_{0 \\leq \\tau \\leq t} \\sum_{|I| \\leq 1} \\big\\| \\nabla_{\\mathcal{Z}}^I h \\big\\|_{L^{\\infty}} \n\t\t& \\leq C \\varepsilon (1 + t)^{-1\/2 + \\updelta} \\leq C \\varepsilon, \\label{E:partialhTUdecayfirstterm} \\\\\n\t\t\\sum_{|I| \\leq 2} \\int_{\\tau = 0}^{t} \n\t\t(1 + \\tau)^{-1} \\big\\| \\nabla_{\\mathcal{Z}}^I h \\big\\|_{L^{\\infty}(D_{\\tau})} \\, d \\tau \n\t\t& \\leq C \\varepsilon \\int_{\\tau = 0}^{\\infty} (1 + \\tau)^{-3\/2 + \\updelta} \\, d \\tau\n\t\t\t\\leq C \\varepsilon. \\label{E:partialhTUdecaythirdterm}\n\t\\end{align}\n\tTo estimate the second term, we use \\eqref{E:mathfrakHTULinfinity} to conclude that for $x \\in D_t,$ we have that\n\t\n\t\\begin{align}\n\t\t(1 + t) |\\mathfrak{H}|_{\\mathcal{T} \\mathcal{N}}\n\t\t\t& \\leq C \\varepsilon (1 + t)^{- 1\/2 + \\updelta} |\\nabla h| \n\t\t\t\\ + \\ C \\varepsilon (1 + t)^{-3\/2 + \\updelta}. \\label{E:partialhTUdecaysecondterm}\n\t\\end{align}\n\tInequality \\eqref{E:partialhTUdecay} now follows from \\eqref{E:partialhTUfirstinequality} - \\eqref{E:partialhTUdecaysecondterm}, and the fact \n\tthat $C \\varepsilon \\int_{0}^t (1 + \\tau)^{-3\/2 + \\updelta} \\, d \\tau \\leq C \\varepsilon.$ Inequality \\eqref{E:partialhdecay} \n\tcan be obtained in a similar fashion using \\eqref{E:mathfrakHLinfinity}.\n\n\\end{proof}\n\n\nTo finish the proof of Proposition \\ref{P:UpgradedDecayhA}, we will use the following Gronwall-type lemma.\n\n\\begin{lemma} \\cite[Slight modification of Lemma 10.7]{hLiR2010} \\label{L:Gronwall} \\textbf{(Gronwall lemma)}\n\tAssume that the continuous functions $b(t) \\geq 0$ and $c(t) \\geq 0$ satisfy\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\tb(t) & \\leq C \\varepsilon \\ + \\ C \\varepsilon \\int_{0}^{t} (1 + \\tau)^{-1 - a} c(\\tau) \\, d \\tau, \n\t\t\t\\label{E:boftGronwall} \\\\\n\t\tc(t) & \\leq C \\varepsilon \\ + \\ C \\varepsilon^2 \\ln(1 + t) \n\t\t\t\\ + \\ C \\varepsilon \\int_{0}^{t} (1 + \\tau)^{-1 - a} c(\\tau) \\, d \\tau \n\t\t\t\\ + \\ C \\int_{0}^{t} (1 + \\tau)^{-1} b^2(\\tau) \\, d \\tau \\label{E:coftGronwall}\n\t\\end{align}\n\t\\end{subequations}\n\tfor some positive constants $a, C$ such that $\\varepsilon < \\frac{a}{4C}$ \n\tand $\\varepsilon < \\frac{2a}{(1 + 4C^2)}.$ Then \n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\tb(t) & \\leq 2 C \\varepsilon, \\label{E:bBound} \\\\\n\t\tc(t) & \\leq 2 C \\varepsilon \\big\\lbrace 1 + a \\ln(1 + t) \\big\\rbrace. \\label{E:cBound}\n\t\\end{align}\n\t\\end{subequations}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\tWe slightly modify the proof of \\cite[Lemma 10.7]{hLiR2010}. Let $T$ be the largest time such that the bounds \n\t\\eqref{E:bBound} - \\eqref{E:cBound} hold. Then inserting these bounds into\n\tthe inequalities \\eqref{E:boftGronwall} - \\eqref{E:coftGronwall}, and using the bound\n\t(and the change of variables $z \\eqdef a \\ln(1 + \\tau)$)\n\t\n\t\\begin{align}\n\t\t\\int_{\\tau=0}^{\\infty} (1 + \\tau)^{-1 - a} \\big\\lbrace1 + a \\ln(1 + \\tau) \\big\\rbrace \\, d\\tau \n\t\t\t\\leq \\int_{\\tau = 0}^{\\infty} (1 + a^{-1} z) e^{- z} \\, dz = 2 a^{-1},\n\t\\end{align}\n\twe deduce that the following inequalities hold for $t \\in [0,T]:$\n\t\n\t\\begin{align}\n\t\tb(t) & \\leq C \\varepsilon \\big\\lbrace 1 + 4 C \\varepsilon a^{-1}   \\big\\rbrace < 2 C \\varepsilon, \\\\\n\t\tc(t) & \\leq C \\varepsilon \\big\\lbrace 1 + 4 C \\varepsilon a^{-1} + (1 + 4C^2) \\varepsilon \\ln(1 + t)  \\big\\rbrace\n\t\t\t< 2 C \\varepsilon \\big\\lbrace 1 + a \\ln(1 + t) \\big\\rbrace.\n\t\\end{align}\n\tSince the above inequalities are a strict improvement of the assumed bounds \\eqref{E:bBound} - \\eqref{E:cBound},\n\twe thus conclude that $T = \\infty.$\n\\end{proof}\n\n\nTo complete the proof of \\eqref{E:partialhTUpointwise} - \\eqref{E:partialhpointwise}, we apply Lemmas \\ref{L:partialhTUpartialhdecay} and \\ref{L:Gronwall} with \n$b(t) \\eqdef (1 + t) \\big\\| |\\nabla h|_{\\mathcal{T} \\mathcal{N}}(t,\\cdot) \\big\\|_{L^{\\infty}}$\nand $c(t) \\eqdef (1 + t) \\big\\| \\nabla h(t,\\cdot) \\big\\|_{L^{\\infty}}.$ This implies\n\\eqref{E:partialhTUpointwise} - \\eqref{E:partialhpointwise} with $(1 + t)$ in place of $(1 + t + |q|).$\nThe additional decay in $|q|$ in \\eqref{E:partialhTUpointwise} and \\eqref{E:partialhpointwise} follows directly from \\eqref{E:weakdecaypartialLinfinity} (the version for the tensor $h$). \\hfill $\\qed$\n \n\n\n\n\n\\subsection{Proof of Proposition \\ref{P:UpgradedDecayh1A}} \\label{SS:Proof of PropositionUpgradedDecayh1A}\n\nWe will prove the proposition using a series of inductive steps. We only prove the estimates for \n$h_{\\mu \\nu}^{(1)}$ and $\\Far_{\\mu \\nu}.$ The estimates for $h_{\\mu \\nu}$ and $H^{\\mu \\nu}$ follow easily from those for \n$h_{\\mu \\nu}^{(1)},$ \\eqref{E:Hintermsofh}, and Lemma \\ref{L:h0decayestimates}. We first prove a technical lemma that will be used during the proof of the proposition.\n\n\\begin{lemma} \\label{L:UpgradedInhomogeneousPointwise} \\textbf{(Pointwise estimates for \nthe $|\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}|$ inhomogeneities)}\n\tSuppose the hypotheses of Proposition \\ref{P:UpgradedDecayhA} hold, and let $\\mathfrak{H}_{\\mu \\nu}$ \n\tbe the inhomogeneous term on the right-hand side of \n\tthe reduced equation \\eqref{E:Reducedh1Summary}. Then if $I$ is any $\\mathcal{Z}-$multi-index with $|I| \\leq \\dParameter ,$ the following\n\tpointwise estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{align}  \\label{E:UpgradedInhomogeneousPointwise}\n\t\t|\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}| \n\t\t& \\leq C \\varepsilon \\sum_{|J| \\leq |I|} (1 + t + |q|)^{-1} \n\t\t\t\\Big(|\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}| + |\\nabla_{\\mathcal{Z}}^J \\Far| \\Big) \\\\\n\t\t& \\ \\ + \\ C \\mathop{\\sum_{|I_1| + |I_2| \\leq |I|}}_{|I_1|, |I_2| \\leq |I|- 1} \n\t\t\t\\Big(|\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}| + |\\Lie_{\\mathcal{Z}}^{I_1} \\Far| \\Big)\n\t\t\t\\Big(|\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h^{(1)}| + |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \\Big) \\notag \\\\\n\t\t& \\ \\ + \\ C \\varepsilon^2 (1 + t + |q|)^{-4}. \\notag\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tLemma \\ref{L:UpgradedInhomogeneousPointwise} follows from \\eqref{E:ZIinhomogeneoushpointwise},\n\t\\eqref{E:partialhTUpointwise}, \\eqref{E:Farupgradedecay}, Lemma \\ref{L:h0decayestimates}, the weak decay estimates of \n\tCorollary \\ref{C:WeakDecay}, \\eqref{E:partialhTUpointwise}, and the fact that $0 < \\updelta < 1\/4.$ We remark that the \n\t$C \\varepsilon^2 (1 + t + |q|)^{-4}$ term arises from the estimate \n\t${|\\nabla\\nabla_{\\mathcal{Z}}^{I_1} h^{(0)}| |\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h^{(0)}| \\leq C \\varepsilon^2 (1 + t + \n\t|q|)^{-4}}.$\n\\end{proof}\n\n\n\nWe are now ready for the proof of the proposition. To prove \\eqref{E:partialZIh1Aupgraded} - \\eqref{E:barpartialZIh1Aupgraded}, we will argue inductively, using the inequalities in the case $|I| \\leq k$ to deduce that they hold in the case $|I| = k+1.$ We also remark that the base case $k=0$ is covered by our argument. \n\\\\\n\n\\noindent \\textbf{Induction step 1: upgraded pointwise decay estimates for $|\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L}\\mathcal{L}}|$\nfor $|I| = k + 1$ and $|\\nabla_{\\mathcal{Z}}^{J} h|_{\\mathcal{L}\\mathcal{T}}|$ for $|J| = k.$}\n\nAs a first step, we will use the wave coordinate condition to upgrade the estimates for $|\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L}\\mathcal{L}}$ for $|J| = k + 1$ and $|\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L}\\mathcal{T}}$ for $|J| = k.$ To this end, we appeal to inequality \\eqref{E:partialZIhLLpluspartialZJhLTLinfinityintegrated}, \nusing inequality \\eqref{E:partialZIh1Aupgraded} for $h$ under the induction hypothesis to bound the integrand, to deduce that\n\n\\begin{align} \\label{E:LieZJLiehLLZJprimeIndependentInductionEstimate}\n\t\\sum_{|I| = k + 1} |\\nabla_{\\mathcal{Z}}^I h|_{\\mathcal{L}\\mathcal{L}}\n\t\\ + \\ \\sum_{|J| = k} |\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{L}\\mathcal{T}} \n \t& \\lesssim \\left\\lbrace \\begin{array}{lr}\n\t    \t\\varepsilon (1 + t + |q|)^{-1 + M_{k-1} \\varepsilon} (1 + |q|)^{- M_{k-1} \\varepsilon}, &  \\mbox{if} \\ q > 0, \\\\\n\t      \\varepsilon (1 + t + |q|)^{-1 + M_{k-1} \\varepsilon} (1 + |q|)^{1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0.\n\t    \\end{array}\n\t  \t\\right.\n\\end{align}\nIn the above estimates, the constant $\\upmu'$ is subject to the restrictions stated in the hypotheses \nof Proposition \\ref{P:UpgradedDecayh1A}. Furthermore, since $H^{\\mu \\nu} = -h^{\\mu \\nu} + O^{\\infty}(|h|^2),$ \\eqref{E:weakdecayLinfinity} implies that the same estimates hold for the tensor $H.$\n\\\\\n\n\n\\noindent \\textbf{Induction step 2: upgraded pointwise decay estimates for $\\big|\\Lie_{\\mathcal{Z}}^I \\Far \\big|,$ $|I| = k+1.$}\n\nLet $\\mathcal{W} \\eqdef \\big\\lbrace (t,x): |x| \\geq 1 + t\/2 \\big\\rbrace \\cap \\big\\lbrace (t,x): |x| \\leq 2t - 1 \\big\\rbrace$\ndenote the ``wave zone'' region. Then for $(t,x) \\nin \\mathcal{W},$ we have that $1 + |q| \\approx 1 + t + |q|.$ Using this fact, for $(t,x) \\nin \\mathcal{W},$ the weak decay estimate \\eqref{E:weakdecaypartialLinfinity} implies that inequality \\eqref{E:partialZIh1Aupgraded} holds for $|\\Lie_{\\mathcal{Z}}^I \\Far|$ in the case $|I| = k + 1.$ Furthermore, by Proposition \\ref{P:FLUTTimproveddecay}, the inequality \\eqref{E:partialZIh1Aupgraded} holds for the null components $\\big|\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far]\\big|,$ $\\big|\\rho[\\Lie_{\\mathcal{Z}}^I \\Far]\\big|,$ and \n$\\big|\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]\\big|$ when $|I| = k + 1.$\n\nIt remains to consider $\\big|\\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far(t,x)]\\big|$ in the case $(t,x) \\in \\mathcal{W}.$ Note that $r \\approx 1 + t + |q| \\approx 1 + t + s$ for $(t,x) \\in \\mathcal{W}.$ We will make use of the weight $\\varpi(q)$ defined in \\eqref{E:decayweight}. Using \\eqref{E:LambdaLieZIualphaEquationGoodqWeights}, Corollary \\ref{C:WeakDecay} (the version for the tensorfield $h$), Proposition \\ref{P:FLUTTimproveddecay}, \\eqref{E:hLTZhLLpointwise}, \\eqref{E:Farupgradedecay}, the induction\nhypothesis, and \\eqref{E:LieZJLiehLLZJprimeIndependentInductionEstimate}, it follows that\n\n\\begin{align} \\label{E:ualphaGoodQweightsReadytobeIntegrated}\n\t\\sum_{|I| \\leq k + 1} \\big|\\nabla_{\\Lambda} \\big(r \\varpi(q) \\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far] \\big)\\big| \n\t& \\leq C \\varepsilon (1 + t + |q|)^{-1} \\sum_{|I| \\leq k + 1} \\big|r \\varpi(q) \\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far] \\big|\n\t\\ + \\ C \\varepsilon (1 + t + |q|)^{-(1 + a)}\n\t\\ + \\ C \\varepsilon^2 (1 + t + |q|)^{-1 + C \\varepsilon},\n\\end{align}\nwhere $0 < a < \\min \\lbrace \\upmu' - \\updelta, \\upgamma - \\updelta - \\upgamma' \\rbrace$  is a fixed constant, and\n$\\Lambda \\eqdef L + \\frac{1}{4} h_{LL} \\uL.$ Note the importance of the independent estimate \n\\eqref{E:hLTZhLLpointwise} for bounding the second, fourth, and fifth sums on the right-hand side of\n\\eqref{E:LambdaLieZIualphaEquationGoodqWeights}, and of the independent estimate \n\\eqref{E:LieZJLiehLLZJprimeIndependentInductionEstimate} (in the case $|I| = k + 1$) \nfor bounding the third sum on the right-hand side of \\eqref{E:LambdaLieZIualphaEquationGoodqWeights}.\n\nLet $\\big(\\tau(\\lambda),y(\\lambda) \\big) $ be the integral curve (as defined in Section \\ref{SSS:ProofofFarUpgradedDecay}) \nof the vectorfield $\\Lambda$ passing through the point $(t,x) = \\big(\\tau(\\lambda_1),y(\\lambda_1) \\big) \\in \\mathcal{W}.$ \nBy the inequality \\eqref{E:hLTZhLLpointwise} for $h_{LL},$ every such integral curve must intersect \nthe boundary of $\\mathcal{W}$ at a point $(t_0,x_0) = \\big(\\tau(\\lambda_0),y(\\lambda_0) \\big)$ lying to the \\emph{past} of $(t,x).$ Using \\eqref{E:hLTZhLLpointwise} again, we have that $\\frac{dt}{d\\lambda} \\approx 1$ along the integral curves, and in the entire region $\\mathcal{W},$ we have that $|y| \\approx \\tau \\approx 1 + |\\tau| \\approx 1 + |\\tau| + \\big||y| - \\tau\\big|.$\nDefine $f(\\lambda) \\eqdef \\sum_{|I| \\leq k + 1} \\Big||y(\\lambda)| \\varpi\\big(q(\\lambda)\\big) \n\\ualpha\\big[\\Lie_{\\mathcal{Z}}^I \\Far\\big(\\tau(\\lambda),y(\\lambda) \\big)\\big]\\Big|,$ where $q(\\lambda) \n\\eqdef |y(\\lambda)| - \\tau(\\lambda).$ Note that $f(\\lambda_1) = \\sum_{|I| \\leq k + 1}\\big|r \\varpi(q) \\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far]\\big|,$ where $q \\eqdef q(\\lambda_1) = |x| - t,$ while the weak decay estimate \\eqref{E:weakdecaypartialLinfinity} implies that $f(\\lambda_0) \\leq C \\varepsilon.$ Integrating inequality \\eqref{E:ualphaGoodQweightsReadytobeIntegrated} and changing variables so that $\\tau$ is the integration variable, we have that\n\n\n\\begin{align} \\label{E:ualphaGoodQweightsGronwallready}\n\t\\overbrace{f(\\lambda_1)}^{f(\\lambda \\circ t)}\n\t& \\leq f(\\lambda_0) \\ + \\ C \\varepsilon \\int_{\\lambda = \\lambda_0}^{\\lambda = \\lambda_1} \n\t\t[1 + \\tau(\\lambda)]^{-1} f(\\lambda) \\, d \\lambda\n\t\t\\ + \\ C \\varepsilon \\int_{\\lambda = \\lambda_0}^{\\lambda = \\lambda_1} [1 + \\tau(\\lambda)]^{-(1 + a)} \\, d \\lambda\n\t\t\\ + \\ C \\varepsilon^2 \\int_{\\lambda = \\lambda_0}^{\\lambda = \\lambda_1} [1 + \\tau(\\lambda)]^{-1 + C \\varepsilon} \n\t\t\t\\, d \\lambda \\\\\n\t& \\leq C \\varepsilon \\ + \\ C \\varepsilon \\int_{\\lambda = \\lambda_0}^{\\lambda = \\lambda_1} \n\t\t\t[1 + \\tau(\\lambda)]^{-1} f(\\lambda) \\, d \\lambda\n\t\t\\ + \\ C \\varepsilon \\int_{\\tau = t_0}^{\\tau = t} (1 + \\tau)^{-(1 + a)} \\, d \\tau\n\t\t\\ + \\ C \\varepsilon^2 \\int_{\\tau = t_0}^{\\tau = t} (1 + \\tau)^{-1 + C \\varepsilon} \\, d \\tau\t\\notag \\\\\n\t& \\leq C \\varepsilon (1 + t)^{C \\varepsilon}\n\t\t\\ + \\ C \\varepsilon \\int_{\\tau = t_0}^{\\tau = t} \n\t\t\t(1 + \\tau)^{-1} f(\\lambda \\circ \\tau) \\, d \\tau. \\notag\n\\end{align}\nApplying Gronwall's lemma to \\eqref{E:ualphaGoodQweightsGronwallready}, we have that\n\n\\begin{align} \\label{E:ualphaGoodqweightsGronwall}\n\tf(\\overbrace{\\lambda \\circ t}^{\\lambda_1}) \n\t& \\leq C \\varepsilon (1 + t)^{C \\varepsilon}\n\t\t\\exp \\Big( C \\varepsilon \\int_{\\tau= t_0}^{\\tau = t} (1 + \\tau)^{-1} \\, d \\tau \\Big) \\, d\\tau \\\\\n\t& \\leq C \\varepsilon (1 + t)^{2 C \\varepsilon}, \\notag\n\\end{align}\nfrom which it easily follows that for $(t,x) \\in \\mathcal{W},$ we have that\n\n\\begin{align} \\label{E:ualphaGoodqweightsinequality}\n\t\\sum_{|I| \\leq k + 1} \\big| \\ualpha[\\Lie_{\\mathcal{Z}}^I \\Far] \\big| & \\leq C \\varepsilon (1 + t)^{-1 + 2 C \\varepsilon} \n\t\\varpi^{-1}(q).\n\\end{align}\nCombining \\eqref{E:ualphaGoodqweightsinequality} and the previous arguments covering $(t,x) \\nin \\mathcal{W}$ \nand the other null components of $\\Lie_{\\mathcal{Z}}^I \\Far,$ we have shown that the estimate \\eqref{E:partialZIh1Aupgraded} holds for $|\\Lie_{\\mathcal{Z}}^I \\Far|$ in the case $|I| = k + 1.$\n\\\\\n\n\n\\noindent \\textbf{Final induction step: upgraded pointwise decay estimates for $|\\nabla\\nabla_{\\mathcal{Z}}^I h|$ and \n$|\\nabla_{\\mathcal{Z}}^I h|,$ $|I| = k + 1.$}\n\nOur first goal is to prove the following estimate in the case $|I| = k + 1:$ \n\n\\begin{align}    \\label{E:boxZIh1ALinfinityUpgraded}\n\t|\\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)}| & \\lesssim\n\t\t\\varepsilon \\sum_{|K| \\leq |I|} (1 + t + |q|)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}|\n\t\t\\ + \\ \\left\\lbrace \\begin{array}{lr}\n\t   \t\\varepsilon^2 (1 + t + |q|)^{-4 + \\updelta} (1 + |q|)^{- \\updelta}, &  \\mbox{if} \\ q > 0, \\\\\n\t   \t\\varepsilon (1 + t + |q|)^{-3}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \\right. \\\\\n\t & \\ + \\ \\left\\lbrace \\begin{array}{lr}\n\t  \t\\varepsilon^2 (1 + t + |q|)^{-2 + 2 M_k \\varepsilon} (1 + |q|)^{-2 - 2 M_k \\varepsilon}, &  \\mbox{if} \\ q > 0, \\\\\n\t   \t\\varepsilon^2 (1 + t + |q|)^{-2 + 2 M_k \\varepsilon} (1 + |q|)^{-1 + 2 \\upmu'}, & \\mbox{if} \\ q < 0.\n\t    \\end{array}\n\t  \\right. \\notag\n\\end{align}\nTo prove \\eqref{E:boxZIh1ALinfinityUpgraded}, we first recall Corollary \\ref{C:boxZIh1ALinfinity}, which\nstates that\n\n\\begin{align} \\label{E:boxZIh1ALinfinityagain}\n\t\t|\\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)}|\t\t \n\t\t\t& \\lesssim |\\nablamod_{\\mathcal{Z}}^I \\mathfrak{H}| \\ + \\ |\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}| \n\t\t\t\t\\ + \\ (1 + t + |q|)^{-1} \\mathop{\\sum_{|K| \\leq |I|}}_{|J| + (|K| - 1)_{+} \\leq |I|} |\\nabla_{\\mathcal{Z}}^J H||\\nabla \n\t\t\t\t\\nabla_{\\mathcal{Z}}^K h^{(1)}| \n\t\t\t\t\\\\\n\t\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\sum_{|K| \\leq |I|}  \n\t\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}| \\Bigg\\lbrace \\mathop{\\sum_{|J| + (|K| - 1)_{+}}}_{\\ \\ \\leq |I|} \n\t\t\t\t|\\nabla_{\\mathcal{Z}}^J H|_{\\mathcal{L} \\mathcal{L}} \n\t\t\t\t\\ + \\ \\underbrace{\\mathop{\\sum_{|J'| + (|K| - 1)_{+}}}_{\\ \\leq |I|-1} |\\nabla_{\\mathcal{Z}}^{J'} H|_{\\mathcal{L} \n\t\t\t\t\\mathcal{T}}}_{\\mbox{Absent if $|I| = 0$}}\n\t\t\t\t\\ + \\ \\underbrace{\\mathop{\\sum_{|J''| + (|K| - 1)_{+}}}_{\\ \\leq |I|-2} |\\nabla_{\\mathcal{Z}}^{J''} H|}_{\\mbox{Absent if \n\t\t\t\t$|I| \\leq 1$ or $|K| = |I|$}} \\Bigg\\rbrace, \\label{E:boxZIh1ALinfinityagainsecondterm}\n\\end{align}\nwhere $(|K|-1)_+ \\eqdef 0$ if $|K| = 0$ and $(|K|-1)_+ \\eqdef |K| - 1$ if $|K| \\geq 1.$\nWe first bound the terms from line \\eqref{E:boxZIh1ALinfinityagainsecondterm},\nconsidering separately the cases $|K| < |I|$ and $|K| = |I| = k + 1.$ For $|K| < |I| = k+1,$ we use \\eqref{E:LieZJLiehLLZJprimeIndependentInductionEstimate} (for the tensorfield $H$) and \\eqref{E:ZIh1Aupgraded}\n(for the tensorfield $H$) under the induction hypotheses to conclude that\n\n\\begin{align}\n\t(1 + |q|)^{-1} \\sum_{|J|\\leq k + 1,|J'|\\leq k,|J''|\\leq k-1} & \\Big(|\\nabla_{\\mathcal{Z}}^J H|_{\\mathcal{L} \\mathcal{L}} \n\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{J'} H|_{\\mathcal{L} \\mathcal{T}} \n\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{J''} H| \\Big) \\\\\n\t& \\lesssim \\left\\lbrace\n\t\t\\begin{array}{lr}\n\t   \t\\varepsilon (1 + t + |q|)^{-1 + M_k \\varepsilon} (1 + |q|)^{-1 - M_k \\varepsilon}, &  \\mbox{if} \\ q > 0, \\\\\n\t   \t\\varepsilon (1 + t + |q|)^{-1 + M_k \\varepsilon} (1 + |q|)^{-1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0. \t\t\n\t      \\notag\n\t   \\end{array}\n\t \\right. \\notag\n\\end{align}\nAlso using \\eqref{E:partialZIh1Aupgraded} under the induction hypotheses to bound $|\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}|,$ it follows that\nall of the terms from \\eqref{E:boxZIh1ALinfinityagainsecondterm} in the case $|K| < |I|$ can be bounded by\nthe last term on the right-hand side of \\eqref{E:boxZIh1ALinfinityUpgraded}.\n\n\nWe now consider the case $|K| = |I| = k+1.$ Since $|J| \\leq 1$ and $|J'| = 0$ in this case, we can use\n\\eqref{E:hLTZhLLpointwise} (for the tensorfield $H$) to deduce the bound\n\n\n\\begin{align}\n\t(1 + |q|)^{-1} \\sum_{|K| = |I|} & \\Bigg\\lbrace |\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}|\n\t\t\\bigg( \\mathop{\\sum_{|J| + (|K| - 1)_{+}}}_{\\ \\ \\leq |I|} |\\nabla_{\\mathcal{Z}}^J H|_{\\mathcal{L} \\mathcal{L}} \n\t\t\\ + \\ \\mathop{\\sum_{|J'| + (|K| - 1)_{+}}}_{\\ \\leq |I|-1} |\\nabla_{\\mathcal{Z}}^{J'} H|_{\\mathcal{L} \\mathcal{T}} \\bigg)   \n\t\t\\Bigg\\rbrace \\\\\n\t& \\lesssim \\varepsilon \\sum_{|K| = |I|} (1 + t + |q|)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}|. \\notag\n\\end{align}\nThus, all of the terms from \\eqref{E:boxZIh1ALinfinityagainsecondterm} in the case $|K| = |I| = k + 1$ can be bounded by\nthe first term on the right-hand side of \\eqref{E:boxZIh1ALinfinityUpgraded}.\n\n\nFor the $|\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}|$ term from the right-hand side of \\eqref{E:boxZIh1ALinfinityagain}, we simply use Lemma \\ref{L:weakdecayLinfinitynablaZISquaregh0}, which shows that $|\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}|$ is bounded by the \nnext to last term on the right-hand side of \\eqref{E:boxZIh1ALinfinityUpgraded}. \n\nTo bound the $|\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}|$ term from the right-hand side of \\eqref{E:boxZIh1ALinfinityagain},\nwe apply Lemma \\ref{L:UpgradedInhomogeneousPointwise}; the first and third terms from the right-hand side of \\eqref{E:UpgradedInhomogeneousPointwise} are manifestly bounded by the right-hand side of\n\\eqref{E:boxZIh1ALinfinityUpgraded}, while the term \n\n\\begin{align*}\n\t\\mathop{\\sum_{|J| + |K| \\leq |I|}}_{|J| \\leq |K| < |I|} \\Big(|\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}| \n\t+ |\\Lie_{\\mathcal{Z}}^J \\Far| \\Big)\n\t\\Big(|\\nabla\\nabla_{\\mathcal{Z}}^K h^{(1)}| + |\\Lie_{\\mathcal{Z}}^K \\Far| \\Big) \n\\end{align*}\t\nfrom the right-hand side of \\eqref{E:UpgradedInhomogeneousPointwise} can be bounded by the last term on the right-hand side of\n\\eqref{E:boxZIh1ALinfinityUpgraded} using the induction hypotheses, since $|J| \\leq |K| \\leq k.$ This completes the proof of \\eqref{E:boxZIh1ALinfinityUpgraded} in the case of $|I| = k + 1.$\n\n\n\nTo obtain the desired upgraded pointwise estimate for $|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|,$ we will estimate the quantity\n\n\\begin{align} \\label{E:nkplusonequantity}\n\tn_{k+1}(t) \\eqdef (1 + t) \\sum_{|I| \\leq k + 1} \\big\\| \\varpi(q) \\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}(t,\\cdot) \\big\\|_{L^{\\infty}},\n\\end{align}\nwhere $\\varpi(q)$ is the weight defined in \\eqref{E:decayweight}. Our goal is to use \nLemma \\ref{E:scalar} with $\\phi \\eqdef \\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)}$ to obtain an integral inequality for $n_{k+1}(t)$ that is amenable to Gronwall's lemma. We begin by estimating the terms on the right-hand side of \\eqref{E:systemdecay}. First, with $a \\eqdef \\min(\\upmu' - \\updelta, \\upgamma - \\updelta - \\upgamma') > 0,$ by the weak decay estimate \\eqref{E:weakdecayLinfinity}, we have that\n\n\\begin{align} \\label{E:ZIh1ZIAweighteddecayestimate}\n\t\\varpi(q)|\\nabla_{\\mathcal{Z}}^I h^{(1)}| & \\lesssim \n\t\\left \\lbrace \\begin{array}{lr}\n   \t\t\\varepsilon (1 + t + |q|)^{-1 + \\updelta} (1 + |q|)^{- \\upgamma} (1 + |q|)^{1 + \\upgamma'}, &  \\mbox{if} \\ q > 0, \\\\\n    \t\\varepsilon (1 + t + |q|)^{-1 + \\updelta} (1 + |q|)^{1\/2} (1 + |q|)^{1\/2 - \\upmu'}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \t\\right., && |I| \\leq \\dParameter - 2 \\\\\n  \t& \\lesssim \\varepsilon (1 + t)^{-a},&& |I| \\leq \\dParameter - 2. \\notag\n\\end{align}\nThis will serve as a suitable bound for estimating the first and fourth terms on the right-hand side of \\eqref{E:systemdecay}.\n\nNext, using \\eqref{E:boxZIh1ALinfinityUpgraded} and the definition \\eqref{E:nkplusonequantity}, \nwe deduce the following pointwise estimate:\n\n\\begin{align} \\label{E:boxZIh1ALinfinityWeightedUpgraded}\n\t\\varpi(q) |\\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)}|\n\t\t\\lesssim (1 + t)^{-2} \\big\\lbrace\\varepsilon n_{k+1} \\ + \\ \\varepsilon^2 (1 + t)^{2M_k \\varepsilon}\n\t\t\\ + \\ \\varepsilon (1 + t)^{-1\/2 - \\upmu'} \\big\\rbrace.\n\\end{align}\nThis will serve as a suitable bound for estimating the third integral on the right-hand side of \\eqref{E:systemdecay}.\n\nWe now apply Corollary \\ref{C:systemdecay}, using \\eqref{E:ZIh1ZIAweighteddecayestimate},\n\\eqref{E:boxZIh1ALinfinityWeightedUpgraded}, and the assumption $k + 1 \\leq \\dParameter - 4$ to deduce that\n\n\\begin{align} \\label{E:nkplus1Gronwallready}\n\tn_{k+1}(t) & \\leq C \\sup_{0 \\leq \\tau \\leq t} \\sum_{|I| \\leq k + 2} \n\t\t\\big\\| \\varpi(q) \\nabla_{\\mathcal{Z}}^I h^{(1)}(t,\\cdot) \\big\\|_{L^{\\infty}} \\\\\n\t& \\ \\ + \\ C \\sum_{|I| \\leq k + 1} \\int_0^t \n\t\t\\varepsilon \\big\\| \\varpi(q) \\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}(\\tau,\\cdot) \\big\\|_{L^{\\infty}} \n\t\t\\ + \\ (1 + \\tau) \\big\\| \\varpi(q) |\\widetilde{\\Square}_g \\nabla_{\\mathcal{Z}}^I h^{(1)}|(\\tau,\\cdot) \n\t\t\\big\\|_{L^{\\infty}(D_{\\tau})} \\, d \\tau \\notag \\\\\n\t& \\ \\ + \\ C \\sum_{|I| \\leq k + 3} \\int_0^t (1 + \\tau)^{-1} \n\t\t\\big\\| \\varpi(q) \\nabla_{\\mathcal{Z}}^I h^{(1)}(\\tau,\\cdot) \\big\\|_{L^{\\infty}(D_{\\tau})} \\, d \\tau \\notag \\\\\n\t& \\leq C \\varepsilon (1 + t)^{-a} \\ + \\ C \\int_{0}^{t} (1 + \\tau)^{-1}  \n\t\t\\Big\\lbrace \\varepsilon n_{k+1}(\\tau) \\ + \\ \\varepsilon^2 (1 + \\tau)^{C \\varepsilon} \n\t\t\\ + \\ \\varepsilon (1 + \\tau)^{-1\/2 - \\upmu'} \n\t\t\t\\ + \\ \\varepsilon (1 + \\tau)^{-a} \\Big\\rbrace \\,d \\tau \\notag \\\\\n\t& \\leq C \\varepsilon \\ + \\ C \\varepsilon(1 + t)^{C\\varepsilon} \n\t\t\\ + \\ C \\varepsilon \\int_0^t (1 + \\tau)^{-1} n_{k+1}(\\tau) \\, d \\tau. \\notag\n\\end{align}\n\nFrom \\eqref{E:nkplus1Gronwallready} and Gronwall's lemma, we conclude that\n$n_{k+1}(t) \\leq 2 C \\varepsilon (1 + t)^{2 C \\varepsilon},$ which proves \\eqref{E:partialZIh1Aupgraded}\nin the case $|I| = k+1.$ As in our proof of Lemma \\ref{L:UpgradedDecayhA}, the estimate \\eqref{E:ZIh1Aupgraded} follows from integrating the bound for $|\\partial_q \\nabla_{\\mathcal{Z}}^I h^{(1)}|$ implied by \\eqref{E:partialZIh1Aupgraded} along the line $\\omega \\eqdef x\/|x| = constant,$ $t + |x| = constant,$ from the hyperplane $t=0$ and using \\eqref{E:weakdecayLinfinity} at $t=0.$ This closes the induction argument. We have completed the proof of Proposition \\ref{P:UpgradedDecayh1A} with the exception of showing that inequality \\eqref{E:barpartialZIh1Aupgraded} holds for \n$|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|,$ $|\\conenabla \\Lie_{\\mathcal{Z}}^I \\Far|,$ $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}},$ and $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}},$ where $|I| \\leq \\dParameter - 5.$ In the next paragraph, we address these inequalities using an argument which is not part of the induction process.\n\\\\\n\n\n\\noindent \\textbf{Upgraded pointwise decay estimates for $|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|,$ \n$|\\conenabla \\Lie_{\\mathcal{Z}}^I \\Far|,$ $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}},$ and \n$|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}},$ $|I| \\leq \\dParameter - 5.$}\n\nWe first note that inequality \\eqref{E:barpartialZIh1Aupgraded} for $|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|$ and \n$|\\conenabla \\Lie_{\\mathcal{Z}}^I \\Far|$ follows from Lemma \\ref{L:PointwisetandqWeightedNablainTermsofZestiamtes}, \\eqref{E:LieZIinTermsofNablaZI}, \\eqref{E:partialZIh1Aupgraded}, and \\eqref{E:ZIh1Aupgraded}.\n\nWe now focus on proving the estimate \\eqref{E:barpartialZIh1Aupgraded} for $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}$ and $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}$ in \\eqref{E:barpartialZIh1Aupgraded}; all of the other estimates of Proposition \\ref{P:UpgradedDecayh1A} have been proved. Recall that $|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}} \\approx |\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far]| + |\\rho[\\Lie_{\\mathcal{Z}}^I \\Far]| + |\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]|.$ We will prove the desired estimate for $|\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far]|$ in detail; the proofs for $|\\rho[\\Lie_{\\mathcal{Z}}^I \\Far]|$ and $|\\sigma[\\Lie_{\\mathcal{Z}}^I \\Far]|$ are similar. \n\nOur proof mirrors the proof of Proposition \\ref{P:FLUTTimproveddecay}, except that we now are able to use the already-proven upgraded estimates of Proposition \\ref{P:UpgradedDecayh1A} in place of the weak decay estimates of Corollary \\ref{C:WeakDecay}. We will use the notation defined in the proof of Proposition \\ref{P:FLUTTimproveddecay}. Using the upgraded \npointwise decay estimates \\eqref{E:partialZIh1Aupgraded} and \\eqref{E:ZIh1Aupgraded}\n(including the versions for the tensorfield $h = h^{(0)} + h^{(1)}$), inequality \\eqref{E:partialqfbound} for $f(t,x) \\eqdef |x|^{-1} \\alpha[\\Lie_{\\mathcal{Z}}^I \\Far(t,x)]$ can be upgraded to  \n\n\\begin{align} \\label{E:nablauLralphaLieZIFarinequality}\n\t|\\partial_q f(t,x)| \n\t& \\leq\n\t\t\\left \\lbrace \\begin{array}{lr}\n\t    \tC_k \\varepsilon (1 + s)^{-3 + C \\varepsilon} (1 + |q|)^{-1 - \\upgamma'}, &  \\mbox{if} \\ q > 0, \\\\\n\t      C_k \\varepsilon (1 + s)^{-3 + C \\varepsilon} (1 + |q|)^{-1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \t\\right., && |I| \\leq \\dParameter - 5.\n\\end{align}\nArguing as in the proof of Proposition \\ref{P:FLUTTimproveddecay}, and using in particular \\eqref{E:fTerminalPointBound},\nwe deduce that\n\n\\begin{align} \\label{E:ralphaLieZIFarinequality}\n\t\\big||x|^{-1} \\alpha[\\Lie_{\\mathcal{Z}}^I \\Far(t,x)]\\big| & \\leq \n\t\tC \\varepsilon (1 + s)^{-3 - (\\overbrace{\\upgamma - \\updelta)}^{> 0}}\n\t\t\\ + \\ \\left \\lbrace \\begin{array}{lr}\n\t    \tC_k \\varepsilon (1 + s)^{-3 + C \\varepsilon} (1 + |q'|)^{- \\upgamma'}, &  \\mbox{if} \\ q' > 0, \\\\\n\t      C_k \\varepsilon (1 + s)^{-3 + C \\varepsilon} (1 + |q'|)^{1\/2 + \\upmu'}, & \\mbox{if} \\ q' < 0,\n\t    \\end{array}\n\t  \t\\right., && |I| \\leq \\dParameter - 5,\n\\end{align}\nfrom which it easily follows that\n\n\\begin{align} \\label{E:alphaPointwiseUpgraded}\n\t\\big|\\alpha[\\Lie_{\\mathcal{Z}}^I \\Far(t,x)]\\big| \n\t& \\leq \n\t\\left \\lbrace \\begin{array}{lr}\n\t    \tC_k \\varepsilon (1 + t + |q|)^{-2 + C \\varepsilon} (1 + |q|)^{ - \\upgamma'}, &  \\mbox{if} \\ q > 0, \\\\\n\t      C_k \\varepsilon (1 + t + |q|)^{-2 + C \\varepsilon} (1 + |q|)^{1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0,\n\t    \\end{array}\n\t  \t\\right., && |I| \\leq \\dParameter - 5.\n\\end{align}\nWe have thus obtained the desired bound \\eqref{E:barpartialZIh1Aupgraded} for \n$\\big| \\alpha[\\Lie_{\\mathcal{Z}}^I \\Far] \\big|.$ \\hfill $\\qed$\n\n\n\n\n\n\n\n\n\\section{Global Existence and Stability} \\label{S:GlobalExistence}\nIn this section, we prove our main stability results. We separate our results into two theorems. The main conclusions are proved in Theorem \\ref{T:MainTheorem}, which is an easy consequence of Theorem \\ref{T:ImprovedDecay}. Theorem \\ref{T:ImprovedDecay}, which concerns the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary}, contains the crux of our bootstrap argument. In this theorem, we make certain assumptions concerning the smallness of the abstract initial data and various pointwise decay estimates for the solution on a local interval of existence $[0,T).$ We then use these assumptions to derive a ``strong'' smallness conclusion for the energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t)$ of the reduced solution on the same interval $[0,T).$ Furthermore, in Section \\ref{S:DecayFortheReducedEquations}, the pointwise decay assumptions of Theorem \\ref{T:ImprovedDecay} were shown to be \\emph{automatic consequences} of the smallness assumptions on the data and the ``weak'' bootstrap assumption \\eqref{E:Bootstrap} for the energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t)$ of the solution, as long as $\\dParameter \\geq 8.$ Consequently, in our proof of Theorem \\ref{T:MainTheorem}, we will be able to appeal to the continuation principle of Proposition \\ref{P:LocalExistence} to conclude that the solution to the reduced equation exists globally in time. Furthermore, this line of reasoning leads to an estimate on the size of $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t),$ which can be used to deduce various decay estimates for the global solution. The wave coordinate condition plays a central role in many of the estimates in this section.\n\n\\noindent \\hrulefill\n\\ \\\\\n\n\\subsection{Statement of the strong-energy-inequality theorem and proof of the global stability theorem}\n\nWe begin by recalling that the norm $E_{\\dParameter;\\upgamma}(0) \\geq 0$ for the abstract initial data is defined by \n\n\\begin{align}\n\tE_{\\dParameter;\\upgamma}^2(0) \n\t& \\eqdef \\| \\underline{\\nabla} \\mathring{\\underline{h}}^{(1)} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{K} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{\\mathfrak{\\Displacement}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2 \n\t\t\\ + \\ \\| \\mathring{\\mathfrak{\\Magneticinduction}} \\|_{H_{1\/2 + \\upgamma}^{\\dParameter}}^2.\n\\end{align}\nWe furthermore recall that the energy $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\geq 0$ for the reduced solution is defined to be\n\n\\begin{align}\n\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}^2(t) & \\eqdef \\underset{0 \\leq \\tau \\leq t}{\\mbox{sup}} \n\t\t\\sum_{|I| \\leq \\dParameter } \\int_{\\Sigma_{\\tau}} \n\t\t\\Big\\lbrace |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|^2 \\Big\\rbrace w(q) \\, d^3 x.\n\\end{align}\n\nIn the above expressions, the weight function $w(q)$ and its derivative $w'(q)$ are defined by\n\\begin{align}\n\tw = w(q) = \\left \\lbrace\n\t\t\\begin{array}{lr}\n    \t1 \\ + \\ (1 + |q|)^{1 + 2 \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      1 \\ + \\ (1 + |q|)^{-2 \\upmu}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \\right., & \\qquad\n  w'(q) = \\left \\lbrace\n\t\t\\begin{array}{lr}\n    \t(1 + 2 \\upgamma)(1 + |q|)^{2 \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      2 \\upmu (1 + |q|)^{-2 \\upmu -1}, & \\mbox{if} \\ q < 0.\n    \\end{array}\n  \\right.\n\\end{align}\nThe constants $\\upmu$ and $\\upgamma$ are subject to the restrictions summarized in Section \\ref{SS:FixedConstants}. The spacetime metric is split into the three pieces\n\n\\begin{subequations}\n\\begin{align}\n\tg_{\\mu \\nu} & = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)}, \\\\\n\th_{\\mu \\nu}^{(0)} & = \\chi\\big(\\frac{r}{t}\\big) \\chi(r) \\frac{2M}{r} \\updelta_{\\mu \\nu},\n\\end{align}\n\\end{subequations}\nwhere the cut-off function $\\chi$ is defined in \\eqref{E:chidef}. Furthermore, by Proposition \\ref{P:SmallNormImpliesSmallEnergy}, if $\\varepsilon$ is sufficiently small and $E_{\\dParameter;\\upgamma}(0) + M \\leq \\varepsilon,$\nthen the initial energy for the reduced solution satisfies\n\\begin{align}\n\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0) & \\lesssim E_{\\dParameter;\\upgamma}(0) + M \\lesssim \\varepsilon.\n\\end{align}\n\n\nWe now state our technical theorem concerning the derivation of a ``strong'' energy inequality.\n\n\\begin{theorem} \t\t\\label{T:ImprovedDecay}\n\t\\textbf{(Derivation of a strong energy inequality)}\n\tLet $\\dParameter \\geq 0$ be an integer. Let $(g_{\\mu \\nu} \\eqdef m_{\\mu \\nu} + \\overbrace{h^{(0)}_{\\mu \\nu} + h^{(1)}_{\\mu \n\t\\nu}}^{h_{\\mu \\nu}}, \\Far_{\\mu \\nu})$ \n\tbe a local-in-time solution of the reduced equations \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} \n\tsatisfying the wave coordinate condition \\eqref{E:wavecoordinategauge1} for $(t,x) \\in [0,T) \\times \\mathbb{R}^3.$  \n\tSuppose also that for some constants $\\upmu', \\upgamma$ satisfying $0 < \\upmu' < 1\/2$ and $0 < \\upgamma < 1\/2,$ for all \n\tvectorfields $Z \\in \\mathcal{Z},$ for all $\\mathcal{Z}-$multi-indices $I$ subject to the restrictions stated below, and for \n\tthe sets $\\mathcal{L} = \\lbrace L \\rbrace,$ $\\mathcal{T} = \\lbrace L, e_1, e_2 \\rbrace,$\n\tand $\\mathcal{N} = \\lbrace {\\uL, L, e_1, e_2} \\rbrace,$ the following pointwise \n\tdecay estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\t|\\nabla h|_{\\mathcal{T} \\mathcal{N}} \\ + \\ (1 + |q|)^{-1}|h|_{\\mathcal{L} \\mathcal{T}}\n\t\t\t\\ + \\ (1 + |q|)^{-1}|\\nabla_Z h|_{\\mathcal{L} \\mathcal{L}} \\ + \\ |\\Far| \n\t\t& \\leq C \\varepsilon (1 + t + |q|)^{-1}, \\label{E:MainTheoremAssumptionStrongLinfinityDecayPrincipalTermCoefficients} \t\t\t\t\n\t\\end{align}\n\t\n\t\\begin{align}\n\t\t|\\nabla\\nabla_{\\mathcal{Z}}^I h| \\ + \\ (1 + |q|)^{-1} |\\nabla_{\\mathcal{Z}}^I h| \\ + \\ |{\\Lie_{\\mathcal{Z}}^I \\Far}|\n\t\t& \\leq \n\t\t\\left \\lbrace \\begin{array}{lr}\n    \tC \\varepsilon (1 + t + |q|)^{-1 + C \\varepsilon} (1 + |q|)^{-1 - C\\varepsilon}, &  \\mbox{if} \\ q > 0, \\\\\n      C \\varepsilon (1 + t + |q|)^{-1 + C \\varepsilon} (1 + |q|)^{-1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \\right., \\qquad |I| \\leq \\lceil \\dParameter\/2 \\rceil, \n  \t\\label{E:MainTheoremAssumptionStrongLinfinityDecay} \\\\\n  |\\conenabla \\nabla_{\\mathcal{Z}}^I h|\n  \\ + \\ (1 + |q|)|\\conenabla \\Lie_{\\mathcal{Z}}^I \\Far|\n  \\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}} \n  \\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}\n  \t& \\leq \n  \t\\left \\lbrace \\begin{array}{lr}\n    \tC \\varepsilon (1 + t + |q|)^{-2 + C \\varepsilon} (1 + |q|)^{- C\\varepsilon}, &  \\mbox{if} \\ q > 0, \\\\\n      C \\varepsilon (1 + t + |q|)^{-2 + C \\varepsilon} (1 + |q|)^{1\/2 + \\upmu'}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \t\\right., \\qquad |I| \\leq \\lceil \\dParameter\/2 \\rceil - 1. \\label{E:MainTheoremAssumptionStrongerLinfinityDecayGoodComponents} \n \t\\end{align}\n\t\\end{subequations}\n\tIn addition, assume that the following smallness conditions on the abstract initial data and ADM mass hold:\n\t\n\t\\begin{align} \\label{E:BootstrapTheoremDataareSmall}\n\t\tE_{\\dParameter;\\upgamma}(0) + M & \\leq \\mathring{\\varepsilon}.\n\t\\end{align}\n\t\n\tThen for any constant $\\upmu$ satisfying $0 < \\upmu < 1\/2 - \\upmu',$ there exist positive constants $\\varepsilon_{\\dParameter},$\n\t$c_{\\dParameter},$ and $\\widetilde{c}_{\\dParameter}$ depending on $\\dParameter ,$ $\\upmu,$ $\\upmu',$ and $\\upgamma$ such that if\n\t$\\mathring{\\varepsilon} \\leq \\varepsilon \\leq \\varepsilon_{\\dParameter},$ then the following energy inequality holds for $t \\in [0,T):$\n\t\n\t\\begin{align} \\label{E:ImprovedEnergyInequality}\n\t\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\leq c_{\\dParameter} (\\mathring{\\varepsilon} + \\varepsilon^{3\/2}) \n\t\t(1 + t)^{\\widetilde{c}_{\\dParameter} \\varepsilon}.\n\t\\end{align}\n\t\n\\end{theorem}\n\n\n\\begin{remark} \\label{R:ImprovedDecay}\n\tBy Lemma \\ref{L:h0decayestimates}, the decompositions $h = h^{(0)} + h^{(1)}$ and $H = H_{(0)} + H_{(1)}$ \n\t(where $H^{\\mu \\nu} \\eqdef (g^{-1})^{\\mu \\nu} - (m^{-1})^{\\mu \\nu}$),\n\tand the fact that $H_{(1)}^{\\mu \\nu} = - h^{(1)\\mu \\nu} \\ + \\ O^{\\infty}(\\big |h^{(0)} + h^{(1)}|^2 \\big),$ it follows that\n\tthe estimates stated in the assumptions of the theorem also hold if we replace $h$ with $h^{(1)},$ $h^{(1)},$ or \n\t$H_{(1)}.$\n\\end{remark}\n\n\n\nWe now  state and (using the results of Theorem \\ref{T:ImprovedDecay}) prove our main global stability theorem.\n\n\\begin{theorem} \\label{T:MainTheorem}\n\t\\textbf{(Global stability of the Minkowski spacetime solution)}\n\tLet $(\\mathring{\\underline{g}}_{jk}  \\delta_{jk} + \\mathring{\\underline{h}}_{jk}^{(0)} + \\mathring{\\underline{h}}_{jk}^{(1)}, \n\t\\mathring{K}_{jk}, \\mathring{\\mathfrak{\\Displacement}_j}, \\mathring{\\mathfrak{\\Magneticinduction}}_j),$ \n\t$(j,k=1,2,3),$ be abstract initial data on the manifold $\\mathbb{R}^3$\n\tfor the Einstein-nonlinear electromagnetic system \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0}\n\tthat satisfy the constraints \\eqref{E:Gauss} - \\eqref{E:DivergenceB0}, and let\n\t$(g_{\\mu \\nu}|_{t=0} = m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)}|_{t=0} + h_{\\mu \\nu}^{(1)}|_{t=0}, \n\t\\partial_t g_{\\mu \\nu}|_{t=0} = \\partial_t h_{\\mu \\nu}^{(0)}|_{t=0} + \\partial_th_{\\mu \\nu}^{(1)}|_{t=0}, \n\t\\Far_{\\mu \\nu}|_{t=0}),$ $(\\mu, \\nu = 0,1,2,3),$ be the corresponding initial data \n\tfor the reduced system \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} as defined in Section \\ref{SS:ReducedData}.\n\tAssume that the abstract initial data are asymptotically flat in the sense that \n\t\\eqref{E:metricdataexpansion} - \\eqref{E:BdecayAssumption} hold. Let $\\dParameter \\geq 8$ be an integer, and let $0 < \\upgamma < 1\/2$ \n\tbe a fixed constant. Then there exist a global system of wave coordinates $(t,x)$ and\n\ta constant $\\varepsilon_{\\dParameter} > 0$ depending on $\\upgamma$ and $\\dParameter $ such that if \n\t$\\varepsilon \\leq \\varepsilon_{\\dParameter},$ and if \n\t\n\t\\begin{subequations}\n\t\\begin{align}\n\t\tE_{\\dParameter;\\upgamma}(0) + M & \\leq \\varepsilon,\n\t\\end{align}\n\t\\end{subequations}\n\tthen the reduced data launch a unique global, classical, geodesically complete solution \\\\\n\t$(g_{\\mu \\nu} \\eqdef m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} + h_{\\mu \\nu}^{(1)}, \\Far_{\\mu \\nu})$ to \n\t\\textbf{both}\\footnote{Of course, we technically mean here that the pair $(h_{\\mu \\nu}^{(1)},\\Far_{\\mu \\nu})$\n\tis a solution to the version \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} of the reduced equations, while the \n\tpair $(g_{\\mu \\nu},\\Far_{\\mu \\nu})$ is a solution to equations \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0}.} the reduced\n\tsystem \\eqref{E:Reducedh1Summary} - \\eqref{E:ReduceddMis0Summary} and the Einstein-nonlinear electromagnetic system \n\t\\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0}. Furthermore, there exists\n\ta constant $0 < \\upmu < 1\/2$ (see Remark \\ref{R:Roleofmu}), \n\tand constants $c_{\\dParameter} > 0,$ $\\widetilde{c}_{\\dParameter} > 0$ depending on $\\upgamma$ and $\\dParameter ,$ \n\tsuch that the solution's energy satisfies the following bound for all $t \\in (-\\infty,\\infty):$\n\t\n\t\\begin{align} \\label{E:GlobalEnergyInequality}\n\t\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\leq c_{\\dParameter} \\varepsilon (1 + |t|)^{\\widetilde{c}_{\\dParameter} \\varepsilon}.\n\t\\end{align}\n\t\n\tIn addition, there exists a constant $C_{\\dParameter} > 0$ depending on $\\upgamma$ and $\\dParameter ,$ such that the \n\tfollowing pointwise decay estimates hold for all $(t,x) \\in (-\\infty,\\infty) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{subequations} \n\t\\begin{align} \\label{E:GlobalStabilityTheoremStrongLinfinityDecayPrincipalTermCoefficients}\n\t\t& (1 + |t| + |q|)^{1 - \\widetilde{c}_{\\dParameter} \\varepsilon}(1 + |q|)^{-1\/2}\n\t\t\t|\\nabla h|_{\\mathcal{L} \\mathcal{T}} \n\t\t\\ + \\ (1 + |t| + |q|)^{1 - \\widetilde{c}_{\\dParameter} \\varepsilon}(1 + |q|)^{-1\/2} |\\nabla \\nabla_Z h|_{\\mathcal{L} \\mathcal{L}}\n\t\t\\ + \\ |\\nabla h|_{\\mathcal{T} \\mathcal{N}} \t\\ + \\ \\big\\lbrace 1 + \\ln(1 + |t|) \\big\\rbrace^{-1}|\\nabla h| \\\\\n\t\t& \\ \\ + \\ (1 + |t| + |q|)^{1 - \\widetilde{c}_{\\dParameter} \\varepsilon}(1 + |q|)^{-3\/2}|h|_{\\mathcal{L} \\mathcal{T}}\n\t\t\t\\ + \\ (1 + |t| + |q|)^{1 - \\widetilde{c}_{\\dParameter} \\varepsilon}(1 + |q|)^{-3\/2}|\\nabla_Z h|_{\\mathcal{L} \\mathcal{L}} \n\t\t\t\\ + \\ |\\Far| \\notag \\\\\n\t\t& \\leq C_{\\dParameter} \\varepsilon (1 + |t| + |q|)^{-1}, \\notag\n\t\\end{align}\n\t\n\t\\begin{align}\n\t\t|\\nabla \\nabla_{\\mathcal{Z}}^I h^{(1)}| \\ + \\ (1 + |q|)^{-1} |\\nabla_{\\mathcal{Z}}^I h^{(1)}| \n\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far|\n\t\t& \\leq \n\t\t\\left \\lbrace \\begin{array}{lr}\n    \tC_{\\dParameter} \\varepsilon (1 + |t| + |q|)^{-1 + \\widetilde{c}_{\\dParameter} \\varepsilon} \n    \t\t(1 + |q|)^{-1 - \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      C_{\\dParameter} \\varepsilon (1 + |t| + |q|)^{-1 + \\widetilde{c}_{\\dParameter} \\varepsilon} (1 + |q|)^{-1\/2}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \\right., \\qquad |I| \\leq \\dParameter - 2, \n  \t\\label{E:GlobalStabilityTheoremStrongLinfinityDecay} \\\\\n  |\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}| \\ + \\ (1 + |q|)|\\conenabla \\Lie_{\\mathcal{Z}}^I \\Far|\n  \\ + \\ |{\\Lie_{\\mathcal{Z}}^I \\Far}|_{\\mathcal{L}\\mathcal{N}} \n  \\ + \\ |{\\Lie_{\\mathcal{Z}}^I \\Far}|_{\\mathcal{T}\\mathcal{T}}\n  \t& \\leq \n  \t\\left \\lbrace \\begin{array}{lr}\n    \tC_{\\dParameter} \\varepsilon (1 + |t| + |q|)^{-2 + \\widetilde{c}_{\\dParameter} \\varepsilon} (1 + |q|)^{- \\upgamma}, &  \\mbox{if} \\ q > 0, \\\\\n      C_{\\dParameter} \\varepsilon (1 + |t| + |q|)^{-2 + \\widetilde{c}_{\\dParameter} \\varepsilon} (1 + |q|)^{1\/2}, & \\mbox{if} \\ q < 0,\n    \\end{array}\n  \t\\right., \\qquad |I| \\leq \\dParameter - 3. \\label{E:GlobalStabilityTheoremStrongerLinfinityDecayGoodComponents} \n \t\\end{align}\n\t\\end{subequations}\n\t\n\t\n\\end{theorem}\n\n\\begin{remark} \\label{R:qdecay}\n\tSome of the $1 + |q|-$decay estimates in inequalities \n\t\\eqref{E:GlobalStabilityTheoremStrongLinfinityDecayPrincipalTermCoefficients} -\n\t\\eqref{E:GlobalStabilityTheoremStrongerLinfinityDecayGoodComponents} are not optimal, and can be improved with additional \n\twork. For example, in \\cite[Section 16]{hLiR2010}, with the help of the fundamental solution of the Minkowski\n\twave operator $\\Square_m,$ the $1 + |q|-$decay estimates \n\t\\eqref{E:GlobalStabilityTheoremStrongLinfinityDecay} - \\eqref{E:GlobalStabilityTheoremStrongerLinfinityDecayGoodComponents}\n\tfor the tensorfield $h^{(1)}$ are strengthened by a power of $1\/2$ in the \n\tinterior region $q < 0.$\n\\end{remark}\n\n\\begin{remark}\n\tProposition \\ref{P:PreservationofWaveCoordianteGauge} shows that the wave coordinate condition\n\t\\eqref{E:wavecoordinategauge1} holds in the domain of classical existence of the solution to the reduced equations; \n\tthis is why the reduced solution also verifies the Einstein-nonlinear electromagnetic equations \\eqref{E:IntroEinstein} - \\eqref{E:IntrodMis0}.\n\\end{remark}\n\n\\begin{comment}\n\\begin{remark}\n\tIt follows from the arguments given in the proof that the constant $\\upgamma$ can be chosen in a such a way that $\\upgamma \n\t\\downarrow 0$ as $\\varepsilon_0 \\downarrow 0.$\n\\end{remark}\n\\end{comment}\n\n\\begin{remark}\n\tA global stability result for the reduced equations under the wave coordinate assumption, \n\twithout regard for the abstract initial data, can be deduced\n\tfrom the smallness of $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(0) + |M|$ (we could even allow for negative $M$!)\n\ttogether with the assumption $\\liminf_{|x| \\to \\infty} |h^{(1)}(0,x)| = 0;$ this latter assumption, which is needed to deduce \n\tthe inequalities \\eqref{E:weakdecayLinfinity} at $t=0,$ is automatically implied by the assumptions of Theorem \n\t\\ref{T:MainTheorem}. \n\\end{remark}\n\n\\begin{proof}\n\tWe only discuss the region of spacetime in which $t \\geq 0;$ the argument for $t \\leq 0$ is similar. \n\tLet us set $E_{\\dParameter;\\upgamma}(0) + M \\eqdef \\mathring{\\varepsilon}.$\n\tBy Proposition \\ref{P:LocalExistence}, we can \n\tchoose constants $\\upgamma', \\upmu, \\upmu',$ and $\\updelta$ subject to the restrictions described in Section \n\t\\ref{SS:FixedConstants} (and in particular depending on $\\upgamma$), and a constant $A_{\\dParameter} > 0$\n\tsuch that if $\\varepsilon \\eqdef A_{\\dParameter} \\mathring{\\varepsilon},$ $A_{\\dParameter}$ is sufficiently large, \n\tand $\\mathring{\\varepsilon}$ is sufficiently small, then there exists a nontrivial spacetime slab \n\t$[0,T) \\times \\mathbb{R}^3$ upon which the solution to the reduced equations exists and satisfies the\n\tenergy bound $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\leq \\varepsilon (1 + t)^{\\updelta}$ \n\tfor $t \\in [0,T).$ We then define\n\t\n\t\\begin{align*}\n\t\tT_* \\eqdef \\sup \\big\\lbrace T & \\mid  \n\t\t\\mbox{the solution exists classically and remains in the regime of hyperbolicity of the reduced equations}, \\\\ \n\t\t\t& \\ \\mbox{and} \\ \\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\leq \\varepsilon (1 + t)^{\\updelta} \\ \\mbox{for} \\ t \\in [0,T) \n\t\t\t\\big\\rbrace.\n\t\\end{align*}\n\tNote that under the above assumptions, we have that $T_* > 0.$ \n\t\n\tWe now observe that the main energy bootstrap assumption \\eqref{E:Bootstrap} is satisfied on $[0,T_*).$\n\tThus, if $\\varepsilon$ is sufficiently small, then by Propositions \\ref{P:UpgradedDecayhA} and \n\t\\ref{P:UpgradedDecayh1A}, all of the hypotheses of Theorem \\ref{T:ImprovedDecay} are necessarily satisfied\n\ton $[0,T_*).$ Here, we are using the fact that $\\lceil \\dParameter\/2 \\rceil \\leq \\dParameter - 4,$ which holds if\n\t$\\dParameter \\geq 8.$ Consequently, the conclusion of that theorem (i.e., estimate \n\t\\eqref{E:ImprovedEnergyInequality}) allows us to deduce that the following energy estimate holds for $t \\in [0,T_*):$\n\t\n\t\\begin{align} \\label{E:ImprovedDecayConclusion}\n\t\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\leq c_{\\dParameter} \\big\\lbrace \\mathring{\\varepsilon} + \\varepsilon^{3\/2} \\big\\rbrace\n\t\t(1 + t)^{\\widetilde{c}_{\\dParameter} \\varepsilon} = c_{\\dParameter} \\bigg\\lbrace \\frac{\\varepsilon}{A_{\\dParameter}} \n\t\t\t+ \\varepsilon^{3\/2} \\bigg\\rbrace (1 + t)^{\\widetilde{c}_{\\dParameter} \\varepsilon}.\n\t\\end{align} \n\tNow if $A_{\\dParameter} > 3 c_{\\dParameter}$ and $\\mathring{\\varepsilon}$ is sufficiently small, then \n\t\\eqref{E:ImprovedDecayConclusion} implies that\n\t\n\t\\begin{align} \\label{E:ImprovedDecayConclusionWithRoom}\n\t\t\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) < \\frac{1}{2} A_{\\dParameter} \\mathring{\\varepsilon} (1 + t)^{A_{\\dParameter} \n\t\t\t\\widetilde{c}_{\\dParameter} \n\t\t\t\\mathring{\\varepsilon}} = \\frac{1}{2} \\varepsilon (1 + t)^{\\widetilde{c}_{\\dParameter} \\varepsilon},\n\t\\end{align}\n\twhich is a strict improvement over the bootstrap assumption assumption \\eqref{E:Bootstrap}.\n\tThus, by \\eqref{E:ImprovedDecayConclusionWithRoom}, the weighted Klainerman-Sobolev inequality \\eqref{E:PhiKlainermanSobolev} \n\t(which, together with \\eqref{E:LieZIinTermsofNablaZI} and the smallness of $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t),$ \n\timplies that the solution remains within the regime of hyperbolicity of the reduced equations),\n\tthe continuation principle of Proposition \\ref{P:LocalExistence},\n\tand the continuity of $\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t),$ it follows that\n\tif $A_{\\dParameter}$ is sufficiently large and $\\mathring{\\varepsilon}$ is sufficiently small, \n\tthen $T_* = \\infty.$ Furthermore, under these assumptions, it is an obvious consequence of this reasoning\n\tthat \\eqref{E:ImprovedDecayConclusionWithRoom} holds for $t \\in [0,\\infty).$ After renaming the constants\n\tin \\eqref{E:ImprovedDecayConclusionWithRoom}, we arrive at \\eqref{E:GlobalEnergyInequality}.\n\t\n\tThe inequalities \\eqref{E:GlobalStabilityTheoremStrongLinfinityDecay} \n\tfollow as in the proof of Corollary \\ref{C:WeakDecay}, but using the strong energy estimate \n\t\\eqref{E:GlobalEnergyInequality} instead of the energy bootstrap assumption \\eqref{E:Bootstrap}. Similarly,\n\tthe inequalities \\eqref{E:GlobalStabilityTheoremStrongLinfinityDecayPrincipalTermCoefficients} follow\t\n\tas in our proof of Proposition \\ref{P:UpgradedDecayhA}, but using the strong energy estimate \n\t\\eqref{E:GlobalEnergyInequality} instead of the energy bootstrap assumption \\eqref{E:Bootstrap}. The inequalities \n\t\\eqref{E:GlobalStabilityTheoremStrongerLinfinityDecayGoodComponents}\n\tfor $|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|$ and $|\\conenabla \\Lie_{\\mathcal{Z}}^I \\Far|$ follow from\n\tLemma \\ref{L:PointwisetandqWeightedNablainTermsofZestiamtes}, \\eqref{E:LieZIinTermsofNablaZI}, and\n\t\\eqref{E:GlobalStabilityTheoremStrongLinfinityDecay}. The inequalities \n\t\\eqref{E:GlobalStabilityTheoremStrongerLinfinityDecayGoodComponents} for \n\t$|{\\Lie_{\\mathcal{Z}}^I \\Far}|_{\\mathcal{L}\\mathcal{N}}$ and \n\t$|{\\Lie_{\\mathcal{Z}}^I \\Far}|_{\\mathcal{T}\\mathcal{T}}$ follow as in our proof of \\eqref{E:FLUTTimproveddecay}, but \n\tusing the strong energy estimate \\eqref{E:GlobalEnergyInequality} instead of the energy bootstrap assumption \n\t\\eqref{E:Bootstrap}.\n\t\n\tBased on these pointwise decay estimates, the geodesic completeness of the spacetime \n\t$(\\mathbb{R}^{1+3},g_{\\mu \\nu} \\eqdef m_{\\mu \\nu} + h_{\\mu \\nu}^{(0)} \n\t+ h_{\\mu \\nu}^{(1)})$ follows as in \\cite[Section 16]{hLiR2005} and \\cite[Section 9]{jL2008}.\n\n\\end{proof}\n\nIt remains to prove Theorem \\ref{T:ImprovedDecay}.\n\n\\begin{center}\\textbf{\\Large Proof of Theorem \\ref{T:ImprovedDecay}} \\end{center}\n\n\\subsection{The main argument in the proof of Theorem \\ref{T:ImprovedDecay}} \\label{SS:MainArgument}\n\nOur goal is to use \\emph{only} the assumptions of Theorem \\ref{T:ImprovedDecay} to\ndeduce (for all sufficiently small non-negative $\\varepsilon,$ and for sufficiently large fixed constants $c_{\\dParameter},\\widetilde{c}_{\\dParameter}$) the ``strong'' energy estimate \\eqref{E:ImprovedEnergyInequality}, which reads\n\n\\begin{align} \\label{E:FundamentalBootstrapMainTheorem}\n\t\\mathcal{E}_{\\dParameter;\\upgamma;\\upmu}(t) \\leq c_{\\dParameter} (\\mathring{\\varepsilon} + \\varepsilon^{3\/2}) \n\t\t(1 + t)^{\\widetilde{c}_{\\dParameter} \\varepsilon}.\n\\end{align}\nThe proof of \\eqref{E:FundamentalBootstrapMainTheorem} is based on a hierarchy of Gronwall-amenable inequalities that arise from careful analysis of the integrals of Proposition \\ref{P:weightedenergy} involving the inhomogeneous terms \n$\\mathfrak{H}_{\\mu \\nu}^{(1;I)}$ and $\\mathfrak{F}_{(I)}^{\\nu}.$ We recall that these inhomogeneous terms are captured by Propositions \\ref{P:InhomogeneousTermsNablaZIh1} and \\ref{P:InhomogeneoustermsLieZIFar}, which state that $\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)}$ and $\\Lie_{\\mathcal{Z}}^I \\Far_{\\mu \\nu}$ are solutions to the following system of equations:\n\n\\begin{subequations}\n\\begin{align}\n\t\\widetilde{\\Square}_{g} \\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)} & = \\mathfrak{H}_{\\mu \\nu}^{(1;I)}, &&\n\t\t(\\mu, \\nu = 0,1,2,3),\n\t\t\\label{E:InhomogeneousTermsNablaZIh1proof} \\\\ \n\t\\nabla_{\\lambda} \\Lie_{\\mathcal{Z}}^I \\Far_{\\mu \\nu} + \\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\nu \\lambda} \n\t\t+ \\nabla_{\\nu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\lambda \\mu} & = 0, && (\\lambda, \\mu, \\nu = 0,1,2,3),\n\t\\label{E:InhomogeneoustermsdFis0proof} \\\\\n\tN^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda} & = \\mathfrak{F}_{(I)}^{\\nu}, &&\n\t\t(\\nu = 0,1,2,3). \\label{E:InhomogeneoustermsdMis0proof}\n\\end{align}\n\\end{subequations}\nMost of the work goes into obtaining suitable estimates for the integrals involving $\\mathfrak{H}_{\\mu \\nu}^{(1;I)}$\nand $\\mathfrak{F}_{(I)}^{\\nu}.$ In order to avoid impeding the flow of the proof, we prove \nmost of the desired inequalities later in this section, after the main argument. For the main part of the argument, we simply quote Corollary \\ref{C:NablaZIh1FundamentalEnergyEstimate} and Corollary \\ref{C:LieZIFarFundamentalEnergyEstimate}, which \nare the key estimates that allow us to apply a suitable version of Gronwall's lemma. We will then return to the proofs of the corollaries, which follow from a large collection of lemmas, each of which involves the analysis of one of the constituent pieces of the integrals involving $\\mathfrak{H}_{\\mu \\nu}^{(1;I)}$ and $\\mathfrak{F}_{(I)}^{\\nu}.$ \n\nWe now proceed to the main argument. Using Proposition \\ref{P:weightedenergy}, Corollary \\ref{C:NablaZIh1FundamentalEnergyEstimate}, and Corollary \\ref{C:LieZIFarFundamentalEnergyEstimate}, we\nhave that\n\n\\begin{align} \\label{E:Mainenergyinequality}\n\t& \\sum_{|I| \\leq k} \n\t\t\\int_{\\Sigma_{t}} \\Big|\\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}}]{\\Lie_{\\mathcal{Z}}^I \\Far} \\Big|^2 w(q) \\,d^3x \n\t\t\\ + \\ \\sum_{|I| \\leq k} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \n\t\t\\bigg\\lbrace |\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}^2 \n\t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}}^2 \\bigg\\rbrace w'(q) \\,d^3x \\, d \\tau \\\\\n\t& \\leq C \\sum_{|I| \\leq k} \\int_{\\Sigma_{0}} \\Big|\\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}}]{\\Lie_{\\mathcal{Z}}^I \n\t\t\\Far} \\Big|^2 w(q) \\,d^3x \n\t\t\\ + \\ C \\varepsilon \\sum_{|I| \\leq k} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \n\t\t\t(1 + \\tau)^{-1} \\Big|\\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}}]{\\Lie_{\\mathcal{Z}}^I \\Far} \\Big|^2 \n\t\t\tw(q) \\, d^3x \\, d \\tau \\notag \\\\\n\t& \\ \\ + \\ C \\sum_{|I| \\leq k} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \n\t\t\\bigg\\lbrace |\\mathfrak{H}^{(1;I)}| |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|  \n\t\t\\ + \\ |(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu}) \\mathfrak{F}_{(I)}^{\\nu} | \\bigg\\rbrace w(q) \\,d^3x \\, d \\tau \\notag \\\\\n\t& \\leq C \\sum_{|I| \\leq k} \\int_{\\Sigma_{0}} \\Big|\\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}}]{\\Lie_{\\mathcal{Z}}^I \n\t\t\\Far} \\Big|^2 w(q) \\,d^3x\n\t\t\\ + \\ C M \\sum_{|I| \\leq k} \\int_{0}^{t} (1 + \\tau)^{-3\/2} \n\t\t\t\\Big(\\sqrt{\\int_{\\Sigma_{\\tau}} |\\nabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) \\, d^3 x} \\Big) \\, d \\tau \n\t\t\\notag \\\\\n\t& \\ \\ + \\ C \\varepsilon \\sum_{|I| \\leq k} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} (1 + \\tau)^{-1} \n\t\t\\Big|\\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}}]{\\Lie_{\\mathcal{Z}}^I \\Far} \\Big|^2 w(q) \\,d^3x \\, d \\tau \\notag \\\\\n\t& \\ \\ + \\ C \\varepsilon \\sum_{|I| \\leq k} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \n\t\t\t\\bigg\\lbrace |\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}^2 \n\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}}^2 \\bigg\\rbrace w'(q)  \\,d^3x \\, d \\tau \\notag \\\\\n\t& \\ \\ + \\ C \\varepsilon  \\underbrace{\\sum_{|J| \\leq k - 1} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} (1 + \\tau)^{- 1 + C \\varepsilon} \n\t\t\t\\Big|\\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}}]{\\Lie_{\\mathcal{Z}}^J \\Far} \\Big|^2 w(q) \\, d^3x \\, d \n\t\t\t\\tau}_{\\mbox{Absent if $k=0$}} \\ + \\ C \\varepsilon^3. \\notag\n\\end{align}\n\nRecalling the definition (where the dependence on $\\upmu,$ $\\upgamma$ is through $w(q)$)\n\n\\begin{align*}\n\t\\mathcal{E}_{k;\\upgamma;\\upmu}^2(t) & \\eqdef \\sup_{0 \\leq \\tau \\leq t} \\sum_{|I| \\leq k} \n\t\t\\int_{\\Sigma_{\\tau}} \\Big\\lbrace |\\nabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|^2 \n\t\t\\Big\\rbrace w(q) \\, d^3x, \n\\end{align*}\nand introducing the quantity $\\mathcal{S}_{k;\\upgamma;\\upmu}(t) \\geq 0,$ which is defined by \n\n\\begin{align}\n\t\\mathcal{S}_{k;\\upgamma;\\upmu}^2(t) & \\eqdef \\sum_{|I| \\leq k} \n\t\t\\int_0^t \\int_{\\Sigma_{\\tau}} \\Big\\lbrace |\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \n\t\t\\Far|_{\\mathcal{L}\\mathcal{N}}^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}}^2 \\Big\\rbrace w'(q) \\, d^3x \\, d \n\t\t\\tau,\n\\end{align}\nit therefore follows from the final inequality of \\eqref{E:Mainenergyinequality} that\n\n\\begin{align} \\label{E:Mainenergyinequalityreexpressed}\n\t\\mathcal{E}_{k;\\upgamma;\\upmu}^2(t) \\ + \\ \\mathcal{S}_{k;\\upgamma;\\upmu}^2(t) \n\t\t& \\leq C \\mathcal{E}_{k;\\upgamma;\\upmu}^2(0) \n\t\t\t\\ + \\ C M \\int_{0}^{t} (1 + \\tau)^{-3\/2} \\mathcal{E}_{k;\\upgamma;\\upmu}(\\tau) \\, d \\tau\n\t\t\t\\ + \\ C \\varepsilon \\int_{0}^{t} (1 + \\tau)^{-1} \\mathcal{E}_{k;\\upgamma;\\upmu}^2(\\tau) \\, d \\tau \\\\\n\t\t& \\ \\ + \\ \\underbrace{C \\varepsilon \\mathcal{S}_{k;\\upgamma;\\upmu}^2(t)}_{\\mbox{absorb into l.h.s.}}\n\t\t\t\\ + \\ C \\varepsilon \\int_{0}^{t} (1 + \\tau)^{- 1 + C \\varepsilon} \\mathcal{E}_{k-1;\\upgamma;\\upmu}^2(\\tau) \\, \n\t\t\td \\tau \\notag \\ + \\ C \\varepsilon^3.\n\\end{align}\n\nFor $\\varepsilon$ sufficiently small, we may absorb the $C \\varepsilon \\mathcal{S}_{k;\\upgamma;\\upmu}^2(t)$ term from \\eqref{E:Mainenergyinequalityreexpressed} into the left-hand side at the expense of increasing all of the constants. We can similarly absorb the term $C M \\int_{0}^{t} (1 + \\tau)^{-3\/2} \\mathcal{E}_{k;\\upgamma;\\upmu}(\\tau) \\, d \\tau$ by using the inequality \\\\\n${C M \\int_{0}^{t} (1 + \\tau)^{-3\/2} \\mathcal{E}_{k;\\upgamma;\\upmu}(\\tau) \\, d \\tau \\leq 1\/2 \\mathcal{E}_{k;\\upgamma;\\upmu}^2(t) + C^2 M^2};$ this inequality follows from the algebraic estimate \\\\\n$C M \\mathcal{E}_{k;\\upgamma;\\upmu}(\\tau)\\leq 1\/4 \\mathcal{E}_{k;\\upgamma;\\upmu}^2(\\tau) + C^2 M^2,$ \nthe integral inequality $\\int_{0}^{t} (1 + \\tau)^{-3\/2} \\, d \\tau \\leq 2,$\nand the fact that $\\mathcal{E}_{k;\\upgamma;\\upmu}^2(\\tau)$ is increasing. If we also use the fact that \n$\\mathcal{E}_{k;\\upgamma;\\upmu}^2(0) \\leq C \\big\\lbrace E_{\\dParameter;\\upgamma}^2(0) + M^2 \\big\\rbrace \\leq \nC \\mathring{\\varepsilon}^2$ (i.e, Proposition \\ref{P:SmallNormImpliesSmallEnergy}), and the inequality \n$M \\leq \\mathring{\\varepsilon},$ then we arrive at the following inequality, valid for all small $\\varepsilon:$\n\n\\begin{align} \\label{E:EkpluSkinequality}\n\t\\mathcal{E}_{k;\\upgamma;\\upmu}^2(t) \\ + \\ \\mathcal{S}_{k;\\upgamma;\\upmu}^2(t) \n\t\\leq C \\big\\lbrace \\mathring{\\varepsilon}^2 + \\varepsilon^3 \\big\\rbrace\n\t\\ + \\ C \\varepsilon  \\int_{0}^t (1 + \\tau)^{-1} \\mathcal{E}_{k;\\upgamma;\\upmu}^2(\\tau) \\, d \\tau \n\t\\ + \\ C \\varepsilon  \\underbrace{\\int_{0}^t (1 + \\tau)^{- 1 + C \\varepsilon} \\mathcal{E}_{k-1;\\upgamma;\\upmu}^2 (\\tau) \\, d \n\t\\tau}_{\\mbox{Absent if $k=0$}}.\n\\end{align}\n\nFor $k=0,$ this implies that\n\\begin{align} \\label{E:Gronwallreadyinequality0}\n\t\\mathcal{E}_{0;\\upgamma;\\upmu}^2(t) \\leq C \\big\\lbrace \\mathring{\\varepsilon}^2 + \\varepsilon^3 \\big\\rbrace   \n\t\\ + \\ c_0 \\varepsilon \\int_{0}^t (1 + \\tau)^{-1} \\mathcal{E}_{0;\\upgamma;\\upmu}^2(\\tau) \\, d \\tau.\n\\end{align}\nFrom \\eqref{E:Gronwallreadyinequality0} and Gronwall's lemma, we conclude that\n\n\\begin{align}\n\t\\mathcal{E}_{0;\\upgamma;\\upmu}^2(t) \\leq C \\big\\lbrace \\mathring{\\varepsilon}^2 + \\varepsilon^3 \\big\\rbrace \n\t(1 + t)^{c_0 \\varepsilon}.\n\\end{align}\n\nInductively using \\eqref{E:EkpluSkinequality}, we therefore derive the following estimate for $k \\geq 1:$ \n\\begin{align} \\label{E:Gronwallreadyinequalityk}\n\t\\mathcal{E}_{k;\\upgamma;\\upmu}^2(t) \\ + \\ \\mathcal{S}_{k;\\upgamma;\\upmu}^2(t) \n\t& \\leq \n\t\tC \\big\\lbrace \\mathring{\\varepsilon}^2 + \\varepsilon^3 \\big\\rbrace \n\t\t\\ + \\ C \\varepsilon  \\int_{0}^t (1 + \\tau)^{-1} \\mathcal{E}_{k;\\upgamma;\\upmu}^2(\\tau) \\, d \\tau \n\t\t\\ + \\ C \\varepsilon \\big\\lbrace \\mathring{\\varepsilon}^2 + \\varepsilon^3 \\big\\rbrace \\int_{0}^t (1 + \\tau)^{-1 + C \n\t\t\\varepsilon} \\, d \\tau \\\\\n\t& \\leq \n\t\tC \\big\\lbrace \\mathring{\\varepsilon}^2 + \\varepsilon^3 \\big\\rbrace \n\t\t\\ + \\ C \\varepsilon  \\int_{0}^t (1 + \\tau)^{-1} \\mathcal{E}_{k;\\upgamma;\\upmu}^2(\\tau) \\, d \\tau. \\notag\n\\end{align}\nFinally, from \\eqref{E:Gronwallreadyinequalityk} and Gronwall's lemma, we deduce that\nif $\\varepsilon$ is sufficiently small, then\n\n\\begin{align}\n\t\\mathcal{E}_{k;\\upgamma;\\upmu}^2(t) \\leq C \\big\\lbrace \\mathring{\\varepsilon}^2 + \\varepsilon^3 \\big\\rbrace \n\t(1 + t)^{c_k \\varepsilon},\n\\end{align}\nwhich closes the induction. Thus, we have shown \\eqref{E:ImprovedEnergyInequality}, which concludes the proof of Theorem \\ref{T:ImprovedDecay}.\n\n\n\n\n\n\n\\subsection{Integral inequalities for the $\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)}$ inhomogeneities}\n\nIn this section, we analyze the integrals in Proposition \\ref{P:weightedenergy} \ncorresponding to the inhomogeneous terms $\\mathfrak{H}_{\\mu \\nu}^{(1;I)}$ in equation \\eqref{E:InhomogeneousTermsNablaZIh1proof}. The main goal\nis to arrive at Corollary \\ref{C:NablaZIh1FundamentalEnergyEstimate}. As opposed to the estimates \nproved in Section \\ref{SS:MainTheoremFarInhomogeneities}, most of the estimates proved in this section are a rather straightforward generalization of the ones proved in \\cite{hLiR2010}; i.e., the estimates involve a similar analysis, but with\nadditional terms arising from the presence of the $\\Far$ terms appearing on the right-hand side of \nthe reduced equation \\eqref{E:Reducedh1Summary}.\n\n\nWe begin with the following lemma, which follows easily from algebraic estimates of the form $|ab| \\lesssim a^2 + b^2.$\n\n\\begin{lemma} \\label{L:MainTheoremH1IInhomogeneousTermNablaZIh1PeterPaul} \\textbf{(Arithmetic-geometric mean inequality)}\nLet $\\mathfrak{H}_{\\mu \\nu}^{(1;I)} = \\nablamod_{\\mathcal{Z}}^I \\mathfrak{H}_{\\mu \\nu} \n- \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square} h_{\\mu \\nu}^{(0)}- \\big\\lbrace \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)} - \\widetilde{\\Square}_{g} \n\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)} \\big\\rbrace$ be the inhomogeneous term on the right-hand side of \\eqref{E:InhomogeneousTermsNablaZIh1}.\nThen the following algebraic inequality holds:\n\n\\begin{align}  \\label{E:MainTheoremH1IInhomogeneousTermNablaZIh1PeterPaul}\n\t|\\mathfrak{H}^{(1;I)}| |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}| \n\t& \\leq \\varepsilon^{-1}(1 + t)|\\nablamod_{\\mathcal{Z}}^I  \\mathfrak{H}|^2 \n\t\t\\ + \\ \\varepsilon^{-1}(1 + t)|\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)} \n\t\t- \\widetilde{\\Square}_{g} \\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)}|^2 \\\\\n\t& \\ \\ + \\ \\varepsilon(1 + t)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 \n\t\t\\ + \\ |\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}| |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|. \\notag\n\\end{align}\t\n\n\\end{lemma}\n\n\nThe next lemma provides a preliminary pointwise estimate for the\n$|\\nablamod_{\\mathcal{Z}}^I  \\mathfrak{H}|$ term on the right-hand side of \\eqref{E:MainTheoremH1IInhomogeneousTermNablaZIh1PeterPaul}. \n\n\\begin{lemma} \\label{L:MainTheoremLieZIh1InhomogeneousAlgebraicEstimate} \\cite[Extension of Lemma 11.2]{hLiR2010} \n\t\\textbf{(Pointwise estimates for the $|\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}|$ inhomogeneities)}\n\tUnder the assumptions of Theorem \\ref{T:ImprovedDecay}, if $I$ is any $\\mathcal{Z}-$multi-index\n\twith $|I| \\leq \\dParameter ,$ and if $\\varepsilon$ is sufficiently small,\n\tthen the following pointwise estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{align} \\label{E:MainTheoremLieZIh1InhomogeneousAlgebraicEstimate}\n\t\t|\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}| \n\t\t& \\lesssim \\varepsilon \\sum_{|J| \\leq |I|}  \n\t\t\t(1 + t)^{-1} \\Big| \\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}}]{\\Lie_{\\mathcal{Z}}^J \\Far} \\Big| \n\t\t\t\\ + \\ \\varepsilon \\sum_{|J| \\leq |I|} (1 + t + |q|)^{-1 + C \\varepsilon} (1 + |q|)^{-1\/2 + \\upmu'} \n\t\t\t|\\conenabla \\nabla_{\\mathcal{Z}}^J h^{(1)}| \\\\\n\t\t& \\ \\ + \\ \\varepsilon^2 \\sum_{|J| \\leq |I|} (1 + t + |q|)^{-1} (1 + |q|)^{-1} |\\nabla_{\\mathcal{Z}}^J h^{(1)}| \n\t\t\t\\ + \\ \\varepsilon \\underbrace{\\sum_{|J'| \\leq |I| - 1} (1 + t)^{- 1 + C \\varepsilon} \n\t\t\t\\Big| \\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}}]{\\Lie_{\\mathcal{Z}}^{J'} \\Far} \\Big|}_{\\mbox{Absent if \n\t\t\t$|I|=0$}} \n\t\t\t\\ + \\ \\varepsilon^2 (1 + t + |q|)^{-4}. \\notag\n\t\\end{align}\t\n\\end{lemma}\n\n\\begin{proof}\n\tBy Proposition \\ref{P:AlgebraicInhomogeneous}, we have that\t\n\t\\begin{align}\n\t\t|\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}| & \\lesssim |(i)| \\ + \\ |(ii)| \\ + \\ |(iii)|,\n\t\\end{align}\n\twhere\n\t\n\t\\begin{align}\n\t\t|(i)| & = \\sum_{|J| + |K| \\leq |I|} |\\nabla\\nabla_{\\mathcal{Z}}^J h|_{\\mathcal{T} \\mathcal{N}} \n\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^K h|_{\\mathcal{T} \\mathcal{N}} \n\t\t\t\\ + \\ |\\conenabla \\nabla_{\\mathcal{Z}}^J h| |\\nabla\\nabla_{\\mathcal{Z}}^K h| \\ + \\ \\underbrace{\\sum_{|J''| + |K''| \\leq \n\t\t\t|I| - 2} |\\nabla\\nabla_{\\mathcal{Z}}^{J''} h| \n\t\t\t|\\nabla \\nabla_{\\mathcal{Z}}^{K''} h|}_{\\mbox{Absent if $|I| \\leq 1$}}, \\\\\n\t\t|(ii)| & = \\sum_{|J| + |K| \\leq |I|} |\\Lie_{\\mathcal{Z}}^{J} \\Far| |\\Lie_{\\mathcal{Z}}^{K} \\Far|, \\\\\n\t\t|(iii)| & = \\sum_{|I_1| + |I_2| + |I_3| \\leq |I|} \n\t\t\t\\Big\\lbrace|\\nabla_{\\mathcal{Z}}^{I_1} h| |\\nabla\\nabla_{\\mathcal{Z}}^{I_2} h| |\\nabla\\nabla_{\\mathcal{Z}}^{I_3}h| \n\t\t\t\\ + \\ |\\nabla_{\\mathcal{Z}}^{I_1} h||\\Lie_{\\mathcal{Z}}^{I_2}\\Far| |\\Lie_{\\mathcal{Z}}^{I_3} \\Far| \n\t\t\t\\ + \\ |\\Lie_{\\mathcal{Z}}^{I_1} \\Far||\\Lie_{\\mathcal{Z}}^{I_2}\\Far| |\\Lie_{\\mathcal{Z}}^{I_3} \\Far| \\Big\\rbrace. \n\t\\end{align}\n\t\n\tThe desired bound for $|(i)|$ was proved in Lemma 11.2 of \\cite{hLiR2010} \n\tusing the decomposition $h = h^{(1)} + h^{(0)},$ and by combining Lemma \\ref{L:h0decayestimates} and the estimates \n\t\\eqref{E:MainTheoremAssumptionStrongLinfinityDecayPrincipalTermCoefficients} - \t\n\t\\eqref{E:MainTheoremAssumptionStrongerLinfinityDecayGoodComponents}. The term (ii) is the main contribution to \n\t$|\\nabla_{\\mathcal{Z}}^I \\mathfrak{H}|$ arising from the presence of non-zero electromagnetic fields.\n\tTo bound $|(ii)|$ by the right-hand side of \\eqref{E:MainTheoremLieZIh1InhomogeneousAlgebraicEstimate}, we consider the cases\n\t$(|J| =\\dParameter , |K|= 0),$ $(|J| = 0, |K| =\\dParameter ),$ $(|J| \\leq \\dParameter -1, |K| \\leq  \\lceil \\dParameter\/2 \\rceil),$ and\n\t$(|J| \\leq \\lceil \\dParameter\/2 \\rceil, |K| \\leq \\dParameter - 1);$ clearly this exhausts all possible cases. \n\tIn the first two cases, we use \\eqref{E:MainTheoremAssumptionStrongLinfinityDecayPrincipalTermCoefficients} to achieve the \n\tdesired bound, while in the last two, we use \\eqref{E:MainTheoremAssumptionStrongLinfinityDecay}. The cubic terms\n\tfrom case $(iii)$ can be similarly bounded using \\eqref{E:MainTheoremAssumptionStrongLinfinityDecay}.\n\t\n\t\n\t\n\\end{proof}\n\nUsing the previous lemma, we now derive the desired integral inequalities corresponding to the\n$\\varepsilon^{-1}(1 + t)|\\nablamod_{\\mathcal{Z}}^I  \\mathfrak{H}|^2$ term on the right-hand side of \\eqref{E:MainTheoremH1IInhomogeneousTermNablaZIh1PeterPaul}.\n\n\\begin{lemma} \\label{L:MainTheoremLieZIh1InhomogeneousIntegralEstimate} \\cite[Extension of Lemma 11.3]{hLiR2010}\n\t\\textbf{(Integral estimates for $\\varepsilon^{-1} (1 + \\tau) |\\nablamod_{\\mathcal{Z}}^I \\mathfrak{H}|^2 w(q)$)}\n\tUnder the assumptions of Theorem \\ref{T:ImprovedDecay}, if $I$ is any \n\t$\\mathcal{Z}-$multi-index with $|I| \\leq \\dParameter ,$ and if $\\varepsilon$ is sufficiently small,\n\tthen the following pointwise estimates hold for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{align} \\label{E:MainTheoremLieZIh1InhomogeneousIntegralEstimate}\n\t\t\\varepsilon^{-1} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} & (1 + \\tau) \n\t\t\t|\\nablamod_{\\mathcal{Z}}^I \\mathfrak{H}|^2 w(q) \\,d^3x \\, d \\tau\t\\\\\n\t\t& \\lesssim  \\varepsilon \\sum_{|J| \\leq |I|}\t\\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \\bigg\\lbrace (1 + \\tau)^{-1} \n\t\t\t\\Big| \\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}}]{\\Lie_{\\mathcal{Z}}^J \\Far} \\Big|^2 w(q) \n\t\t\t\\ + \\ |\\conenabla \\nabla_{\\mathcal{Z}}^J h^{(1)}|^2 w'(q) \\bigg\\rbrace \\,d^3x \\, d \\tau \\notag \\\\\n\t\t& \\ \\ + \\ \\underbrace{\\varepsilon \\sum_{|J'| \\leq |I| - 1}\t\\int_{0}^{t} \\int_{\\Sigma_{\\tau}} (1 + \\tau)^{-1 + \n\t\t\tC\\varepsilon} \\Big| \\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}}]{\\Lie_{\\mathcal{Z}}^{J'} \\Far} \\Big|^2 w(q) \n\t\t\t\\,d^3x \\, d \\tau}_{\\mbox{Absent if $|I|=0$}} \\ + \\ \\varepsilon^3. \\notag\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tAfter squaring both sides of \\eqref{E:MainTheoremLieZIh1InhomogeneousAlgebraicEstimate}, \n\tmultiplying by $\\varepsilon^{-1} (1 + t) w(q),$ using the inequality \\\\\n\t${(1 + |q|)^{-1} (1 + q_-)^{2\\upmu} w(q) \\lesssim w'(q)}$ (i.e., inequality \\eqref{E:weightinequality})\n\tand the fact that $\\upmu + \\upmu' < 1\/2,$  and integrating, the only terms that are not manifestly bounded by the right-hand side\n\tof \\eqref{E:MainTheoremLieZIh1InhomogeneousIntegralEstimate} are\n\t\n\t\\begin{align}\n\t\t\\varepsilon^3 \\sum_{|J| \\leq |I|} \\int_{0}^{t} \\int_{\\Sigma_t} \n\t\t(1 + \\tau)^{-1} (1 + |q|)^{-2} |\\nabla_{\\mathcal{Z}}^J h^{(1)}|^2 w(q) \\, d^3x \\, d \\tau.\n\t\\end{align}\n\tThe desired bound for these terms can be achieved with the help of the Hardy inequalities of Proposition \\ref{P:Hardy},\n\twhich imply that\n\n\t\\begin{align}\n\t\t\\int_{\\Sigma_t} (1 + \\tau)^{-1}(1 + |q|)^{-2} \n\t\t\t|\\nabla_{\\mathcal{Z}}^J h^{(1)}|^2 w(q) \\, d^3x\n\t\t\t\\lesssim \\int_{\\Sigma_t} (1 + \\tau)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}|^2 w(q) \\, d^3 x.\n\t\\end{align}\t\n\n\t\n\\end{proof}\n\n\nWe now derive the desired the desired integral inequalities corresponding to the \n$|\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}| |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|$\nterm on the right-hand side of \\eqref{E:MainTheoremH1IInhomogeneousTermNablaZIh1PeterPaul}.\n\n\n\\begin{lemma} \\label{L:mathfrakH0partialZIhAproductestimate} \\cite[Lemma 11.4]{hLiR2010}\n\\textbf{(Integral estimates for $|\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}| |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}| w(q)$)} Let $M$ be the ADM mass. Under the assumptions of Theorem \\ref{T:ImprovedDecay}, if $I$ is a $\\mathcal{Z}-$multi-index satisfying $|I| \\leq \\dParameter ,$ and if $\\varepsilon$ is sufficiently small,\nthen the following integral inequality holds for $t \\in [0,T):$\n\n\\begin{align} \\label{E:mathfrakH0partialZIhAproductestimate}\n\t\\int_{0}^{t} \\int_{\\Sigma_{\\tau}} | \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}| \n\t|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}| w(q) \\,d^3x \\, d \\tau \n\t& \\lesssim M  \\sum_{|J| \\leq |I|} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} (1 + \\tau)^{-2} \n\t\t|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) \\, d^3x \\, d \\tau  \\\\\n\t& \\ \\ + \\ M \\sum_{|J| \\leq |I|} \\int_{0}^{t} (1 + \\tau)^{-3\/2} \\bigg( \n\t\\sqrt{\\int_{\\Sigma_{\\tau}} \n\t\t\t| \\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) \\, d^3 x} \\bigg) \\, d \\tau.  \\notag\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tWe first use the Cauchy-Schwarz inequality for integrals to obtain\n\t\\begin{align} \\label{E:CauchySchwarzmathfrakH0partialZIhAproductestimate}\n\t\t\\int_{0}^{t} \\int_{\\Sigma_{\\tau}} | & \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}| \n\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}| w(q) \\,d^3x \\, d \\tau \\\\\n\t\t& \\leq \t\\int_{0}^{t} \\Bigg\\lbrace \\bigg(\\int_{\\Sigma_{\\tau}} |\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g \n\t\t\th^{(0)}|^2 w(q) \\, d^3 x\\bigg)^{1\/2} \n\t\t\t\\times \\int_{\\Sigma_{\\tau}} \\bigg(|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) \\, d^3x \\bigg)^{1\/2} \n\t\t\t\\Bigg\\rbrace \\, d \\tau. \t\t\\notag\n\t\\end{align}\n\tFurthermore, under the present assumptions, the previous proof of inequality \n\t\\eqref{E:weakdecayLinfinitynablaZISquaregh0MoreGeneral} remains valid. Thus,\n\tusing \\eqref{E:weakdecayLinfinitynablaZISquaregh0MoreGeneral} and the Hardy inequalities of Proposition \\ref{P:Hardy}, it \n\tfollows that\n\t\n\t\\begin{align} \\label{E:mathfrakH0Squaredintegratedestimate}\n\t\t\\int_{\\Sigma_t} |\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_g h^{(0)}|^2 w(q) \\, d^3 x \n\t\t\\lesssim M^2(1 + t)^{-3} \n\t\t\\ + \\ M^2 (1 + t)^{-4} \\sum_{|J| \\leq |I|} \n\t\t\t\\int_{\\Sigma_t} |\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}|^2 w(q) \\, d^3 x.\n\t\\end{align}\n\tThe estimate \\eqref{E:mathfrakH0partialZIhAproductestimate} now follows from \n\t\\eqref{E:CauchySchwarzmathfrakH0partialZIhAproductestimate}, \\eqref{E:mathfrakH0Squaredintegratedestimate}\n\tand the inequalities $\\sqrt{|a| + |b|} \\lesssim \\sqrt{|a|} + \\sqrt{|b|},$ $|ab| \\lesssim a^2 + b^2.$\n\t\n\\end{proof}\n\n\nThe following integral estimate for the commutator term $\\varepsilon^{-1}(1 + t) \\big|\\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)} - \\widetilde{\\Square}_{g} (\\nabla_{\\mathcal{Z}}^I h_{\\mu \\nu}^{(1)}) \\big|^2$ on the right-hand side of \\eqref{E:MainTheoremH1IInhomogeneousTermNablaZIh1PeterPaul} was proved in \\cite{hLiR2010}. Its lengthy proof is similar to our proof of Lemma \\ref{L:MainTheoremFarCommutatorIntegralEstimate} below, and we don't bother to repeat it here.\n\n\n\\begin{lemma} \\label{L:NablaZIBoxCommutatorIntegrated}  \\cite[Lemma 11.5]{hLiR2010}\n\t\\textbf{(Integral estimates for $\\varepsilon^{-1} \\big| \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)} \n\t\t\t- \\widetilde{\\Square}_{g} \\nabla_{\\mathcal{Z}}^I h_{\\mu\\nu}^{(1)}\\big|^2 w(q)$)}\n\tUnder the assumptions of Theorem \\ref{T:ImprovedDecay}, if $I$ is a $\\mathcal{Z}-$multi-index satisfying\n\t$1 \\leq |I| \\leq \\dParameter ,$ and if $\\varepsilon$ is sufficiently small, then \n\tthe following integral inequality holds for $t \\in [0,T):$\n\t\n\t\\begin{align} \\label{E:NablaZIBoxCommutatorIntegrated}\n\t\t\\varepsilon^{-1} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} & (1 + \\tau) \n\t\t\t\\big| \\nablamod_{\\mathcal{Z}}^I \\widetilde{\\Square}_{g} h_{\\mu \\nu}^{(1)} \n\t\t\t- \\widetilde{\\Square}_{g} \\nabla_{\\mathcal{Z}}^I h_{\\mu\\nu}^{(1)}\\big|^2 w(q) \\,d^3x \\, d \\tau \\\\\n\t\t& \\lesssim \\varepsilon \\sum_{|J| \\leq |I|} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \\bigg\\lbrace (1 + \\tau)^{-1} \n\t\t\t| \\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}|^2 w(q) + |\\conenabla \\nabla_{\\mathcal{Z}}^J h^{(1)}|^2 w'(q) \n\t\t\t\\bigg\\rbrace \\,d^3x \\, d \\tau \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|J'| \\leq |I| - 1} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} (1 + \\tau)^{-1 + C \\varepsilon}\n\t\t\t| \\nabla\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}|^2 w(q) \\, d^3x \\, d \\tau \\ + \\ \\varepsilon^3. \\notag\n\t\\end{align}\n\\end{lemma}\n\n\\hfill $\\qed$\n\n\nCombining Lemmas \\ref{L:MainTheoremH1IInhomogeneousTermNablaZIh1PeterPaul}, \\ref{L:MainTheoremLieZIh1InhomogeneousIntegralEstimate}, \\ref{L:mathfrakH0partialZIhAproductestimate}, and \\ref{L:NablaZIBoxCommutatorIntegrated}, we arrive at the following corollary.\n\n\n\\begin{corollary} \\label{C:NablaZIh1FundamentalEnergyEstimate}\n\tUnder the assumptions of Theorem \\ref{T:ImprovedDecay}, if $0 \\leq k \\leq \\dParameter $ and\n\t$\\varepsilon$ is sufficiently small, then \n\tthe following integral inequality holds for $t \\in [0,T):$\n\t\n\t\\begin{align} \\label{E:NablaZIh1FundamentalEnergyEstimate}\n\t\t\\sum_{|I| \\leq k} \n\t\t\t\\int_{0}^{t} \\int_{\\Sigma_{\\tau}} |\\mathfrak{H}^{(1;I)}| |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}| \\,d^3x \\, d \\tau\n\t\t& \\lesssim M \\sum_{|I| \\leq k} \\int_{0}^{t} (1 + \\tau)^{-3\/2} \n\t\t\t\\Big(\\sqrt{\\int_{\\Sigma_{\\tau}} |\\nabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) \\, d^3 x} \\Big) \\, d \\tau\t\\\\\n\t\t& \\ \\ + \\ \\varepsilon  \\sum_{|I| \\leq k} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}}  (1 + \\tau)^{-1} \n\t\t\t\\Big|\\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}}]{\\Lie_{\\mathcal{Z}}^I \\Far} \\Big|^2 w(q) \n\t\t\t\\,d^3x \\, d \\tau \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon  \\sum_{|I| \\leq k} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} \n\t\t\t\\Big( |\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 + |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}^2 \n\t\t\t+ |\\Lie_{\\mathcal{Z}}^J \\Far|_{\\mathcal{T} \\mathcal{T}}^2 \\Big) w'(q)  \\, d^3x \\, d \\tau \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon  \\underbrace{\\sum_{|J| \\leq k - 1} \\int_{0}^{t} \\int_{\\Sigma_{\\tau}} (1 + \\tau)^{-1 + C \n\t\t\t\\varepsilon} \\Big|\\myarray[{\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}}]{\\Lie_{\\mathcal{Z}}^J \\Far} \\Big|^2 w(q) \n\t\t\t\\, d^3x \\, d \\tau}_{\\mbox{Absent if $k=0$}} \\ + \\ \\varepsilon^3. \\notag\n\t\\end{align}\n\n\\end{corollary}\n\nThis completes our analysis of the integral inequalities for the $h_{\\mu \\nu}^{(1)}$ inhomogeneities. \n\n\\hfill $\\qed$\n\n\n\n\n\n\\subsection{Integral inequalities for the $\\Lie_{\\mathcal{Z}}^I \\Far_{\\mu \\nu}$ inhomogeneities} \\label{SS:MainTheoremFarInhomogeneities}\n\nIn this section, we analyze the integrals in Proposition \\ref{P:weightedenergy} \ncorresponding to the inhomogeneous terms $\\mathfrak{F}_{(I)}^{\\nu}$ in equation \\eqref{E:InhomogeneoustermsdMis0proof}. The main goal is to arrive at Corollary \\ref{C:LieZIFarFundamentalEnergyEstimate}. \n\nWe begin with the following lemma, which provides pointwise estimates for the wave coordinate-controlled quantities\n$|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|_{\\mathcal{L} \\mathcal{L}}$ and \n$|\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}|_{\\mathcal{L} \\mathcal{T}}$ for $|I| \\leq \\dParameter $ and $|J| \\leq \\dParameter - 1.$ \nThese pointwise estimates will be used to help to derive suitable integrated estimates later in this section.\n\n\n\\begin{lemma} \\label{L:NablaZIh1LLh1TLMainTheoremWaveCoordianteAlgebraicEstimate}\n\t\\textbf{(Pointwise estimates for $\\sum_{|I| \\leq k} |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|_{\\mathcal{L} \\mathcal{L}}\n\t\t\\ + \\ \\sum_{|J| \\leq k - 1} |\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}|_{\\mathcal{L} \\mathcal{T}}$)}\n\tUnder the assumptions of Theorem \\ref{T:ImprovedDecay}, if $0 \\leq k \\leq \\dParameter $\n\tand $\\varepsilon$ is sufficiently small, then the following pointwise inequality holds for\n \tfor $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\n\t\\begin{align}  \\label{E:NablaZIh1LLh1TLMainTheoremWaveCoordianteAlgebraicEstimate}\n\t\t\\sum_{|I| \\leq k}|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|_{\\mathcal{L} \\mathcal{L}}\n\t\t& \\ + \\ \\underbrace{\\sum_{|J| \\leq k - 1} |\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}|_{\\mathcal{L} \\mathcal{T}}}_{\\mbox{Absent \n\t\tif $k = 0$}} \n\t\t\t \\\\\n\t\t& \\lesssim \\sum_{|I| \\leq k} |\\conenabla \\nabla_{\\mathcal{Z}}^{I} h^{(1)}| \n\t\t\t\\ + \\ \\varepsilon (1 + t + |q|)^{-2} \\chi_0(1\/2 \\leq r\/t \\leq 3\/4) \\ + \\ \\varepsilon^2 (1 + t + |q|)^{-3} \\notag \\\\\n\t\t& \\ \\\t+ \\ \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-1 + C \\varepsilon}(1 + |q|)^{1\/2 + \n\t\t\\upmu'}|\\nabla\\nabla_{\\mathcal{Z}}^{I} h^{(1)}| \\notag \\\\ \n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-1 + C \\varepsilon}(1 + |q|)^{-1\/2 + \\upmu'} \n\t\t\t|\\nabla_{\\mathcal{Z}}^{I} h^{(1)}| \\notag \\\\\n\t\t& \\ \\ + \\ \\underbrace{\\sum_{|J'| \\leq k - 2} |\\nabla\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}|}_{\\mbox{absent if $k \\leq 1$}}, \t\n\t\t\\notag \n\t\t\\end{align}\n\twhere $\\chi_0(1\/2 \\leq z \\leq 3\/4)$ is the characteristic function of the interval $[1\/2,3\/4].$\n\t\t\n\n\n\n\\end{lemma}\n\n\\begin{proof}\n\tLemma \\ref{L:NablaZIh1LLh1TLMainTheoremWaveCoordianteAlgebraicEstimate} follows \n\tfrom Lemma \\ref{L:NablaZIh1LLh1LTwaveCoordinateAlgebraicEstimate} (for the tensorfield $h_{\\mu \\nu}^{(1)}$) \n\tand the pointwise decay assumptions \\eqref{E:MainTheoremAssumptionStrongLinfinityDecay}\n\tfor $h_{\\mu \\nu}^{(1)}.$\n\\end{proof}\n\n\n\n\n\\begin{lemma} \\label{L:MainTheoremEnergyInhomogeneousTermAlgebraicEstimate}\n\t\\textbf{(Pointwise estimates for $|(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu})\\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}_{(I)}^{\\nu}|$)}\n\t\tLet $\\mathfrak{F}_{(I)}^{\\nu} = \\Liemod_{\\mathcal{Z}}^I \n\t\t\\mathfrak{F}^{\\nu} + \\Big\\lbrace N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I\\Far_{\\kappa \\lambda} \n\t\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big)\\Big\\rbrace$ be the \n\t\tinhomogeneous term \\eqref{E:LiemodZIdifferentiatedEOVInhomogeneousterms} in the equations of variation \\eqref{E:EOVdMis0} \n\t\tsatisfied by $\\dot{\\Far} \\eqdef \\Lie_{\\mathcal{Z}}^I \\Far.$ Under the assumptions of Theorem \\ref{T:ImprovedDecay}, \n\t\tif $0 \\leq k \\leq \\dParameter $ and $\\varepsilon$ is sufficiently small, then the following pointwise inequality holds for \n\t\t$(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{align} \\label{E:MainTheoremEnergyInhomogeneousTermAlgebraicEstimate}\n\t\t\\sum_{|I| \\leq k} |(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu})\\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}_{(I)}^{\\nu} |\n\t\t\t& \\lesssim \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|^2\n\t\t\t\t\\ + \\ \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 \\\\\n\t\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} (1 + |q|)^{-1}(1 + q_-)^{-2\\upmu} |\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 \n\t\t\t\t\\notag \\\\\n\t\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq |k|} (1 + |q|)^{-1}(1 + q_-)^{-2\\upmu} \n\t\t\t\t\\big(|\\Lie_{\\mathcal{Z}}^I  \\Far|_{\\mathcal{L} \\mathcal{N}}^2 + |\\Lie_{\\mathcal{Z}}^I  \\Far|_{\\mathcal{T} \n\t\t\t\t\\mathcal{T}}^2 \\big). \\notag\n\t\t\\end{align}\n\t\n\\end{lemma}\n\n\n\n\\begin{proof}\n\nWe first recall the decomposition \\eqref{E:LiemodZIdifferentiatedEOVInhomogeneousterms} of $\\mathfrak{F}_{(I)}^{\\nu}:$\n\n\\begin{align}\n\t \\mathfrak{F}_{(I)}^{\\nu} & = \\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu}\n\t\t\t\\ + \\ \\Big\\lbrace N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I\\Far_{\\kappa \\lambda}\n\t\t\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big)\\Big\\rbrace,\n\t\t\t\t&& (\\nu = 0,1,2,3). \\label{E:LiemodZIdifferentiatedEOVInhomogeneoustermsagain}\n\\end{align}\n\nNow using \\eqref{E:LieZIFarNullFormInhomogeneousTermAlgebraicEstimate} with $X_{\\nu} \\eqdef \\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu},$\ntogether with the decomposition $h = h^{(0)} + h^{(1)}$ and the $h^{(0)}-$decay estimates of Lemma \\ref{L:h0decayestimates}, it follows that\n\n\\begin{align} \\label{E:MainTheoremEnergyInhomogeneousTermPreliminaryAlgebraicEstimate}\t\n\t\\sum_{|I| \\leq k} |(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu}) \\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu}|\n\t& \\lesssim \\mathop{\\sum_{|I| \\leq k}}_{|I_1| + |I_2| \\leq |I|} \n\t\t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\conenabla \\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}|\n\t\t|\\Lie_{\\mathcal{Z}}^{I_2}\\Far| \\\\\n\t& \\ \\ + \\ \\mathop{\\sum_{|I| \\leq k}}_{|I_1| + |I_2| \\leq |I|} \n\t\t|\\Lie_{\\mathcal{Z}}^I \\Far| \n\t\t|\\nabla\\nabla_{\\mathcal{Z}}^{I_1}h^{(1)}|\n\t\t\\big(|\\Lie_{\\mathcal{Z}}^{I_2}\\Far|_{\\mathcal{L}\\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^{I_2}\\Far|_{\\mathcal{T}\\mathcal{T}} \\big)\\notag \\\\\n\t& \\ \\ + \\ \\mathop{\\sum_{|I| \\leq k}}_{|I_1| + |I_2| + |I_3| \\leq |I|}\n\t\t|\\Lie_{\\mathcal{Z}}^I \\Far|\n\t\t|\\nabla\\nabla_{\\mathcal{Z}}^{I_1}h^{(1)}| \n\t\t|\\Lie_{\\mathcal{Z}}^{I_2}\\Far| \n\t\t|\\Lie_{\\mathcal{Z}}^{I_3}\\Far| \\notag \\\\\n\t& \\ \\ + \\ \\mathop{\\sum_{|I| \\leq k}}_{|I_1| + |I_2| + |I_3| \\leq |I|} \n\t\t|\\Lie_{\\mathcal{Z}}^I \\Far|\n\t\t|\\nabla_{\\mathcal{Z}}^{I_1}h^{(1)}| \n\t\t|\\nabla\\nabla_{\\mathcal{Z}}^{I_2}h^{(1)}| \n\t\t|\\Lie_{\\mathcal{Z}}^{I_3}\\Far| \\notag \\\\\n\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^{I_1}h^{(1)}|^2 \\notag \\\\\n\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-1} (1 + |q|)^{-2} |\\nabla_{\\mathcal{Z}}^{I_1}h^{(1)}|^2 \\notag \\\\\n\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} (1 + t + |q|)^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|^2. \\notag\n\\end{align}\n\nInequality \\eqref{E:MainTheoremEnergyInhomogeneousTermAlgebraicEstimate} now follows from \nthe assumptions of Theorem \\ref{T:ImprovedDecay}, \\eqref{E:MainTheoremEnergyInhomogeneousTermPreliminaryAlgebraicEstimate}, \nand repeated application of algebraic inequalities of the form $|ab| \\lesssim \\varsigma a^2 + \\varsigma^{-1} b^2.$ As an example, \nwe consider the term $|\\Lie_{\\mathcal{Z}}^I \\Far| |\\nabla\\nabla_{\\mathcal{Z}}^{I_1}h^{(1)}|\n|\\Lie_{\\mathcal{Z}}^{I_2}\\Far|_{\\mathcal{L}\\mathcal{N}}$ in the case that $|I_1| \\leq |I| \\leq \\lceil \\dParameter\/2 \\rceil$ \n(such an inequality must be satisfied by either $|I_1|$ or $|I_2|$).\nThen with the help of \\eqref{E:MainTheoremAssumptionStrongLinfinityDecay} \nand the fact that $\\upmu + \\upmu' < 1\/2,$ it follows that if $\\varepsilon$ is sufficiently small, then\n\n\\begin{align}\n\t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\nabla\\nabla_{\\mathcal{Z}}^{I_1}h^{(1)}|\n\t|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{L}\\mathcal{N}}\n\t& \\lesssim \\varepsilon (1 + t + |q|)^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|^2 \n\t\t\\ + \\ \\varepsilon^{-1} (1 + t + |q|) |\\nabla\\nabla_{\\mathcal{Z}}^{I_1}h^{(1)}|^2\n\t\t|\\Lie_{\\mathcal{Z}}^{I_2}\\Far|_{\\mathcal{L}\\mathcal{N}}^2 \\\\\n\t& \\lesssim \\varepsilon(1 + t + |q|)^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|^2  \n\t\t\\ + \\ \\varepsilon(1 + |q|)^{-1} (1 + q_-)^{-2\\upmu} |\\Lie_{\\mathcal{Z}}^{I_2}\\Far|_{\\mathcal{L}\\mathcal{N}}^2. \\notag \n\\end{align}\nWe now observe that the right-hand side of the above inequality is manifestly bounded by the right-hand side of \\eqref{E:MainTheoremEnergyInhomogeneousTermAlgebraicEstimate}.\n\n\n\\end{proof}\n\n\\begin{lemma} \\label{L:MainTheoremFarInhomogeneousTermIntegralEstimate}\n\t\\textbf{(Integral estimates for $\\big|(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu})\\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu} \\big| \n\tw(q)$)} \n\tUnder the assumptions of Lemma \\ref{L:MainTheoremEnergyInhomogeneousTermAlgebraicEstimate},\n\tif $0 \\leq k \\leq \\dParameter $ and\n\t$\\varepsilon$ is sufficiently small, then the following integral inequality holds for\n\t$t \\in [0,T):$\n\n\t\\begin{align} \\label{E:MainTheoremFarInhomogeneousTermIntegralEstimate}\n\t\t\\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}}\n\t\t\\big|(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu})\\Liemod_{\\mathcal{Z}}^I \\mathfrak{F}^{\\nu} \\big| w(q) \\, d^3x \\, d \\tau \n\t\t& \\lesssim \\varepsilon \\sum_{|I| \\leq k} \n\t\t\t\\int_0^t \\int_{\\Sigma_{\\tau}} (1 + \\tau)^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|^2 w(q) \\, d^3x \\, d \\tau \\\\\n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} (1 + \\tau)^{-1} \n\t\t\t|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) \\, d^3x \\, d \\tau \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w'(q) \\, d^3x \\, d \\tau \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t\\big(|\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L} \\mathcal{N}}^2 \n\t\t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T} \\mathcal{T}}^2  \\big) w'(q) \\, d^3x \\, d \\tau.   \\notag\n\t\\end{align}\n\n\\end{lemma}\n\n\\begin{proof}\n\tInequality \\eqref{E:MainTheoremFarInhomogeneousTermIntegralEstimate}\n\tfollows from multiplying inequality \\eqref{E:MainTheoremEnergyInhomogeneousTermAlgebraicEstimate} by $w(q),$\n\tintegrating, and using the fact that \\\\\n\t$(1 + |q|)^{-1}(1 + q_-)^{-2\\upmu} w(q) \\lesssim w'(q).$\n\\end{proof}\n\n\n\n\n\\begin{lemma} \\label{L:MainTheoremFarCommutatorAlgebraicEstimate}\n\t\\textbf{(Pointwise estimates for $\\Big|(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu}) \n\t\t\\Big\\lbrace N^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}\n\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big) \\Big\\rbrace \\Big|$)}\n\t\n\tLet $N^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}\n\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big)$ be the inhomogeneous\n\tcommutator term \\eqref{E:LiemodZINnablaFarCommutatorTerms} in the equations of \n\tvariation \\eqref{E:EOVdMis0} satisfied by $\\dot{\\Far}_{\\mu \\nu} \\eqdef \\Lie_{\\mathcal{Z}} \\Far_{\\mu \\nu}.$ Under the \n\tassumptions of Theorem \\ref{T:ImprovedDecay}, if $1 \\leq k \\leq \\dParameter $ and $\\varepsilon$ is sufficiently small, then the \n\tfollowing pointwise inequality holds for $(t,x) \\in [0,T) \\times \\mathbb{R}^3:$\n\t\n\t\\begin{align} \\label{E:MainTheoremFarCommutatorAlgebraicEstimate}\n\t\t\\sum_{|I| \\leq k} \\Big|(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu}) \n\t\t& \\Big\\lbrace N^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}\n\t\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big) \\Big\\rbrace \\Big|\t\\\\\n\t\t& \\lesssim \\varepsilon (1 + t + |q|)^{-1} \\sum_{|I| \\leq |k|}|\\Lie_{\\mathcal{Z}}^I \\Far|^2 \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon (1 + t + |q|)^{-1}(1 + |q|)^{-2} \\sum_{|I| \\leq |k|} |\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon (1 + |q|)^{-1}(1+ q_-)^{-2 \\mu} \n\t\t\t\\sum_{|I| \\leq |k|} \\big( |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}^2\n  \t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}^2 \\big) \\notag \\\\\n  \t& \\ \\ + \\ \\varepsilon (1 + t + |q|)^{-1 + C \\varepsilon} (1 + |q|)^{-(2 + C \\varepsilon)} (1+ q_-)^{-2 \\upmu}\n  \t\t\\sum_{|I| \\leq k} |\\nabla_{\\mathcal{Z}}^I h^{(1)}|_{\\mathcal{L}\\mathcal{L}}^2  \\notag \\\\\n  \t& \\ \\ + \\ \\varepsilon (1 + t + |q|)^{-1 + C \\varepsilon} (1 + |q|)^{-(2 + C \\varepsilon)} (1+ q_-)^{-2 \\upmu}\n  \t\t\\sum_{|J| \\leq k - 1}  \n  \t\t|\\nabla_{\\mathcal{Z}}^J h^{(1)}|_{\\mathcal{L}\\mathcal{T}}^2  \\notag \\\\\n  \t& \\ \\ + \\ \\varepsilon (1 + t + |q|)^{-1 + C \\varepsilon} (1 + |q|)^{-2}\n  \t\t\\underbrace{\\sum_{|J'| \\leq k - 2}\t|\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}|^2}_{\\mbox{absent if $k = 1$}} \\notag \\\\\n  \t& \\ \\\t+ \\ \\varepsilon (1 + t + |q|)^{-1 + C \\varepsilon} \\sum_{|J| \\leq k - 1} |\\Lie_{\\mathcal{Z}}^J \\Far|^2. \\notag\n\t\\end{align}\n\t\n\\end{lemma}\n\n\n\n\\begin{proof}\n\tUsing inequality \\eqref{E:EnergyInhomogeneousTermAlgebraicEstimate} with $X_{\\nu} \\eqdef \\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu},$\n\tthe pointwise decay assumptions of Theorem \\ref{T:ImprovedDecay}, together with the decomposition $h = h^{(0)} + h^{(1)}$ and \n\tthe $h^{(0)}$ decay estimates of Lemma \\ref{L:h0decayestimates}, it follows that\n\t\n\t\\begin{align} \\label{E:MainTheoremEnergyInhomogeneousTermAlgebraicEstimateMultiplied}\n\t\t\\sum_{|I| \\leq k} \n\t\t\\Big|\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu} & \\Big\\lbrace N^{\\# \\mu \\nu \\kappa \n\t\t\t\\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I\\Far_{\\kappa \\lambda}\n\t\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big) \\Big\\rbrace \\Big|\t\\\\\n\t\t& \\lesssim (1 + |q|)^{-1} \\mathop{\\sum_{|I| \\leq k, |I'| \\leq k}}_{|J| \\leq 1} \n\t\t\t\t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\nabla_{\\mathcal{Z}}^{I'} h^{(1)}|_{\\mathcal{L}\\mathcal{L}} |\\Lie_{\\mathcal{Z}}^J \\Far| \n\t\t\t\\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\sum_{|I| \\leq k, |I'| \\leq k}}_{|J| \\leq 1} \n\t\t\t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\nabla_{\\mathcal{Z}}^J h^{(1)}|_{\\mathcal{L}\\mathcal{L}} |\\Lie_{\\mathcal{Z}}^{I'} \\Far| \n\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\sum_{|I| \\leq k} |\\Lie_{\\mathcal{Z}}^I \\Far|^2 |h|_{\\mathcal{L}\\mathcal{T}} \n\t\t\t\\ + \\ (1 + |q|)^{-1} \\mathop{\\mathop{\\sum_{|I|\\leq k}}_{|I_1| + |I_2| \\leq k + 1}}_{|I_1|, |I_2| \\leq k}\n\t\t\t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}| \n\t\t\t\\big(|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{L} \\mathcal{N}} + |\\Lie_{\\mathcal{Z}}^{I_2} \\Far|_{\\mathcal{T} \t\n\t\t\t\\mathcal{T}} \\big) \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon(1 + t + |q|)^{-1} \\sum_{|I|\\leq k} |\\Lie_{\\mathcal{Z}}^I \\Far|^2 \n\t\t\t\\ + \\ \\varepsilon(1 + t + |q|)^{-1} (1 + |q|)^{-2} \\sum_{|I|\\leq k} |\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 \\notag \\\\ \n\t\t& \\ \\ + \\ (1 + t + |q|)^{-1} \\mathop{\\mathop{\\sum_{|I|\\leq k}}_{|I_1| + |I_2| \\leq k + 1}}_{|I_1|, |I_2| \\leq k}\n\t\t\t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \n\t\t\t\\ + \\ (1 + |q|)^{-1} \\mathop{\\mathop{\\sum_{|I|\\leq k}}_{|I_1| + |I_2| \\leq k + 1}}_{|I_1| \\leq k-1, |I_2| \\leq k - 1}\n\t\t \t\t|\\Lie_{\\mathcal{Z}}^I \\Far||\\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}|_{\\mathcal{L}\\mathcal{L}} \n\t\t \t\t|\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\mathop{\\sum_{|I|\\leq k}}_{|I_1| + |I_2| \\leq k}}_{|I_1| \\leq k - 1, |I_2| \\leq k - 1}\n\t\t \t|\\Lie_{\\mathcal{Z}}^I \\Far||\\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}|_{\\mathcal{L}\\mathcal{T}} |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \n\t\t \\ \\ + \\ (1 + |q|)^{-1} \\underbrace{\\mathop{\\mathop{\\sum_{|I|\\leq k}}_{|I_1| + |I_2| \\leq k - 1}}_{|I_1| \\leq k - 2, |I_2| \n\t\t\t\\leq k - 1} |\\Lie_{\\mathcal{Z}}^I \\Far||\\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}| \n\t\t\t|\\Lie_{\\mathcal{Z}}^{I_2} \\Far|}_{\\mbox{absent if $k = 1$}} \\notag \\\\\n\t\t& \\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\mathop{\\sum_{|I|\\leq k}}_{|I_1| + |I_2| + |I_3| \\leq k + 1}}_{|I_1|, |I_2|, |I_3| \\leq k}\n\t\t \t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}| |\\nabla_{\\mathcal{Z}}^{I_2} h^{(1)}| \n\t\t \t|\\Lie_{\\mathcal{Z}}^{I_3}\\Far| \\notag \\\\\n\t\t & \\ \\ + \\ (1 + |q|)^{-1} \\mathop{\\mathop{\\sum_{|I|\\leq k}}_{|I_1| + |I_2| + |I_3| \\leq k + 1}}_{|I_1|, |I_2|, |I_3| \\leq k}\n\t\t \t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\nabla_{\\mathcal{Z}}^{I_1} h^{(1)}| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \n\t\t \t\t|\\Lie_{\\mathcal{Z}}^{I_3}\\Far| \n\t\t \t\\\t+ \\ (1 + |q|)^{-1} \\mathop{\\mathop{\\sum_{|I|\\leq k}}_{|I_1| + |I_2| + |I_3| \\leq k + 1}}_{|I_1|, |I_2|, |I_3| \\leq k}\n\t\t\t\t|\\Lie_{\\mathcal{Z}}^I \\Far| |\\Lie_{\\mathcal{Z}}^{I_1} \\Far| |\\Lie_{\\mathcal{Z}}^{I_2} \\Far| \n\t\t\t\t|\\Lie_{\\mathcal{Z}}^{I_3}\\Far|. \\notag\n\t\\end{align}\n\tWe remark that the $\\varepsilon (1 + t + |q|)^{-1} \\sum_{|I|\\leq k} |\\Lie_{\\mathcal{Z}}^I \\Far|^2$ and \n\t$\\varepsilon (1 + t + |q|)^{-1} (1 + |q|)^{-2} \\sum_{|I|\\leq k} |\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2$ sums on the right-hand \n\tside of \\eqref{E:MainTheoremEnergyInhomogeneousTermAlgebraicEstimateMultiplied} account for all of the terms containing a \n\tfactor $\\nabla_{\\mathcal{Z}}^J h^{(0)}$ for some $J.$\n\tInequality \\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate} now follows from \n\t\\eqref{E:MainTheoremEnergyInhomogeneousTermAlgebraicEstimateMultiplied}, the pointwise decay assumptions of Theorem \n\t\\ref{T:ImprovedDecay} (including the implied estimates for $h^{(1)}$), and simple algebraic estimates of the form $|ab| \\lesssim \\varsigma a^2 + \n\t\\varsigma^{-1} b^2$ (as in the proof of \\eqref{E:MainTheoremEnergyInhomogeneousTermPreliminaryAlgebraicEstimate}).\n\t\n\\end{proof}\n\n\n\n\n\\begin{lemma} \\label{L:MainTheoremFarCommutatorIntegralEstimate}\n\t\\textbf{(Integral estimates for $\\Big|(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu}) \n\t\t\\Big\\lbrace N^{\\# \\mu \\nu \\kappa \\lambda} \\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I \\Far_{\\kappa \\lambda}\n\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big) \\Big\\rbrace \\Big|$)}\n\tUnder the assumptions of Lemma \\ref{L:MainTheoremEnergyInhomogeneousTermAlgebraicEstimate}, if \n\t$1 \\leq k \\leq \\dParameter $ and $\\varepsilon$ is sufficiently small,\n\tthen the following integral inequality holds for $t \\in [0,T):$ \n\t\n\t\\begin{align} \\label{E:MainTheoremFarCommutatorIntegralEstimate}\n\t\t\\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \\Big|(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu}) \n\t\t& \\Big\\lbrace N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I\\Far_{\\kappa \\lambda}\n\t\t\t- \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \\lambda}\\big) \\Big\\rbrace \\Big|\n\t\t\tw(q) \\, d^3x \\, d \\tau \\\\\n\t\t& \\lesssim \\varepsilon \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t\t(1 + \\tau)^{-1} |\\Lie_{\\mathcal{Z}}^I \\Far|^2 w(q) \\, d^3x \\, d \\tau \n\t\t\t\\ + \\ \\varepsilon \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t\t(1 + \\tau)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) \\, d^3x \\, d \\tau\n\t\t\t\t\\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t\\Big\\lbrace |\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2\n\t\t\t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}^2\n  \t\t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}^2 \\Big\\rbrace w'(q) \\, d^3x \\, d \\tau \\notag \\\\\n  \t& \\ \\ + \\ \\underbrace{\\varepsilon \\sum_{|J'| \\leq k - 2} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t\t(1 + \\tau + |q|)^{-1 + C \\varepsilon} |\\nabla\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}|^2 w(q) d^3x \\, d \\tau}_{\\mbox{absent if $k = 1$}} \n\t\t\t\t \\notag \\\\\n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|J| \\leq k - 1} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t(1 + \\tau + |q|)^{-1 + C \\varepsilon} |\\Lie_{\\mathcal{Z}}^J \\Far|^2 w(q) d^3x \\, d \\tau \n\t\t\t\\ + \\ \\varepsilon^2. \\notag\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tWe begin by multiplying by both sides of \\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate} by $w(q)$ and integrating\n\t$\\int_0^t \\int_{\\Sigma_{\\tau}} d^3x \\, d \\tau.$ The integrals corresponding to the first and last sums on the right-hand \n\tside of \\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate} are manifestly bounded by the first \n\tand penultimate sums on the right-hand side of \\eqref{E:MainTheoremFarCommutatorIntegralEstimate}. Using also the fact that \n\t$(1 + |q|)^{-1}(1+ q_-)^{-2 \\upmu} w(q) \\lesssim w'(q),$ the integral corresponding to the\n\tthird sum on the on the right-hand side of \\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate} is bounded by the third sum \n\ton the right-hand side of \\eqref{E:MainTheoremFarCommutatorIntegralEstimate}.\n\t\n\tTo bound the integral corresponding to the second sum on the right-hand side of \n\t\\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate}, we simply use the Hardy inequalities of Proposition \\ref{P:Hardy}\n\tto derive the inequality\n\t\n\t\\begin{align}\n\t\t\\sum_{|I| \\leq k} & \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t(1 + \\tau + |q|)^{-1}(1 + |q|)^{-2} |\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) d^3x \\, d \\tau \\\\\n\t\t& \\lesssim \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t(1 + \\tau + |q|)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) d^3x \\, d \\tau. \\notag\n\t\\end{align}\n\tAfter multiplying by $\\varepsilon,$ the right-hand side of the above inequality \n\tis manifestly bounded by the right-hand side of \\eqref{E:MainTheoremFarCommutatorIntegralEstimate}.\n\tUsing the same reasoning, we obtain the following bound for the integral \n\tcorresponding to the sixth sum on the right-hand side \n\tof \\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate}:\n\t\n\t\\begin{align} \\label{E:Hardybound}\n\t\t\\sum_{|J'| \\leq k - 2} & \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t(1 + \\tau + |q|)^{-1 + C \\varepsilon}(1 + |q|)^{-2} |\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}|^2 w(q) d^3x \\, d \\tau \\\\\n\t\t& \\lesssim \\sum_{|J'| \\leq k - 2} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t(1 + \\tau + |q|)^{-1 + C \\varepsilon} |\\nabla\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}|^2 w(q) d^3x \\, d \\tau. \\notag\n\t\\end{align}\n\tWe then multiply \\eqref{E:Hardybound} by $\\varepsilon$ and observe that the right-hand side of the resulting inequality \n\tis manifestly bounded by the right-hand side of \\eqref{E:MainTheoremFarCommutatorIntegralEstimate}.\n\t\n\t\n\tTo address the fourth and fifth sums on the right-hand side of \\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate},\n\twe will make use of the weight $\\widetilde{w}(q),$ which is defined by\n\t\n\t\\begin{align}\n\t\t\\widetilde{w}(q) \\eqdef \\min \\big\\lbrace w'(q), (1 + t + |q|)^{-1 + C \\varepsilon} w(q) \\big\\rbrace.\n\t\\end{align}\n\tWe note that by \\eqref{E:weightinequality}, the following inequality is satisfied:\t\n\n\t\\begin{align}\n\t\t\\widetilde{w}(q) & \\lesssim (1 + |q|)^{-1} w(q). \\label{E:widetildewInequality1} \n\t\\end{align}\n\t\n\tWith the help of Lemma \\ref{L:NablaZIh1LLh1TLMainTheoremWaveCoordianteAlgebraicEstimate},\n\t\\eqref{E:widetildewInequality1}, and the Hardy inequalities of Proposition \\ref{P:Hardy}, we estimate the integral \n\tcorresponding to the fourth sum on the right-hand side of \\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate} as follows:\n\t\n\t\\begin{align} \\label{E:MainTheoremFarCommutatorIntegratedEstimate}\n\t\t\\sum_{|I| \\leq k} & \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t(1 + t + |q|)^{-1 + C \\varepsilon} (1 + |q|)^{-(2 + C \\varepsilon)} (1+ q_-)^{-2 \\upmu}\n  \t\t|\\nabla_{\\mathcal{Z}}^I h^{(1)}|_{\\mathcal{L}\\mathcal{L}}^2 w(q) d^3x \\, d \\tau \t\\\\\n  \t& \\lesssim \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t|\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|_{\\mathcal{L}\\mathcal{L}}^2   \n  \t\t\\widetilde{w}(q) d^3x \\, d \\tau \\notag \\\\\n  \t& \\lesssim \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w'(q) d^3x \\, d \\tau \\notag \\\\\n  \t& \\ \\ + \\ \\varepsilon^2 \\int_0^t \\int_{\\Sigma_{\\tau}} (1 + \\tau + |q|)^{-4} \\chi_0^2(1\/2 < r\/t < 3\/4) w'(q) d^3x \\, d \\tau \t\n  \t\t\\notag \\\\\n  \t& \\ \\ + \\ \\varepsilon^4 \\int_0^t \\int_{\\Sigma_{\\tau}} (1 + \\tau + |q|)^{-6} w'(q) d^3x \\, d \\tau\n  \t\t\\ + \\ \\varepsilon^2 \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t(1 + \\tau)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) d^3x \\, d \\tau \\notag \\\\\n  \t& \\ \\ + \\ \\varepsilon^2 \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t(1 + \\tau + |q|)^{-1} (1 + |q|)^{-2}|\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) d^3x \\, d \\tau \\notag \\\\\n  \t& \\ \\ + \\ \\underbrace{\\sum_{|J'| \\leq k - 2} \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t(1 + \\tau + |q|)^{-1 + C \\varepsilon} |\\nabla\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}|^2 w(q) d^3x \\, d \\tau}_{\\mbox{absent if \n  \t\t$k = 1$}} \\notag \\\\\n  \t& \\lesssim \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t|\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w'(q) d^3x \\, d \\tau\n  \t \t\\ + \\ \\sum_{|I| \\leq k}  \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t(1 + \\tau)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) d^3x \\, d \\tau \\notag \\\\\n  \t& \\ \\ + \\ \\underbrace{\\sum_{|J'| \\leq k - 2} \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t(1 + \\tau)^{-1 + C \\varepsilon} |\\nabla\\nabla_{\\mathcal{Z}}^{J'} h^{(1)}|^2 w(q) d^3x \\, d \\tau}_{\n  \t\t\t\\mbox{absent if $k = 1$}}\n  \t\t\\ + \\ \\varepsilon^2, \\notag\n\t\\end{align}\n\twhere to pass to the lass inequality, we have again used Proposition \\ref{P:Hardy}\n\tto estimate \\\\\n\t$\\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t(1 + \\tau + |q|)^{-1} (1 + |q|)^{-2}|\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) d^3x \\, d \\tau$\n  \t\t$\\lesssim \\sum_{|I| \\leq k}  \\int_0^t \\int_{\\Sigma_{\\tau}} \n  \t\t(1 + \\tau)^{-1} |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 w(q) d^3x \\, d \\tau.$\n\tAfter multiplying both sides of \\eqref{E:MainTheoremFarCommutatorIntegratedEstimate} by $\\varepsilon,$ the resulting\n\tright-hand side is manifestly bounded by the right-hand side of \\eqref{E:MainTheoremFarCommutatorIntegralEstimate}\n\tas desired. The integral corresponding to the fifth sum on the right-hand side of \n\t\\eqref{E:MainTheoremFarCommutatorAlgebraicEstimate} can be bounded through the same reasoning.\n\t\n\t\n\\end{proof}\n\n\nCombining Lemma \\ref{L:MainTheoremFarInhomogeneousTermIntegralEstimate} and Lemma \\ref{L:MainTheoremFarCommutatorIntegralEstimate}, we arrive at the following corollary.\n\n\\begin{corollary} \\label{C:LieZIFarFundamentalEnergyEstimate}\n\t\tLet $\\mathfrak{F}_{(I)}^{\\nu} = \\Liemod_{\\mathcal{Z}}^I \n\t\t\\mathfrak{F}^{\\nu} + \\Big\\lbrace N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu} \\Lie_{\\mathcal{Z}}^I\\Far_{\\kappa \n\t\t\\lambda} - \\Liemod_{\\mathcal{Z}}^I \\big(N^{\\# \\mu \\nu \\kappa \\lambda}\\nabla_{\\mu}\\Far_{\\kappa \n\t\t\\lambda}\\big)\\Big\\rbrace$ be the \n\t\tinhomogeneous term \\eqref{E:LiemodZINnablaFarCommutatorTerms} in the equations of \n\t\tvariation \\eqref{E:EOVdMis0} satisfied by $\\dot{\\Far}_{\\mu \\nu} \\eqdef \\Lie_{\\mathcal{Z}} \\Far_{\\mu \\nu}.$ Under the \n\t\tassumptions of Theorem \\ref{T:ImprovedDecay}, if $0 \\leq k \\leq \\dParameter $ and $\\varepsilon$ is sufficiently small, then \n\t\tthe following integral inequality holds for $t \\in [0,T):$\n\n\t\t\\begin{align} \\label{E:LieZIFarFundamentalEnergyEstimate}\n\t\t\\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} |(\\Lie_{\\mathcal{Z}}^I \\Far_{0 \\nu}) \\mathfrak{F}_{(I)}^{\\nu} |\n\t\t\tw(q) \\, d^3x \\, d \\tau \n\t\t& \\lesssim \\varepsilon \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t(1 + \\tau)^{-1} \\Big\\lbrace |\\nabla\\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 \n\t\t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|^2 \\Big\\rbrace w(q) \\, d^3x \\, d \\tau \\\\\n\t\t& \\ \\ + \\ \\varepsilon \\sum_{|I| \\leq k} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t\\Big\\lbrace |\\conenabla \\nabla_{\\mathcal{Z}}^I h^{(1)}|^2 \n\t\t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{L}\\mathcal{N}}^2\n  \t\t+ |\\Lie_{\\mathcal{Z}}^I \\Far|_{\\mathcal{T}\\mathcal{T}}^2 \\Big\\rbrace w'(q) \\, d^3x \\, d \\tau \\notag \\\\\n\t\t& \\ \\ + \\ \\underbrace{\\varepsilon \\sum_{|J| \\leq k - 1} \\int_0^t \\int_{\\Sigma_{\\tau}} \n\t\t\t(1 + \\tau)^{-1 + C \\varepsilon} \\Big\\lbrace |\\nabla\\nabla_{\\mathcal{Z}}^J h^{(1)}|^2 \n\t\t\t+ |\\Lie_{\\mathcal{Z}}^J \\Far|^2 \\Big\\rbrace w(q) \\, d^3x \\, d \\tau}_{\\mbox{absent if $k = 0$}} \\notag \\\\\n\t\t& \\ + \\ \\varepsilon^3. \\notag\n\t\t\\end{align}\n\t\t\n\\end{corollary}\n\n\\hfill $\\qed$\n\n\n\\section*{Acknowledgments}\nI would like to thank Igor Rodnianski for delivering an especially illuminating set of lectures on the work\n\\cite{hLiR2010} at Princeton University during Spring 2009. I offer thanks to Michael Kiessling for introducing me to his work \n\\cite{mK2004a}, \\cite{mK2004b} on nonlinear electromagnetism, to Sergiu Klainerman for suggesting that I write the precursor \\cite{jS2010a} to the present article, and to A. Shadi Tahvildar-Zadeh for introducing me to the ideas of \\cite{dC2000}. I would also like to thank Mihalis Dafermos, Shadi Tahvildar-Zadeh, and Willie Wong for the useful comments and helpful discussion they provided. I am appreciative of the support offered by the University of Cambridge and Princeton University during the writing of this article. \n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is commonly believed that an understanding of the dynamics of gravity and\nthe structure of space-time at shortest distances requires an explicit quantum\ntheory for gravity. The well-known fact that the perturbative quantisation\nprogram for gravity in four dimensions faces problems has raised the suspicion\nthat a consistent formulation of the theory may require a radical deviation\nfrom the concepts of local quantum field theory, $e.g.$ string theory.\nIt remains an interesting and open challenge to prove, or falsify, that a\nconsistent quantum theory of gravity cannot be accommodated for within the\notherwise very successful framework of local quantum field theories.\n\nSome time ago Steven Weinberg added a new perspective to this problem by\npointing out that a quantum theory of gravity in terms of the metric field may\nvery well exist, and be renormalisable on a non-perturbative level, despite\nit's notorious perturbative non-renormalisability \\cite{Weinberg}. This\nscenario, since then known as ``asymptotic safety'', necessitates an\ninteracting ultraviolet fixed point for gravity under the renormalisation\ngroup (RG)\n\\cite{Weinberg,Litim:2006dx,Niedermaier:2006ns,NiedermaierReuter,Percacci:2007sz}. If\nso, the high energy behaviour of gravity is governed by near-conformal scaling\nin the vicinity of the fixed point in a way which circumnavigates the virulent\nultraviolet (UV) divergences encountered within standard perturbation theory.\nIndications in favour of an ultraviolet fixed point are based on\nrenormalisation group studies in four and higher dimensions\n\\cite{Reuter:1996cp,Souma:1999at,Lauscher:2001ya,Lauscher:2002mb,Reuter:2001ag,Percacci:2002ie,Litim:2003vp,Bonanno:2004sy,Fischer:2006fz,Fischer:2006at,DL06,Codello:2007bd,Machado:2007ea},\ndimensional reduction techniques \\cite{Forgacs:2002hz,Niedermaier:2002eq},\nrenormalisation group studies in lower dimensions\n\\cite{Weinberg,Gastmans:1977ad,Christensen:1978sc,Kawai:1992fz,Aida:1996zn},\nfour-dimensional perturbation theory in higher derivative gravity\n\\cite{Codello:2006in}, large-$N$ expansions in the matter fields\n\\cite{Percacci:2005wu}, and lattice simulations\n\\cite{Hamber:1999nu,Hamber:2005vc,Ambjorn:2004qm}.\n\nIn this contribution, we review the key elements of the asymptotic safety\nscenario (Sec.~\\ref{AS}) and introduce renormalisation group techniques\n(Sec.~\\ref{RG}) which are at the root of fixed point searches in quantum\ngravity. The fixed point structure in four (Sec.~\\ref{FP}) and\nhigher dimensions (Sec.~\\ref{ED}), and the phase diagram of gravity\n(Sec.~\\ref{PD}) are discussed and evaluated in the light of the underlying\napproximations. Results are compared with recent lattice simulations\n(Sec.~\\ref{Lattice}), and phenomenological implications are indicated\n(Sec.~\\ref{pheno}). We close with some conclusions (Sec.~\\ref{conclusions}).\n  \n\n\\section{Asymptotic Safety}\n\\label{AS}\nWe summarise the basic set of ideas and assumptions of asymptotic safety as\nfirst laid out in \\cite{Weinberg} (see\n\\cite{Litim:2006dx,Niedermaier:2006ns,NiedermaierReuter,Percacci:2007sz} for\nreviews).  The aim of the asymptotic safety scenario for gravity is to provide\nfor a path-integral based framework in which the metric field is the carrier\nof the fundamental degrees of freedom, both in the classical and in the\nquantum regimes of the theory. This is similar in spirit to effective field\ntheory approaches to quantum gravity \\cite{Donoghue:1993eb}. There, a\nsystematic study of quantum effects is possible without an explicit knowledge\nof the ultraviolet completion as long as the relevant energy scales are much\nlower than the ultraviolet cutoff $\\Lambda$ of the effective theory, with\n$\\Lambda$ of the order of the Planck scale (see \\cite{Burgess:2003jk} for a\nrecent review).\n\nThe asymptotic safety scenario goes one step further and assumes that the\ncutoff $\\Lambda$ can in fact be removed, $\\Lambda\\to\\infty$, and that the\nhigh-energy behaviour of gravity, in this limit, is characterised by an\ninteracting fixed point.  It is expected that the relevant field\nconfigurations dominating the gravitational path integral at high energies are\npredominantly ``anti-screening'' to allow for this limit to become\nfeasible. If so, it is conceivable that a non-trivial high-energy fixed point\nof gravity may exist and should be visible within $e.g.$ renormalisation group\nor lattice implementations of the theory, analogous to the well-known\nperturbative high-energy fixed point of QCD.  Then the high-energy behaviour\nof the relevant gravitational couplings is ``asymptotically safe'' and\nconnected with the low-energy behaviour by finite renormalisation group flows.\nThe existence of a fixed point together with finite renormalisation group\ntrajectories provides for a definition of the theory at arbitrary energy\nscales.\n\nThe fixed point implies that the high-energy behaviour of gravity is\ncharacterised by universal scaling laws, dictated by the residual high-energy\ninteractions. No a priori assumptions are made about which invariants are the\nrelevant operators at the fixed point. In fact, although the low-energy\nphysics is dominated by the Einstein-Hilbert action, it is expected that (a\nfinite number of) further invariants will become relevant, in the\nrenormalisation group sense, at the ultraviolet fixed point.\\footnote{For\n  {infrared} fixed points, universality considerations often simplify the task\n  of identifying the set of relevant, marginal and irrelevant operators. This\n  is not applicable for interacting {ultraviolet} fixed points.} Then, in\norder to connect the ultraviolet with the infrared physics along some\nrenormalisation group trajectory, a finite number of initial parameters have\nto be fixed, ideally taken from experiment. In this light, classical general\nrelativity would emerge as a ``low-energy phenomenon'' of a fundamental\nquantum field theory in the metric field.\n\n\nWe illustrate this scenario with a discussion of the renormalisation group\nequation for the gravitational coupling $G$, following \\cite{Litim:2006dx}\n(see also \\cite{Niedermaier:2006ns,NiedermaierReuter}). Its canonical\ndimension is $[G]=2-d$ in $d$ dimensions and hence negative for $d>2$.  It is\ncommonly believed that a negative mass dimension for the relevant coupling is\nresponsible for the perturbative non-renormalisability of the theory.  We\nintroduce the renormalised coupling as $G(\\mu)=Z^{-1}_G(\\mu)\\, G$, and the\ndimensionless coupling as $g(\\mu)=\\mu^{d-2}\\,G(\\mu)$; the momentum scale $\\mu$\ndenotes the renormalisation scale. The graviton wave function renormalisation\nfactor $Z_G(\\mu)$ is normalised as $Z_G(\\mu_0)=1$ at \n$\\mu=\\mu_0$ with $G(\\mu_0)$ given by Newton's coupling constant $G_N=6.67428\n\\cdot 10^{-11}\\s0{m^3}{{\\rm kg}\\,s^2}$.  The graviton anomalous dimension\n$\\eta$ related to $Z_G(\\mu)$ is given by $\\eta=-\\mu\\s0{{\\rm d}}{{\\rm d}\\mu}\n\\ln Z_G$. Then the Callan-Symanzik equation for $g(\\mu)$ reads\n\\begin{equation}\\label{dg}\n\\beta_g\\equiv\\mu\\frac{{\\rm d}g(\\mu)}{{\\rm d}\\mu}\n=(d-2+\\eta)g(\\mu)\\,.\n\\end{equation}\nHere we have assumed a fundamental action for gravity which is local in the\nmetric field. In general, the graviton anomalous dimension $\\eta(g,\\cdots)$ is\na function of all couplings of the theory including matter fields. The RG\nequation \\eq{dg} displays two qualitatively different types of fixed\npoints. The non-interacting (gaussian) fixed point corresponds to $g_*=0$\nwhich also entails $\\eta=0$. In its vicinity with $g(\\mu_0)\\ll 1$, we have\ncanonical scaling since $\\beta_g=(d-2)g$, and\n\\begin{equation}\n\\label{Gauss}\nG(\\mu) = G(\\mu_0)\n\\end{equation}\nfor all $\\mu<\\mu_0$. Consequently, the gaussian regime corresponds to the\ndomain of classical general relativity. In turn, \\eq{dg} can display an\ninteracting fixed point $g_*\\neq 0$ in $d>2$ if the anomalous dimension takes\nthe value $\\eta(g_*,\\cdots)=2-d$; the dots denoting further gravitational and\nmatter couplings.  Hence, the anomalous dimension precisely counter-balances\nthe canonical dimension of Newton's coupling $G$.  This structure is at the\nroot for the non-perturbative renormalisability of quantum gravity within a\nfixed point scenario.\\footnote{Integer values for anomalous dimensions are\n  well-known from other gauge theories at criticality and away from their\n  canonical dimension. In the $d$-dimensional $U(1)$+Higgs theory, the abelian\n  charge $e^2$ has mass dimension $[e^2]=4-d$, with $\\beta_{e^2}=(d-4+\\eta)\\,\n  e^2$. In three dimensions, a non-perturbative infrared fixed point at\n  $e^2_*\\neq 0$ leads to $\\eta_*=1$ \\cite{Bergerhoff:1995zq}. The fixed point\n  belongs to the universality class of conventional superconductors with the\n  charged scalar field describing the Cooper pair. The integer value $\\eta_*\n  =1$ implies that the magnetic field penetration depth and the Cooper pair\n  correlation length scale with the same universal exponent at the phase\n  transition \\cite{Bergerhoff:1995zq,Herbut:1996ut}.  In Yang-Mills theories\n  above four dimensions, ultraviolet fixed points with $\\eta=4-d$ and\n  implications thereof have been discussed in\n  \\cite{Kazakov:2002jd,Gies:2003ic,Morris:2004mg}.}  Consequently, at an\ninteracting fixed point where $g_*\\neq0$, the anomalous dimension implies the\nscaling\n\\begin{equation}\\label{G-asym}\nG(\\mu)= \\frac{g_*}{\\mu^{d-2}}\n\\end{equation}\nfor the dimensionful gravitational coupling. In the case of an ultraviolet\nfixed point $g_*\\neq 0$ for large $\\mu$, the dimensionful coupling $G$ becomes\narbitrarily small in its vicinity. This is in marked contrast to\n\\eq{Gauss}. Hence, \\eq{G-asym} indicates that gravity weakens at the onset of\nfixed point scaling.  Nevertheless, at the fixed point the theory remains\nnon-trivially coupled because of $g_*\\neq 0$. The weakness of the coupling in\n\\eq{G-asym} is a dimensional effect, and should be contrasted with $e.g.$\nasymptotic freedom of QCD in four dimensions where the dimensionless\nnon-abelian gauge coupling becomes weak because of a non-interacting\nultraviolet fixed point. In turn, if \\eq{G-asym} corresponds to a non-trivial\ninfrared fixed point for $\\mu\\to 0$, the dimensionful coupling $G(\\mu)$ grows\nlarge. A strong coupling behaviour of this type would imply interesting long\ndistance modifications of gravity. \n\nAs a final comment, we point out that asymptotically safe gravity is expected\nto become, in an essential way, two-dimensional at high energies.\nHeuristically, this can be seen from the dressed graviton propagator whose\nscalar part, neglecting the tensorial structure, scales as ${\\cal G}(p^2)\\sim\np^{-2(1-\\eta\/2)}$ in momentum space. Here we have evaluated the anomalous\ndimension at $\\mu^2\\approx p^2$. Then, for small $\\eta$, we have the standard\nperturbative behaviour $\\sim p^{-2}$. In turn, for large anomalous dimension\n$\\eta\\to 2-d$ in the vicinity of a fixed point the propagator is additionally\nsuppressed $\\sim (p^2)^{-d\/2}$ possibly modulo logarithmic corrections. After\nFourier transform to position space, this corresponds to a logarithmic\nbehaviour for the propagator ${\\cal G}(x,y)\\sim \\ln(|x-y|\\mu)$, characteristic\nfor bosonic fields in two-dimensional systems.\n\n\\section{Renormalisation Group}\\label{Implications}\n\\label{RG}\nWhether or not a non-trivial fixed point is realised in quantum gravity can be\nassessed once explicit renormalisation group equations for the scale-dependent\ngravitational couplings are available. To that end, we recall the set-up of\nWilson's (functional) renormalisation group (see\n\\cite{Litim:1998yn,Litim:1998nf,Bagnuls:2000ae,BTW,JP,Pawlowski:2005xe,Gies:2006wv}\nfor reviews), which is used below for the case of quantum gravity. Wilsonian\nflows are based on the notion of a cutoff effective action $\\Gamma_k$, where\nthe propagation of fields $\\phi$ with momenta smaller than $k$ is suppressed.\nA Wilsonian cutoff is realised by adding $\\Delta\nS_k=\\s012\\int\\varphi(-q)\\,R_k(q)\\,\\varphi(q)$ within the Schwinger functional\n\\begin{equation}\\label{Z}\n\\ln\\, Z_k[J]=\\ln \\int [D\\varphi]_{\\rm ren.}\\exp\\left(-S[\\varphi]\n-\\Delta S_k[\\varphi]\n+\\int J\\cdot \\varphi\n\\right)\\,\n\\end{equation}\nand the requirement that $R_k$ obeys (i) $R_k(q)\\to 0$ for $k^2\/q^2\\to 0$,\n(ii) $R_k(q)> 0$ for $q^2\/k^2\\to 0$, and (iii) $R_k(q)\\to\\infty$ for\n$k\\to\\Lambda$ (for examples and plots of $R_k$, see \\cite{Litim:2000ci}). Note\nthat the Wilsonian momentum scale $k$ takes the role of the renormalisation\ngroup scale $\\mu$ introduced in the previous section. Under infinitesimal\nchanges $k\\to k-\\Delta k$, the Schwinger functional obeys $\\partial_t\\ln\nZ_k=-\\langle\\partial_t\\Delta S_k\\rangle_J$; $t=\\ln k$. We also introduce its\nLegendre transform, the scale-dependent effective action\n$\\Gamma_k[\\phi]=\\sup_J\\left(\\int J\\cdot \\phi -\\ln Z_k[J]\\right)-\\s012\\int \\phi\nR_k\\phi$, $\\phi=\\langle \\varphi\\rangle_J$. It obeys an exact functional\ndifferential equation introduced by Wetterich \\cite{Wetterich:1992yh}\n\\begin{equation}\\label{ERG} \n\\partial_t \\Gamma_k=\n\\frac{1}{2} {\\rm Tr} \\left({\\Gamma_k^{(2)}+R_k}\\right)^{-1}\\partial_t R_k\\,,\n\\end{equation} \nwhich relates the change in $\\Gamma_k$ with a one-loop type integral over the\nfull field-dependent cutoff propagator. Here, the trace ${\\rm Tr}$ denotes an\nintegration over all momenta and summation over all fields, and\n$\\Gamma_k^{(2)}[\\phi](p,q)\\equiv \\delta^2\\Gamma_k\/\\delta\\phi(p)\\delta\\phi(q)$.\nA number of comments are in order:\n\\begin{itemize}\n\\item[$\\bullet$] {\\bf Finiteness and interpolation property.} By construction,\n  the flow equation \\eq{ERG} is well-defined and finite, and interpolates\n  between an initial condition $\\Gamma_\\Lambda$ for $k\\to \\Lambda$ and the\n  full effective action $\\Gamma\\equiv\\Gamma_{k=0}$. This is illustrated in\n  Fig.~\\ref{Vergleich}. The endpoint is independent of the regularisation,\n  whereas the trajectories $k\\to \\Gamma_k$ depend on it.\n\\item[$\\bullet$] {\\bf Locality.} The integrand of \\eq{ERG} is peaked for field\n  configurations with momentum squared $q^2\\approx k^2$, and suppressed for\n  large momenta [due to condition (i) on $R_k$] and for small momenta [due to\n    condition (ii)]. Therefore, the flow equation is essentially local in\n  momentum and field space \\cite{Litim:2000ci,Litim:2005us}.\n\\item[$\\bullet$] {\\bf Approximations.} Systematic approximations for\n  $\\Gamma_k$ and $\\partial_t\\Gamma_k$ are required to integrate\n  \\eq{ERG}. These include (a) perturbation theory, (b) expansions in powers of\n  the fields (vertex functions), (c) expansion in powers of derivative\n  operators (derivative expansion), and (d) combinations thereof. The\n  iterative structure of perturbation theory is fully reproduced to all\n  orders, independently of $R_k$ \\cite{Litim:2001ky,Litim:2002xm}. The\n  expansions (b) - (d) are genuinely non-perturbative and lead, via \\eq{ERG}, to\n  coupled flow equations for the coefficient functions. Convergence is then\n  checked by extending the approximation to higher order.\n\\item[$\\bullet$] {\\bf Stability.} The stability and convergence of\n  approximations is, additionally, controlled by $R_k$\n  \\cite{Litim:2000ci,Litim:2001up}. Here, powerful optimisation techniques are\n  available to maximise the physics content and the reliability through\n  well-adapted choices of $R_k$\n  \\cite{Litim:2000ci,Litim:2001up,Litim:2005us,Pawlowski:2003hq,Pawlowski:2005xe}. These\n  ideas have been explicitly tested in $e.g.$~scalar \\cite{Litim:2002cf} and\n  gauge theories \\cite{Pawlowski:2003hq}.\n\\item[$\\bullet$] {\\bf Symmetries.} Global or local (gauge\/diffeomorphism)\n  symmetries of the underlying theory can be expressed as Ward-Takahashi\n  identities for $n$-point functions of $\\Gamma$. Ward-Takahashi identities\n  are maintained for all $k$ if the insertion $\\Delta S_k$ is compatible with\n  the symmetry. In general, this is not the case for non-linear symmetries\n  such as in non-Abelian gauge theories or gravity. Then the requirements of\n  gauge symmtry for $\\Gamma$ are preserved by either (a) imposing modified\n  Ward identities which ensure that standard Ward identities are obeyed in the\n  the physical limit when $k\\to 0$, or by (b) introducing background fields\n  into the regulator $R_k$ and taking advantage of the background field\n  method, or by (c) using gauge-covariant variables rather than the gauge\n  fields or the metric field \\cite{Branchina:2003ek}. For a discussion of\n  benefits and shortcomings of these options see\n  \\cite{Litim:1998nf,Pawlowski:2005xe}. For gravity, most implementations\n  presently employ option (b) together with optimisation techniques to control\n  the symmetry \\cite{Fischer:2006fz,Fischer:2006at}.\n\n\\item[$\\bullet$] {\\bf Integral representation.} The physical theory described\n  by $\\Gamma$ can be defined without explicit reference to an underlying\n  path integral representation, using only the (finite) initial condition\n  $\\Gamma_\\Lambda$, and the (finite) flow equation\n  \\eq{ERG}\n\\begin{figure}[t]\n\\begin{center}\n  \\unitlength0.001\\hsize\n\\begin{picture}(1000,260)\n\\put(50,250){{a)}\n\\put(500,250){{b)}\n\\put(220,220){$S$}\n\\put(680,220){$\\Gamma_*$}\n\\put(210,130){$k\\partial_k\\Gamma_{k}$}\n\\put(690,130){$k\\partial_k\\Gamma_{k}$}\n\\put(180,-20){$\\Gamma_{0}\\approx\\Gamma$}\n\\put(660,-20){$\\Gamma_{0}\\approx\\Gamma_{\\rm EH}$}\n{}\\hskip.05\\hsize\n\\includegraphics[width=.42\\hsize]{Flow.eps\n\\hskip.05\\hsize\n\\includegraphics[width=.42\\hsize]{FlowGravity.eps}\n\\end{picture}\n\\vskip.5cm\n\\caption{\\label{Vergleich} Wilsonian flows for scale-dependent effective\n  actions $\\Gamma_k$ in the space of all action functionals (schematically);\n  arrows point towards smaller momentum scales and lower energies $k\\to 0$.\n  {a)} Flow connecting a fundamental classical action $S$ at high energies\n  in the ultraviolet with the full quantum effective action $\\Gamma$ at low\n  energies in the infrared (``top-down''). {b)} Flow connecting the\n  Einstein-Hilbert action at low energies with a fundamental fixed point\n  action $\\Gamma_*$ at high energies (``bottom-up'').}\n\\end{center}\n\\end{figure}\n\\begin{equation}\\label{integral}\n\\Gamma=\\Gamma_\\Lambda+\n\\int_\\Lambda^0\n\\s0{dk}{k}\\s012\\,\n{\\rm Tr} \\left({\\Gamma_k^{(2)}+R_k}\\right)^{-1}\\partial_t R_k\\,.\n\\end{equation}\nThis provides an implicit regularisation of the path integral\nunderlying \\eq{Z}. It should be compared with the standard representation\nfor $\\Gamma$ via a functional integro-differential equation\n\\begin{equation}\\label{DSE}\ne^{-\\Gamma}\n=\\int [D\\varphi]_{\\rm ren.}\\exp\n\\left(-S[\\phi+\\varphi]\n+\\int \\0{\\delta\\Gamma[\\phi]}{\\delta\\phi}\\cdot\\varphi\\right)\n\\end{equation}\nwhich is at the basis of $e.g.$ the hierarchy of Dyson-Schwinger equations.\n\\item[$\\bullet$] {\\bf Renormalisability.} In renormalisable theories, the\n  cutoff $\\Lambda$ in \\eq{integral} can be removed, $\\Lambda\\to\\infty$, and\n  $\\Gamma_\\Lambda\\to \\Gamma_*$ remains well-defined for arbitrarily short\n  distances. In perturbatively renormalisable theories, $\\Gamma_*$ is given by\n  the classical action $S$, such as in QCD.  In this case, illustrated in\n  Fig.~\\ref{Vergleich}a), the high energy behaviour of the theory is simple,\n  given mainly by the classical action, and the challenge consists in deriving\n  the physics of the strongly coupled low energy limit.  In perturbatively\n  non-renormalisable theories such as quantum gravity, proving the existence\n  (or non-existence) of a short distance limit $\\Gamma_*$ is more difficult.\n  For gravity, illustrated in Fig.~\\ref{Vergleich}b), experiments indicate\n  that the low energy theory is simple, mainly given by the Einstein Hilbert\n  theory. The challenge consists in identifying a possible high energy fixed\n  point action $\\Gamma_*$, which upon integration matches with the known\n  physics at low energies.  In principle, any $\\Gamma_*$ with the above\n  properties qualifies as fundamental action for quantum gravity. In\n  non-renormalisable theories the cutoff $\\Lambda$ cannot be removed. Still,\n  the flow equation allows to access the physics at all scales $k<\\Lambda$\n  analogous to standard reasoning within effective field theory\n  \\cite{Burgess:2003jk}.\n\\item[$\\bullet$] {\\bf Link with Callan-Symanzik equation.} The well-known\n  Callan-Symanzik equation describes a flow $k\\s0d{dk}$ driven by a mass\n  insertion $\\sim k^2\\phi^2$. In \\eq{ERG}, this corresponds to the choice\n  $R_k(q^2)=k^2$, which does not fulfill condition (i). Consequently, the\n  corresponding flow is no longer local in momentum space, and requires an\n  additional UV regularisation. This highlights a crucial difference between\n  the Callan-Symanzik equation and functional flows \\eq{ERG}. In this light,\n  the flow equation \\eq{ERG} could be interpreted as a functional\n  Callan-Symanzik equation with {momentum-dependent} mass term insertion\n  \\cite{Symanzik:1970rt}.\n\\end{itemize}\n\nNow we are in a position to implement these ideas for quantum gravity\n\\cite{Reuter:1996cp}. A Wilsonian effective action for gravity $\\Gamma_k$\nshould contain the Ricci scalar $R(g_{\\mu\\nu})$ with a running gravitational\ncoupling $G_k$, a running cosmological constant $\\Lambda_k$ (with canonical\nmass dimension $[\\Lambda_k]=2$), possibly higher order interactions in the\nmetric field such as powers, derivatives, or functions of $e.g.$ the Ricci\nscalar, the Ricci tensor, the Riemann tensor, and, possibly, non-local\noperators in the metric field. The effective action should also contain a\nstandard gauge-fixing term $S_{\\rm gf}$, a ghost term $S_{\\rm gh}$ and matter\ninteractions $S_{\\rm matter}$. Altogether,\n\\begin{equation}\\label{EHk}\n\\Gamma_k=\n\\int d^dx \\sqrt{\\det g_{\\mu\\nu}}\\left[\n\\0{1}{16\\pi G_k}\\left(-R+2\\Lambda_k \\right)\n+\\cdots+S_{\\rm gf}+S_{\\rm gh}+ S_{\\rm matter}\\right]\\,,\n\\end{equation}\nand explicit flow equations for the coefficient functions such as $G_k$,\n$\\Lambda_k$ or vertex functions, are obtained by appropriate projections after\ninserting \\eq{EHk} into \\eq{ERG}.  All couplings in \\eq{EHk} become running\ncouplings as functions of the momentum scale $k$.  For $k$ much smaller than\nthe $d$-dimensional Planck scale $M_*$, the gravitational sector is\nwell approximated by the Einstein-Hilbert action with $G_k\\approx G_{k=0}$,\nand similarily for the gravity-matter couplings.  At $k\\approx M_*$ and above,\nthe RG running of gravitational couplings becomes important. This\nis the topic of the following sections.\n\nA few technical comments are in order: To ensure gauge symmetry within this\nset-up, we take advantage of the background field formalism and add a\nnon-propagating background field $\\bar g_{\\mu\\nu}$\n\\cite{Reuter:1996cp,Litim:1998nf,Reuter:1993kw,Litim:1998qi,Freire:2000bq,Litim:2002hj,Pawlowski:2001df}. This\nway, the extended effective action $\\Gamma_k[g_{\\mu\\nu},\\bar g_{\\mu\\nu}]$\nbecomes gauge-invariant under the combined symmetry transformations of the\nphysical and the background field. A second benefit of this is that the\nbackground field can be used to construct a covariant Laplacean $-\\bar D^2$,\nor similar, to define a mode cutoff at momentum scale $k^2=-\\bar D^2$. This\nimplies that the mode cutoff $R_k$ will depend on the background fields. The\nbackground field is then eliminated from the final equations by identifying it\nwith the physical mean field. This procedure, which dynamically readjusts the\nbackground field, implements the requirements of ``background independence''\nfor quantum gravity. For a detailed evaluation of Wilsonian background field\nflows, see \\cite{Litim:2002hj}. Finally, we note that the operator traces\n${\\rm Tr}$ in \\eq{ERG} are evaluated using heat kernel techniques. Here,\nwell-adapted choices for $R_k$ \\cite{Litim:2000ci,Litim:2001up} lead to\nsubstantial algebraic simplifications, and open a door for systematic fixed\npoint searches, which we discuss next.\n\n\n\n\n\n\\section{Fixed Points}\n\\label{FP}\nIn this section, we discuss the main picture in a simple approximation which\ncaptures the salient features of an asymptotic safety scenario for gravity,\nand give an overview of extensions. We consider the Einstein-Hilbert theory\nwith a cosmological constant term and employ a momentum cutoff $R_k$ with the\ntensorial structure of \\cite{Lauscher:2001ya} and variants thereof, an\noptimised scalar cutoff $R_k(q^2)\\sim(k^2-q^2)\\theta(k^2-q^2)$\n\\cite{Litim:2000ci,Litim:2001up}, and a harmonic background field gauge with\nparameter $\\alpha$ in a specific limit introduced in \\cite{Litim:2003vp}.  The\nghost wave function renormalisation is set to $Z_{C,k}=1$, and the effective\naction is given by \\eq{EHk} with $S_{\\rm matter}=0$.  In the domain of\nclassical scaling $G_k$ and $\\Lambda_k$ are approximately constant, and\n\\eq{EHk} reduces to the conventional Einstein-Hilbert action in $d$ euclidean\ndimensions. The dimensionless renormalised gravitational and cosmological\nconstants are\n\\begin{equation}\\label{glk}\n\\begin{array}{l}\ng=k^{d-2}\\, G_k\\, \\equiv k^{d-2}\\, Z^{-1}_{G}(k)\\ \\bar G\\ ,\n\\quad\\quad\n\\lambda=\\,k^{-2}\\, \\Lambda_k\\ \n\\end{array}\n\\end{equation}\nwhere it is understood that $g$ and $\\lambda$ depend on $k$.  Then the coupled\nsystem of $\\beta$-functions is\n\\begin{eqnarray}\n\\partial_t\\lambda\\equiv\\beta_{\\lambda}(\\lambda,g)&=&\n-2\\lambda +\\frac{g}{2}\\,d\\,(d+2)\\,(d-5)\n-d(d+2)g\\, \\frac{(d-1)g\n+\\frac{1}{d-2}(1-4\\frac{d-1}{d}\\lambda)}{2g-\\frac{1}{d-2}(1-2\\lambda)^2}\n\\label{beta-l-inf}\n\\\\\n\\label{beta-g-inf}\n\\partial_t g\\equiv\\beta_{ g}(\\lambda,g)&=&\n(d-2)g+\\frac{2(d-2)(d+2)g^2}{2(d-2)g-(1-2\\lambda)^2}\\,.\n\\end{eqnarray}\nWe have rescaled $g\\to g\/c_d$ with $c_d=\\Gamma(\\s0d2+2)(4\\pi)^{d\/2-1}$ to\nremove phase space factors. This does not alter the fixed point structure. The\nscaling $g\\to g\/(384\\pi^2)$ reproduces the $4d$ classical force law in the\nnon-relativistic limit \\cite{Robinson:2006yd}.  For the anomalous dimension,\nwe find\n\\begin{eqnarray}\n\\label{eta-inf}\n\\eta(\\lambda, g;d)&=&     \n\\0{(d+2)\\, g}{ g- g_{\\rm bound}(\\lambda)}\\,,\n\\quad\\quad\ng_{\\rm bound}(\\lambda;d)\n=\n\\0{\\left(1-2\\lambda\\right)^2}{2(d-2)}\\,.\n\\end{eqnarray}\n\\begin{table}[t]\n\\begin{displaymath}\n\\begin{array}{c|c|c|c\n&\n\\quad\\quad\\theta'\\quad\\quad&\n\\quad\\quad \\theta''\\quad\\quad &\n\\quad\\quad {\\rm ref.}\\quad\\quad \n\\\\ \\hline\n\\quad a)\\quad&\n1.1-2.3 & \n2.5-7.0 & \n\\cite{Lauscher:2001ya}\n\\\\\nb)&\n1.4-2.0 & \n2.4-4.3 & \n\\cite{Litim:2003vp}\n\\\\\nc)&\n1.5-1.7 & \n3.0-3.2 & \n\\cite{Fischer:2006fz}\n\\\\\n\\end{array}\n\\end{displaymath}\n\\caption{ \\label{tEH-4d}The variation of $4d$ scaling exponents\n  $\\theta_{1,2}=\\theta'\\pm i\\theta''$ in the Einstein-Hilbert theory with the\n  gauge fixing parameter $\\alpha$ and the cutoff function $R_k$. Results\n  indicate the range covered under $a)$ partial variation of both $\\alpha$ and\n  $R_k$, $b)$ full $\\alpha$-variation with optimised $R_k$, and $c)$ full\n  $R_k$-variation and optimisation in Feynman gauge ($\\alpha=1$).  In all\n  cases the fixed point is stable. The variation with $R_k$, amended by\n  stablity considerations \\cite{Litim:2000ci,Litim:2001up}, is weaker than the\n  $\\alpha$-variation.}\n\\end{table}\nThe anomalous dimension vanishes for vanishing gravitational coupling, and for\n$d=\\pm2$.  At a non-trivial fixed point the vanishing of $\\beta_g$ implies\n$\\eta_*=2-d$, and reflects the fact that the gravitational coupling is\ndimensionless in two dimensions.  At $g=g_{\\rm bound}$, the anomalous\ndimension $\\eta$ diverges. The full flow \\eq{ERG} is finite (no poles) and\nwell-defined for all $k$, as are the full $\\beta$-functions derived from\nit. Therefore the curve $g=g_{\\rm bound}(\\lambda)$ limits the domain of\nvalidity of the approximation.\n\nWe first consider the case $\\lambda=0$ and find two fixed points, the gaussian\none at $g_*=0$ and a non-gaussian one at $g_*=1\/(4d)<g_{\\rm bound}(0)$, which\nare connected under the renormalisation group.  The universal eigenvalue\n$\\partial \\beta_g\/\\partial g|_*=-\\theta$ at the fixed point are $\\theta=2-d$\nat the gaussian, and\n\\begin{equation} \\label{theta0-NG}\n\\theta=2d\\ \\0{d-2}{d+2} \n\\end{equation}\nat the non-gaussian fixed point.  \n\nNext we allow for a non-vanishing dynamical\ncosmological constant term $\\Lambda_k\\neq 0$ in \\eq{EHk}. The coupled system\nexhibits the gaussian fixed point $(\\lambda_*,g_*)=(0,0)$ with eigenvalues\n$-2$ and $d-2$.\n\\begin{table}[t]\n\\begin{displaymath}\n\\begin{array}{c|c|c|c|c|c|c}\n&\n\\quad i \\quad\n&\\theta'\n&\\quad \\theta''\\quad \n&\\quad \\theta_3\\quad \n&\\quad \\theta_4\\quad\n&\\quad {\\rm ref.}\\quad \n\\\\ \\hline\n\\quad a)\\quad & 2& \\quad 2.1 - 3.4 \\quad &\\quad 3.1 - 4.3\\quad   \n&\\quad 8.4 - 28.8\\quad   &-\n&\n\\cite{Lauscher:2002mb}\n\\\\\nb)&2 &1.4  &2.8  &25.6  &-\n& \n\\cite{DL06}\n\\\\\nc)&\n2 &1.7  &3.1  &3.5  & -\n&\n\\cite{DL06}\n\\\\\nd)&\n2 &1.4  &2.3  &26.9  & -\n& \n\\cite{Codello:2007bd}\n\\\\\ne)&\n3 &2.7  &2.3  &2.1  & -4.2  \n&\n\\cite{Codello:2007bd}\n\\\\\nf)&\n4 &2.9  &2.5  &1.6  & -3.9  \n&\n\\cite{Codello:2007bd}\n\\\\\ng)&\n5 &2.5  &2.7  &1.8  & -4.4 \n&\n\\cite{Codello:2007bd}\n\\\\\nh)&\n6 &2.4  &2.4  &1.5  &  -4.1 \n&\n\\cite{Codello:2007bd}\n\\\\\n\\end{array}\n\\end{displaymath}\n\\caption{\\label{tRn-4d}The variation of $4d$ scaling exponents with the order\n  $i$ of the expansion, including the invariants $\\int\\sqrt{\\det g_{\\mu\\nu}}$\n  and $\\int\\sqrt{\\det g_{\\mu\\nu}} R^i$ from $i=2\\cdots 6$, using $a)$ Feynman\n  gauge under partial variations of $R_k$; $b)$ and $c)$ Feynman gauge and\n  optimised $R_k$; $d)-h)$ Landau-deWitt background field gauge with optimised\n  $R_k$.}\n\\end{table}\nNon-trivial fixed points of \\eq{beta-l-inf} and \\eq{beta-g-inf} are found as\nfollows: For non-vanishing $\\lambda\\neq 0$, we find a non-trivially vanishing\n$\\beta_g$ for $g=g_0(\\lambda)$, with $g_0(\\lambda)=\\s01{4d}(1-2\\lambda)^2$.\nNote that $g_0(\\lambda)<g_{\\rm bound}(\\lambda)$ for all $d>2$.  Evaluating\n\\eq{beta-l-inf} for $g=g_0(\\lambda)$, we find\n$\\beta_\\lambda(\\lambda,g_0(\\lambda))=\n\\s014(d-4)(d+1)(1-2\\lambda)^2-2d\\,\\lambda+\\s0d2\\,.  $ The first term vanishes\nin $d=4$ dimensions. Consequently, we find a unique ultraviolet fixed point\n$\\lambda_*=\\s014$ and $g_*=\\s01{64}$. In $d> 4$, the vanishing of\n$\\beta_\\lambda$ leads to two branches of real fixed points with either\n$\\lambda_*>\\s012$ or $\\s012>\\lambda_*>0$. Only the second branch corresponds\nto an ultraviolet fixed point which is connected under the renormalisation\ngroup with the correct infrared behaviour. This can be seen as follows: At\n$\\lambda=\\s012$, we find $\\eta=d+2>0$. On a non-gaussian fixed point, however,\n$\\eta<0$. Furthermore, $g$ cannot change sign under the renormalisation group\nflow \\eq{beta-g-inf}. Consequently, $\\eta$ cannot change sign either. Hence,\nto connect a fixed point at $\\lambda>\\s012$ with the gaussian fixed point at\n$\\lambda=0$, $\\eta$ would have to change sign at least twice, which is\nimpossible. Therefore, we have a unique physically relevant solution given by\n\\begin{equation}\\label{FP-d}\n\\lambda_*=\n\\frac{d^2-d-4-\\sqrt{2d(d^2-d-4)}}{2(d-4)(d+1)}\\,,\\quad\\quad\ng_*=\n\\0{(\\sqrt{d^2-d-4}-\\sqrt{2d})^2}{2(d-4)^2(d+1)^2}\\,.\n\\end{equation}\nAn interesting property of this system is that the scaling exponents\n$\\theta_1$ and $\\theta_2$ -- the eigenvalues of the stability matrix\n$\\partial\\beta_i\/\\partial g_j$ $(g_1\\equiv g,g_2\\equiv\\lambda)$ at the fixed\npoint -- are a complex conjugate pair, $\\theta_{1,2}=\\theta'\\pm i\\theta''$ with\n$\\theta'=\\s053$ and $\\theta''=\\s0{\\sqrt{167}}{3}$ in four dimensions. The\nreason for this is that the stability matrix, albeit real, is not\nsymmetric. Complex eigenvalues reflect that the interactions at the fixed\npoint have modified the scaling behaviour of the underlying operators\n$\\int\\sqrt{\\det g_{\\mu\\nu}}R$ and $\\int\\sqrt{\\det g_{\\mu\\nu}}$. This pattern\nchanges for lower and higher dimensions, where eigenvalues are real\n\\cite{Litim:2006dx}.\n\nAt this point it is important to check whether the fixed point structure and\nthe scaling exponents depend on technical parameters such as the gauge fixing\nprocedure or the momentum cutoff function $R_k$, see Tab.~\\ref{tEH-4d} and\n~\\ref{tRn-4d}. For the Einstein-Hilbert theory in $4d$, results are summarised\nin Tab.~\\ref{tEH-4d}. The $\\alpha$-dependence of the $\\beta$-functions is\nfairly non-trivial, $e.g.$\n\\cite{Reuter:1996cp,Lauscher:2001ya,Litim:2003vp}. It is therefore noteworthy\nthat scaling exponents only depend mildly on variations thereof. Furthermore,\nthe $R_k$-dependence is smaller than the dependence on gauge fixing\nparameters. We conclude that the fixed point is fully stable and\n$R_k$-independent for all technical purposes, with the presently largest\nuncertainty arising through the gauge fixing sector.  In Tab.~\\ref{tRn-4d}, we\ndiscuss the stability of the fixed point under extensions beyond the\nEinstein-Hilbert approximation, including higher powers of the Ricci scalar\nboth in Feynman gauge \\cite{Lauscher:2002mb,DL06} and in Landau-DeWitt gauge\n\\cite{Codello:2007bd}. Once more, the fixed point and the scaling exponents\ncome out very stable. Furthermore, starting from the operator $\\int \\sqrt{\\det\n  g_{\\mu\\nu}}R^3$ and higher, couplings become irrelevant with negative\nscaling exponents \\cite{Codello:2007bd,Machado:2007ea}. This is an important\nfirst indication for the set of relevant operators at the UV fixed point being\nfinite. Finally, we mention that the stability of the fixed point under the\naddition of non-interacting matter fields has been confirmed in\n\\cite{Percacci:2002ie}.\n\n  \\begin{table}\n\\begin{center}\n\\begin{tabular}{c|ccccccc}\n$d$  & 5 & 6 & 7 & 8 & 9 & 10\n\\\\\n\\hline\n$\\theta'$ \n&2.69 -- 3.11&4.26 -- 4.78&6.43 -- 6.89&8.19 -- \n                   9.34&10.5 -- 12.1&13.1 -- 15.2\n\\\\\n$\\theta''$\n&4.54 -- 5.16&6.52 -- 7.46&8.43 -- 9.46&10.3 -- \n                   11.4&12.1 -- 13.2&13.9 -- 15.0\n\\\\\n$|\\theta|$\n&5.31 -- 6.06&7.79 -- 8.76&10.4 -- 11.6&13.2 -- \n                   14.7&16.1 -- 17.9&19.1 -- 21.3\n\\\\\n\\end{tabular}\n\\end{center}\n\\caption{The variation of scaling exponent with dimensionality, gauge fixing\n  parameters (using either Feynman gauge, or harmonic background field gauge\n  with $0\\leq \\alpha \\leq 1$), and the regulator $R_k$; data from\n  \\cite{Fischer:2006fz,Fischer:2006at}. The $R_k$-variation, covering various\n  classes of cutoff functions, is on the level of a few percent and smaller\n  than the variation with $\\alpha$.}\n\\label{tEH-d}\n    \\end{table}\n\n\\section{Extra Dimensions}\n\\label{ED}\n\nIt is interesting to discuss fixed points of quantum gravity specifically in\nmore than four dimensions. The motivation for this is that, first of all, the\ncritical dimension of gravity -- the dimension where the gravitational\ncoupling has vanishing canonical mass dimension -- is two. For any dimension\nabove the critical one, the canonical dimension is negative. Hence, from a\nrenormalisation group point of view, the four-dimensional theory is not\nspecial. Continuity in the dimension suggests that an ultraviolet fixed point,\nif it exists in four dimensions, should persist towards higher dimensions.\nMore generally, one expects that the local structure of quantum fluctuations,\nand hence local renormalisation group properties of a quantum theory of\ngravity, are qualitatively similar for all dimensions above the critical one,\nmodulo topological effects for specific dimensions.  Secondly, the dynamics of\nthe metric field depends on the dimensionality of space-time. In four\ndimensions and above, the metric field is fully dynamical. Hence, once more,\nwe should expect similarities in the ultraviolet behaviour of gravity in four\nand higher dimensions.  Interestingly, this pattern is realised in the results\n\\cite{Litim:2003vp}, see the analytical fixed point \\eq{FP-d}. An extended\nsystematic search for fixed points in higher-dimensional gravity for general\ncutoff $R_k$ has been presented in \\cite{Fischer:2006fz,Fischer:2006at}, also\ntesting the stability of the result against variations of the gauge fixing\nparameter (see Tab.~\\ref{tEH-d}). The variation with $R_k$,\nammended by stability considerations, is smaller than the variation with\n$\\alpha$. We conclude from the weak variation that the fixed point indeed\npersists in higher dimensions. Further studies including higher derivative\noperators confirm this picture \\cite{DL06}. This structural stability also\nstrengthens the results in the four-dimensional case, and supports the view\nintroduced above. A phenomenological application of these findings in\nlow-scale quantum gravity is discussed below (see Sec.~\\ref{pheno}).\n\n\n    \\begin{figure}\n\\begin{center}\n\\unitlength0.001\\hsize\n\\begin{picture}(700,480)\n\\put(750,20){\\Large $\\lambda_{4d}$}\n\\put(-70,410){\\Large $g_{4d}$}\n\\includegraphics[width=.7\\hsize]{Flowsd4a0Smallb1.eps}\n\\end{picture}\n\\caption{ The phase diagram for the running gravitational coupling\n  $g_{4d}$ and the cosmological constant $\\lambda_{4d}$ in four dimensions. The\n  Gaussian and the ultraviolet fixed point are indicated by dots (red). The\n  separatrix connects the two fixed points (full green line). \nThe full (red) line indicates the bound $g_{\\rm bound}(\\lambda)$ where\n  $1\/\\eta=0$. Arrows indicate the direction of the RG flow with decreasing\n  $k\\to 0$.}\n\\label{PD4}\n\\end{center}\n\\end{figure}\n\n\\section{Phase Diagram}\n\\label{PD}\nIn this section, we discuss the main characteristics of the phase portrait of\nthe Einstein-Hilbert theory \\cite{Reuter:2001ag,Litim:2003vp} (see\nFig.~\\ref{PD4}).  Finiteness of the flow \\eq{ERG} implies that the line\n$1\/\\eta=0$ cannot be crossed.  Slowness of the flow implies that the line\n$\\eta=0$ can neither be crossed (see Sec.~\\ref{FP}).  Thus, disconnected\nregions of renormalisation group trajectories are characterised by whether $\ng$ is larger or smaller $ g_{\\rm bound}$ and by the sign of $g$. Since $\\eta$\nchanges sign only across the lines $\\eta=0$ or $1\/\\eta=0$, we conclude that\nthe graviton anomalous dimension has the same sign along any trajectory. In\nthe physical domain which includes the ultraviolet and the infrared fixed\npoint, the gravitational coupling is positive and the anomalous dimension\nnegative. In turn, the cosmological constant may change sign on trajectories\nemmenating from the ultraviolet fixed point.  Some trajectories terminate at\nthe boundary $g_{\\rm bound}(\\lambda)$, linked to the present\napproximation. The two fixed points are connected by a separatrix.  The\nrotation of the separatrix about the ultraviolet fixed point reflects the\ncomplex nature of the eigenvalues. At $k\\approx M_{\\rm Pl}$, the flow displays\na crossover from ultraviolet dominated running to infrared dominated\nrunning. The non-vanishing cosmological constant modifies the flow mainly in\nthe crossover region rather than in the ultraviolet.  In the infrared limit,\nthe separatrix leads to a vanishing cosmological constant $\\Lambda_k=\\lambda_k\nk^2\\to 0$ and is interpreted as a phase transition boundary between\ncosmologies with positive or negative cosmological constant at large\ndistances. Trajectories in the vicinity of the separatrix lead to a\npositive cosmological constant at large scales and are, therefore, candidate\ntrajectories for realistic cosmologies \\cite{cosmology}.  This picture agrees\nvery well with numerical results for a sharp cut-off flow\n\\cite{Reuter:2001ag}, except for the location of the line $1\/\\eta=0$ which is\nnon-universal. Similar phase diagrams are found in higher dimensions\n\\cite{Fischer:2006fz,Fischer:2006at}.\n\n\n\\section{Lattice}\n\\label{Lattice}\nLattice implementations for gravity in four dimensions have been put forward\nbased on Regge calculus techniques \\cite{Hamber:1999nu,Hamber:2005vc} and\ncausal dynamical triangulations \\cite{Ambjorn:2004qm}.  In the Regge calculus\napproach, a critical point which allows for a lattice continuum limit has been\ngiven in \\cite{Hamber:1999nu} using the Einstein Hilbert action with fixed\ncosmological constant. A scaling exponent has been measured in the\nfour-dimensional simulation based on varying Newton's coupling to the critical\npoint, with $\\partial_g\\beta_g|_*=-\\s01\\nu$. The result reads $\\nu\\approx\n\\s013$, and should be contrasted with the RG result $\\nu=1\/\\theta=\\s038$\n\\cite{Litim:2003vp} as discussed in Sec.~\\ref{FP}. In the large-dimensional\nlimit, geometrical considerations on the lattice lead to the estimate\n$\\nu=\\s01{d-1}$ \\cite{Hamber:2005vc}, a behaviour which is in qualitative\nagreement with the explicit RG fixed point result $\\nu=\\s01{2d}$ in the\ncorresponding limit, see \\eq{theta0-NG}.\n\nWithin the causal dynamical triangulation approach, global aspects of quantum\nspace-times have been assessed in \\cite{Ambjorn:2004qm}. There, the effective\ndimensionality of space-time has been measured as a function of the length\nscale by evaluating the return probability of random walks on the triangulated\nmanifolds. The key result is that the measured effective dimensionality\ndisplays a cross-over from $d\\approx 4$ at large scales to $d\\approx 2$ at\nsmall scales of the order of the Planck scale. This behaviour compares nicely\nwith the cross-over of the graviton anomalous dimension $\\eta$ under the\nrenormalisation group (see Sec.~\\ref{AS}), and with renormalisation group\nstudies of the spectral dimension (see\n\\cite{Niedermaier:2006ns,NiedermaierReuter,Lauscher:2005qz}). These findings\ncorroborate the claim that asymptotically safe quantum gravity behaves, in an\nessential way, two-dimensional at short distances.\n\n\\section{Phenomenology}\n\\label{pheno}\nThe phenomenology of a gravitational fixed point covers the physics of black\nholes \\cite{blackholes}, cosmology\n\\cite{cosmology,Bonanno:2007wg,Bentivegna:2003rr}, modified dispersion\nrelations \\cite{Girelli:2006sc}, and the physics at particle colliders\n\\cite{Litim:2007iu,Hewett:2007st,Koch:2007yt}.  In this section we concentrate\non the later within low-scale quantum gravity models \\cite{add,aadd}.  There,\ngravity propagates in $d=4+n$ dimensional bulk whereas matter fields are\nconfined to a four-dimensional brane. The four-dimensional Planck scale\n$M_{\\rm Pl}\\approx 10^{19}$~GeV is no longer fundamental as soon as the $n$\nextra dimensions are compact with radius $\\sim L$. Rather, the\n$d=4+n$-dimensional Planck mass $M_*$ sets the fundamental scale for gravity,\nleading to the relation $M^2_{\\rm Pl}\\sim M^2_{*} (M_*\\,L)^{n}$ for the\nfour-dimensional Planck scale.  Consequently, $M_*$ can be significantly lower\nthan $M_{\\rm Pl}$ provided $1\/L\\ll M_*$.  If $M_*$ is of the order of the\nelectroweak scale, this scenario lifts the hierarchy problem of the standard\nmodel and opens the exciting possibility that particle colliders could\nestablish experimental evidence for the quantisation of gravity\n\\cite{grw,tao,virtual_kk}.\n\n\\begin{figure}\n\\begin{picture}(1000,20)\n\\end{picture}\n\\unitlength0.001\\hsize\\includegraphics[width=.4\\hsize]{FlowExtra.eps}\n\\vskip-.325\\hsize\n\\hskip.5\\hsize\n\\includegraphics[width=.49\\hsize]{p2.eps}\n\\vskip-.025\\hsize\n\\hskip.5\\hsize\n\\includegraphics[width=.49\\hsize]{p1.eps}\n\\begin{picture}(1000,5)\n\\put(880,320){{$|\\eta|$}}\n\\put(865,265){{$n=1$}}\n\\put(665,340){{$n=7$}}\n\\put(880,150){{\\large $\\displaystyle \\frac{G_k}{G_0}$}}\n\\put(130,390){{a)}\\ \\ schematically}\n\\put(660,390){{b)}\\ \\ numerically}\n\\put(20,340){$\\ln g$}\n\\put(270,310){UV (fixed point)}\n\\put(155,200){IR\\ \\ \\ \\ \\  ($d=4+n$)}\n\\put(20,80){IR\\ \\ \\ \\ \\ \\ \\ \\  ($d=4$)}\n\\put(410,45){$\\ln k$}\n\\put(740,20){$\\ln k$}\n\\put(230,20){$\\ln M_*$}\n\\put(130,20){$\\ln 1\/L$}\n\\end{picture}\n\\caption{ The scale-dependence of the gravitational coupling in a scenario\n  with large extra dimension of size $\\sim L$ with fundamental Planck scale\n  $M_*$ and $M_*L\\gg 1$. The fixed point behaviour in the deep ultraviolet\n  enforces a softening of gravitational coupling (see text). {a)} In the\n  infrared (IR) regime where $|\\eta|\\ll 1$, the coupling $g=G_k\\,k^{d-2}$\n  displays a crossover from 4-dimensional to $(4+n)$-dimensional classical\n  scaling at $k\\approx 1\/L$.  The slope ${\\rm d}\\ln g\/{\\rm d}\\ln k\\approx d-2$\n  measures the effective number of dimensions. At $k\\approx M_*$, a\n  classical-to-quantum crossover takes place from $|\\eta|\\ll1$ to $\\eta\\approx\n  2-d$ (schematically). {b)} Classical-to-quantum crossover at the respective\n  Planck scale for $G_k$ and the anomalous dimensions $\\eta$ from numerical\n  integrations of the flow equation; $d=4+n$ dimensions with $n=1,\\cdots,7$\n  from right to left.}\n\\label{RunningG}\n\\end{figure}\n\nThe renormalisation group running of the gravitational coupling in this\nscenario has been studied in \\cite{Fischer:2006fz,Fischer:2006at,Litim:2007iu}\nand is summarised in Fig.~\\ref{RunningG}. The main effects due to a fixed\npoint at high energies set in at momentum scales $k\\approx M_*$, where the\ngravitational coupling displays a cross-over from perturbative scaling\n$G(k)\\approx$ const.~to fixed point scaling $G(k)\\approx g_*\nk^{2-d}$. Therefore we expect that signatures of this cross-over should be\nvisible in scattering processes at particle colliders as long as these are\nsensitive to momentum transfers of the order of $M_*$.\n\nWe illustrate this at the example of dilepton production through virtual\ngravitons at the Large Hadron Collider (LHC) \\cite{Litim:2007iu}.  To lowest\norder in canonical dimension, the dilepton production amplitude is generated\nthrough an effective dimension--8 operator in the effective action, involving\nfour fermions and a graviton~\\cite{grw}.  Tree--level graviton exchange is\ndescribed by an amplitude ${A} = {S}\\cdot {T}$, where ${T} = T_{\\mu\\nu}\nT^{\\mu\\nu} - \\frac{1}{2+n} T_\\mu^\\mu T_\\nu^\\nu$ is a function of the\nenergy-momentum tensor, and\n\\begin{equation} \\label{S}\n{S}= \\frac{2\\pi^{n\/2}}{\\Gamma(n\/2)} \\; \n  \\frac{1}{M_*^{4}} \\;   \n\\int_0^\\infty \\frac{d m}{M_*} \\; \\left(\\frac{m}{M_*}\\right)^{n-1}\\, \n{\\cal G}(s,m)\n\\end{equation}\nis a function of the scalar part ${\\cal G}(s,m)$ of the graviton\npropagator~\\cite{grw,gps}.  The integration over the Kaluza-Klein masses $m$,\nwhich we take as continuous, reflects that gravity propagates in the\nhigher-dimensional bulk. If the graviton anomalous dimension is small, the\npropagator is well approximated by ${\\cal G}(s,m)=(s+m^2)^{-1}$. This\npropagator is used within effective theory settings, and applicable if the\nrelevant momentum transfer is $\\ll M_*$. In this case, \\eq{S} is ultraviolet\ndivergent for $n\\ge 2$ due to the Kaluza-Klein modes\n\\cite{grw}. Regularisation by an UV cutoff leads to a power-law dependence of\nthe amplitude ${S}\\sim M_*^{-4}( {\\Lambda}\/{M_*} )^{n-2}$ on the cutoff\n$\\Lambda$.  In a fixed point scenario, the behaviour of $S$ is improved due to\nthe non-trivial anomalous dimension $\\eta$ of the graviton, $e.g.$\n\\eq{eta-inf}. Evaluating $\\eta$ at momentum scale $k^2\\approx s+m^2$, we are\nlead to the dressed propagator ${\\cal\n  G}(s,m)\\approx\\frac{M_*^{n+2}}{(s+m^2)^{n\/2+2}}$ in the vicinity of an UV\nfixed point. The central observation is that \\eq{S} becomes finite even in the\nUV limit of the integration. An alternative matching has been adapted in\n\\cite{Hewett:2007st,Koch:2007yt}, based on the substitution $G(k)\\to\nG(\\sqrt{s})$ in \\eq{S}, setting $G=M_*^{2-d}$. In that case, however, \\eq{S}\nremains UV divergent due to the Kaluza-Klein modes. We conclude that the large\nanomalous dimension in asymptotically safe gravity provides for a finite\ndilepton production rate.\n\n\n\\begin{figure*}[t]\n\\begin{picture}(1000,112)\n\\unitlength0.001\\hsize\n\\put(80,250){{a)}\\ \\ effective theory}\n\\put(550,250){{b)}\\ \\ renormalisation group}\n\\includegraphics[width=.33\\textwidth]{plehn.eps}\n\\includegraphics[width=.66\\textwidth]{plehnRG.eps}\n\\end{picture}\n\\vskip-.3cm\n\\caption{The $5\\sigma$ discovery contours in $M_D$ at the LHC ($d=4+n)$, as a\n  function of a cutoff $\\Lambda$ on $E_{\\rm parton}$ for an assumed integrated\n  luminosity of $10{\\rm fb}^{-1}$ ($100{\\rm fb}^{-1}$). a) Effective theory:\n  the sensitivity to the cutoff $\\Lambda$ is\n  reflected in the $M_D$ contour; plot from \\cite{gps}. b) Renormalisation\n  group: the limit $\\Lambda\\to\\infty$ can be performed, and the leveling-off\n  at $M_D\\approx \\Lambda$ reflects the gravitational fixed\n  point, thin lines show a $\\pm$10\\% variation in the transition scale; plot\n  from \\cite{Litim:2007iu}.  }\n\\label{fig:discovery}\n\\vskip-.3cm\n\\end{figure*}\n\nIn Fig.~\\ref{fig:discovery} we show the discovery potential in the fundamental\nPlanck scale at the LHC, and compare effective theory studies \\cite{gps} with\na gravitational fixed point \\cite{Litim:2007iu}. In either case the minimal\nsignal cross sections have been computed for which a $5\\sigma$ excess can be\nobserved, taking into account the leading standard model backgrounds and\nassuming statistical errors. This translates into a maximum reach $M_D$ for\nthe fundamental Planck scale $M_*$.  To estimate uncertainties in the RG\nset-up, we allow for a 10\\% variation in the scale where the transition\ntowards fixed point scaling sets in.  Consistency is checked by introducing an\nartificial cutoff $\\Lambda$ on the partonic energy \\cite{gps}, setting the\npartonic signal cross section to zero for $E_{\\rm parton}>\\Lambda$.  It is\nnicely seen that $M_D$ becomes independent of $\\Lambda$ for $\\Lambda\\to\\infty$\nwhen fixed point scaling is taken into account.\n\n\\section{Conclusions}\n\\label{conclusions}\nThe asymptotic safety scenario offers a genuine path towards quantum gravity\nin which the metric field remains the fundamental carrier of the physics even\nin the quantum regime. We have reviewed the ideas behind this set-up in the\nlight of recent advances based on renormalisation group and lattice studies.\nThe stability of renormalisation group fixed points and scaling exponents\ndetected in four- and higher-dimensional gravity is remarkable, strongly\nsupporting this scenario.  Furthermore, underlying expansions show good\nnumerical convergence, and uncertainties which arise through approximations\nare moderate.  If the fundamental Planck scale is as low as the electroweak\nscale, signs for the quantisation of gravity and asymptotic safety could even\nbe observed in collider experiments. It is intriguing that key aspects of\nasymptotic safety are equally seen in lattice studies. It will be interesting\nto evaluate these links more deeply in the future.  Finally, asymptotically\nsafe gravity is a natural set-up which leads to classical general relativity\nas a ``low energy phenomenon'' of a fundamental quantum field theory in the\nmetric field.\n\n\\section*{Acknowledgements}\nI thank Peter Fischer and Tilman Plehn for collaboration on the topics\ndiscussed here, and the organisers for their invitation to a very stimulating\nworkshop.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Motivation and Significance}\n\nThe Earth is facing unprecedented climatic, geomorphologic, environmental or anthropogenic changes, which require global scale, long term observation with Earth Observation (EO) sensors. SAR sensors, due to their observation capability during day and night and independence on atmospheric effects, are the only EO technology to insure global and continuous observations. Meanwhile, the SAR observations of Sentinel-1 satellites in the frame of the European Copernicus program, are worldwide freely and openly accessible. This is immensely enlarging the SAR Data Science and applications, covering a multitude of areas as: urbanization, agriculture, forestry, geology, tectonics, oceanography, polar surveys, or biomass estimation, only to enumerate a few. Copernicus Open Access Hub provides more than 457.59 PB data of satellites covering the Earth  for more than 570,000 users all around the world. \\footnote{https:\/\/scihub.copernicus.eu\/reportsandstats\/}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=8.5cm]{fig_intro.pdf}\n\\caption{A simple illustration of how SAR images the world (Stripmap Mode). SAR society is facing the big data challenge but with limited ground truth. In the meanwhile, the knowledge of SAR is equally important. This is also the motivation of the physical layers in this paper.}\n\\label{fig_intro}\n\\end{figure}\n\nSAR is a pioneer technology in the field of computational sensing and imaging, of which the imaging mechanism is totally different from optical sensors. A radar instrument carried by an airborne or spaceborne platform illuminates the scene by side-looking or forward-looking, which allows to discriminate objects in the range direction. As the platform moving along its track, the SAR sensor is constantly transmitting a sequence of chirp signals and receiving echos reflected from objects on the ground, as depicted in Fig. \\ref{fig_intro}. When recording all individual acquisitions with a short physical antenna and mathematically combining them into a synthetic image, a much larger synthesized aperture is formed. This allows high capacity to distinguish objects in azimuth despite a physically small antenna \\cite{moreira2013tutorial}. A high resolution \"image\" can be processed by applying SAR focusing principle, e.g., matching filtering \\cite{meta2007signal}. \n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=14cm]{fig_intro2.png}\n\\caption{\\textbf{a} The conventional data-driven paradigm for intelligent SAR image understanding based on deep neural networks and the proposed paradigm shift integrated and interacted with physical knowledge of SAR. \\textbf{b} A bridge can be imaged as multiple bright lines, similar as a couple of bridges imaged in the other SAR image, depending on the observation parameters and orientations. This positions the load and outmost difficulty of SAR image understanding. \\textbf{c} The multipath scattering formation in the SAR image.}\n\\label{fig_overview}\n\\end{figure*}\n\nA deluge of SAR sensors have increased the data availability for various SAR applications. The allure of data-driven learning stems from the ability of automatically extracting abstract features from large data volumes \\cite{amrani2021new,amrani2018sar,amrani2017efficient,amrani2017deep}, and therefore, many deep learning studies for SAR applications have been developed in recent years \\cite{Zhu2021,rs11131532,HUGHES2020166,Xiang2022}. Current popular paradigm predominantly follows the blue part in Fig. \\ref{fig_overview} (a), where SAR image data is all that is required to operate an intelligent network. In addition to data, however, the physical model and principles of SAR sensor should not be neglected. In the upper example of Fig. \\ref{fig_overview} (b), A bridge over a placid river that is illuminated perpendicular to its primary orientation appears as many brilliant lines, resembling the lower SAR image in which several bridges are imaged from a different viewing angle. The phenomena can be explicable by multi-path scattering \\cite{zhangCharacteristicsMultipathScattering2015,soergelExtractionBridgeFeatures2007}, as illustrated in Fig. \\ref{fig_overview} (c). Apart from the direct scattering from the bridge, the double bounce reflection between the bridge and water or vice versa occurs at the corner reflector spanned from the smooth vertical bridge facets facing the sensor and the water surface. In addition, the triple-bounce reflection and maybe some five-path scattering would happen between the horizontal plane of bridge and water surface. Thus, SAR image implies the causality of multi-path scattering phenomena and object characteristics. This positions the load of SAR image understanding, and the outmost challenge of data science, as new and particular paradigm of Artificial Intelligence (AI).\n\nSo far, some researches have discussed the paradigm that attempts to bring scientific knowledge and data science models together, applied to a broad range of research themes such as partial differential equation solving \\cite{karniadakisPhysicsinformedMachineLearning2021a} and Earth sciences \\cite{karpatneTheoryGuidedDataScience2017a,karpatneMachineLearningGeosciences2019,Reichstein2019}. In particular for SAR community, however, this topic has rarely been systematically analyzed and illustrated. Thus, we aim to prospect the hybrid modeling paradigm for intelligent SAR image understanding, where deep learning is integrated and interacted with SAR physical models and principles, to achieve explainability, physics awareness, and trustworthiness.\n\n\n\\begin{figure*}[!htb]\n\\centering\n\\includegraphics[width=12.5cm]{fig_physicallayer1.png}\n\\caption{Physical layer (i): Sensor and Platform. \\textbf{a}: The moving platform creates Doppler variations and synthesizes large virtual aperture; PolSAR transmits and receives diverse polarized wave, and SAR polarimetric characteristics are depicted. \\textbf{b}: Based on the physics behind the platform and sensor, the physical layer produces SAR specific representations such as sub-aperture synthesis image and polarimetric feature, with specified physical parameters.}\n\\label{fig_physicallayer1}\n\\end{figure*}\n\nExplainable AI is a broad concept. A scientific understanding of explainability is the capacity to clarify the results in the context of domain knowledge. The algorithms still remain a black box. A different approach is the algorithmic explainability. This is constructed such that the results of the used model can be described algorithmically. To obtain a higher degree of explainability, we aim at the synergy of the paradigms: \\textit{algorithmic and scientific explainability}. Algorithmic explainability lies in the guarantee of transparency to understand how the machine learning algorithm works by participation of SAR physical models and principles. Scientific explainability ensures the physical consistency of AI output, as well as learning of trustworthy results with physical meaning.\n\nTo ground this, we first lay out a representation of SAR physical layer in the context of SAR domain knowledge, as presented in Section \\ref{sec:phylayer}. Further, we describe how to integrate and interact them with popular neural networks to build a hybrid and translucent model for SAR applications using illustrative examples, demonstrated in Section \\ref{sec:hybrid}. The perspective of trustworthy models and supplementary explanation for SAR community are discussed in Section \\ref{sec:trust} and \\ref{sec:explanation}. The conclusion and perspectives are finally given in Section \\ref{sec:conclusion}.\n\n\\section{SAR Physical Layers}\n\\label{sec:phylayer}\n\nOther than the neural network layers equipped with a number of learnable parameters, SAR physical layers are ones embedded with physical knowledge of SAR, well-established, interpretable, and supported by domain theories. The concept of \"physical layer\" apart from \"neural network layer\" arose in literature \\cite{Reichstein2019} to make the model more physically realistic. As motivated in Fig. \\ref{fig_intro}, three SAR physical layers are highlighted specific for SAR applications in this paper, i.e., (i) sensor and platform: referring to antenna characteristics and moving satellite\/aircraft, (ii) imaging system: figuring image formation with focusing process and (iii) scattering signature: reflecting the physical properties of terrain and objects. \n\n\n\\subsection{Sensor and Platform}\n\nFig. \\ref{fig_physicallayer1} demonstrates the physical layer of sensor and platform that indicates the physics behind the SAR acquisition principle, such as aperture synthesizing with moving platform and various characteristics of antenna.\n\nExisting spaceborne EO SAR missions work in a monostatic or quasi-monostatic configuration. The simplest illumination mode of a SAR system is the stripmap mode in which the antenna pointing direction is constant throughout the acquisition, as shown in Fig. \\ref{fig_physicallayer1} \\textbf{a}. The moving platform leads to a sliding Doppler spectrum that impacts the complex SAR image. Knowing the behaviour of the Doppler centroid to create sub-looks is essential for exploiting look angle diversity of the input data, especially for very high-resolution SAR images.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=16cm]{fig_physicallayer2.png}\n\\caption{Physical layer (ii): Image Formation. \\textbf{a.} Targets are characterized by sliding bandpass filtering in the Fourier domain. \\textbf{b.} On the basis of image formation principle and target scattering model, the physical layer generates the rich target description with physical meaning.}\n\\label{fig_physicallayer2}\n\\end{figure*}\n\nIt is well-known that in high-resolution SAR image where the signals are performed over a broad bandwidth and wide angular aperture, the targets are no longer isotropic and non-dispersive. Instead, it is more plausible to infer that the target's backscattering is dependent on illumination angle and frequencies \\cite{triaDiscriminatingRealObjects2007}. The sub-aperture processing can be applied to analyze the target scattering variations. Fig. \\ref{fig_physicallayer1} \\textbf{b} gives an example of a synthesized pseudo color SAR image via sub-aperture processing. The complex-valued SAR image is first transformed to the azimuth spectral domain by a one-dimensional Fourier transform. Then, the full Doppler spectrum is equally split into three intervals, named sub-apertures or sub-looks, each containing 1\/3 range of azimuth angles. Finally, the three sub-apertures are transformed back to time-domain using an inverse Fourier transform, coded as the R, G, and B channels, respectively. Red, Green, and Blue targets respond mainly on the first, second, and third sub-looks, respectively, whilst gray targets indicate that they respond equivalently in different sub-looks. The pseudo-colored image well demonstrates the particular behavior of some targets. Given the precise knowledge of the parameters related to Doppler variations (e.g., orbit, azimuth steering rate, radiation pattern, incidence angle), the physical layer can generate sub-look data deterministically and there is no need to design a neural network that should learn to create sub-looks from various types of SAR training data. \n\n\nSensor characteristics, such as polarization, interferometry and tomography, construct physical layer as well. Fig. \\ref{fig_physicallayer1} \\textbf{b} presents a Pauli pseudo RGB image, where R, G, and B channels are formed with $|HH-VV|^2$, $2|HV|^2$, and $|HH+VV|^2$, respectively, indicating the polarimetric relation. Several physical layers can be stacked to represent rich physics of SAR sensor and platform. Early in literature \\cite{ferro-famil_scene_2003}, the diversity in the polarimetric features with the azimuthal look angle was exploited. Thus, the moving platform and polarimetric sensor are both characterized. Similarly, the stacked physical layers can represent polarimetric and interferometric properties of PolInSAR data, or any other combinations.\n\n\n\n\\subsection{Imaging System}\n\nThe second physical layer we suppose delineates the physics behind SAR image formation with an imaging system. The selected exemplars are illustrated in Fig. \\ref{fig_physicallayer2}.\n\nA pulse-based radar or a frequency modulated continuous wave (FMCW) radar is usually used in a SAR system, where a range profile is obtained for each transmitted\/received waveform, either by range compression in the case of a pulse-based radar or by applying a Fourier transform to the beat signal in the case of an FMCW radar \\cite{anghel2014short}. By a coherent processing of the range profiles, the azimuth focusing process outputs a SAR image representing a two-dimensional complex reflectivity map of the illuminated area. SAR processing, taking a simple point target as example, aims to collect the dispersed signal energy in range and azimuth into a single pixel. Many traditional imaging algorithms are in terms of a Fourier synthesis framework \\cite{oliver1989synthetic}, as such, Fourier transform provides a specific physical meaning for SAR image. This kind of physical layer assists AI model to better depict the target scattering beyond the \"image\" domain. \n\n\n\n\nFig. \\ref{fig_physicallayer2} (a) first shows a simple time-frequency analysis of target with short-time Fourier transform \\cite{spigai2008time,singh2011sar}, characterizing the backscattering intensity variations in 2-D range and azimuth frequency domain. Four kinds of backscattering behaviors observed in SAR were defined in literature \\cite{Spigai2011}, related to different objects shown in Fig. \\ref{fig_physicallayer2}. In the high-resolution case (wide bandwidth chirp signal and broad angular aperture), the complex amplitude of a target is frequency and aspect dependent \\cite{triaDiscriminatingRealObjects2007}. Thus, the image formation can be extended to four dimension (called hyperimage) with wavelet transform, providing a concise physically relevant description of target scattering. This frequency and angular energy response pattern is proved useful for discriminating different scatterers, offering valuable prior information to AI model, depicted in Fig. \\ref{fig_physicallayer2} \\textbf{b}.\n\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=16cm]{fig_physicallayer3.png}\n\\caption{Physical layer (iii): Scattering Signatures of Objects. \\textbf{a}. The Golden Gate Bridge revealing multipath scattering characteristics in a Gaofen-3 quad-pol SAR image, and a typical single building representing different scattering regions in a high-resolution (1m) SAR image \\cite{Chen2021}. \\textbf{b}. The scattering mechanisms indicated by the H\/$\\alpha$ plane for full-polarized SAR data point out the land-use and land-cover classes \\cite{Yonezawa2012}. \\textbf{c}. The physical layer describes the relationship and reasoning between the scattering characteristics seen in the SAR image and the object's features, such as its shape, structure, or even semantics.}\n\\label{fig_physicallayer3}\n\\end{figure*}\n\n\\subsection{Scattering Signatures of Objects}\n\nThirdly, we introduce the physical layer regarding the scattering signatures of objects, in which the causality of target characteristics and scattering behaviors is involved.\n\nFor optical images, what you see is what you receive, that is, the objects depicted on the optical image are in accord with human cognition. Targets in SAR images are reflected by scattering characteristics, yet they include a wealth of physical information that the human eye cannot immediately identify. Fig. \\ref{fig_physicallayer3} \\textbf{a} shows example of two typical SAR targets of bridge and building. The scattering phenomenon that shows several parallel lines over the river can be interpreted as single, double, and multiple scattering of the bridge based on the domain knowledge. The building, with scattering signatures of layover, shadow, single and secondary scattering in the high-resolution SAR image, can also be reflected as only layover and shadow \\cite{Chen2021}, depending on the building orientation and shape. Similar research by Ferro et al.  \\cite{Ferro2011} investigated the relationship between double bounce and the orientation of buildings in VHR SAR images. Fig. \\ref{fig_physicallayer3} \\textbf{b} demonstrates the relations between the scattering mechanism of H\/$\\alpha$ plane and the semantics of land-cover and land-use classes \\cite{Yonezawa2012}. Likewise, one can deduce the scattering center position and the specific shape of distributed target from a SAR image by applying some scattering models \\cite{Potter1997}.\n\nThe conventional data-driven convolutional neural network can capture the image contents as we \"see\" in the SAR image, whereas it is not equipped with the ability to \"interpret\" the scattering phenomenon as we discussed before. This indicates the knowledge gap between SAR scattering signatures and human vision cognition. The physical layer delivering semantic understanding behind the SAR scattering signature permits a more thorough interpretation of the SAR image. As shown in Fig. \\ref{fig_physicallayer3} \\textbf{c}, the physical layer defines the association between the scattering characteristics of a SAR image and the object's qualities, such as shape, structure, or semantics. It can be written as an objective function or a regularization term that constrains the training of neural networks. This will improve the intelligence of AI model to master some causality between scattering signatures and the object nature.\n\n\\section{Hybrid Modeling with SAR Physical Layers}\n\\label{sec:hybrid}\n\nThe integration and interaction of neural network layers and physical layers construct the hybrid modeling for SAR image interpretation. In view of algorithmic explainability, the explainable physical models and domain knowledge improves the transparency. For scientific explainability, the hybrid modeling ensures the physical meaning of output in physical layers and the prediction can maintain the physical consistency. In this section, we demonstrate several hybrid modeling approaches with SAR physical layer to achieve explainability and physics awareness.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=13cm]{fig_physicallayer1_eg.pdf}\n\\caption{Our recent work Deep SAR-Net (DSN) \\cite{HUANG2020179} for SAR image classification can be regarded as a typical example of inserting the physical layers into a deep model.}\n\\label{fig_physicallayer1_eg}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=15cm]{fig_physicallayer2_eg.png}\n\\caption{\\textbf{a}. Unsupervised HDEC-TFA method \\cite{Huang2020}. It automatically discovered the radar spectrogram patterns more than the four defined in \\cite{Spigai2011} with deep neural networks. \\textbf{b}. Learning the polarimetric features from single-polarized SAR image, supervised by Entropy-Alpha-Anisotropy generated from full-pol data \\cite{zhao2019contrastive}. \\textbf{c}. The physical layers in \\textbf{a} and \\textbf{b} play the role of input transform (blue) and supervision generation (green) in hybrid modeling. In addition, the physical layer (red) can act as feedback to restrict learning and produce physically consistent outcomes.}\n\\label{fig_physicallayer2_eg}\n\\end{figure*}\n\n\\subsection{Insert for Substitution}\n\nThe introduced physical layer can be inserted in a deep neural network for substitution, extracting explainable and meaningful features, either as the input of a DNN or fused with DNN features in intermediate layers. A common way is to insert a physical layer into the input layer to obtain the polarimetric features for PolSAR image classification, including the elements of coherency matrix, Pauli decomposition features, etc \\cite{Zhang2017,vinayaraj_transfer_2020}. Similarly, the sub-aperture images are generated as the input for target detection \\cite{lei_feature_2021}. The other usage of physical layer is for feature fusion, where the features obtained by well-established physical model and deep neural networks are combined \\cite{Zhang2020,feng_sar_2021}.\n\nOur recent work, a deep learning framework named Deep SAR-Net (DSN) \\cite{HUANG2020179}, addressed both aspects that inserts the physical layer into the input and the intermediate position of deep model. As shown in Fig. \\ref{fig_physicallayer1_eg}, DSN was proposed for classifying SAR images with complex values. Instead of the entire data-driven method, i.e. the complex-valued convolutional neural networks (CV-CNN), the designed DSN encompasses three shallow neural network modules and two physical layers. The first physical layer generates the high-dimensional radar spectrogram based on time-frequency analysis. The second one handles the features of the 2-D projection along the frequency axises \\cite{singh2011sar} to maintain the location constraint, making it possible to be fused with spatial features from intensity image. DSN outperformed CV-CNN especially with limited labeled training data, and had a remarkable performance in discriminating the man-made target scenes compared with the traditional CNN. It demonstrates the Fourier process on single-look complex SAR image embedded the knowledge like synthesizing the antenna well characterizes the physical property of SAR target, and the usages of physical layer cut down unnecessary parameters in neural network layers to improve the model performance with limited ground truth. \n\n\n\n\n\n\\subsection{Compensation for Imperfect Knowledge with Feedback}\n\n\nIn condition of unknown\/inconclusive physical models or incomplete knowledge, it is difficult to extract perfect physical parameters or physical scattering characteristics of SAR via model-based methods. For instance, obtaining the polarimetric features from dual-pol, or even single-polarized SAR image. Thus, the physical layer interacted with deep neural network take effect.\n\n\\subsubsection{Target Character Identification}\n\n\\hspace*{\\fill}\n\nSome researches have analyzed the energy response pattern in frequency dimensions of target varied in SAR image, and discussed the nonstationary targets \\cite{ferro-famil_scene_2003,ovarlezAnalysisSARImages2003}. Spigai et al. \\cite{Spigai2011} pointed out four canonical targets with a rough definition shown in Fig. \\ref{fig_physicallayer2} a. However, it remains unknown for many complicated scene and objects. Fig. \\ref{fig_physicallayer2_eg} show our related work of using physical layer and deep neural network for compensation of imperfect knowledge. The first is the unsupervised hierarchical deep embedding clustering based on time-frequency analysis (HDEC-TFA) \\cite{Huang2020}, which was proposed to automatically characterize the radar spectrogram (or the sub-band scattering pattern defined in \\cite{Huang2020}) basically in urban area, discovering the various scattering pattern more than the four specific classes defined in \\cite{Spigai2011}. It offered a new perspective to describe the physical properties of single-polarized SAR. Furthermore, we used two stacked physical layers to obtain the polarimetric and time-frequency patterns and analyzed with deep neural network in reference \\cite{Huang2020IGARSS}.\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=13cm]{fig_HDECTFAeg.pdf}\n\\caption{The SOLEIL synchrotron in France and the surrounding buildings with different shapes are depicted in the Gaofen-3 SAR image. Both the GD-Wishart \\cite{Ratha2018} result on Quad-Polarization SAR data and the HDEC-TFA \\cite{Huang2020} result on HH channel single-polarized SAR can capture the special scattering characteristics of the objects.}\n\\label{fig_HDECTFAeg}\n\\end{figure*}\n\nFig. \\ref{fig_HDECTFAeg} demonstrates the result compared with the polarimetric physical model. The SOLEIL synchrotron in France, shown as the round building in the Google Earth remote sensing image, is surrounded by three different shapes of buildings. The HDEC-TFA method can capture the special characteristics of the architectures even in single HH channel SAR image, as much as the physical model based method GD-Wishart \\cite{Ratha2018} on quad-pol SAR. Some other man-made targets examples characterized by time-frequency model with neural networks are given in \\cite{HuangEUSAR2021}. Our experiments in \\cite{Huang2020} demonstrated the trained model varies with different imaging conditions since the sub-band scattering pattern is influenced by several imaging parameters, which should be taken into consideration when transferring the AI model to other situations.\n\n\\subsubsection{Polarimetric Parameter Extraction}\n\n\\hspace*{\\fill}\n\nBy transmitting and receiving waves that are both horizontally and vertically polarized, the full-pol SAR image captures abundant physical characteristics of the imaged objects that can lead to various physical parameters. In contrast, single-pol and dual-pol SAR data are less informative for physical feature extraction. If only one polarization channel is obtained, one cannot derive the other polarization channels in principle. Once the objects are known, i.e., once the characteristics of targets such as geometry, surface roughness, etc, are identified, deep learning can be employed to transfer the knowledge learned from physical models to reconstruct the physical parameters of objects. As shown in Fig. \\ref{fig_physicallayer2_eg} b, Zhao et al. \\cite{zhao2019contrastive} proposed a complex-CNN model to learn physical parameters (entropy $H$ and $\\alpha$ angle) with transfer learning from single-pol and dual-pol SAR data, supervised by features obtained with a physical layer. Some similar studies include but not limit to \\cite{song_radar_2018,qu_study_2021}. Song et al. \\cite{song_radar_2018} addressed \"radar image colorization\" issue to reconstruct the polarimetric covariance matrix with a designed deep neural network, where the supervision was also generated with a physical layer.\n\n\nWhen training a data-driven deep neural network, some physical consistencies may not be guaranteed. The authors pointed that the reconstructed covariance matrix may not be positive semi-definite \\cite{song_radar_2018}, and they proposed an algorithm to correct it. In this case, the additional physical layer embedded prior constraint acts as post-processing to revise the physically inconsistent result of DNNs. Furthermore, this type of physical layer is suggested to provide feedbacks during training, as demonstrated in Fig. \\ref{fig_physicallayer2_eg} c, the red part. The feedback of physical layer aims to prevent the model from learning the physical inconsistency. \n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=8.5cm]{fig_ssl.pdf}\n\\caption{The SAR physical layer can be integrated in a self-supervised learning framework to guide the neural network training without ground truth. \\textbf{a}. The physical layer generates various modalities of SAR image using well-established physical models, such as sub-aperture images, different polarimetric features, etc. The self-supervised learning can be conducted with contrastive learning paradigm. \\textbf{b}. The physical layer produces a physical representation of image, serving as a guided signal that drives the neural network to learn a similar representation.}\n\\label{fig_ssl}\n\\end{figure}\n\n\\subsubsection{SAR Image Generation\/Simulation}\n\n\\hspace*{\\fill}\n\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=12cm]{fig_pgn.pdf}\n\\caption{A physics guided network was proposed \\cite{Huang2021IGARSS,huang2022physically} where a novel deep learning paradigm and loss function were designed to associate the SAR scattering characteristics with image semantics.}\n\\label{fig_physicallayer3_eg}\n\\end{figure*}\n\nThis paradigm can be popularized to other SAR applications. SAR target image generation (or simulation) based on deep generative model (such as variational auto-encoder \\cite{hu_feature_2021} and generative adversarial network \\cite{guo2017}) has attracted much attention in recent years. The generated SAR images are expected to be used as data supplements to support target identification. The authenticity and interpretability of current deep SAR image generation is a substantial obstacle that has a significant impact on subsequent tasks \\cite{malmgren2017improving}. Many latest studies input important physical parameters into the deep generative model or use them as supervision at the output layer, such as depression angle and target orientation, that facilitated more reliable outcomes \\cite{song_sar_2019,oh_peacegan_2021}. We consider them physical layer as shown in Fig. \\ref{fig_physicallayer2_eg} c, the green and blue part.\n\nCoupling the physical layer as a feedback in neural network for SAR image generation has yet to be explored. When generative model produces a pseudo SAR image, a physical layer will be applied to verify whether it is consistent with the knowledge base of SAR, e.g. physical parameters derived from a well-established model. If not, the current generative model will revise the pseudo result to minimize the inconsistency. There are some examples to learn from in the field of fluid simulation \\cite{chu2021,Xie2018}. As such, the physical layer is used for constructing physical inconsistency as a feedback that explicitly constrain the generative model to fulfill some quantitative conditions, so as to guarantee authenticity. Referred to some latest studies in other fields, physical model as a feedback or constraint in the loop of deep learning is also applied to under water image enhancement \\cite{zhouUnderwaterImageEnhancement2022} and seismic impedance inversion \\cite{Wang2022physics}.\n\n\\subsection{Self Supervised Learning Guidance}\n\n\nSelf supervised learning has been attracted much attention in recent years, since it can help reduce the required amount of labeling. One can pre-train a model on unlabeled data and fine-tune it on a small labeled dataset. It offers great opportunity for SAR community where big data volume is available while the ground truth is usually difficult to obtain. There is a remarkable potential for SAR physical layer to apply for self-supervised learning.\n\nAs shown in Fig. \\ref{fig_ssl}, two self-supervised learning paradigms are given. The physical layer helps to establish a pretext task for SAR image. In Fig. \\ref{fig_ssl} \\textbf{a}, different SAR image representations are generated by physical layer, for instance, the sub-aperture images, various polarimetric feature images, etc. As similar to SimCLR \\cite{pmlr-v119-chen20j} that conducted the contrastive learning based on data-augmentation, or NPID \\cite{Wu_2018_CVPR} that learned the optimal feature via instance-level discrimination, the surrogate task can be built to form a self-supervised learning. An illustrative example is in reference \\cite{Ren2021}.\n\nFig. \\ref{fig_ssl} \\textbf{b} illustrates a second line of thought, which we refer to as physics guided learning. Firstly, the physical layer is used for generating meaningful physical representations, like scattering mechanisms (physical layer (i) and (ii) can both achieve this). Meanwhile, the neural network extracts hierarchical spatial features from SAR image. The crucial point is how to establish a connection between physical properties and image features. We propose to exploit physical layer (iii) to reveal relationships and thereby design an objective function for self-supervised learning.\n\nOur recent work \\cite{Huang2020IGARSS,Huang2021IGARSS,huang2022physically} details the paradigm in Fig. \\ref{fig_ssl} \\textbf{b}. A physics guided network (PGN) for SAR image feature learning was proposed as shown in Fig. \\ref{fig_physicallayer3_eg}. First, a physical layer is deployed at the beginning, where the physical scattering properties are derived. Based on the crucial assumption that SAR image features and the abstract physical scattering mechanisms should share common attributes in semantic level, a surrogate task was established via the other physical layer that defines a loss function. The inspiration is from reference \\cite{radu}, which indicated the abstract topic mixture on scattering properties and the high-level image features are with similar semantics. Thus, we built the relation between the image semantics and SAR scattering characteristics. A novel objective function was designed to instruct self-supervised learning guided by physical scattering mechanisms.\n\nThe advantages of this kind of learning paradigm lie in two aspects. First, the training process takes all labeled and unlabeled data so that the learned features generalize well in test set. Second, the guidance of physical information leads to physics awareness of features learned by neural networks, i.e., the DNN feature maintains physical consistency. In a word, the prior physical knowledge is embedded in the neural network. The experiments in \\cite{huang2022physically} verified this quantitatively and qualitatively. \n\n\\begin{figure*}\n     \\centering\n     \\begin{subfigure}[b]{0.43\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{fig_featurevis_a.png}\n         \\caption{}\n     \\end{subfigure}\n     ~\n     \\begin{subfigure}[b]{0.43\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{fig_featurevis_b.png}\n         \\caption{}\n     \\end{subfigure}\n     \\caption{Visualization of physics guided signals on test data by t-sne. (a) Different colors represent semantic labels of sea-ice. (b) The physics guided signals are grouped into eight clusters, where each color indicates samples with similar physical scattering properties.}\n     \\label{fig_featurevis}\n\\end{figure*}\n\n\nAdditionally, the outputs of the physically interpretable deep model can be further explained, which in turn inspires algorithm improvement. We illustrate with an example of sea-ice classification in polar area \\cite{Huang2021IGARSS}. The physics guided learning is driven by physical signals that reflect the scattering properties of SAR image. The guided physical signals are visualized with t-sne in Fig. \\ref{fig_featurevis}, where different colors in (a) represent semantic labels of sea-ice and each color in (b) indicates samples with similar physical scattering properties. One characteristic that can be seen is that young ice and water bodies have extremely similar physical representations, which would impede semantic discrimination. It can explain the physics guided learning result in \\cite{Huang2021IGARSS} that about 23\\% test samples in water bodies class were predicted as young ice. The explanation will motivate us to improve the algorithm by, for instance, relaxing the physical constraints between the two classes.\n\nSimilarly, a very recent work \\cite{feng2022electromagnetic} was proposed for SAR target recognition inspired by our work\\cite{huang2022physically}. The authors proposed a CNN under the guidance of SAR target physical model, attributed scattering center (ASC), to extract the significant target features, that were successively injected into the classification network for more robust and interpretable results.\n\n\n\\section{Trustworthy Modeling}\n\\label{sec:trust}\n\n\\subsection{Why Trustworthy Modeling Needed}\n\nThe results obtained by applying AI techniques in SAR processing can be validated using in situ measurements of known targets. For example, a common approach for calibration\/validation of SAR data is to employ an electronic target (transponder) that receives a signal, applies a controllable time delay, and transmits the delayed signal towards the receiver of the bistatic\/monostatic system. Such a target can be used to validate results related to deformation measurements (e.g., atmospheric corrections) or polarimetric analysis. \n\nSome real world applications of SAR requires the measurement of reliability and uncertainty. One example is the sea-ice classification in the untraversed polar regions where the ice is always promptly changeable, that would result in the difficulty for annotation and the lack of reference data. In this case, the predictions in unknown polar areas obtained by AI model need to be trusted by humans. Strong robustness and plausible degree of confidence of ML system prediction are equally as important as its accuracy.\n\nFig. \\ref{fig_trustworthyexample} \\textbf{a} indicates building orientations have a great impact on polarization orientation angles \\cite{xiang2016unsupervised} and scattering mechanisms \\cite{Ratha2018}. The zoomed-in region mainly contains ortho buildings buildings where $\\phi_1$ is close to $0^\\circ$ and orientated buildings with a larger $\\phi_2$. The polarization orientation angles of ortho buildings are obviously smaller than those of oriented buildings. Ortho built-up areas mainly depict double scattering (DS) and mixed scattering (MS) where the double scattering dominates. The oriented buildings are with volume scattering (VS). Fig. \\ref{fig_trustworthyexample} \\textbf{b} shows limited robustness of recognition performance as the angle of test data varies when training with a small range of angles. A trustworthy model is expected to perceive SAR scattering variations with a variety of physical parameters and be perturbation-tolerant.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=17cm]{fig_trustworthyexample.png}\n\\caption{A trustworthy model should perceive the SAR scattering variations with a variety of physical parameters and be perturbation-tolerant. \\textbf{a} Differently oriented buildings reflect various polarization orientation angles and scattering mechanisms in a PolSAR image. \\textbf{b} SAR targets vary violently with orientation angles. When training with a small range of angles, limited robustness of recognition performance is observed as the angle of test data varies.}\n\\label{fig_trustworthyexample}\n\\end{figure*}\n\n\n\n\\subsection{Trustworthy Modeling with Uncertainty Quantification}\n\nThe development of Bayesian deep learning \\cite{wilson2020bayesian} has caught much attention in recent years, where the posterior distribution over parameters are obtained instead of the point estimation. A crucial property of the Bayesian method is its ability to quantify uncertainty, to the benefit of constructing trustworthy model.\n\nIn the case of Fig. \\ref{fig_trustworthyexample} \\textbf{b}, the performance of deep neural networks drops dramatically when testing SAR targets of very different orientation angles with training data. The model is over-confident about some uncertain data that cannot be perceived by frequentist method. Bayesian deep neural network, instead, is able to calibrate the output score and measure the uncertainty of the prediction. Some recent studies applied Bayesian deep learning for SAR sea-ice segmentation \\cite{Hartmann2021,Saberi2022,Asadi2021}, as well as target discrimination \\cite{Blomerus2021}. The generated uncertainty map can serve as a guideline for the experts in annotation and improve trust between users and the model. Some approximation strategies of Bayesian deep neural network, such as Monte Carlo Dropout \\cite{gal2016dropout} and Deep ensembles \\cite{lakshminarayananSimpleScalablePredictive2017}, are promising for different SAR applications.\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=13cm]{fig_uncertaintyresult.pdf}\n\\caption{The SAR ship detection result on AIR-SARShip-1.0 dataset \\cite{R19097}, obtained by the detection deep learning algorithm FCOS \\cite{tian2019fcos}. Many false alarms appear in the detection result, due to the limited training data and the interference of complex scattering. The prediction uncertainty is estimated by MC-Dropout \\cite{gal2016dropout} and the uncertain results are discard to achieve a better performance.}\n\\label{fig_uncertaintyresult}\n\\end{figure*}\n\nWe give an example of SAR ship detection for demonstration. The limited labeled training data, and the interference of complex scattering from target itself or the inshore background, would strongly restricted the detection performance. The ship detection result on some selected SAR images from AIR-SARShip-1.0 dataset \\cite{R19097}, obtained by FCOS detection algorithm \\cite{tian2019fcos}, are shown in the first row of Fig. \\ref{fig_uncertaintyresult}. Compared with the ground truth annotation in the third row, the detection result appears many false alarms. It is crucial to estimate the model uncertainty, which is basically brought by inadequate training data, to evaluate the reliability of SAR ship detection model and provide more trustworthy predictions.\n\nWhen we apply the Monte Carlo (MC) Dropout training strategy to approximate the Bayesian inference \\cite{gal2016dropout}, it captures the uncertainty from the existing deep model for SAR ship detection. The results with high uncertainty and very low classification scores are discarded, with only the trustworthy predictions preserved. The results are shown in the second row of Fig. \\ref{fig_uncertaintyresult}, where the false alarms are evidently reduced. In the fourth and fifth SAR image, the localization uncertainty of two large ships visualized with circles around the corner of the predicted bounding box is relatively high. Intuitively, we can infer the reason for the weak capability of the trained model in detecting such kind of targets is probably the lack of the large size ships in the training set. The feedback from the uncertainty estimation should further inspire the follow-up studies to improve the algorithm and build trustworthier models.\n\n\n\\section{Supplementary Explanations}\n\\label{sec:explanation}\n\nBeyond the hybrid and trustworthy modeling, extra explanations and other interpretable models are as well required to assist with developing more transparent AI model for SAR. The explainable artificial intelligence (XAI) techniques, such as gradient based, attention based, and occlusion based explanation methods, are helpful to demonstrate the effectiveness of integrating physical layers to achieve explainability.\n\n\nThe transparent machine learning models, such as linear regression, decision trees, and Bayesian models, are interpretable \\cite{BARREDOARRIETA202082}. The algorithm itself provides explanations, for example, Latent Dirichlet Allocation (LDA) builds a three-level hierarchical Bayesian model to describe the underlying relationship among document-topic-word. That is, the document can be explained with a set of topic, where each topic in turn, is represented by a distribution over words. Karmakar et al. \\cite{karmakar2020feature} used the LDA model for SAR image data mining to generate the topic compositions and group them into semantic classes, which were fused with domain knowledge obtained by active learning from experts. The transparent model can be also integrated in a deep learning framework to approach the explainability. Huang et al. \\cite{Huang2021IGARSS,huang2022physically} applied the LDA model to generate the physical attributes representation as the guided physics signals, rather than directly using the physical scattering characteristic labels to train the physics guided network. That is because the learned physics-aware features are expected to the benefit of semantic label prediction, but the semantic gap actually exists between the physical scattering characteristics and the semantic annotation. Consequently, the LDA model enables the guided signals to gain the abstract semantics and be explained with physical scattering properties.\n\nThe other purpose for approaching the explainability lies in the applications of transfer learning. The manual annotation in SAR domain is difficult and the deficiency of labeled data basically restricts the development of data-driven methods. Facing a wide variety of launched SAR platforms with various frequency bands and resolutions, as well as other multi-spectral, hyper-spectral, optical remote sensing sensors, it is of vital importance for elucidating the transferability of ML models among inhomogeneous data. Arrieta et al. \\cite{BARREDOARRIETA202082} indicated the transferability is one of the goals toward reaching the explainability. Although many researches have explored different deep transfer learning methods in SAR domain \\cite{Huang2017,malmgren2017improving,rostami2019deep}, the inner transfer mechanisms of deep learning model still need explanation of insight. An insufficient understanding of the model may mislead the user toward inappropriate design of algorithm and fatal consequences, i.e. the negative transfer. Based on SAR target recognition, we proposed to analyze the transferability of features in DNN, which contributed to explaining what, where, and how to transfer more effectively for SAR images \\cite{huangwhat}. The inspiration also motivates the follow-up studies, including the SAR-specific pretrained model \\cite{Huang2021GRSL}, the application in detection task \\cite{an2021transitive}, and the interpretability analysis of deep learning model in radar image \\cite{LI2021}.\n\n\n\n\n\\section{Conclusion and Perspectives}\n\\label{sec:conclusion}\n\nIn this paper, we prospect an AI paradigm shift for SAR applications that is explainable, physics aware and trustworthy. To ground this, SAR physical layers embedded with domain knowledge are introduced, which are supposed to be integrated and interacted with neural networks for hybrid modeling. Some illustrative examples are provided to demonstrate the general patterns, showing algorithmic and scientific explainability. In addition, we emphasize the importance and approaches of trustworthy modeling with Bayesian deep learning, as well as illustrating some other techniques such as interpretable machine learning method, explainable techniques, and model transferability, that would assist with developing more transparent AI model for SAR. In fact, this field belonging to interdisciplinary research is still largely undeveloped. To our best knowledge, such approaches have not been formulated in the past years. So far, only some plain attempts have been made. Significant questions and challenges remain, e.g., the feasible representation of SAR physical layer, the optimized form of physical constraint, and hybrid modeling optimization.\n\nCurrently there are several smart sensing techniques in the SAR community that can be exploited as pre-processing steps of data fed into DNNs, e.g., multi-aperture focusing in bistatic configurations \\cite{rosu2020multiaperture}, monostatic\/bistatic tomography, polarimetric decomposition, deformation time series. The outputs of these techniques can expose features that probably cannot be directly extracted by a DNN, especially when using a small training data set.  The newly introduced AI paradigms can apply to the broad class of coherent imaging systems. A few examples can be enumerated: computer tomography, THz imaging, echographs in medicine or industrial applications, sonar or seismic observations in Earth sciences, or radio-telescope data in astrophysics.\n\n\n\n\n\n\n\n\n \\section*{Acknowledgment}\n\nThis work was supported by the National Natural Science Foundation of China under Grant 62101459, China Postdoctoral Science Foundation under Grant BX2021248, the Fundamental Research Funds for the Central Universities under Grant G2021KY05104, and a grant of the Romanian Ministry of Education and Research, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2020-2120, within PNCDI III. We would like to thank the associate editor and the anonymous reviewers for their great contribution to this article.\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{aba:sec1}\nA diabaryon is a bound state (or resonance) with a baryon number $B=2$.\nThe deuteron, made of a proton and a neutron,\n is only a stable dibaryon observed in Nature so far.\n Thus it is interesting to ask whether other dibaryons exist in Nature or not.\n \n A dibaryon  can be classified in the flavor SU(3) representation as\n\\begin{eqnarray}\n{\\bf 8} \\otimes {\\bf 8} &=& {\\bf 27} \\oplus {\\bf 8_s} \\oplus {\\bf 1} \\oplus \\overline{\\bf 10} \\oplus {\\bf 10} \\oplus {\\bf 8_a}\n\\end{eqnarray}\nfor the octet--octet baryons, where a deuteron belongs to the $\\overline{\\bf 10}$ representation, while a $H$--dibaryon was predicted in the  {\\bf 1}\nrepresentation\\cite{Jaffe:1976yi}, and was recently studied in lattice QCD\\cite{Inoue:2010es,Inoue:2011ai,Inoue:2010hs,Beane:2010hg,Francis:2018qch}.\nClassifications including decuplet  (${\\bf 10}$) baryons are\n\\begin{eqnarray}\n{\\bf 10} \\otimes {\\bf 8} &=& {\\bf 25} \\oplus {\\bf 8} \\oplus {\\bf 10} \\oplus {\\bf 27}, \n\\end{eqnarray}\nwhere $N\\Omega$ and $N\\Delta$ dibaryons were predicted in ${\\bf 8}$ and ${\\bf 27}$ representations, respectively\\cite{Goldman:1987ma,Oka:1988yq}, and\n\\begin{eqnarray}\n{\\bf 10} \\otimes {\\bf 10} &=& {\\bf 28} \\oplus {\\bf 27} \\oplus {\\bf 35} \\oplus \\overline{\\bf 10}, \\end{eqnarray}\nwhere $\\Omega\\Omega$ was  predicted in the ${\\bf 28}$ representation\\cite{Zhang:1997ny},\nwhile $\\Delta\\Delta$ dibaryon were predicted in the $\\overline{\\bf 10}$ representation\\cite{Dyson:1964xwa,Kamae:1976at}, \nwhose candidate $d^*(2380)$ has indeed been observed\\cite{Adlarson:2011bh}.\nNote however that only $\\Omega$ is a stable decuplet baryon against strong decays.\n \n\\section{HAL QCD potential method}\nA fundamental quantity in the HAL QCD method\\cite{Ishii:2006ec,Aoki:2009ji,Aoki:2012tk} is an equal-time Nambu--Bethe--Salpeter\n(NBS) wave function in the center of mass system, \nwhich is given for a two nucleon system as\n\\begin{eqnarray}\n\\varphi_{\\bf k} ({\\bf r} ) &=& \\langle 0 \\vert N({\\bf r}\/2,0) N(-{\\bf r}\/2,0)\\vert NN, W_{\\bf k}\\rangle,  \n\\end{eqnarray}\nwhere $\\vert NN, W_{\\bf k}\\rangle$ is a two-nucleon eigenstate in QCD having the relative momentum ${\\bf k}$ and the center of mass energy $W_{\\bf k} = 2\\sqrt{{\\bf k}^2 + m_N^2}$ \nwith the nucleon mass $m_N$, and $N(x)$ with $x=({\\bf x},t)$ is the nucleon operator.\nIn our study, we usually take the local operator in terms of quark fields as $N(x) = \\epsilon^{abc}( u^T_a(x) C\\gamma_5 d_b (x) ) q_c(x)$, where $u_a$ ($d_a$) is a up (down) quark field with color $a$ while $q = u$ ($d$) corresponds to a proton (neutron), and $C=\\gamma_2\\gamma_4$ is a charge conjugation matrix\nacting on spinor indices. A choice of the nucleon operator is a part of the definition (or scheme) for the potential. \nIn the HAL QCD method, we restrict the total energy below the lowest inelastic threshold as\n$W_{\\bf k} <  W_{\\rm th} \\equiv 2 m_N + m_\\pi$ with a pion mass $m_\\pi$, so that\nonly the elastic $NN$ scatterings can occur.\n\nIt can be shown\\cite{Lin:2001ek,Aoki:2005uf} that an asymptotic behavior of the NBS wave function at large $ r\\equiv \\vert {\\bf r}\\vert$ is given by\n\\begin{eqnarray}\n\\varphi_{\\bf k} ({\\bf r}) &\\simeq & \\sum_{l,m} C_l \\frac{\\sin(kr -l\\pi\/2+\\delta_l(k))}{kr} Y_{lm}(\\Omega_{\\bf r}), \\quad k\\equiv\\vert{\\bf k}\\vert, \n\\end{eqnarray}\nwhere $Y_{lm}$ is a spherical harmonic function for the solid angle of ${\\bf r}$ ($\\Omega_{\\bf r}$).\nFor simplicity, we ignore the spin of nucleons here, but you can find a complete formula in Refs.~\\cite{Ishizuka:2009bx,Aoki:2009ji}.  It is important to note that $\\delta_l(k)$ is the phase of the $S$-matrix in QCD for the partial wave with the angular momentum $l$, which is encoded in the asymptotic behavior of the NBS wave function, similar to the scattering phase shift of the scattering wave in quantum mechanics. \n\nUsing this property,  we define the energy--independent {\\it potential}\nwith derivatives from NBS wave functions as  \n\\begin{eqnarray}\n\\left[ E_{\\bf k} - H_0\\right] \\varphi_{\\bf k}({\\bf r}) &=& V({\\bf r}, \\nabla) \\varphi_{\\bf k}({\\bf r}),\n\\quad E_{\\bf k} =\\frac{\\bf k^2}{m_N}, \\ H_0 =\\frac{-\\nabla^2}{m_N},\n\\label{eq:def_pot}\n\\end{eqnarray}\nfor $W_{\\bf k} < W_{\\rm th}$.\nThe potential at the next-to-leading order (NLO) takes a form as\n\\begin{eqnarray}\nV({\\bf r},\\nabla) &=& V_0(r) + V_\\sigma (r) ({\\bf \\sigma}_1\\cdot {\\bf \\sigma}_2)\n+ V_{\\rm T}(r) S_{12} + V_{\\rm LS}(r) {\\bf L}\\cdot {\\bf S} + O\\left(\\nabla^2\\right), ~~~\n\\end{eqnarray}\nwhere ${\\bf \\sigma}_i$ is a spin operator acting on the $i$-th nucleon, \n$S_{12} = 3({\\bf \\sigma}_1\\cdot {\\bf \\hat r}) ({\\bf \\sigma}_2\\cdot {\\bf \\hat r})\n- ({\\bf \\sigma}_1\\cdot {\\bf \\sigma}_2)$ with ${\\bf \\hat r} ={\\bf r}\/r$ is the tensor operator, \n${\\bf L} ={\\bf r}\\times \\nabla$ and ${\\bf S} =({\\bf \\sigma}_1 +{\\bf \\sigma}_2)\/2$.\nThe Schr\\\"odinger equation with the potential $ V({\\bf r}, \\nabla)$ give a correct QCD {\\it phase shift} $\\delta_l(k)$,  since NBS wave functions are solutions to the equation  by construction.\nNote that non-relativistic approximation is not employed here since\nthe Klein--Gordon operator reduces to the Helmholtz operator for a given center of mass energy as $ -\\Box - m^2 =(W_k\/2)+\\nabla^2 -m^2 = {\\bf k}^2 +\\nabla^2$.\n\nWe determine local functions such as $V_X(r)$ ($X=0,\\sigma, {\\rm T}, {\\rm LS}$) order by order.\nFor example, the leading order (LO) potential $V_0^{\\rm LO}(r) \\equiv V_0(r) + V_\\sigma (r) ({\\bf \\sigma_1}\\cdot {\\bf \\sigma_2}) + V_{\\rm T}(r) S_{12}$ can be approximately determined from one NBS wave function $\\varphi_{\\bf k}$ as\n\\begin{eqnarray}\nV_0^{\\rm LO}(r; \\varphi_{\\bf k}) &=& \\frac{[E_{\\bf k}  -H_0] \\varphi_{\\bf k}({\\bf x})}{ \\varphi_{\\bf k}({\\bf x})}, \n\\end{eqnarray}\nwhere an argument $\\varphi_{\\bf k}$ of the potential represents an input for its determination.\nIf $V_0^{\\rm LO}(r; \\varphi_{\\bf k})  \\simeq V_0^{\\rm LO}(r; \\varphi_{\\bf q})$\nfor  $\\vert{\\bf k}\\vert <  \\vert{\\bf q}\\vert$, it turns out that the LO approximation is good at  ${\\bf p}$ \nin $\\vert{\\bf k}\\vert \\le  \\vert{\\bf p}\\vert   \\le\\vert{\\bf q}\\vert$. \nIf $V_0^{\\rm LO}(r; \\varphi_{\\bf k})  \\not= V_0^{\\rm LO}(r; \\varphi_{\\bf q})$, on the other hand, \nthe NLO term can be determined from two equations given by\n\\begin{eqnarray}\n\\left[E_{\\bf k}  -H_0\\right] \\varphi_{\\bf p}({\\bf x}) &=& \\left[ V_0^{\\rm NLO}(r) + V_{\\rm LS}^{\\rm NLO}(r)\n{\\bf L}\\cdot{\\bf S}\\right] \\varphi_{\\bf p}({\\bf x}), \\quad {\\bf p} ={\\bf k}, {\\bf q}, \n\\end{eqnarray}\nwhere a superscript NLO represent the order of the approximation to determine these terms.\nWe can continue this procedure to increase accuracy of the determination.\nOnce the potential is approximately obtained, physical observables such as scattering phase shift can be extracted. \n\nIn lattice QCD, a NBS wave function is extracted from a 4-pt correlation function as\n\\begin{eqnarray}\nF({\\bf r},t) &\\equiv & \\langle 0 \\vert N({\\bf r}\/2,t) N(-{\\bf r}\/2,t) \\bar{\\cal J}_{NN}(0) \\vert 0\\rangle \n=\\sum_n A_n \\varphi_{{\\bf k}_n}({\\bf r}) e^{-W_{{\\bf k}_n} t} +\\cdots \\nonumber \\\\\n&\\simeq & A_0 \\varphi_{{\\bf k}_0}({\\bf r}) e^{-W_{{\\bf k}_0} t}, \\quad t\\rightarrow\\infty,\n\\end{eqnarray}\nwhere $ \\bar{\\cal J}_{NN}(t)$ is an operator which creates two-nucleon elastic states at $t$ with\nan overlap factor $A_n = \\langle NN, W_{{\\bf k}_n}\\vert  \\bar{\\cal J}_{NN}(0) \\vert 0\\rangle$, \nan ellipsis represents contributions form inelastic states, and $W_{{\\bf k}_0}$ is an energy of the $NN$ ground state.\nIn practice it is very difficult to take a large $t$ due to a bad signal-to-noise ration for two baryons,\nbut a use of smaller $t$ may introduce large systematic errors due to contaminations from elastic excited states to the grand state, which is a very serious problem for the conventional method\\cite{Iritani:2016jie,Aoki:2016dmo,Iritani:2017rlk,Aoki:2017byw,Iritani:2018zbt,Iritani:2018vfn}.\n\nIn Ref.~\\cite{HALQCD:2012aa}, an improved method to extract potentials has been proposed.\nWe define the normalized 4-pt function as\n\\begin{eqnarray}\nR({\\bf r},t) &\\equiv&\\frac{F({\\bf r},t)}{G_N(t)^2} = \\sum_n \\bar A_n \\varphi_{{\\bf k}_n}({\\bf r}) e^{-\\Delta W_{{\\bf k}_n} t} +\\cdots \\nonumber , \\quad \\Delta W_{{\\bf k}_n} = W_{{\\bf k}_n} -2 m_N, \n\\end{eqnarray}\nwhere $G_N(t)$ is a nucleon 2-pt function at rest, which behaves as $Z e^{- m_N t}$ as long as \ninelastic contributions to the 2-pt function can be neglected, and $\\bar A_n = A_n\/Z^2$.\nSince all NBS wave functions, $\\phi_{{\\bf k}_n}$, below inelastic threshold satisfy the same Shr\\\"odinger equation (\\ref{eq:def_pot}),  we obtain\n\\begin{eqnarray}\n\\left\\{ -H_0 -\\frac{\\partial}{\\partial t} +\\frac{1}{4m_N^2} \\frac{\\partial^2}{\\partial t^2}\\right\\}\nR({\\bf r},t) &=& V({\\rm r}, \\nabla) R({\\bf r},t) \\simeq V_0^{\\rm LO}({\\rm r}) R({\\bf r},t),~~~\n\\end{eqnarray}\nwhere we use a relation $\\Delta W_{\\bf k} = {\\bf k}^2\/m_N -(\\Delta W_{\\bf k})^2\/(4m_N^2)$, and\nwe need to\ntake a moderately large $t$ satisfying  $W_{\\rm th} t \\gg 1$ to ignore inelastic contributions. \nNote that $V({\\bf r}, \\nabla)$ extracted from the above equation should be $t$ independent.\nTherefore, \nthe $t$ dependence for the LO potential  $V_0^{\\rm LO}({\\rm r})$, for example,\nindicates either an existence of inelastic contributions or contributions from higher order terms in the derivative expansion. \n \n\\section{Dibaryons at the almost physical pion mass}\n\\begin{figure}[bt]\n\\centering\n \\includegraphics[width=0.48\\textwidth]{Figs\/POT.pdf}\n   \\includegraphics[width=0.48\\textwidth]{Figs\/BEdash.pdf}\n \\caption{\n (Left) The $\\Omega^-\\Omega^-$ potential $V(r)$ in the $^1S_0$ channel\n at $t\/a =16,17,18$,  in $2+1$ flavor QCD at almost physical pion mass. \n (Right) The binding energy of the $\\Omega^-\\Omega^-$ system and the root-mean-square distance between two $\\Omega^-$'s\n without and without the Coulomb repulsion (blue solid diamond and red solid triangle, respectively).\n Taken from \\citep{Gongyo:2017fjb}.  \n }\n \\label{fig:OmegaOmega}\n \\end{figure} \nAs an application of the HAL QCD potential method, we presents some results on dibaryons.\n \nIn our studies on dibayons, we employ (2+1)-flavor gauge configurations generated on\na $L^3\\times T = 96^3\\times 96$ lattice with the RG-improved Iwasaki gauge action and\nnon-perturbatively $\\mathcal{O}(a)$-improved Wilson quark action,\nat $a\\simeq 0.085$ fm (thus $La\\simeq 8.1$ fm) with $(m_\\pi, m_K, m_N) \\simeq (146, 525, 955)$ MeV. which correspond to the almost physical point.\n\nWe first consider the $\\Omega^-\\Omega^-$ system in the $^1S_0$ channel,\nwhich belongs to  the {\\bf 28} representation~\\citep{Gongyo:2017fjb}.\n\n\nIn Fig.~\\ref{fig:OmegaOmega} (Left), we show $\\Omega^-\\Omega^-$ potentials at $t\/a=16,17,18$,\nwhich has qualitative features similar to the central potentials for $NN$.\nWe notice, however, that its repulsion is weaker and attraction is shorter-ranged\nthan the $NN$ case.\nWith this potential, we obtain one shallow bound state,\nwhose binding energy is shown in Fig.~\\ref{fig:OmegaOmega} (Right) as a function of the root-mean-square distance,  with and  without Coulomb repulsion between $\\Omega^-\\Omega^-$\nas $\\alpha\/r$, denoted by red circle and blue square, respectively. \nSuch a bound state may be \nsearched experimentally by two-particle correlations in future relativistic heavy-ion collisions~\\citep{Morita:2019rph}.\n\nWe next consider the $N\\Omega^-$ system with $S=-3$ in the $^5S_2$ channel,\nwhich belongs to the {\\bf 8} representation~\\citep{Iritani:2018sra}.\nAt the almost physical pion mass, $N\\Omega$($^5$S$_2$) may couple to\n{$D$}-wave octet-octet channels below the $N\\Omega$ threshold such as $\\Lambda\\Xi$ and $\\Sigma\\Xi$. We thus assume that such couplings are small.\n\n\\begin{figure}[bt]\n\\centering\n \\includegraphics[width=0.48\\textwidth]{Figs\/pot_nomega_t11_14.png}\n   \\includegraphics[width=0.48\\textwidth]{Figs\/RMS_BE_w_Coulomb.pdf}\n \\caption{\n (Left) The  $N\\Omega$ potential $V_C(r)$ in the $^5S_2$ channel\n   at $t\/a=11,12,13,14$, \n   with the same lattice setup for\n   $\\Omega\\Omega$.\n (Right) The binding energy and the root-mean-square distance \n for the $n\\Omega^-$ (red open circle) and \n $p\\Omega^-$ (blue open square).\n Taken from \\citep{Iritani:2018sra}. \n }\n \\label{fig:NOmega}\n\\end{figure} \nIn Fig.~\\ref{fig:NOmega} (Left), we plot the $N\\Omega^-$ potential at $t\/a=11$--$14$, \nshowing attraction at all distances without repulsive core. \nThus there is a chance to form a bound state, and indeed  one bound state is found to exist in this channel.\nFig.~\\ref{fig:NOmega} (Right) shows  the binding energy as a function of the the root-mean-square distance for $n\\Omega^-$ with no Coulomb interaction (red) and $p\\Omega^-$ with  Coulomb attraction (blue). \nThese binding energies are found to be much smaller than $B=18.9(5.0)(^{+12.1}_{-1.8})$ MeV at heavier pion mass $m_\\pi = 875$ MeV~\\citep{Etminan:2014tya}.\n  Such a $N\\Omega$ state \n  can be searched through two-particle correlations in relativistic nucleus-nucleus collisions~\\citep{Morita:2019rph}, if indeed exists.\n  Actually, some indications in experiments were  recently reported~\\citep{STAR:2018uho},\n  and more will be expected to come.\n\n\\begin{figure}[tbh]\n\\centering\n \\includegraphics[width=0.6\\textwidth]{Figs\/comp_ere_param_dibaryons.pdf}\n \\caption{The ratio of the effective range and the scattering length $r_{\\rm eff}\/a_0$ as a function of $r_{\\rm eff}$\n     for $\\Omega\\Omega (^1S_0)$ (blue open diamond) and $N\\Omega(^5S_2)$ (red open circle)\n   obtained in lattice QCD, as well as for $NN(^3S_1)$ (purple open up-triangle) and \n   $NN(^1S_0)$ (green open down-triangle) in experiments.\n Taken from  \\citep{Iritani:2018sra}.\n  }\n \\label{fig:Unitary}\n\\end{figure} \nFrom potentials we obtain for $\\Omega\\Omega$($^1$S$_0$) and $N\\Omega$($^{{5}}$S$_2$),\nwe calculate the scattering length $a_0$ and the effective range $r_{\\rm eff}$ for these systems.\nFig.~\\ref{fig:Unitary} shows the ratio $r_{\\rm eff}\/a_0$ as a function of $r_{\\rm eff}$  for \n$\\Omega\\Omega$($^1$S$_0$) (blue diamond) and $N\\Omega$($^{{5}}$S$_2$) (red circle) obtained in lattice QCD near {the} physical pion mass, together with the experimental values for $NN$($^3$S$_1$) (deuteron, purple up-triangle) and\n$NN$($^1$S$_0$) (di-neutron, green down-triangle). \nFor all cases, $\\vert r_{\\rm eff}\/a_0\\vert$ is small, indicating\nthat these systems are located close to the unitary limit.\nIt will be interesting to understand \nwhy dibaryons or dibaryon candidates appear in the unitary region \nnear the physical pion mass.\n\n\\section*{Acknowledgments} \nThis work is supported in part by the Grant-in-Aid of the Japanese Ministry of Education, Sciences and Technology, Sports and Culture (MEXT) for Scientific Research (Nos. JP16H03978, JP18H05236),  \nby a priority issue (Elucidation of the fundamental laws and evolution of the universe) to be tackled by using Post ``K\" Computer, and by Joint Institute for Computational Fundamental Science (JICFuS). \n\n\\bibliographystyle{ws-procs9x6}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nSince the 1950s, high-level programming languages have resulted in\norders-of-magnitude productivity improvements compared to\nmachine-level coding. This feat has been a great enabler of the\ncomputing revolution, during a time when computer memories and\nconceptual program complexity have steadily grown at exponential\nrates. The history of computing is testament to language designers'\nand implementers' accomplishments: of the 53 Turing awards to present\n(from 1966 to 2018) a full 16 have been awarded for contributions to\nprogramming languages or compilers.\\footnote{The count is based on\n  occurrences of ``programming language(s)'' or ``compiler(s)'' in the\n  brief citation text of the award, also including Richard Hamming who\n  is cited for ``automatic coding systems'' (i.e., the L2 precursor of\n  Fortran).  Notably, the number does not include John McCarthy or\n  Dana Scott, who are well-known for languages contributions yet the\n  terms do not appear in the citation.}\n\n\nAt this time, however, the next steps in programming language\nevolution are hard to discern. Large productivity improvements will\nrequire a Kuhnian \\emph{paradigm shift} in languages. A change of paradigm, in the\nKuhn sense, is a drastic reset of our understanding and nomenclature.\nIt is no surprise that we are largely ineffective at predicting its\nonset, its nature, or its principles.\n\nDespite this conceptual difficulty, the present paper is an attempt to\npeer behind the veil of next-paradigm programming languages. I happen\nto believe (on even days of the month) that a change of paradigm is\nimminent and all its technical components are already here. But you do\nnot have to agree with me on either point---after all, the month also\nhas its odd days.\n\nReasonable people may also differ on the possible catalysts of such a\nparadigm shift. Will it be machine learning and statistical\ntechniques~\\cite{43146}, trained over vast data sets of code instances?  Will it be\nprogram synthesis techniques~\\cite{DBLP:journals\/ftpl\/GulwaniPS17},\nemploying symbolic reasoning and complex\nconstraint solving? Will it be mere higher-level language design\ncombined with technology trends, such as vast computing power and\nenormous memories?\n\nRegardless of one's views, I hope to convince the reader that\nthere is reasonable clarity on \\emph{some} features that\nnext-paradigm programming languages will have, \\emph{if} they\never dominate. Similarly, there is reasonable clarity on\nwhat changes next-paradigm programming languages will\ninduce in the tasks of everyday software development.\n\nFor a sampling of the principles I will postulate and their\ncorollaries, consider the following conjectures:\n\n\\begin{itemize}[leftmargin=12pt,itemsep=2pt]\n  \n\\item Next-paradigm programming languages will not display on the\n  surface the computational complexity of their calculations.  Large\n  changes in asymptotic complexity (e.g., from \n  $O(n^4)$ to $O(n^2)$) will be effected by the language\n  implementation. The language will not have loops or explicit\n  iteration. The programmer will often opt for worse asymptotic\n  complexity factors, favoring solution simplicity and countering\n  performance problems by limiting the inputs of algorithms (e.g.,\n  applying an expensive computation only locally)\n  or accepting approximate results.\n\n\\item Next-paradigm programming languages will need a \\emph{firm\n  mental grounding}, in order to keep program development\n  manageable. This grounding can include: a well-understood cost model;\n  a simple and clear model on how new code can or cannot affect the\n  results of earlier code; a natural adoption of parallelism without\n  need for concurrency reasoning; and more.\n  \n\\item Development with next-paradigm programming languages will be\n  significantly different from current software development.  Minute\n  code changes will have tremendous impact on the output and its\n  computation cost. Incremental development will be easier.  Testing\n  and debugging will be as conceptually involved as coding. Formal\n  reasoning will be easier, but less necessary.\n\\end{itemize}\n\nIn addition to postulating such principles, the goal of the paper\nis to illustrate them.\nI will use examples from real, deployed code, written (often by\nme) in a declarative language---Datalog. My experience in declarative\nprogramming is a key inspiration for most of the observations of the\npaper. It is also what makes the conjectures of the paper\n``real''. All of the elements I describe, even the most surprising,\nare instances I have encountered in programming practice. I begin\nwith this personal background before venturing to further\nspeculation.\n\n\n\n\\vspace{0.5cm}\n\\epigraph{I've seen things, that's why I'm seeing things.}{\\textit{---me}}\n\n\\section{Where I Come From}\n\nIn the past decade, I have had the opportunity to write declarative,\nlogic-based code of uncommon volume and variety, under stringent\nperformance requirements. This experience underlies my speculation on\nthe properties of next-paradigm programming languages.\n\n\n\\paragraph{Declarative code---lots of it.} Most of my research (and\nthe vast majority of my personal coding effort) in the past decade has\nbeen on declarative program analysis~\\cite{10.1007\/978-3-642-24206-9_14}. My\ngroup and a growing number of external collaborators have implemented\nlarge, full-featured static analysis frameworks for Java\nbytecode~\\cite{Bravenboer09}, LLVM bitcode~\\cite{10.1007\/978-3-662-53413-7_5},\nPython~\\cite{tensor19}, and Ethereum VM\nbytecode~\\cite{gigahorse,Grech2018oopsla}. The frameworks have been the\necosystem for a large number of new static analysis algorithms,\nleading to much new research in the area.\n\nThese analysis frameworks are written in the Datalog language.\nDatalog is a bottom-up variant of Prolog, with similar syntax.\n``Bottom-up'' means that no search is performed to find solutions to\nlogical implications---instead, all valid solutions are computed, in parallel.\nThis makes the language much more declarative than Prolog: reordering\nrules, or clauses in the body of a rule, does not affect the output.\nAccordingly, computing all possible answers simultaneously means that\nthe language has to be limited to avoid possibly infinite\ncomputations. Construction of new objects (as opposed to new\ncombinations of values) is, therefore, outside the core language and,\nin practice, needs to be strictly controlled by the programmer. These\nfeatures will come into play in later observations and conjectures.\n\nThe Datalog language had been employed in static program analysis long\nbefore our\nwork~\\cite{repsdb,DBLP:conf\/aplas\/WhaleyACL05,pods\/LamWLMACU05,codequest}.\nHowever, our frameworks are\ndistinguished by being almost entirely written in Datalog: not just\nquick prototypes or ``convenient'' core computations are expressed as\ndeclarative rules, but the complete, final, and well-optimized version\nof the deployed code, as well as much of the scaffolding of the\nanalysis.  As a result, our analysis frameworks are possibly the\nlargest Datalog programs ever written, and among the largest pieces of\ndeclarative code overall. For instance, the Doop codebase \\cite{Bravenboer09}\ncomprises several thousands of Datalog rules, or tens of thousands of\nlines of code (or rather, of logical specifications). This may seem like\na small amount of code, but, for logical rules in complex mutual\nrecursion, it represents a daunting amount of complexity. This\ncomplexity captures core static analysis algorithms, language\nsemantics modeling (including virtually the entire complexity of\nJava), logic for common frameworks and dynamic behaviors, and more.\n\n\n\\paragraph{Emphasis on performance, including parallelism.}\nThe Datalog code I have contributed to in the past decade aims for\nperformance at least equal to a manually-optimized imperative\nimplementation. That is, every single rule is written with a clear\ncost model in mind. The author of a declarative rule knows, at least\nin high-level terms, how the rule will be evaluated: in how many\nnested loops and in what order, with what indexing structures, with\nwhich incrementality policy for faster convergence when recursion is\nemployed. Optimization directives are applied to achieve maximum\nperformance. Shared-memory parallelism is implicit, but the programmer\nis well aware of which parts of the evaluation parallelize well and\nwhich are inherently sequential. In short, although the code is very\nhigh-level, its structure is anything but random, and its performance\nis not left to chance. Maximum effort is expended to encode\nhighest-performing solutions purely declaratively.\n\n\n\\paragraph{Many domains.} My experience in declarative programming\nextends to several domains, although static analysis of programs has\nbeen the primary one. Notably, I have served as consultant, advisor,\nand academic liaison for LogicBlox Inc., which developed the Datalog\nengine~\\cite{Aref:2015:DIL:2723372.2742796} used in Doop until\n2017. The company built a Datalog platform comprising a language\nprocessor, JIT-like back-end optimizer, and specialized database\nserving as an execution environment. All applications on the platform\nwere developed declaratively---even UI frameworks were built out of a\nfew externally-implemented primitives and otherwise entirely\nlogic-based rules. The company enjoyed a lucrative acquisition,\nmostly based on the value of its deployed applications and\ncustomers. All applications were in the domain of retail \nprediction---about as distant from static analysis as one can imagine.\nFor a different example, an algorithm for dynamic race detection\ndeveloped in the course of my\nresearch~\\cite{cprace} was implemented\nin Datalog and all experiments were over the Datalog\nimplementation. An imperative implementation would be substantially\nmore involved and was never successfully completed in the course of\nthe research.\n\n\n\\paragraph{``Declarative languages aren't good for this.''} A repeat\npattern in my involvement with declarative languages has been to\nencode aspects of functionality that were previously thought to\nrequire a conventional (imperative or functional) implementation.  The\npossibility of the existence of the Doop framework itself was under\nquestion a little over a decade ago---e.g.,\nLhot\\'{a}k~\\cite{Lhotak:2006:PAU} writes: \\emph{``[E]ncoding all the\n  details of a complicated program analysis problem [...] purely in\n  terms of subset constraints [i.e., Datalog] may be difficult or\n  impossible.''}  In fact, writing all analysis logic in Datalog has\nbeen a guiding principle of Doop---extending even to parts of the\nfunctionality that might be \\emph{harder} to write declaratively. Very\nfew things have turned out to be truly harder---it is quite surprising\nhow unfamiliarity with idioms and techniques can lead to a fundamental\nrejection of an approach as ``not suitable''. Most recently, we got\ngreat generality and scalability benefits from encoding a decompiler\n(from very-low-level code) declaratively~\\cite{gigahorse}, replacing a previous imperative\nimplementation~\\cite{vandal}---a task that the (highly expert) authors\nof the earlier decompiler considered near-impossible.\n\n\n\\paragraph{Design declarative languages.} Finally, I have had\nthe opportunity to see declarative languages not just from the\nperspective of a power user and design advisor, but also from that of\na core designer and implementer~\\cite{pql, deal}. This dual view has\nbeen essential in forming my understanding of the principles and\neffects of next-paradigm languages.\n\n\n\n\n\\vspace{0.5cm}\n\n\\newpage\n\n\\epigraph{[A]ll programming languages seem very similar to each other. They all have variables, and arrays, a few loop constructs, functions, and some arithmetic constructs.}{\\textit{Patrick S. Li~\\cite{li16}}}\n\n    \n\\section{Principles of Next-Paradigm Languages}\n\nBefore going on, I will emphasize again that the reader does not need\nto agree with me on the usefulness of declarative\nlanguages. Next-paradigm programming languages could be based on any\nof several potential technologies---e.g., perhaps on machine learning\nand statistical techniques, or on SMT solvers and symbolic reasoning.\nRegardless of the technology, however, I think that some elements are\nnear-inevitable and these are the ones I am trying to postulate as\n``principles''. I will illustrate these principles with examples from\ndeclarative programming, because that's the glimpse of the future I've\nhappened to catch. But other glimpses may be equally (or more) valid.\n\n\n\\subsection{Programming Model: Cost}\n\n\\begin{principle}[Productivity and Performance Tied Together]\nIf a language can give orders-of-magnitude improvements in productivity \\\\\n\\textbf{then}\nits implementation has the potential for orders-of-magnitude changes in performance.\n\\label{principle:performance}\n\\end{principle}\n\nLarge variations in both productivity and performance are functions of\na language being \\emph{abstract}. Neither is possible with the\ncurrent, ultra-\\emph{concrete} mainstream languages. If one needs to\nexplicitly specify ``loops and arrays'', neither large productivity\ngains, nor large performance variations are possible. Instead, the\nlanguage implementation (or ``compiler'' for short\\footnote{A language\n  implementation consists of an interpreter or compiler (ahead-of-time\n  or just-in-time) and a runtime system. The term ``compiler'',\n  although not always accurate, seems to encompass most of the\n  concepts we are used to in terms of advanced language\n  implementations.}) of a high-productivity, next-paradigm language\nwill likely be able to effect orders-of-magnitude performance\ndifferences via dynamic or static optimization. For performance\nvariability of such magnitude, the asymptotic complexity of the\ncomputation will also likely change.\n\n\\begin{corollary}\n  Programs $\\neq$ \n  Algorithms $+$\n  Data Structures. \\\\\n  \\emph{Instead:}\\\\\n  Compiler(Program) $=$ Algorithms $+$ Data Structures.\n\\end{corollary}\n\nPrograms in next-paradigm languages will likely \\emph{not} be the sum\nof algorithms and data structures, contradicting Wirth's famous\nequality. Instead, programs will be specifications---carefully written\nto take into account an execution model that includes a search process\n(done by the compiler) over the space of implementations. Major\nalgorithmic elements and key data structure decisions will be\ndetermined automatically by this search. The compiler will be a mere\nfunction from programs to concrete implementations, consisting of\nalgorithms and data structures.\n\n\\paragraph{Example: Choice of Algorithm.} Language optimization that can affect the\nasymptotic complexity of the computation is hardly new. Relational\nquery optimization is a prime realistic example.\\footnote{Relational\n  database languages, such as SQL, are a limited form of declarative\n  programming. Due to the simplified setting and commercial success,\n  many ideas we discuss have originated in that domain.}  In our\nsetting, we can revisit it, with Datalog syntax, before building\nfurther on it.  A common static analysis rule, responsible for\ninterpreting calls as assignments from actual to formal arguments, is\nshown below:\n\n\\noindent\\begin{minipage}{\\columnwidth}\n\\begin{datalogcode}\nAssign(formal, actual) :-\n  CallGraphEdge(invocation, method),\n  FormalParam(index, method, formal),\n  ActualParam(index, invocation, actual).\n\\end{datalogcode}\n\\end{minipage}\n\nThe logic just says that if we have computed a call-graph edge from\ninstruction \\sv{invocation} to a \\sv{method}, then the $i$-th\n(\\sv{index}) actual argument of the call is assigned to the $i$-th\nformal parameter of the method.\n\nIn terms of declarative computation, this rule is evaluated via a\nrelational join of the current contents of (conceptual) tables\n\\sv{CallGraphEdge}, \\sv{FormalParam}, and \\sv{ActualParam}. But it is\nup to the compiler to decide whether to start the join from table\n\\sv{CallGraphEdge} or one of the others. This decision may be informed\nby dynamic statistics---i.e., by current knowledge of the size of each\nof the three tables and of the past selectivity of joining each two\ntables together. It could well be that our input consists of\noverwhelmingly zero-argument functions. Thus, the join of\n\\sv{CallGraphEdge} and \\sv{FormalParam} will be small. It is wasteful\n(up to asymptotically so) to start by iterating over all the contents\nof \\sv{CallGraphEdge}, only to discover that most of them never\nsuccessfully match a method with an entry in table\n\\sv{FormalParam}. Instead, the join may be much quicker if one starts\nfrom functions that do take arguments, i.e., from table\n\\sv{FormalParam}. The LogicBlox Datalog engine~\\cite{Aref:2015:DIL:2723372.2742796}\nperforms precisely this kind of dynamic, online query optimization,\nbased on relation sizes and expected selectivities.\n\n\\paragraph{Example: Choice of Data Structures.}\nData structure choice is already standard practice in relational\nlanguages. For instance, the Souffl{\\'e}\\xspace{}~\\cite{Jordan16} implementation\nof Datalog automatically infers when to add indexes to existing\ntables, so that all rule executions are\nfast~\\cite{DBLP:journals\/pvldb\/SuboticJCFS18}. In our earlier example, Souffl{\\'e}\\xspace{} will\nadd an index (i.e., a B-tree or trie) over table \\sv{FormalParam},\nwith the second column, \\sv{method}, as key, and similarly for\n\\sv{ActualParam}, with either column as key. Then, if computation\nstarts from an exhaustive traversal of \\sv{CallGraphEdge}, only the\nmatching subsets of the other two tables will be accessed, using the\nindex to identify them. We illustrate below, by denoting the partial,\nindexed traversal by a $\\Pi$ prefix on the accessed-by-index tables,\nand by underlining the variables bound by earlier clauses during the\nevaluation:\n\n\\noindent\\begin{minipage}{\\columnwidth}\n\\begin{datalogcode}\nAssign(formal, actual) :-\n  CallGraphEdge(invocation, method),\n  (*$\\Pi$*)FormalParam(index, (*\\underline{\\texttt{method}}*), formal),\n  (*$\\Pi$*)ActualParam((*\\underline{\\texttt{index}}*), (*\\underline{\\texttt{invocation}}*), actual).\n\\end{datalogcode}\n\\end{minipage}\n\nNote that such choice of data structure is not based on local\nconstraints, but on all uses of the table, in any rule in a\n(potentially large) program. However, per our discussion of trends, it\nis typically fine for the compiler to maintain an extra data\nstructure, if this will turn an exhaustive traversal into an indexed\ntraversal, even if the benefit arises in very few rules.\n\nGenerally, I believe it is almost a foregone conclusion that\nnext-paradigm programming languages will perform automatic data\nstructure selection. The language will likely only require the\nprogrammer to declare data and will then automatically infer efficient\nways to access such data, based on the structure of the\ncomputation. Both technology trends and data structure evolution\nconspire to make this scenario a near certainty:\n\n\\begin{itemize}[leftmargin=12pt,itemsep=2pt]\n\\item Although many potential data structures exist, a\n  logarithmic-complexity, good-locality, ordered structure (such as a\n  B-tree or trie) offers an excellent approximation of most realistic\n  data traversals. Both random access and ordered access are\n  asymptotically fast, and constant factors are\n  excellent. (Accordingly, most scripting languages with wide adoption\n  in recent decades have made a standard ``map'' their primary\n  data type.)\n  If one adds a union-find tree, abstracted behind an ``equivalence\n  class'' data type, there may be nearly nothing more that a\n  high-productivity language will need for the vast majority of\n  practical tasks.\n\n  Of course, the are glaring exceptions to such broad\n  generalizations---e.g., there is no provision for probabilistic data\n  structures, such as bloom filters, cryptographically secure\n  structures, such as Merkle trees, or other classes of\n  structures essential for specific domains. However, the use of such\n  structures is substantially less frequent. Additionally, a theme for\n  next-paradigm languages will be escaping the language easily---as I\n  argue later (Section~\\ref{sec:others}).\n  \n\\item Adding an extra data structure vs. not adding a data structure\n  is no longer a meaningful dilemma, under current memory and speed\n  trends. The cost of additional ways to organize data only grows\n  linearly, while the speed benefit can be asymptotic. Therefore, when\n  in doubt, adding an extra B-tree or trie over a set of data is an\n  easy decision.\n\\end{itemize}\n\n\n\n\n\n\\paragraph{Example: Auto-Incrementalization.} Another realistic example\nof asymptotic complexity improvements offered routinely in declarative\nlanguages is automatic incrementalization. Our earlier example rule\nis, in practice, never evaluated as a full join of tables\n\\sv{CallGraphEdge}, \\sv{FormalParam}, and \\sv{ActualParam}. The reason\nis that other rules in a typical program analysis will use the\nresulting relation, \\sv{Assign}, in order to infer new call-graph\nedges (e.g., in the case of virtual calls). This makes the computation\nof \\sv{Assign} mutually recursive with that of \\sv{CallGraphEdge}.\nTherefore, the rule will be evaluated incrementally, for each stage\nof recursive results. The rule, from the viewpoint of the Datalog compiler\nlooks like this:\n\n\\begin{datalogcode}\n(*$\\Delta$*)Assign(formal, actual) :-\n  (*$\\Delta$*)CallGraphEdge(invocation, method),\n  FormalParam(index, method, formal),\n  ActualParam(index, invocation, actual).\n\\end{datalogcode}\n\nThis means that the new-stage (denoted by the $\\Delta$ prefix)\nresults of \\sv{Assign} are computed by joining only the newly-derived\nresults for \\sv{CallGraphEdge}. Tuples in \\sv{CallGraphEdge} that\nexisted in the previous recursive stage do not need to be considered,\nas they will never produce results not already seen. (The other two\ntables involved have their contents fixed before this recursive\nfixpoint.)  In practice, such automatic incrementalization has been a\nmajor factor in making declarative implementations highly\nefficient---often much faster than hand-written solutions, since\nincrementalization in the case of complex recursion is highly\nnon-trivial to perform by hand.\n\nIncrementalization also exhibits complex interplay with other\nalgorithmic optimizations. For instance, the latest delta of a\ntable is likely smaller than other relations, in which case the exhaustive\ntraversal of a join should start from it.\n\n\n\\begin{corollary}[Cheapest is hardest.]\n\\label{cor:cheap}\n  ``Easy'' in terms of (sequential) computational complexity may mean\n  ``hard'' to express efficiently in next-paradigm languages.\n\\end{corollary}\n\nThe shortcomings of next-generation languages may be more evident in\nthe space where human ingenuity has produced incredibly efficient\nsolutions, especially in the low-end of the computational complexity\nspectrum (i.e., linear or near-linear algorithms). In the\nultra-efficient algorithm space, there is much less room for automated\noptimization than in more costly regions of the complexity\nhierarchy.\\footnote{This general conjecture may be easily violated in\n  specialized domains where symbolic search already \\emph{beats} human\n  ingenuity. E.g., program synthesis has already exhibited remarkable\n  success in producing optimal algorithms based on bitwise\n  operators~\\cite{Gulwani:2011:SLP:1993498.1993506}.}\n\n\\paragraph{Example: Depth-First Algorithms and Union-Find Structures.}\nCurrent declarative languages are markedly bad at expressing (without\nasymptotic performance loss) efficient algorithms based on depth-first\ntraversal.  For instance, declarative computation of\nstrongly-connected components in directed graphs is asymptotically\nless efficient than Tarjan's algorithm.  Also, union-find trees cannot be\nreplicated and need special-purpose coding.\n\nGenerally, algorithms that are hard to parallelize (e.g., depth-first\nnumbering is $P$-hard) and data structures that heavily employ\nimperative features (both updates and aliasing) are overwhelmingly the\nones that are a bad fit for declarative programming. It is reasonable\nto speculate that this observation will generalize to any\nnext-paradigm programming language. After all, a high-productivity\nlanguage will need to be abstract, whereas imperative structures and\nnon-parallelizable algorithms rely on concrete step ordering and\nconcrete memory relationships (i.e., aliasing). If this speculation\nholds, it is a further argument for the inevitability of next-paradigm\nprogramming languages. In most foreseeable technological futures,\nparallelism and non-random-access memory are much more dominant than\nsequential computation and a shared, random-access memory space.  The\nalgorithms that will dominate the future are likely amenable to\ngeneral automatic optimization in a high-productivity language.\n\n\n\\begin{corollary}[Even Asymptotics May Not Matter]\n  Asymptotically sub-optimal computations may become dominant,\n  for limited, well-supervised deployment.\n\\end{corollary}\n\nAsymptotic performance degradation factors are impossible to ignore,\nsince they typically turn a fast computation into an ultra-slow or\ninfeasible one. However, in next-paradigm languages, a programmer may\nroutinely ignore even asymptotic factors and favor ultra-convenient\nprogramming. To avoid performance degradation in a realistic setting,\nthe applicability of inefficient computations may be limited to a\nlocal setting, or approximate results may be acceptable~\\cite{DBLP:conf\/pldi\/CarbinKMR12}.\n\n\\paragraph{Example: Inefficient Graph Computations.} In Datalog code\nI have often favored quadratic, cubic, or worse solutions, as long as\nthey are applied only locally or other constraints ensure efficient\nexecution. Graph concepts offer generic examples. (In practice the\ncomputation is rarely about a literal graph, but binary relations are\noften convenient viewed in graph terms.) For instance, I have often\nused code that computes all pairs of predecessors of a graph node,\ngenerically written as:\n\n\\begin{datalogcode}\nBothPredecessors(pred1, pred2, next) :-\n  Edge(pred1, next),\n  Edge(pred2, next),\n  pred1 != pred2.\n\\end{datalogcode}\n\nAs long as the in-degree of the graph is bounded, the ``wasteful''\nall-pairs concept costs little to compute and can be quite handy\nto have cached.\n\nSimilarly, a wasteful but convenient concept is that of\ndirected graph reachability without going through a given node:\n\n\\begin{datalogcode}\nReachableExcluding(node, node, notInPath) :-\n  IsNode(node),\n  IsNode(notInPath),\n  node != notInPath.\n  \nReachableExcluding(source, target, notInPath) :-\n  Edge(source, target),\n  IsNode(notInPath),\n  source != notInPath,\n  target != notInPath.\n\nReachableExcluding(source, target, notInPath) :-\n  ReachableExcluding(source, interm, notInPath),\n  Edge(interm, target),\n  target != notInPath.\n\\end{datalogcode}\n\nNote that the computation is worst-case bounded only by a $n^4$\npolynomial, for $n$ graph nodes---e.g., the last rule enumerates\nnear-all possible node 4-tuples, \\sv{source}, \\sv{target},\n\\sv{interm}, and \\sv{notInPath}.\n\nWritten as above, the computation would be infeasible for any but the\nsmallest graphs. However, if we limit our attention to a local\nneighborhood (for whatever convenient definition, since this pattern\napplies in several settings) the computation is perfectly feasible,\nand, in fact, common in production code:\n\n\\begin{datalogcode}\nReachableExcluding(node, node, notInPath) :-\n  InSameNeighborhood(node, notInPath),\n  node != notInPath.\n\nReachableExcluding(source, target, notInPath) :-\n  Edge(source, target),\n  InSameNeighborhood(source, target),\n  InSameNeighborhood(source, notInPath),\n  source != notInPath,\n  target != notInPath.\n\nReachableExcluding(source, target, notInPath) :-\n  ReachableExcluding(source, interm, notInPath),\n  Edge(interm, target),\n  InSameNeighborhood(source, target),\n  target != notInPath.\n\\end{datalogcode}\n\nGenerally, I believe that programmers will be quite inventive in\nreshaping a problem in order to employ ultra-high-level but\ninefficient computations. Coding simplicity and correctness clarity\nare excellent motivators for questioning whether a full, exact answer\nis strictly needed.\n\n\n\\begin{corollary}[Implicit Parallelism]\n  In any high-productivity setting, parallelism will be pervasive but implicit.\n\\end{corollary}\n\nA next-paradigm programming language, offering orders-of-magnitude\nproductivity improvements, will very likely heavily leverage\nparallelism, yet completely hide it! There is no doubt that\nshared-memory concurrency correctness is among the thorniest\nprogramming challenges in existence. High-productivity and explicit\nsynchronization, of any form, are very unlikely to be compatible.\nHigh levels of abstraction also seem to mesh well with automatic data\npartitioning and replication solutions, as does the earlier observation\nabout sacrificing even asymptotic optimality on the altar of programming\nproductivity.\n\n\\paragraph{Example: Implicit Shared-Memory Parallelism.} Declarative languages\nalready auto-parallelize programs. All the Datalog logic we have seen\nis executed by a modern engine (e.g., Souffl{\\'e}\\xspace{}) in parallel, by\nmultiple threads over partitionings of the input data. A join\noperation is naturally massively parallel, so, the larger the input\ndata, the more parallelism one can easily attain. \n\n\n\\subsection{Programming Model: Semantic Invariants}\n\nIn addition to a cost model, a next-paradigm PL programmer's mental\nmodel should include semantic guarantees.\n\n\\begin{principle}[Need For Firm Mental Grounding]\nThe programming model of next-paradigm languages will offer strong\nsemantic guarantees (about what code can do and how new code can affect old).\n\\label{principle:grounding}\n\\end{principle}\n\nA language that will offer orders-of-magnitude improvements in\nproductivity will necessarily enable the programmer to express more\nwith less. A highly concise specification will yield a detailed\noptimized implementation, with the compiler playing a huge role in\nsearching the space of potential algorithms and data\nstructures. Keeping one's sanity will not be trivial. A one-word (or\neven one-character) change has the potential to completely alter\nprogram output or its performance. I am not giving examples from my\ndeclarative programming experience because anecdotes don't do justice\nto the magnitude of the issue. After all, small changes with large\neffects can also arise in conventional programming practice. However,\nwhat is currently an extreme case will be common, everyday experience\nin next-paradigm languages. Changes that the programmer considers\ncompletely innocuous may result in vastly different program\nimplementations.\n\nFaced with such complexity, the language will need to provide firm,\nsanity-keeping semantic guarantees. What can these guarantees be?  It\nis hard to speculate on specifics without knowing the exact technology\nbehind a language. Most likely, semantic invariants will guarantee\nwhat the program can or cannot do, what effect code in one part of the\nprogram can have on others, how the program output can change once new\ncode is added, etc. Current examples give a glimpse of the\npossibilities.\n\n\\paragraph{Example: Monotonicity.}\nThe top sanity-preserving semantic invariant in Datalog is\nmonotonicity. Computation is (dominantly) monotonic---other rules can\nonly \\emph{add} to the output of a rule, never invalidate previous\noutputs. This gives excellent properties of code understanding via\nlocal inspection. It also helps with understanding the updated program\nsemantics upon a code change---code addition is overwhelmingly the\nmost common change in program development. Consider the earlier\nexample of an \\sv{Assign} inference from established instances of\n\\sv{CallGraphEdge}. To understand the rule, we have never needed to\nwonder about either other rules that inform \\sv{Assign} or the rules\nthat establish \\sv{CallGraphEdge}, even if those themselves employ\n\\sv{Assign}. Also, we have never needed to wonder about the evaluation\norder of this rule relative to any others. The rule works like a pure\nlogical inference precisely because of the monotonicity property. The\nsame holds for the three rules we employed for\n\\sv{ReachableExcluding}: we have not needed the definition of any rule\nin order to understand the others.\\footnote{In reality, Datalog\n  programs typically use ``stratified negation'', allowing\n  non-monotonic operators (i.e., negation and arbitrary aggregation)\n  over predicates defined in a previous evaluation stratum.  This\n  means that extra rules \\emph{can} produce fewer results, but only if\n  these extra rules affect a predicate that is conceptually ``more\n  primitive'' than the extra results. The programmer needs to have a\n  stratification of the program logic in mind. This is a fairly\n  manageable mental burden (given compiler support for tracking\n  violations), even for large code bases.}\n\n\n\n\n\\paragraph{Example: Termination.}\nA guarantee of program termination is another semantic invariant of\nthe Datalog language. Core Datalog does not include operators for\ninventing new values, therefore all computation can at most combine\ninput values. This results in a polynomial number of possible\ncombinations, more and more of which are computed monotonically during\nprogram evaluation. Therefore, a program is guaranteed to terminate.\n\nOf course, in practice strict guarantees of termination are impossible\nor impractical. A guarantee of termination makes a language sub-Turing\ncomplete. Practical implementations of Datalog allow creating new\nvalues, thus no ironclad guarantee of termination exists. However, it\nis valuable to distinguish a core guarantee, applicable to the\nmajority of the code, from an exception that can be effected only by\nuse of a well-identified language feature.\\footnote{``Pure''\n  functional languages do something similar. They certainly permit\n  side effects (e.g., I\/O), but encapsulate them behind a\n  well-identified interface---a monad.}  It is much easier to reason\nabout the possible new values invented via constructors than have\npotentially non-terminating computations possibly lurking behind every\ncomputation in the language.\n\n\n\\subsection{``Known'' Principles? \\\\\n  Abstraction, Extensibility, Modularity, Interoperability}\n\\label{sec:others}\n\nIt is interesting to speculate on new design principles of\nnext-paradigm languages, but what about current, well-established\nlanguage principles?  These include, at the very least:\n\n\\begin{itemize}[leftmargin=12pt,itemsep=2pt]\n\\item (module) abstraction\/growing a language: packaging recurring\n  functionality in a reusable, parameterizable\n  module~\\cite{Steele:1998:GL:346852.346922};\n\\item language extensibility: the marriage of powerful module\n  abstraction with syntactic configurability;\n\\item modularity: having mechanisms for keeping parts\nof the code isolated, only visible through specific, identified\ninterfaces;\n\\item multi-paradigm interoperability: easy interfacing between\n  languages so that programming tasks can be expressed in terms\n  well-suited to the computation at hand.\n\\end{itemize}\n\nThere is no doubt that current, established principles of good\nlanguage design will play a big role in next-paradigm language design\nas well. However, these principles on their own are not enough to get\nus to a next-paradigm language. Furthermore, the benefits the\nestablished principles afford may be only second-order effects. They\nmay pale compared to the chief benefit of a next-paradigm language:\nlevels of abstraction high enough to yield orders-of-magnitude\nproductivity improvements. In fact, the first next-paradigm\nprogramming languages may well be lacking in module abstraction,\nextensibility, or modularity. Later incarnations will likely benefit\nmore from these principles, as have existing languages.\n\nThe one principle that seems a priori indispensable for next-paradigm\nlanguages is that of multi-paradigm interoperability. I already\nconjectured that next-paradigm programming languages will not be good\nat everything---e.g., see Corollary~\\ref{cor:cheap}. Escaping the\nlanguage is then inevitable, even if it happens rarely. The language\nescape mechanism should not break the fundamental high-productivity\nabstraction. Instead, it can encapsulate, behind a clear interface,\ncomputations best optimized by manual, lower-level coding. There are\nmechanisms of this nature in common current use---e.g., uninterpreted\nfunctions\/external functors.\n\n\\paragraph{Diminishing Returns.}\nThe need to escape a language raises an interesting question. How can\nhigh productivity be sustained, if parts of the coding need to happen\nin a conventional setting? The law of diminishing returns dictates\nthat dramatically speeding up one part of software development will\nonly move the bottleneck to a different part. If, say, the core of\napplication logic currently takes $70\\%$ of programming effort and is\nmade $10$x more efficient, with other tasks staying the same, then\n$37\\%$ of the original effort is still required---the overall benefit\nis merely a factor of $2.7$x.  Tasks that may not benefit much from\nnext-paradigm languages include low-level coding, as well as bespoke\ncode for UI\/storage\/other interfacing with the environment.\n\nIndeed, this is a constraint that can limit the large benefits of\nhigh-productivity languages to some kinds of development tasks but not\nothers. However, I believe the effect will be largely mitigated by\nmore conventionally forms of productivity enhancement: domain-specific\nlanguages and reusable modules (i.e., module\nabstraction). Furthermore, many of the non-core tasks of program\ndevelopment parallelize a lot more easily than the core application\nlogic itself. Building a UI, a service wrapper, or integration APIs\nmay require effort, but the effort can be more easily split among\nmultiple programmers. Therefore, even though the overall effort may\nindeed see smaller (than orders-of-magnitude) improvements, other\nmetrics of development productivity, such as end-to-end development\ntime, may improve more, by dedicating more programmers to the task.\n\n\n\\vspace{0.5cm}\n\n\\epigraph{We have programs that are vastly powerful but also vastly mysterious, meaning small changes can badly destabilize the system and we don't yet know how to talk about debugging or maintenance.}{\\textit{Jan-Willem Maessen, in reference to Sculley et al.~\\cite{43146}}}\n\n\\section{Changes to Development Workflows}\n\nAn informal axiom of the Software Engineering community\nis that the choice of programming language does not fundamentally\nchange the software engineering \\emph{process}. This may be true in\nthe sense that process stages (e.g., requirements analysis,\narchitecture, design, coding, testing, verification) remain the same\nconceptual entities. However, the relative effort and emphasis of each\nstage may change dramatically with high-productivity, high-abstraction\nlanguages.  The \\emph{practice} of software development will change.\n\n\\begin{principle}[Workflows Will Change]\n  Next-paradigm programming languages will change well-established\n  patterns in current programming workflow.\n\\label{principle:workflow}\n\\end{principle}\n\nOrders-of-magnitude productivity improvements will, very likely,\ndisrupt the established workflow of program development. Code will be\nmuch more terse and abstract, resembling a formal specification.\nSmall changes will have a huge impact on both functionality and\nperformance. It is hard to fully grasp precisely how today's common\npractices will evolve, so I will speculate modestly, straying little\nfrom observations from my own coding practices.\n\n\\begin{corollary}[Incremental Development]\n  In next-paradigm languages, it will be much easier to develop programs\n  incrementally, and to continue from where one has left off.\n\\end{corollary}\n\nThis observation may seem pedestrian, but it has been one of the most\nstriking revelations in my everyday experience with declarative\nprogramming. The higher level of abstraction means that one can rely\non highly-powerful concepts without ever looking at their\ndefinitions. Specifically for Datalog development, monotonicity in the\nlanguage evaluation model means that developing more functionality is\na natural extension of what is already there. However, the experience\nof incremental development will likely generalize to all higher-level\nprogramming settings. Programming with high-level specifications is\nnaturally incremental. For one, at any point in development, a partial\nspecification yields a program. The program's outputs are incomplete,\nbut they immediately suggest what is missing and how more work can get\nthe program closer to the desired task. Adding extra features\ninteracts in a predictable way with earlier functionality.\n\n\\paragraph{Example: An Interactive Graphical Application.} Imagine\ncreating an interactive graphical application (e.g., a video game or a\ndrawing program) in a language with high degrees of abstraction. Such\na language will likely accept a logical specification of \\emph{what}\nshould be displayed, \\emph{where} and \\emph{when}, without encoding at\nall the \\emph{how}. Whether a control should have an effect at a\ncertain point, whether a graphical element is visible or obstructed,\nhow the display adjusts to changes of the environment, etc., are all\nelements that the language implementation should address, and not the\nprogram itself.\n\nDevelopment in such a language is naturally incremental. It is\nstraightforward to start with specifications of elements that appear\non the screen, under some control conditions (which may involve\ntiming). There is no need to specify yet what the timing or the\ncontrols are---just to declare them. It is easy to add sample inputs\nfor these and see graphical elements displayed---the incomplete\nspecification already yields a working program. Making progress on any\nfront is incremental and straightforward. One can add more graphical\nelements, or more detail over current elements, or complex\nspecifications to define user control, or a mechanism for storing\ndata, or any other desired functionality. All new functionality should\ninteract cleanly with existing functionality. The language\nimplementation will resolve conflicts (sometimes with the programmer's\nhelp, in case of hard conflicts) and will produce vastly different\nlow-level programs as the specification becomes more and more\ncomplete.\n\n\n\\begin{corollary}[Testing Changes]\n  In next-paradigm languages, testing will be a larger and deeper part\n  of the programmer's workflow.\n\\end{corollary}\n\nWhen programming effectively becomes a collaboration between a\ncreative human and a compiler with vast abilities in exploring an\nimplementation space, the role of testing will change dramatically.\nThe programmer will write highly concise code that produces very\ncomplex outputs. Continuous checking of assumptions against the\nimplemented model will be necessary. The programmer may be spending\nmuch more time running code in complex settings than writing it.\n\nThe task of testing and debugging will also be conceptually\nharder. Testing may be as complicated as writing the code, and indeed\nwriting testing specifications may be obligatory. The difficulty\narises because efficiency in execution necessarily means that most\nintermediate results will never be part of the output. This\ncomplicates debugging enormously. For instance, the mechanism of\ntime-travel debugging (e.g., see\nReference~\\cite{Barr:2014:TAT:2660193.2660209}), which has captured\nthe programming community's imagination in recent years, works only\nwhen the space of intermediate values of a computation is not that\nmuch larger from the space of final values. This is not the case for\nabstract programs. Both the language implementation and the program\nitself may be collapsing a much larger space of values in order to get\na single output~\\cite{DBLP:conf\/datalog\/KohlerLS12}.\n\n\\paragraph{Example: Paths.} A pedestrian but illustrative example is\nthat of a declarative ``transitive closure'' computation: compute when\nthere is a path between two nodes, given direct edges as input. The\nrecursive rule for this is:\n\n\\begin{datalogcode}\nPath(source, target) :-\n  Edge(source, interm),\n  Path(interm, target).\n\\end{datalogcode}\n\nThis computation only stores the fact that there \\emph{is} a path, but\nnot how this path was established: the \\sv{interm} value is dropped.\nKeeping all \\sv{interm} values will make the cost of computing and\nstoring the \\sv{Path} relation combinatorially larger: $O(n^3)$\ninstead of $O(n^2)$.\\footnote{If one wants to be pedantic, in the\n  worst case, the cost of computing \\texttt{Path} is $O(n^3)$\n  anyway. But in the sparse graphs that arise in practice, the\n  computation is typically $O(n^2)$ if intermediate nodes do not need\n  to be kept and $O(n^3)$ if they do.}  In practice, this increase is\noften prohibitive. Consider that the transitive closure computation\nshown in the above rule is the simplest possible recursive computation\nin a logical specification. Most real specifications that employ\nrecursion will be much more complex, with several intermediate values\nused to derive results but not memorized. Therefore, completely\nroutine computations become intractable if it becomes necessary to\nfully trace how the result was derived.\n\n\n\\begin{corollary}[Different Balance of Formal Reasoning and Coding]\n  For programs in next-paradigm languages, formal proofs will be\n  easier.  Yet they will also be less necessary.\n\\end{corollary}\nA program that is effectively an executable specification removes some\nof the need for formal verification. The value of verification in\ncurrent coding stems partly from the assurance of formal reasoning and\npartly from specifying the computation in a completely different\nformulation, unencumbered by implementation constraints. In a\nnext-paradigm language, the level of abstraction of the program will\nlikely be much closer to that of a non-(efficiently-)executable\nspecification. There will likely still be a need for formal reasoning,\nto establish properties of the program with confidence and in full\ngenerality. Such reasoning will be easier just by virtue of the\nsmaller gap between the two specifications.\n\n\n\\section{Conclusions}\n\nNext-paradigm programming languages will need a revolutionary change\nin level of abstraction, if they are to ever realize large\nproductivity improvements. Such a change will necessarily have many\nrepercussions on the role of the compiler, on the programmer's mental\nmodel, and on development patterns. In this paper, I tried to identify\nthese changes, looking over the future through the misty glass of the\npresent. Necessarily, I only present the view of the future from where\nI currently stand. Others will likely find it too speculative or too\nmyopic---and that's fine. But we need a conversation about\nnext-paradigm programming languages and I hope to help start it.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}