{"text":"\\subsection*{Acknowledgements}}\n\\newcommand{\\thismonth}{\\ifcase\\month\\or\n January\\or February\\or March\\or April\\or May\\or June\\or\n July\\or August\\or September\\or October\\or November\\or December\\fi\n \\space\\number\\year}\n\\newcommand{\\sideremark}[1]{\\marginpar{\\small #1}}\n\\DeclareSymbolFont{script}{U}{eus}{m}{n}\n\\DeclareSymbolFontAlphabet{\\mathscr}{script}\n\\DeclareMathSymbol{\\EuWedge}{0}{script}{\"5E}\n\\DeclareMathAlphabet{\\mathrmsl}{OT1}{cmr}{m}{sl}\n\\newcommand{\\symb}[2]{\\newcommand{#1}{{\\mathit{#2}}}}\n\\newcommand{\\rssymb}[2]{\\newcommand{#1}{{\\mathrmsl{#2}}}}\n\\newcommand{\\calsymb}[2]{\\newcommand{#1}{{\\mathcal{#2}}}}\n\\newcommand{\\bbsymb}[2]{\\newcommand{#1}{{\\mathbb{#2}}}}\n\\newcommand{\\liealg}[2]{\\newcommand{#1}{{\\mathfrak{#2}}}}\n\\newcommand{\\liealr}[2]{\\renewcommand{#1}{{\\mathfrak{#2}}}}\n\\newcommand{\\lieoper}[2]{\\newcommand{#1}{\\mathop\n 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\n\\newcommand{\\Lie}{\\cL} \n\\newcommand{\\bdy}{\\partial} \n\\newcommand{\\del}{\\partial} \n\\newcommand{\\dbar}{\\overline\\partial} \n\\newcommand{\\dbyd}[1]{\\del\/\\del{#1}} \n\\newcommand{\\Proj}{\\mathrmsl{P}} \n\\newcommand{\\Gr}{\\mathrmsl{Gr}} \n\\newcommand{\\RP}[1]{\\R\\Proj^{#1}} \n\\newcommand{\\CP}[1]{\\C\\Proj^{#1}} \n\\newcommand{\\HP}[1]{\\HQ\\Proj^{#1}} \n\\newcommand{\\OP}[1]{\\OC\\Proj^{#1}} \n\\newcommand{\\half}{\\tfrac12} \n\\newcommand{\\Cinf}{\\mathrm{C}^\\infty} \n\\newcommand{\\const}{\\mathrm{const.\\ }} \n\\newcommand{\\st}{\\mathrel{|}} \n\\newcommand{\\ie}{\\textit{i.e.}} \n\\newcommand{\\empt}{\\varnothing} \n\\newcommand{\\sub}{\\subseteq} \n\\renewcommand{\\d}{{\\mathrmsl{d}}} \n\\rsoper\\dimn{dim} \n\\rsoper\\rank{rank} \n\\rsoper\\degree{deg} \n\\rsoper\\kernel{ker}\\rsoper\\image{im} \n\\rsoper\\alt{alt} \\rsoper\\sym{sym} \n\\rsoper\\Ad{Ad} \\rsoper\\ad{ad} \n\\rsoper\\CoAd{CoAd} \\rsoper\\coad{coad} \n\\rsoper\\trace{tr} \\rsoper\\trfree{tf} \n\\rsoper\\detm{det} \n\\rsoper\\Vol{Vol} \n\\rssymb\\vol{vol} \n\\rssymb\\iden{id} \n\\liealg{\\f}{f} \\liealg{\\g}{g} \\liealg{\\h}{h} \\liealg{\\n}{n} \\liealg{\\m}{m}\n\\liealg{\\p}{p} \\liealg{\\q}{q} \\liealr{\\t}{t} \\liealg{\\z}{z}\n\\newcommand{\\ccon}{\\theta\n\\newcommand{\\trl}{\\zeta\n\\newcommand{\\mathbin{\\raise1pt\\hbox{$\\scriptstyle\\bigcirc$}}}{\\mathbin{\\raise1pt\\hbox{$\\scriptstyle\\bigcirc$}}}\n\\newcommand{\\mathscr D}{\\mathscr D}\n\\newcommand{{\\boldsymbol x}}{{\\boldsymbol x}}\n\\newcommand{\\ell}{\\ell}\n\\newcommand{\\tabv}[2]{$\\vcenter{\\hbox{\\strut$#1$}\\hbox{\\strut{$#2$}}}$}\n\\newcommand{\\tabvvv}[3]{$\\vcenter{\\hbox{\\strut$#1$}\\hbox{\\strut{$#2$}}%\n\\hbox{\\strut{$#3$}}}$}\n\\begin{document}\n\\title[Subriemannian metrics and the metrizability of parabolic\ngeometries]{Subriemannian metrics\\\\\nand the metrizability of parabolic geometries}\n\\date{\\today}\n\\author{David M.J. Calderbank}\n\\address{Mathematical Sciences\\\\ University of Bath\\\\\nBath BA2 7AY\\\\ UK.}\n\\email{D.M.J.Calderbank@bath.ac.uk}\n\\author{Jan Slov\\smash{\\'a}k}\n\\address{Department of Mathematics and Statistics\\\\\nMasaryk University\\\\ Kotl\\'a\\v rsk\\'a 2\\\\ 611 37 Brno\\\\ Czech Republic.}\n\\email{slovak@math.muni.cz}\n\\author{Vladim\\smash{\\'\\i}r Sou\\smash{\\v c}ek}\n\\address{Mathematical Institute\\\\ Charles University\\\\ Sokolovsk\\'a 83\\\\\n186 75 Praha 8\\\\ Czech Republic.}\n\\email{soucek@karlin.mff.cuni.cz}\n\\begin{abstract}\nWe present the linearized metrizability problem in the context of parabolic\ngeometries and subriemannian geometry, generalizing the metrizability problem\nin projective geometry studied by R. Liouville in 1889. We give a general\nmethod for linearizability and a classification of all cases with irreducible\ndefining distribution where this method applies. These tools lead to natural\nsubriemannian metrics on generic distributions of interest in geometric control\ntheory.\n\\end{abstract}\n\\thanks{The authors thank the Czech Grant Agency, grant nr. P201\/12\/G028,\nfor financial support.}\n\\maketitle\n\n\n\\section{Introduction}\n\nMany areas of geometric analysis and control theory deal with distributions on\nsmooth manifolds, i.e., smooth subbundles of the tangent bundle. Let $\\cH\\leq\nTM$ be such a distribution of rank $n$ on a smooth $m$-dimensional manifold\n$M$. A smooth curve $c\\colon [a,b] \\to M$ ($a\\leq b\\in\\R$) is called\n\\emph{horizontal} if it is tangent to $\\cH$ at every point, i.e., for every\n$t\\in [a,b]$, the tangent vector $\\dot c(t)$ to $c$ at $c(t)\\in M$ belongs to\n$\\cH$. It is well known that, at least locally, any two points $x,y\\in\nM$ can be connected by a horizontal curve $c$ if and only if $\\cH$ is\n\\emph{bracket-generating} in the sense that any tangent vector can be obtained\nfrom iterated Lie brackets of sections of $\\cH$.\n\nThis paper is concerned with bracket-generating distributions arising in\n\\emph{parabolic geometries}~\\cite{CS}, which are Cartan--Tanaka geometries\nmodelled on homogeneous spaces $G\/P$ where $G$ is a semisimple Lie group and\n$P\\leq G$ a parabolic subgroup. On a manifold $M$ equipped with such a\nparabolic geometry, each tangent space is modelled on the $P$-module $\\g\/\\p$,\nand the socle $\\h$ of this $P$-module (the sum of its minimal nonzero\n$P$-submodules) induces a bracket-generating distribution $\\cH$ on $M$. Simple\nand well-known examples include projective geometry and (Levi-nondegenerate)\nhypersurface CR geometry: in the former case, $\\g\/\\p$ is irreducible and so\n$\\cH=TM$, but in the latter case $\\cH$ is the corank one contact distribution\nof the hypersurface CR structure.\n\nA more prototypical example for this paper is when $\\cH\\leq TM$ is generic of\nrank $n$ and corank $\\frac12 n(n-1)$, i.e., $m=\\frac12 n(n+1) = n+\\frac12\nn(n-1)$, and $[\\Gam(\\cH),\\Gam(\\cH)]=\\Gam(TM)$. In this case the Lie bracket on\nsections of $\\cH$ induces an isomorphism $\\Wedge^2\\cH\\cong TM\/\\cH$ and the\ndistribution is said to be \\emph{free}. Any such manifold is a parabolic\ngeometry where $G=\\SO(V)$ with $\\dim V=2n+1$ and $P$ is the stabilizer of a\nmaximal ($n$-dimensional) isotropic subspace $U$ of $V$~\\cite{DS}. Then\n$\\g\/\\p$ has socle $\\h\\cong U^*\\otimes (U^\\perp\/U)$ with quotient isomorphic to\n$\\Wedge^2 \\h$, and $\\h\\leq \\g\/\\p$ induces the distribution $\\cH\\leq TM$ on\n$M$.\n\nWhile parabolic geometry is the main tool for the present work, our motivation\nis subriemannian geometry, which concerns the following notion~\\cite{Mont}.\n\n\\begin{defn} Consider an $m$-dimensional manifold $M$ with a given smooth\ndistribution $\\cH\\leq TM$ of constant rank $n$. A\n(pseudo-)Riemannian metric $g$ on $\\cH$ is called a \\emph{horizontal}\nor \\emph{subriemannian metric on $M$}.\n\\end{defn}\nHorizontal metrics are important in both geometric analysis and control\ntheory. Among the horizontal curves joining two points, it may be important to\nfind those which are optimal in some sense, for example those of shortest\nlength with respect to a horizontal metric. Horizontal metrics also allow for\nthe definition of a hypo-elliptic sublaplacian~\\cite{JL}, allowing methods of\nharmonic analysis to be applied. However, this raises the question: what is a\ngood choice of horizontal metric?\n\nFor the distribution $\\cH$ on a parabolic geometry, there is a natural\ncompatibility condition that can be imposed. Indeed, one of the key features\nof such a geometry is that it admits a canonical class of connections,\ncalled \\emph{Weyl connections}, which form an affine space modelled on\nthe space of $1$-forms.\n\n\\begin{defn} A horizontal metric on the distribution $\\cH\\leq TM$ induced\nfrom a parabolic geometry $M$ is \\emph{compatible} if it is covariantly\nconstant in horizontal directions with respect to some Weyl connection on $M$.\nWe say $M$ is \\emph{\\textup(locally\\textup) metrizable} if there exists\n(locally) a compatible horizontal metric.\n\\end{defn}\n\nThe metrizability problem has been studied for several classes of parabolic\ngeometry with $\\cH=TM$, in particular, the case of real projective. These\nexamples exhibit several interesting features, which we seek to generalize to\nall parabolic geometries---in particular to those with $\\cH\\neq TM$.\n\nFirst, whereas the metrizability condition appears to be highly nonlinear, it\nlinearizes when viewed as a condition on the inverse metric on $\\cH^*$\nmultiplied by a suitable power of the horizontal volume form. Secondly, this\nlinear equation is highly overdetermined, with a finite dimensional solution\nspace. Hence parabolic geometries admitting such horizontal metrics are rather\nspecial. This has been used to extract detailed information about the\nstructure of the geometry~\\cite{BDE,CEMN,DM,EM,Frost,Liouville,Sinjukov}.\n\nIf $\\h$ is the socle of $\\g\/\\p$, it is not generally the case that $S^2\\h$ is\nirreducible---indeed $\\h$ itself need not be irreducible. In order to\ngeneralize the studied examples, we introduce a condition on $P$-submodules\n$B\\leq S^2 \\h$ containing nondegenerate elements, which we call the\n\\emph{algebraic linearization condition} (ALC). Our first main result\n(Theorem~\\ref{alt}) justifies this terminology by showing that for parabolic\ngeometries and $P$-submodules $B\\leq S^2\\h$ satisfying the ALC, there is a\nbijection between compatible horizontal metrics and nondegenerate solutions of\nan overdetermined first order \\emph{linear} differential equation. (In fact,\nif $\\h$ is not irreducible we need a technical extra condition, which we call the\n\\emph{strong} ALC.)\n\nOur second main result (Theorem~\\ref{main}) is a complete classification of\nall parabolic geometries and all $P$-submodules $B\\leq S^2\\h$ such that $\\h$\nis irreducible and $B$ satisfies the ALC. The classification exhibits two\nnicely counterbalancing features. On the one hand, among parabolic geometries\nwith irreducible socle, those admitting $P$-submodules $B\\leq S^2\\h$\nsatisfying the ALC are rare. On the other hand, the list of examples is quite\nlong: we state the classification using three tables containing 14 infinite\nfamilies and 6 exceptional cases. Many of these examples invite further study\n(see e.g.~\\cite{P}).\n\nThe structure of the paper is as follows. In section~\\ref{s:bg} we briefly\noutline the main notions and tools of parabolic geometry, referring\nto~\\cite{CS} for details, but concentrating on examples. We also establish the\nlocal metrizability of the homogeneous model. In section~\\ref{s:mlp}, we\ndescribe the linearization principle and prove Theorem~\\ref{alt}. We give\nexamples, and in particular show how explicit formulae can be obtained not\nonly for the homogeneous model, but also for so-called \\emph{normal\n solutions}. Section~\\ref{s:class} is devoted to the main classification\nresult. We conclude by giving examples (Theorem~\\ref{more}) where the socle is\nnot irreducible.\n\n\n\\section{Background and motivating examples}\\label{s:bg}\n\nWe work throughout with real smooth manifolds $M$, real Lie groups $P$ and\nreal Lie algebras $\\p$ (e.g., we view $\\GL(n,\\C)$ as a real Lie group and\n$\\gl(n,\\C)$ as a real Lie algebra).\n\nA (real or complex) \\emph{$P$-module} $W$ is a finite dimensional (real or\ncomplex) vector space carrying a representation $\\rho_W\\colon P\\to \\GL(W)$;\n$W$ is then also a $\\p$-module, where $\\p$ is the Lie algebra of $P$, i.e., it\ncarries a representation $\\tilde\\rho_W \\colon\\p\\to\\gl(W)$. We write $\\xi\\mathinner{\\raise2pt\\hbox{$\\centerdot$}}\nw$ for $\\tilde\\rho_W(\\xi)(w)$. The \\emph{nilpotent radical} of $\\p$ is the\nintersection $\\n$ of the kernels of all simple $\\p$-modules. It is an ideal in\n$\\p$ and the quotient $\\p_0:=\\p\/\\n$ is reductive. We let $P_0:=P\/\\exp\\n$ be\nthe corresponding quotient group with Lie algebra $\\p_0$. Any $P$-module $W$\nhas a filtration\n\\begin{equation}\\label{eq:alg-filt}\n0=W^{(0)}\\noroot{}\\edyn\\;\\;\n\\dyn \\noroot{}\\llink<\\root{}\\link\\root{}\\dots\\root{1}\\edyn$&\n$\\so(2\\ell+1,\\C)\\;\\; \\ell\\geq 2$ &$2k, 2k+k(k-1)$\\\\\n\\hline\n$G_2^{h}$&$\\dyn \\noroot{}\\lllink<\\root{1\\strut}\\edyn\\;\\;\n\\dyn \\root{1}\\lllink>\\noroot{}\\edyn$& $G_2^\\C$&$4,6,10$\\\\\n\\hline\n\\end{tabular}\n\\smallbreak\n\\caption{Complex geometries with hermitian $B$}\\label{t:hermitian}\n\\end{table}\n\\begin{table}[!ht]\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\nCase& Diagram $\\Delta_\\ell$ for $\\p,B$ & Real simple $\\g$ & Growth \\\\\n\\hline\n$A_\\ell^{1,1}$&$\\dyn \\noroot{}\\link\\root{}\\dots\\root{2\\strut}\\edyn$&\n$\\sgl(\\ell+1,\\R)\\;\\; \\ell\\geq 2$ & $\\ell$\\\\\n\\hline\n$A_\\ell^{1,2}$&$\\dyn\\root{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{}\\link\\root{1\\strut}\\link\\root{}\\edyn$& \n\\tabv{\\sgl(\\ell+1,\\R),\\,\\sgl(p+1,\\HQ)}\n{\\ell=2p+1 ,\\, p\\geq2}&$4p$\\\\\n\\hline\n$B_\\ell^{1,k}$&$\\dyn\\root{2\\strut}\\link\\root{}\n\\dots\\root{}\\link\\nodroot{k\\geq2}\\link\\root{}\\dots\\root{}\\llink>\\root{}\\edyn$&\n\\tabv{\\so(p,q),\\;k\\leq p\\leq q}{p+q=2\\ell+1}&\\tabv{d=k(2\\ell-2k+1),}\n{n=d+\\frac12 k(k-1)}\\\\\n\\hline\n$B_{\\ell}^{1,\\ell}$&$\\dyn \\root{2\\strut}\\link\\root{}\\dots\\root{}\\llink>\\noroot{}\\edyn$&\n$\\so(\\ell,\\ell+1)\\; \\ell\\geq 2$&$k,k+\\frac12 k(k-1)$\\\\\n\\hline\n$C_4^{1,2}$&$\\dyn \\root{}\\link\\noroot{}\\link\\root{}\\llink<\\root{1\\strut}\\edyn$&\n\\tabv{\\symp(8,\\R)}{\\symp(2,2)\\;\\;\\symp(1,3)}&$8,11$\\\\\n\\hline \n$C_\\ell^{1,k}$&$\\dyn\\root{}\\link\\root{1\\strut}\\dots\n\\root{}\\link\\nodroot{k=2j\\geq 4}\\link\\root{}\\dots\\root{}\\llink<\\root{}\\edyn$&\n\\tabv{\\symp(2\\ell,\\R)\\;\\;\\;\\symp(p,q)}{\\quad\\;\\ell=p+q,\\; k \\leq p\\leq q}\n&\\tabv{d=k(2\\ell-2k),}{n=d+\\frac12 k(k+1)}\\\\\n\\hline\n$D_\\ell^{1,k}$&$\\dyn \\root{2\\strut}\\link\\root{}\n\\dots\\root{}\\link\\nodroot{k\\geq 2}\\link\\root{}\n\\dots\\root{}\\rootupright{}\\rootdownright{}\\edyn$&\n\\tabv{\\so(p,q)\\qquad\\quad \\so^*(2\\ell)}\n{\\begin{matrix}2\\ell=p+q\\\\\nk\\leq p\\leq q\\end{matrix}\\quad \\begin{matrix}k=2j\\\\ k\\leq \\ell-2\\end{matrix}}\n&\\tabv{d=k(2\\ell-2k),}{n=d+\\frac12 k(k-1)}\\\\\n\\hline\n$E_6^{1,1}$& $\\dyn\\noroot{}\\link\\root{}\\link\\root{}\\rootdown{}\n\\link\\root{}\\link\\root{1\\strut}\\edyn$& $E_{6(6)}$, $E_{6(-26)}$ &$16$\\\\\n\\hline\n$G_2^{1,1}$&$\\dyn \\noroot{}\\lllink<\\root{2\\strut}\\edyn$& $G_{2(2)}$&$2,3,5$\\\\\n\\hline\n\\end{tabular}\n\\smallbreak\n\\caption{Real geometries with absolutely irreducible $\\h$}\n\\label{t:absirred}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\nCase& Diagram $\\Delta_\\ell$ for $\\p,B$ & Real simple $\\g$ & Growth \\\\\n\\hline\n$A_3^{2,1}$&$\\dyn \\noroot{}\\link\\root{2\\strut}\\link\\noroot{}\\edyn$&\n$\\su(1,3),\\;\\su(2,2)$&$4,5$\\\\\n\\hline\n$A_\\ell^{2,k}$&$\\dyn\\root{1\\strut}\\dots\\root{}\\link\\nodroot{k\\geq 2}\\link\\root{}\n\\dots\\root{}\\link\\nodroot{\\ell-k}\\link\\root{}\\dots\\root{1\\strut}\\edyn$&\n\\tabv{\\su(p,q),\\;k\\leq p \\leq q}{\\ell=p+q-1\\geq 4}\n&\\tabv{d=2k(\\ell-2k+1),}{n=d+k^2}\\\\\n\\hline\n$A_\\ell^{2,h}$&\n$\\begin{matrix}\n\\dyn\\root{}\\link\\noroot{}\\link\\root{}\\link\\root{\\strut1}\n\t\\dots\\root{}\\link\\noroot{}\\link\\root{}\\edyn \\\\\n\t\\oplus\\\\\n\t\\dyn\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\n\t\\link\\root{}\\link\\noroot{}\\link\\root{}\n\t\\edyn\n\\end{matrix}$&\n\\tabv{\\su(p,q),\\;2\\leq p\\leq q}{\\ell=p+q-1\\geq 6}&$4(\\ell-3),4(\\ell-2)$\\\\\n\\hline\n$A_{2k+1}^{2,s}$&\n$\\begin{matrix}\n\\dyn\\root{}\\link\\root{\\strut1}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\link\\noroot{}\\link\\root{}\\dots\\root{}\\link\\root{}\\edyn\\\\\n \\oplus\\\\\n\\dyn\\root{}\\link\\root{}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\link\\root{}\\edyn \n\\end{matrix}$&\n\\tabvvv{\\su(k,k+2),}{\\su(k+1,k+1)}{\\ell=2k+1\\geq 7}\n &$4k,4k+k^2$\n\\\\\n\\hline\n$A_{2k}^{2,s}$&$\\begin{matrix}{\\dyn\\root{2\\strut}\\link\\root{}\\dots\\root{}\\link\n\\noroot{}\\link\\noroot{}\\link\\root{}\\dots\\root{}\\link\\root{}\\edyn}\\\\\n\\oplus\\\\\n{\\dyn \\root{}\\link\\root{}\\dots\\root{}\\link\\noroot{}\\link\\noroot{}\n\\link\\root{}\\dots\\root{}\\link\\root{\\smash{2}}\\edyn}\\end{matrix}$\n& \\tabv{\\su(k,k+1)}{\\ell=2k\\geq 4}&$2k,2k+k^2$\\\\\n\\hline\n$D_\\ell^{2,s}$&$\\dyn \\root{2}\\link\\root{}\\dots\\root{}\n\\norootupright{}\\norootdownright{}\\edyn$&\n\\tabv{\\so(\\ell-1,\\ell+1)}{\\so^*(2\\ell),\\;\\ell=2j+1}&\\tabv{d=2(\\ell-1),}\n{d+\\frac12(\\ell-1)(\\ell-2)}\\\\\n\\hline\n$D_\\ell^{2,h}$&$\\dyn \\root{}\\link\\root{1}\\dots\\root{}\n\\norootupright{}\\norootdownright{}\\edyn$&\n\\tabv{\\so(\\ell-1,\\ell+1)}{\\so^*(2\\ell),\\; \\ell=2j+1}&\\tabv{d=2(\\ell-1),}\n{d+\\frac12(\\ell-1)(\\ell-2)}\\\\\n\\hline\n$E_6^{2,h\\vphantom{{}^2}}$&$\\dyn\\noroot{}\\link\\root{}\\link\\root{}\\rootdown{1\\strut}\n\\link\\root{}\\link\\noroot{}\\edyn$ & $E_{6(2)}$&$16,24$\\\\\n\\hline\n\\end{tabular}\n\\smallbreak\n\\caption{Real geometries with $\\h$ not absolutely irreducible}\n\\label{t:red}\n\\end{table}\n\\end{thm}\n\n\\begin{proof}[Outline of Proof]\nIn the gradings of the complex algebras $\\g$ corresponding to parabolic\ngeometries, the number of irreducible components of $\\h^*$ is equal to the\nnumber of crosses in the Dynkin diagram describing the chosen parabolic\nsubalgebra. However, in the real forms of $\\g$, there might be complex or\nquaternionic components giving rise to two components in the\ncomplexification. These two complex components have to be either conjugate (in\nthe complex case) or isomorphic (in the quaternionic case).\n\nThe latter observation reduces our quest to diagrams with two crosses placed\nin a symmetric way. Indeed, more than two crosses cannot result in one\ncomponent, while asymmetric positions of the crosses inevitably yield two\ncomplex components which are neither conjugate nor isomorphic. Moreover,\nhaving two components in the complexified $\\h$, we may ignore the symmetric\nproducts of the individual parts in $S^2\\h$, because there cannot be any\nnondegenerate metrics there.\n\nWe first dispense with the case that $\\g$ is complex but $B$ is not, so that\n$B\\otimes\\C$ is irreducible in $\\g\\otimes\\C\\cong\\g\\oplus\\g$ and the diagram\nfor $(\\p,B)$ is invariant under the automorphism exchanging the two components\nof the Dynkin diagram. Thus $B\\otimes\\C=\\h_\\alpha\\otimes\\h_\\beta$ where\n$\\h\\otimes\\C=\\h_\\alpha\\oplus\\h_\\beta$. Now the ALC is satisfied provided\n$\\h_\\alpha\\otimes \\h_\\alpha^*$ (and hence also $\\h_\\beta\\otimes\\h_\\beta^*$)\nhas precisely two irreducible components as a representation of a component of\n$\\p_0\\otimes\\C$. Only the (dual) defining representations in type A have this\nproperty, and so $\\g$ must have type $A,B$ or $G$, where the nodes crossed\nin $\\g\\otimes\\C$ are end nodes corresponding to short simple roots. The\npossibilities are listed in Table~\\ref{t:hermitian}, covering the following\nthree cases:\n\n\\begin{case}[$A_{\\ell}^h$] The c-projective geometries may be equipped with\ndistinguished hermitian metrics.\n\\end{case}\n\\begin{case}[$B_{\\ell}^h$] The almost complex version of a free distribution of\nrank $k$, may be equipped with distinguished hermitian metrics.\n\\end{case}\n\\begin{case}[$G_2^h$] The almost complex version of the\n$(2,3,5)$-distributions may be equipped with distinguished hermitian metrics. \n\\end{case}\n\nWe analyse the remaining real cases with irreducible $\\h$ by\nthe Dynkin type of $\\g$ in the following sections.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{main} when $\\g$ has type $A_\\ell$}\n\nThe case $\\ell=1$ is trivial, so we assume $\\ell\\geq 2$, and first consider the\ncase of a single crossed node. If the crossed node is one of the ends of the\nDynkin diagram, the only real $\\g$ is the split form, $\\h$ and $S^2\\h$ are\nirreducible, and $B=S^2\\h$ satisfies the ALC: when $\\ell=2$,\n\\[\nB\\simeq \\dyn \\noroot{}\\link\\root{2}\\edyn \\qquad \\h^*\\otimes B \\simeq\n\\dyn\\noroot{}\\link\\root{3}\\edyn\\oplus \\dyn\\noroot{}\\link\\root{1}\\edyn\n\\]\nand when $\\ell\\geq 3$,\n\\begin{equation*}\nB\\simeq \\dyn \\noroot{}\\link\\root{}\\dots\\root{2}\\edyn\\qquad\n\\h^*\\otimes B \\simeq \\dyn\\noroot{}\\link\\root{1}\\dots\\root{2}\\edyn\\oplus \n\\dyn\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn.\n\\end{equation*}\nThese examples can be summarized in the following statement.\n\n\\begin{case}[$A^{1,1}_\\ell$] Here $\\g=\\sgl(\\ell+1,\\R)$, $\\ell\\geq2$, $\\h\\cong\\R^\\ell$\nand $B=S^2\\h$. This is the most classical case of projective structures on\n$\\ell$-dimensional manifolds $M$, and nondegenerate sections of $\\cB$ are\ninverse to arbitrary pseudo-Riemannian metrics on $M$.\n\\end{case}\n\nSuppose next that the cross is adjacent to one end of the diagram, with\n$\\ell\\geq 3$. We then have $S^2\\h= B\\oplus B'$, where\n\\begin{gather*}\n\\h\\simeq\\dyn\\root{1}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn \\qquad\n\\h^*\\simeq \\dyn\\root{1}\\link\\noroot{}\\link\\root{1}\\dots\\root{}\\edyn\\\\\nB\\simeq\\dyn\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\link\\root{}\\edyn\n\\;(\\ell\\geq 4)\\qquad\nB'\\simeq\\dyn\\root{2}\\link\\noroot{}\\link\\root{}\\dots\\root{2}\\edyn\n\\end{gather*}\nand $B$ is trivial for $\\ell=3$ (when $\\h\\cong \\h^*$). The tensor product\n$\\h^*\\otimes B'$ decomposes into four irreducible components, except for the\nreal form $\\su(2,2)$ when $\\ell=3$, in which case there are only three\ncomponents. In any case, $B'$ does not satisfy the ALC.\n\nIn order for $B$ to have nondegenerate elements, $\\ell$ must be odd, and for\n$\\ell=2p+1\\geq 5$, $\\h^*\\otimes B\\simeq\n\\dyn\\root{1}\\link\\noroot{}\\link\\root{1}\\link\\root{}\n\\dots\\root{1}\\link\\root{}\\edyn \\oplus\n\\dyn\\root{1}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn$; thus the ALC holds\nfor $B$.\n\n\\begin{case}[$A^{1,2}_\\ell$] For each $\\ell=2p+1\\geq 5$, there are two real forms.\nWhen $\\g\\simeq \\sgl(2p+2,\\R)$, the geometries are the almost grassmannian\nstructures on manifolds $M$ of dimension $4p$, modelled on the grassmannian of\n$2$-planes in $\\R^{2p}$. The tangent bundle $TM$ is identified with a tensor\nproduct $E\\otimes F$, where $\\rank E=2$, $\\rank F=2p$, and the nondegenerate\nmetrics in $\\cB$ are tensor products of area forms on $E$ and symplectic forms\non $F$. When $\\g\\simeq \\sgl(p,\\HQ)$, the geometries are almost quaternionic\ngeometries, where $TM$ is a quaternionic vector bundle, and the nondegenerate\nmetrics in $\\cB$ are the (real parts of) quaternionic hermitian forms.\n\\end{case}\n\nWhen the cross is further from the ends of the diagram, we have $S^2\\h=\nB\\oplus B'$ with\n\\begin{gather*}\nB\\simeq\\dyn\\root{}\\link\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{1}\\link\\root{}\\edyn\\qquad\nB'\\simeq\\dyn\\root{2}\\dots\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{2}\\edyn.\n\\end{gather*} \nand there are too many components in both $\\h^*\\otimes B$ and $\\h^*\\otimes\nB'$ to satisfy the ALC.\n\nWe now turn to cases with two crossed nodes, related by the diagram\nautomorphism of $A_\\ell$. First suppose the crossed nodes are the endpoints.\nIn order to have nontrivial $B$ we must have $\\ell\\geq3$, in which case\n$S^2\\h= B\\oplus B'\\oplus B''$ where\n\\begin{gather*}\n\\h\\simeq \\dyn\\noroot{}\\link\\root{1}\\dots\\root{}\\link\\noroot{}\\edyn\\oplus\n\\dyn\\noroot{}\\link\\root{}\\dots\\root{1}\\link\\noroot{}\\edyn \\simeq \\h^*\\\\\nB\\simeq \\dyn\\noroot{}\\link\\root{2}\\link\\noroot{}\\edyn \\text{ or }\n\\dyn\\noroot{}\\link\\root{1}\\link\\root{}\n\\dots\\root{}\\link\\root{1}\\link\\noroot{}\\edyn\\qquad\nB'\\simeq \\dyn\\noroot{}\\link\\root{2}\\dots\\root{}\\link\\noroot{}\\edyn\n\\oplus\\dyn\\noroot{}\\link\\root{}\\dots\\root{2}\\link\\noroot{}\\edyn\n\\end{gather*}\nand $B''$ is trivial. Clearly $\\h^*\\otimes B'$ has too many irreducible\ncomponents to satisfy the ALC, no matter which real form we consider.\n\nIt remains to consider $B$, first in the case $\\ell=3$, where the possible\nreal forms (with $\\h$ irreducible) are $\\su(2,2)$ and $\\su(1,3)$. Then\n\\[\n\\h^*\\otimes B\\simeq \\bigl(\\,\\dyn\\noroot{}\\link\\root{3}\\link\\noroot{}\\edyn\n\\oplus \\dyn\\noroot{}\\link\\root{3}\\link\\noroot{}\\edyn\\,\\bigr)\n\\oplus\n\\bigl(\\,\\dyn\\noroot{}\\link\\root{1}\\link\\noroot{}\\edyn\n\\oplus \\dyn\\noroot{}\\link\\root{1}\\link\\noroot{}\\edyn\\,\\bigr)\n\\]\nand the ALC is satisfied, since these are complexifications of two complex\ncomponents for the real form in question. However, for $\\ell\\geq 4$, we find\nthat the product $\\h^*\\otimes B$ leads to complexifications with three\ncomplex components, so the ALC is not satisfied. \n \n\\begin{case}[$A^{2,1}_3$] Here $\\g$ is\n\t$\\su(2,2)$ or $\\su(1,3)$, and $M$ has a CR structure, i.e., a contact distribution\n$\\cH$ equipped with a complex structure. The Levi form induces the\nclass of trivial parallel hermitian metrics (the Weyl connections\ncorresponding to the contact forms leave parallel both the complex structure\nand the symplectic form, thus also the associated metric, and the\nmetrizability problem is trivial as in the conformal case). However, we now\nsee that there may also be interesting compatible subriemannian metrics on\n$\\cH\\leq TM$ which are hermitian and tracefree with respect to the\nLevi form.\n\\end{case}\n\nNow suppose the crosses are not placed at the ends, say the left one at the\n$k$-th position, $2\\leq k$. Thus we consider the real forms $\\su(p,q)$ with\n$k\\leq p\\leq q$. We have\n\\begin{gather*}\n\\h \\simeq \n\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{1}\\link\\noroot{}\\link\\root{}\\dots\\root{}\\edyn \\oplus\n\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\\root{1}\n\\dots\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn \n\\\\\n\\h^* \\simeq \n\\dyn\\root{}\\dots\\root{1}\\link\\noroot{}\\link\\root{1}\n\\dots\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{}\\edyn \\oplus\n\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{1}\\link\\noroot{}\\link\\root{1}\\dots\\root{}\\edyn \n\\end{gather*}\nfor $\\ell>2k$ and\n\\begin{gather*}\n\\h \\simeq \n\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\noroot{}\n\\link\\root{}\\dots\\root{}\\edyn \\oplus\n\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\\noroot{}\n\\link\\root{}\\dots\\root{1}\\edyn \n\\\\\n\\h^* \\simeq\n\\dyn\\root{}\\dots\\root{1}\\link\\noroot{}\\link\\noroot{}\n\\link\\root{}\\dots\\root{}\\edyn \\oplus\n\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\\noroot{}\n\\link\\root{1}\\dots\\root{}\\edyn \n\\end{gather*}\nfor $\\ell=2k$. In particular, we have $S^2\\h\\supset B$ where\n\\begin{equation*}\nB\\simeq \\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn,\n\\end{equation*}\nwhich admits nondegenerate metrics and satisfies the ALC, with\n\\begin{align*}\n\\h^*\\otimes B&\\simeq\n\\bigl(\\,\\dyn\\root{1}\\dots\\root{1}\\link\\noroot{}\\link\\root{1}\n\\dots\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn \\oplus\n\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{1}\\link\\noroot{}\\link\\root{1}\\dots\\root{1}\\edyn\\,\\bigr)\\\\\n&\\;{}\\oplus\\bigl(\\,\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\\root{1}\n\\dots\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn\\oplus\n\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{1}\\link\\noroot{}\\link\\root{}\\dots\\root{}\\edyn\\, \\bigr)\\\\\n\\text{or}\\qquad\n\\h^*\\otimes B &\\simeq \\bigl(\\,\\dyn\\root{1}\\dots\\root{1}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn \\oplus\n\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{1}\\dots\\root{1}\\edyn\\,\\bigr)\n\\\\\n&\\;{}\\oplus\\bigl(\\,\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn\\oplus\n\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{}\\dots\\root{}\\edyn\\, \\bigr).\n\\end{align*}\n\n\\begin{case}[$A^{2,k}_\\ell$] Here $\\g\\simeq \\su(p,q)$ with nodes $k$\nand $\\ell+1-k$ crossed, where $2\\leq k\\leq p \\leq q, p+q=\\ell+1$. In these\ngeometries, $\\cH\\cong E\\otimes F$, where $E$ is a complex vector bundle of\nrank $k$, and the rank $(\\ell-2k+1)$ complex vector bundle $F$ comes with a\nhermitian form of signature $(p-k,q-k)$. The corank of $\\cH\\leq TM$ is $k^2$,\nand the metrics on $\\cH$ are the products of hermitian metrics on $E$ with the\ngiven ones on $F$. When $\\ell=2k$ (i.e., $F$ has rank $1$), $\\g=\\su(k,k+1)$\nwith the nodes $k,k+1$ are crossed. These are the free CR geometries with\ncomplex structure on $\\cH$ studied in \\cite{SchmalzS} (where it is also\nexplained how complex structure arises on $\\cH$).\n\\end{case}\n\nThe remaining components of $S^2\\h$ do not satisfy the ALC, except in special\ncases $k=2$, $2k=\\ell$ and $2k+1=\\ell$. In particular, when $k=2$,\n\\begin{equation*}\nB'\\simeq \\dyn\\root{}\\link\\noroot{}\\link\\root{}\\link\\root{1}\n\\dots\\root{}\\link\\noroot{}\\link\\root{}\\edyn \n\\oplus\n\\dyn\\root{}\\link\\noroot{}\\link\\root{}\\dots\\root{1}\n\\link\\root{}\\link\\noroot{}\\link\\root{}\n\\edyn\n\\end{equation*}\nsatisfies the ALC (and is nontrivial for $\\ell\\geq 6$).\n\n\\begin{case}[$A^{2,h}_\\ell$] Here $\\g\\simeq \\su(p,q)$ with nodes $2$\nand $\\ell-1$ crossed, where $2\\leq p\\leq q$ and $\\ell=p+q-1\\geq 6$. In this\ngeometry, $\\cH\\cong E\\otimes F$, where $E$ is a complex vector bundle of\nrank $2$, and $F$ is a complex vector bundle of rank $\\ell-3$. The\ncorank of $\\cH\\leq TM$ is $4$. The eligible metrics are the\ncomplex symmetric bilinear forms of the form of tensor product of two exterior\nforms.\n\\end{case}\t\n\nWhen $2k=\\ell$, we obtain $S^2\\h= B\\oplus B'$ where\n\\begin{gather*}\nB'= \\begin{matrix}\n\\dyn\\root{2}\\dots\\root{}\\link\\noroot{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{}\\edyn\\oplus{}\\\\\n\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\\noroot{}\\link\\root{}\n\\dots\\root{2}\\edyn\\quad\\end{matrix}\n\\end{gather*}\nwhich admits nondegenerate metrics, and satisfies the ALC, with\n\\begin{gather*}\n\\h^*\\otimes B' \\simeq \\bigl(\\,\\dyn\\root{2}\\dots\\root{1}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{}\\dots\\root{}\\edyn \\oplus\n\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{1}\\dots\\root{2}\\edyn\\,\\bigr)\\oplus{}\\\\\n\\qquad\\qquad\\quad \\bigl(\\,\\dyn\\root{2}\\dots\\root{}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{1}\\dots\\root{}\\edyn \\oplus\n\\dyn\\root{}\\dots\\root{1}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{}\\dots\\root{2}\\edyn\\,\\bigr)\\oplus{}\\\\\n\\qquad\\qquad\\;\\,\\bigl(\\,\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{}\\dots\\root{}\\edyn\\oplus\n\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\n\\noroot{}\\link\\root{}\\dots\\root{1}\\edyn\\, \\bigr).\n\\end{gather*}\n\\begin{case}[$A^{2,s}_{2k}$] This case is again the free CR geometry,\nwith $\\g=\\su(k,k+1)$, but the eligible metrics are the complex bilinear\nmetrics on $\\cH$.\n\\end{case}\nSimilarly, when $\\ell=2k+1$ with the $k$-th and $(k+2)$-nd nodes crossed,\n\\begin{equation*}\nB'\\simeq \\dyn\\root{}\\link\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\link\\noroot{}\\link\\root{}\\dots\\root{}\\edyn \\oplus\n\\dyn\\root{}\\dots\\root{}\\link\\noroot{}\\link\\root{}\n\\link\\noroot{}\\link\\root{}\\dots\\root{1}\\link\\root{}\\edyn \n\\end{equation*}\nsatisfies the ALC.\t\n\\begin{case}[$A^{2,s}_{2k+1}$] Here $\\ell=2k+1,$ $\\g$ is $\\su(k,k+2),$\nor $\\su(k+1,k+1),$ with nodes $k$ and $k+2$ crossed. In this geometry,\n$\\cH\\cong E\\otimes F$, where $E$ is a complex vector bundle of\nrank $k$, and $F$ is a complex vector bundle of rank $2$. The\ncodimension of $\\cH\\leq TM$ is $k^2$. The eligible metrics are the\ncomplex symmetric bilinear forms of the form of tensor product of two exterior\nforms.\n\\end{case}\nWe have now exhausted all possibilities, completing the proof in type A.\n\n\\subsection{Proof of Theorem~\\ref{main} when $\\g$ has type $B_\\ell$}\n\nIn the type $B$ case, there are no complex or quaternionic modules to\nconsider, so the irreducible cases have one cross only. The unique grading of\nlength one is odd dimensional conformal geometry. In dimension three we then\nhave\n\\[\n\\h^* \\simeq \\dyn\\noroot{}\\llink>\\root{2}\\edyn \\simeq \\h\\qquad\nS^2\\h\\simeq \\dyn\\noroot{}\\llink>\\root{4}\\edyn\\oplus \n\\dyn\\noroot{}\\llink>\\root{}\\edyn.\n\\] \nThe trivial representation in $S^2\\h$ corresponds to the trivial case of\nmetrics in the conformal class, which are excluded from our classification,\nand choosing $B$ to be the other component leads to three components in\n$B\\otimes \\h^*$, so the ALC fails. Similarly, for conformal geometries of\ndimensions $2\\ell-1\\geq 5$ we obtain\n\\[\n\\h^* \\simeq \\dyn\\noroot{}\\link\\root{1}\\dots\\root{}\\llink>\\root{}\\edyn \n\\simeq \\h\\qquad\nS^2\\h\\simeq \\dyn\\noroot{}\\link\\root{2}\\dots\\root{}\\llink>\\root{}\\edyn\n\\oplus \\dyn\\noroot{}\\link\\root{}\\dots\\root{}\\llink>\\root{}\\edyn.\n\\] \nAs before, the trivial summand is excluded, and the other component fails the\nALC.\n\nWe turn now to Lie contact geometries, with the second node crossed. For\n$B_3$,\n\\[\n\\h^* \\simeq \\dyn\\root{1}\\link\\noroot{}\\llink>\\root{2}\\edyn \n\\simeq \\h\\qquad\nS^2\\h= B\\oplus B'\\oplus B''\\simeq \n\\dyn\\root{2}\\link\\noroot{}\\llink>\\root{}\\edyn\n\\oplus\\dyn\\root{}\\link\\noroot{}\\llink>\\root{2}\\edyn\n\\oplus\\dyn\\root{2}\\link\\noroot{}\\llink>\\root{4}\\edyn.\n\\] \nHere, $B\\otimes \\h^* = \\dyn\\root{3}\\link\\noroot{}\\llink>\\root{2}\\edyn\\oplus\n\\dyn\\root{1}\\link\\noroot{}\\llink>\\root{2}\\edyn$ and satisfies the ALC. The\nother choices lead to too many components. For $B_\\ell$ with $\\ell\\geq 4$, we\nhave instead\n\\begin{gather*}\n\\h^* \\simeq\n\\dyn\\root{1}\\link\\noroot{}\\link\\root{1}\\dots\\root{}\\llink>\\root{}\\edyn \n\\simeq \\h \\qquad S^2\\h= B\\oplus B'\\oplus B'' \\\\ \nB\\simeq\n\\dyn\\root{2}\\link\\noroot{}\\link\\root{}\\dots\\root{}\\llink>\\root{}\\edyn\\qquad\nB'\\simeq \\dyn\\root{2}\\link\\noroot{}\\link\\root{2}\\dots\\root{}\\llink>\n\\root{}\\edyn\\qquad\nB''\\simeq \\dyn\\root{}\\link\\noroot{}\\link\\root{}\n\\link\\root{1}\\dots\\root{}\\llink>\\root{}\\edyn,\n\\end{gather*}\nexcept that when $\\ell=4$, $B''=\\dyn\\root{}\\link\\noroot{}\\link\\root{}\n\\llink>\\root{2}\\edyn$. Now we check that $B'\\otimes \\h^*$ has six components,\n$B''\\otimes\\h^*$ has three components, but the ALC is again satisfied by $B$.\nLie contact geometries exist for $\\g=\\so(p,q)$ with $2\\leq p\\leq q$; $\\h$ is the\ntensor product of defining representations $\\R^2$ of $\\sgl(2,\\R)$ and\n$\\R^{p+q-4}$ of $\\so(p-2,q-2)$, and $B$ is the tensor product of a symmetric\nform on $\\R^2$ and the defining inner product of signature $(p-2,q-2)$ on\n$\\R^{p+q-4}$. See \\cite[\\S4.2.5]{CS} for more details on these\ngeometries.\n\nNext we consider $B_\\ell$ with the cross on $k$-th position, $3\\leq k\\leq\n\\ell-1$; the outcome is quite similar to the Lie contact case. For $k\\neq\n\\ell-1$, $S^2\\h= B \\oplus B'\\oplus B''$, where\n\\begin{gather*}\n\\h^*\\simeq\\dyn\\root{}\\dots\\root{1}\\link\\noroot{}\\link\\root{1}\\dots\n\\root{}\\llink>\\root{}\\edyn \\qquad\n\\h\\simeq\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{1}\\dots\n\\root{}\\llink>\\root{}\\edyn\\\\\nB\\simeq\\dyn\\root{2}\\dots\\root{}\\link\\noroot{}\\link\\root{}\\dots\n\\root{}\\llink>\\root{}\\edyn\\\\\nB'\\simeq\\dyn\\root{2}\\dots\\root{}\\link\\noroot{}\\link\\root{2}\\dots\n\\root{}\\llink>\\root{}\\edyn\\qquad\nB''\\simeq\n\\dyn\\root{}\\link\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{}\\link\\root{1}\n\\dots\\root{}\\llink>\\root{}\\edyn\\\\\n\\h^*\\otimes B\\simeq \\dyn\\root{2}\\dots\\root{1}\\link\\noroot{}\\link\\root{1}\\dots\n\\root{}\\llink>\\root{}\\edyn \\oplus\n\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{1}\\dots\n\\root{}\\llink>\\root{}\\edyn,\n\\end{gather*}\nso $B$ satisfies the ALC, but $B'$ and $B''$ do not. If $k=\\ell-1$,\n$S^2\\h=B\\oplus B'\\oplus B''$ with\n\\begin{gather*}\n\\h^* \\simeq \\dyn\\root{}\\dots\\root{1}\\link\\noroot{}\\llink>\\root{2}\\edyn\n\\qquad \\h \\simeq \\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\llink>\\root{2}\\edyn\\\\\nB\\simeq \\dyn\\root{2}\\dots\\root{}\\link\\noroot{}\\llink>\\root{}\\edyn\\qquad\nB'\\simeq \\dyn\\root{2}\\dots\\root{}\\link\\noroot{}\\llink>\\root{4}\\edyn\\qquad\nB''\\simeq \\dyn\\root{}\\link\\root{1}\n\\dots\\root{}\\link\\noroot{}\\llink>\\root{2}\\edyn\n\\end{gather*}\nand again, $B$ satisfies the ALC, but $B'$ and $B''$ do not. These\n$|2|$-graded geometries are modelled on the flag variety of isotropic\n$k$-planes and exist for the real forms $\\so(p,q)$ with $k\\leq p\\leq q$. We have\n$\\h\\cong \\R^k\\otimes \\R^{p+q-k}$ and $B$ corresponds to the tensor product of\na symmetric form on $\\R^k$ with the defining inner product on $\\R^{p+q-k}$.\n\\begin{case}[$B^{1,k}_\\ell$] Here $\\g\\simeq \\so(p,q)$ with $k\\leq p\\leq q$\nand $p+q=2\\ell+1$, and the geometries come equipped with the identification of\nthe horizontal distribution $\\cH\\leq TM$ with the tensor product\n$E\\otimes F$, where $E$ has rank $k$ and $F$ carries a metric of signature\n$(p-k,q-k)$. The corank of $\\cH\\leq TM$ is $\\frac12 k(k-1)$.\nThe metrics in $B$ are the tensor products of symmetric nondegenerate forms on\n$E$ and the given metric on $F$.\n\\end{case}\n\nFinally, we arrive at the cross at the very end. For $B_\\ell$ with $\\ell\\geq\n2$, we have\n\\begin{gather*}\n\\h^*\\simeq\\dyn\\root{}\\dots\\root{1}\\llink>\\noroot{}\\edyn\\qquad\n\\h\\simeq\\dyn\\root{1}\\dots\\root{}\\llink>\\noroot{}\\edyn\\qquad\nB= S^2\\h \\simeq \\dyn\\root{2}\\dots\\root{}\\llink>\\noroot{}\\edyn\\\\\n\\h^*\\otimes B \\simeq \\dyn\\root{3}\\llink>\\noroot{}\\edyn\\oplus\n\\dyn\\root{1}\\llink>\\noroot{}\\edyn (\\ell=2)\\qquad\n\\h^*\\otimes B \\simeq\n\\dyn\\root{2}\\dots\\root{1}\\llink>\\noroot{}\\edyn\n\\oplus \n\\dyn\\root{1}\\dots\\root{}\\llink>\\noroot{}\\edyn (\\ell\\geq 3),\n\\end{gather*}\nand the ALC is satisfied.\n\\begin{case}[$B^{1,\\ell}_\\ell$] Here $\\g$ is the split form $\\so(\\ell,\\ell+1)$.\nThe geometries are the well known free distributions, cf.~\\cite{DS}, with rank\n$\\ell$ horizontal distribution $\\cH\\leq TM$ of corank\n$\\frac12\\ell(\\ell-1)$. The metrics in $B$ are all nondegenerate metrics on\n$\\cH$.\n\\end{case}\n\n\\subsection{Proof of Theorem~\\ref{main} when $\\g$ has type $C_\\ell$}\n\nAs with type $B_\\ell$, we only have to consider cases with a single crossed\nnode. We begin with the first node crossed, corresponding to the well known\ncontact projective structures, with\n\\[\n\\h^*\\simeq\\dyn \\noroot{}\\link\\root{1}\\dots\\root{}\\llink<\\root{}\\edyn\\simeq\\h\\;;\n\\]\nwe have discussed the lowest dimension three already as the $B_2$ case, which\ncoincides with the free distribution of rank two. For $\\ell\\geq 3$, the picture\nchanges since\n\\begin{gather*}\nS^2\\h\\simeq \\dyn\n\\noroot{}\\link\\root{2}\\dots\\root{}\\llink<\\root{}\\edyn\\simeq B\\\\\nB\\otimes \\h^* \\simeq\n\\dyn\\noroot{}\\link\\root{3}\\dots\\root{}\\llink<\\root{}\\edyn\n\\oplus \\dyn \\noroot{}\\link\\root{2}\\link\\root{1}\\dots\\root{}\\llink<\\root{}\\edyn\n\\oplus \\dyn \\noroot{}\\link\\root{1}\\dots\\root{}\\llink<\\root{}\\edyn\n\\end{gather*}\nand thus the ALC fails.\n\nMoving on to the second node, we obtain another well known family of examples:\nthe quaternionic contact geometries (for $\\g\\cong\\symp(p,\\ell-p)$, $1\\leq p\\leq\n\\ell\/2$) or their split analogues (for $\\g\\cong\\symp(2\\ell,\\R)$)---see\n\\cite[\\S4.3.3]{CS}. For $\\ell=3$, we have\n\\begin{gather*}\n\\h^*\\simeq\\dyn \\root{1}\\link\\noroot{}\\llink<\\root{1}\\edyn\\simeq\\h\\qquad \nS^2\\h=B' \\oplus B''\\quad\\text{with}\\quad\nB'\\simeq \\dyn \\root{2}\\link\\noroot{}\\llink<\\root{2}\\edyn\n\\end{gather*}\nand $B''$ trivial, while for $\\ell\\geq 4$, we have\n\\begin{gather*}\n\\h^*\\simeq\\dyn \\root{1}\\link\\noroot{}\\link\\root{1}\\dots\n\\root{}\\llink<\\root{}\\edyn\\simeq \\h \\qquad S^2\\h=B \\oplus B'\\oplus B''\\\\\nB\\simeq \\dyn \\root{}\\link\\noroot{}\\link\\root{}\\llink<\\root{1}\\edyn \\quad\n\\text{or}\\quad \\dyn\\root{0}\\link\\noroot{}\\link\\root{}\\link\\root{1}\\dots\n\\root{}\\llink<\\root{}\\edyn\n\\qquad B'\\simeq\\dyn \\root{2}\\link\\noroot{}\\link\\root{2}\n\\dots\\root{}\\llink<\\root{}\\edyn\n\\end{gather*}\nand $B''$ trivial. Since $\\h^*\\otimes B'$ decomposes into four components,\nthere are only nontrivial possibilities for $\\ell\\geq 4$. For $\\ell=4$,\n\\begin{gather*}\n\\h^*\\otimes B \\simeq\n\\dyn \\root{1}\\link\\noroot{}\\link\\root{1}\\llink<\\root{1}\\edyn \n\\oplus\n\\dyn \\root{1}\\link\\noroot{}\\link\\root{1}\\llink<\\root{}\\edyn\n\\end{gather*}\nand so the ALC holds for $B$, but for $\\ell\\geq 5$, $ \\h^*\\otimes B$ has three\nirreducible components, and the ALC is not satisfied.\n\n\\begin{case}[$C^{1,2}_4$] Here the possible real Lie algebras\nare $\\symp(8,\\R)$, $ \\symp(2,2)$, or $\\symp(1,3)$, with the second node\ncrossed. In the first case, the geometries come equipped with the\nidentification of the horizontal distribution $\\cH\\leq TM$ with the\ntensor product $E\\otimes F$, where $E$ is rank $2$ and the rank $4$ vector\nbundle $F$ comes with a symplectic form. The eligible metrics in $B$ are the\ntensor products of a area form on $E$ and the given symplectic form on $F$. In\nthe quaternionic cases, $\\cH$ is quaternionic and the eligible metrics in $B$\nare quaternionic hermitian forms.\n\\end{case}\n\nLet us next suppose that the $k$-th node is crossed for $3\\leq k\\leq \\ell-2$.\nThen\n\\begin{gather*}\n\\h^*\\simeq\\dyn\\root{}\\dots\\root{1}\\link\\noroot{}\\link\\root{1}\\dots\\root{}\n\\llink<\\root{}\\edyn\n\\qquad \\h\\simeq\n\\dyn\\root{1}\\dots\\root{}\\link\\noroot{}\\link\\root{1}\\dots\\root{}\n\\llink<\\root{}\\edyn\\\\\nS^2\\h\\simeq B \\oplus B'\\oplus B''\\qquad\nB\\simeq \\dyn\\root{}\\link\\root{1}\\dots\\root{}\\link\\noroot{}\n\\link\\root{}\\dots\\root{}\\llink<\\root{}\\edyn\\\\\nB'\\simeq\\dyn\\root{2}\\dots\\root{}\\link\\noroot{}\\link\\root{2}\\dots\\root{}\n\\llink<\\root{}\\edyn\\qquad\nB''\\simeq \\dyn\\root{}\\link\\root{1}\\dots\\root{}\\link\\noroot{}\n\\link\\root{}\\link\\root{1}\\dots\\root{}\\llink<\\root{}\\edyn\\\\\n\\h^*\\otimes B \\simeq\n\\dyn\\root{}\\link\\root{1}\\dots\\root{1}\\link\\noroot{}\\link\\root{1}\\dots\n\\root{}\\llink<\\root{}\\edyn \\oplus \\dyn\\root{1}\\dots\n\\root{}\\link\\noroot{}\\link\\root{1}\\dots\\root{} \\llink<\\root{}\\edyn\n\\end{gather*}\nand so $B$ satisfies the ALC, but the other components do not. The relevant\nmetrics are again tensor products of an exterior form on the rank $k$\nauxiliary bundle $E$ and the given symplectic form on $F$ (where the\nhorizontal distribution is identified with $E\\otimes F$). These geometries\nare available for the split form $\\symp(2\\ell,\\R)$ and, if $k$ is even then\nalso for the real forms $\\symp(p,q)$, $k\\leq p$ 16.5 \\\\\n\\hline\n & 2 msec & 4 msec & 10 msec & 20 msec & 40 msec \\\\\n\\hline\nHigh Loop & R$_{int}$ $<$ 10 & 10 $<$ R$_{int}$$<$12 & 12 $<$ R\n$_{int}$$<$14 & 14 $<$ R$_{int}$$<$16.5 & R $>$ 16.5 \\\\\n\\hline\n & 2 msec & 4 msec & 10 msec & 20 msec & 40 msec \\\\\n\\hline\n\\end{tabular}\n\\caption{\\footnotesize In this table are listed the \nintegration times used for the two WFS with respect to the integrated \nmagnitude of the both references asterism. The values are tuned to the \nstatistical characteristics of the conjugated planes. }\\label{table:3}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|}\n\\hline\nLayer ID & Layers Altitude $[$m$]$ & Cn2 fraction & Wind $[$$^{m}$\/\n$_{s}$$]$ \\\\\n\\hline\n1 & 0 & 0.65 & 6.6 \\\\\n\\hline\n2 & 1800 & 0.08 & 12.4 \\\\\n\\hline\n3 & 3200 & 0.12 & 8.0 \\\\\n\\hline\n4 & 5800 & 0.03 & 33.7 \\\\\n\\hline\n5 & 7400 & 0.03 & 23.2 \\\\\n\\hline\n6 & 13100 & 0.08 & 22.2 \\\\\n\\hline\n7 & 15800 & 0.01 & 8.0 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\footnotesize Here are listed the atmospheric \nparameters used in the simulations. For each layer an outer-scale of 20 \nm has been considered. The isoplanatic angle for the overall atmosphere \nis about 15 arcsec at the 2.2$\\mu m$ pass band. This model is not the most \nrecent one where there is a bit more turbulence power in the ground \nlayer (67\\% instead of 65\\%). In this study a 0\".73 seeing in V Band and \n0\".66 seeing in R band were considered.}\\label{table:4}\n\\end{center}\n\\end{table}\n\n\\subsection{Optimization test}\nThe integration times used for both loops were set according to the \nintegrated magnitude of the asterism in the 6 arcmin annular FoV and in \nthe central 2 arcmin FoV, respectively for the Ground and the High WFS \n(see Table~\\ref{table:3}). This solution to set this important \ncouple of parameters is correct only for a first order approach. In \nfact, for example, it neglects the effect of the different illumination \nof the sub-apertures in the High WFS due to the references position and \ndifferent brightness. A smarter analysis should take into account a \nfine-tuning of the frame rates for the two WFSs. But an optimization \nprocedure to be applied to all the asterisms considered it's not \nfeasible because in this case the overall number of simulations \nperformed will increase too much. In other cases described so far\\cite{LNPDRLOSDA,MADLOSDA} we considered a grid of possible values for the two frame rates in \norder to taking into account the different combinations of the two \nintegration times. Here we considered a small $3\\times3$ grid with values \naround to the ones specified in Table~\\ref{table:3} (these \ndepending on the integrated magnitude) ranging from 25$\\%$ less and 25$\\%$ \nmore the two values taking into account for the simulation performed \nyet. We optimized the integration time only for a small set of the \nasterisms (20) used in the three 1$\\times$1 square degree fields. The results \nof this optimization compared to the non-optimized data allow \nextrapolating the optimized SR values for all the simulated cases. \n\n\\section{Data analysis}\n\\subsection{CPU time and Workstation}\nWe found in the catalogue 40000 useful stars over the 3 sky-fields \nconsidered. We analysed 3072 regions, running 2000 simulations on the \nfound asterisms. Each simulation took 3-6 hours of CPU time according to \nthe CPU clock (at most we used 2000 MHz). \n\nThe overall CPU time used was of 330days, the most spent on the Arcetri \nBeowulf cluster with 16 nodes, each one equipped with two CPUs. The LOST \ncode is not parellelized yet, but we ran ``in parallel\" different \nsimulations on different nodes at the same time.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=2.8in]{SPIE5490-941}}\n\\caption{\\footnotesize{In this plot the ``+\" sign defines the direction where the SR is computed in all the simulated cases, the step of this square grid is 24arcsec. The two circles represent the 1arcmin field and the 2arcmin. The correction was applied in the 2 arcmin FoV. In addition to these $6\\times6$ directions the SR data were computed also in the Natural Guide Stars positions.}}\\label{fig:2}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=4.0in]{SPIE5490-942}}\n\\caption{\\footnotesize{This picture shows a map of the analytical SR found for the Galactic Plane case. Each square covers 101$\\times$101arcsec region. If there is a lack of data or no good asterisms to drive the adaptive system this square is black. The color bar on the right indicates the value of the SR. For example: on the left of this map there is a very bright star that saturates all the plates taken into account for the catalogue preparation: in fact a circular hole without stars appears in our analysis. The analytical SR gives only an idea of the possible SR achievable because it does not take into account neither the distribution of the stars in the 6 and 2 arcmin FoV nor the turbulence profile.}}\\label{fig:3}\n\\end{figure}\n\nWe developed an IDL procedure to manage the different steps described in \nthe section above. Moreover this found the best asterism for each couple \nof 2arcmin and 6arcmin fields; the best integration times for the two \nloops and it computed an analytical SR over the 3 1$\\times$1 deg$^{2}$ \nfields considering the asterism and integration times found before.\n\n\\subsection{Data analysis}\nFor each simulation LOST computed the SR values over the 2arcmin FoV. A \n6$\\times$6 grid of SR evolution data was retrieved for each of the 32$\\times$32 positions in the three 1-degree field (see Figure~\\ref{fig:2}). Using these data the averages SR were computed on the 1arcmin circle \nand the 2-arcmin FoV. Moreover the on axis direction SR was taken into \naccount averaging the 4 ``probe\" stars close to the centre of the field \n(see Figure~\\ref{fig:2}). \n\n\n\nWe optimize the MCAO system parameters looking for the loops closure and \nthe robustness of the correction and we do not optimize the system in \norder to achieve high SR (more than 60\\%). In fact considering only the \neffect of the spatial sampling of the metapupils for the ground (8$\\times$8) \nand for the high loop ($7\\times7$) the best SR achievable is between 55\\%-60\\% \nwith the atmospheric parameters here considered.\n\nEach simulation performed has an iteration step of 2 msec for a total of \n250 iterations that gives an overall time of 0.5 seconds. The length of \nthese simulations is not enough to estimate a representative long \nexposure SR because of the effect of the bootstrap, so we assumed the \nmaximum SR achieved during the run as reference for our analysis \n(the SR values we consider take into account the tip-tilt residual \nalso). Analyzing the SR data of the 3 different 1 square degree fields \nconsidered, we drew different SR maps according to the 3 FoV sizes we \nassumed. These cases can be seen as representative of different \ninstruments:\n\\begin{itemize}\n\\item A camera with few arcsec FoV;\\\\\n\\item An instrument with 1 arcmin FoV;\\\\\n\\item An instrument, or more instruments mounted on the same system \ncovering the corrected 2arcmin field.\n\\end{itemize}\n\n\\subsubsection{Few arcsec FoV Case}\\label{section:a3b2c1}\nWe assumed mounted at the focus of the telescope a camera with a small FoV of few arcsec centered in the optical axis direction. In this case the SR must be uniform because the FoV is smaller than the isoplanatic patch size. So we considered representative for this case the value of the SR obtained for the 4 ``probe\" stars more close to the centre of the FoV (see Figure~\\ref{fig:2}). Figure~\\ref{fig:4} presents the results in terms of sky coverage VS threshold SR:\n\\begin{figure}\n\\centerline{\\includegraphics[width=3.6in]{SPIE5490-943}}\n\\caption{\\footnotesize{This picture shows the sky coverage results for the three galactic latitude cases taken into account and relative to the on axis direction only. The functions plotted here represent the percentage of the simulated case where at least the SR showed in abscissa was achieved. Dotted line represents the North Galactic Pole; the dashed one refers to the South Galactic Pole and the solid to the Galactic Anti-centre. The percentage is relative to the 32$\\times$32 directions considered for each galactic field.}}\\label{fig:4}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\includegraphics[width=3.6in]{SPIE5490-944}}\n\\caption{\\footnotesize{This picture shows the results for the central 1arcmin FoV case for the three galactic latitudes taken into account. The functions plotted here represent the percentage where at least the SR showed in abscissa was achieved. Dotted line represents the North Galactic Pole; the dashed one refers to the South Galactic Pole and the solid to the Galactic Anti-centre.}}\\label{fig:5}\n\\end{figure}\n\nIn the 98$\\%$ of the cases taken into account the SR on axis was higher \nthan 10$\\%$, while percentages of 48$\\%$ and 25$\\%$ were retrieved \nrespectively for the North and the South Galactic poles. For the low \nGalactic latitudes in half of the cases considered the SR on axis was \nhigher than 40\\%. \n\n\\subsubsection{1 arcmin FoV case}\\label{section:a3b2c2}\nNow we describe the sky coverage analysis for an instrument with 1arcmin \nFoV centered in the axis direction (the same axis relative to the \ncorrection applied by the adaptive system). For this case the \nrepresentative SR is the average SR over the central one arcmin computed \nby the simulations. The Figure~\\ref{fig:5} shows the results \nrelative to this field size: for the low galactic latitude (the Galactic \nAnticentre) in the 98\\% of the directions considered the average SR was \nhigher than 10\\%, while the same values for the North and South Poles \nwere 38\\% and 17\\% respectively.\n\n\\subsubsection{2 arcmin FoV case}\\label{section:a3b2c3}\nIn this last case we supposed several instruments (or a unique big \ncamera) observing the whole region corrected by the MCAO system (we \nconsidered a corrected FoV of 2 arcmin). As in the 1arcmin case we took \ninto account the average SR, but now over the 2arcmin FoV. \n\nIn the the results are presented: considering a 10\\% threshold for the \nSR as condition to define the coverage we found sky coverage of 99\\% for \nGalactic plane, 25\\% and 13\\% respectively for the North and South Galactic poles. \n\nWe want to stress that the sky-coverage values within the 3 FoV sizes \nchanges of a factor $\\sim 2$ for the galactic poles while it is un-changed \nfor the Galactic anticentre (see Figure~\\ref{fig:7}). This \ndifferent behaviour depends on the different stars density of the two \ngalactic regions. The poles are poor of stars with respect to the low \ngalactic latitudes: this translates in a less number of reference stars \nfor the galactic poles and so a lower uniformity for the correction with \nrespect to the galactic plane where it is quite easy to cover \nhomogenously the corrected 2arcmin FoV with natural guide stars. \n\n\\begin{figure}\n\\centerline{\\includegraphics[width=3.6in]{SPIE5490-945}}\n\\caption{\\footnotesize{This picture shows the results in the overall corrected 2 arcmin FoV and for the three galactic latitude cases analyzed. The functions plotted here are representing the percentages where at least the SR showed in abscissa was achieved. Dotted line represents the North Galactic Pole; the dashed one refers to the South Galactic Pole and the solid to the Galactic Anti-centre. The SR here considered is average SR over the corrected field of 2 arcmin.}}\\label{fig:6}\n\\end{figure}\n\n\\subsubsection{Dealing to a different definition}\nNow we want to analyse the results presented in the previous sections \n(\\ref{section:a3b2c1}, \\ref{section:a3b2c2} and \n\\ref{section:a3b2c3}) according to a different definition of sky \ncoverage, using, for example, that one given in the {reference\\cite{2003SPIE.4839..566M}}, and that \nwe discussed also above (section~\\ref{section:_Ref76797586}). We assume \nas reference for the infinite SNR SR the maximum SR achieved in each of \nthe 3 cases of Field of View considered that are: $\\sim$0.6 for the few \narcsec field of view case, $\\sim$0.5 for the 1 arcmin case and $\\sim$0.4 for the \n2 arcmin FoV. Using these values we found the SR$_{50}$ thresholds \n(50\\% of the SR relative to the infinite SNR case) for each of these \ncases: 0.3, 0.25 and 0.2 respectively. Applying these thresholds instead \nof the 10\\% one, we found coverage of 90\\% for the low galactic \nlatitudes case and between 7\\%and 15\\% for the galactic poles. Even if \nreferring to different wavelengths these values agree with the ones \ngiven in the references\\cite{mfov,2003SPIE.4839..566M} (here we considered correction in the K \nband while it was the R band in the {reference\\cite{2003SPIE.4839..566M}}).\n\\begin{figure}\n\\centerline{\\includegraphics[width=3.6in]{SPIE5490-946}}\n\\caption{\\footnotesize{This figure shows the percentage of sky-coverage with respect to a threshold SR. Dashed line represents the coverage with respect to the on axis direction SR; the dotted one refers to the average SR over 1 arcmin FoV and the solid line to the SR averaged over the 2arcmin corrected field. All the 3 curves refer to the analysis performed on the 1$\\times$1 square degree field centered in the North Galactic Pole.}}\\label{fig:7}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\includegraphics[width=4.0in,height=3.2in]{SPIE5490-947}}\n\\caption{\\footnotesize{This figure shows the results for the Galactic Anti-Centre in the on axis case. Different colors indicate different SR values.}}\\label{fig:8}\n\\end{figure}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|}\n\\hline\n & North Galactic Pole & South Galactic Pole & Galactic Anticentre \\\\\n\\hline\nOn Axis(Few arcsec FoV) & 48 \\% & 25 \\% & 99 \\% \\\\\n\\hline\n1 arcminFoV & 38 \\% & 17 \\% & 97 \\% \\\\\n\\hline\n2 arcminFoV & 25 \\% & 13 \\% & 99 \\% \\\\\n\\hline\n\\end{tabular}\n\\caption{In this table are summarized the results \nfor the different cases analyzed. A limit SR of 10\\% was assumed.}\\label{table:5}\n\\end{center}\n\\end{table}\n\n\\section{Example of SCIENCE-COVERAGE: cluster of galaxies at high \nred-shift }\n\n\nWhat we said and stressed about different sky-coverage for different FoV \nbecomes important when we consider the possible astronomical \napplications. For example we took as possible astronomical target the \ncluster of galaxies. As everybody knows their apparent dimension changes \nwith their distance, but because of the structure and evolution of the \nuniverse this depend also by cosmological parameters. \n\nThe angular size of clusters is related to the red-shift (z) then for \nthese objects the sky-coverage is a function of the z. We plotted the \nangular size taken from the reference with respect to the redshift\\cite{2004A&A...417...13E} in \nFigure~\\ref{fig:9}.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=4.0in]{ettori.eps}}\n\\caption{\\footnotesize{This figure shows the angular size with respect of the red-shift according to the data in the {reference\\cite{2004A&A...417...13E}}. We plotted a dashed line to a radius of 1 arcmin, corresponding to the 2arcmin FoV case we considered in the sky-coverage analysis. Following our results the clusters at $z \\sim0.9$ have sky coverage of 25\\% (at North Galactic Pole).}}\\label{fig:9}\n\\end{figure}\n\n\n\n\n\nThis plot says that high-z clusters have higher sky-coverage. If the \ncluster has dimension bigger than 2~arcmin then more \ncontiguous asterisms are needed to cover its entire dimension and, in this case sky \ncoverage decreases. \n\n\\section{Conclusions}\n\n\nWe analyse the sky coverage problem in the case of a specific Layer \nOriented Multiple Field of View system. We showed basic relationship \nbetween sky coverage and field dimension to be studied and presented a \nscientific case. We analyse the definition of sky coverage and we \nstressed that it must be related to the performance requested to the \nadaptive system and the class of objects to be studied with the \nscientific instrument to be used. In particular we set a reasonable \nthreshold to the 10\\% in SR as condition to define if there is sky \ncoverage or not. We showed that for low galactic latitudes the \ncorrection is feasible about everywhere while at the galactic poles the \ncoverage decreases, but down to reasonable values (20\\%-40\\%) to justify \nthe use of this natural guide stars technique also for high galactic \nlatitudes targets.\n\n \n\n\\section*{ACKNOWLEDGEMENTS}\n\n\n\nThanks to M. Le Louarn for the atmosphere profile of the Cerro Paranal \n(Chile), and to A. Puglisi as ``problem-solver\" regarding the Beowulf \ncluster.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\nQuasinormal ringing is the principal gravitational-wave signature of\nthe final black hole after a binary merger. This is described by a\nspectrum of complex quasinormal frequencies $\\omega_{lmn}$, which are\nuniquely specified in linear perturbation theory by the mass and spin\nof the Kerr background~\\cite{KokkotasSchmidt1999,Nollert:1999ji,Berti2009}. Precise measurement of these frequencies\ntherefore characterizes the background~\\cite{Echeverria89} and moreover constrains\ndeviations from general relativity (with more than one mode, or when\ncombined with other measurements)~\\cite{Dreyer:2003bv,Berti:2005ys,Brito:2018rfr,LIGOScientific:2021sio}. Although data today already hint at\nmodes beyond the fundamental~\\cite{Isi:2019aib,Cotesta:2022pci,Finch:2022ynt,Capano:2021etf}, future observations with sensitive\ndetectors are sure to enable detailed spectroscopy~\\cite{Berti:2005ys,Bhagwat:2021kwv,Ota:2019bzl}.\n\nTo interpret future observations, however, it will be necessary to\nunderstand quasinormal mode interactions. The ringdown follows a highly\nnonlinear phase (the merger) and although numerical calculations indicate that a sum of\nmodes may be sufficient to represent the gravitational-wave emission~\\cite{Giesler:2019uxc,Mourier:2020mwa,Chen:2022dxt},\nit is not clear that this corresponds to a full nonlinear\ndescription. Indeed, nonlinear ringdown effects have been identified in numerical simulations of binary mergers~\\cite{Mitman:2022qdl,Cheung:2022rbm} as well as in anti-de Sitter black holes~\\cite{Bantilan:2012vu,Sberna:2021eui}.\nIn other contexts (e.g., perturbations of large\nanti-de Sitter black holes) quasinormal modes can interact and even\nbecome turbulent~\\cite{Green:2013zba,Adams:2013vsa}. The point of this paper is to introduce some tools that may be helpful when \ndeveloping a theory of quasinormal mode interactions. \n\nCompared to normal modes, quasinormal modes do not in general form in a \nstraightforward sense a\ncomplete ``basis'' of solutions to the linearized field equations. In fact, black\nhole perturbations are only described by quasinormal modes for an\nintermediate time period in their evolution; at early times they are\ndescribed by a free propagation piece, and at\nlate times by a power law tail~\\cite{Price1972a,Leaver1986b,Ching:1995tj}. \nThe spatial wavefunction of a (decaying) quasinormal mode also\n\\emph{diverges} at the bifurcation surface and at spatial\ninfinity. This makes it hard to write down canonical (conserved) $L^2$-type inner products based on the usual Cauchy-surfaces\nof Kerr.\\footnote{Note however that one may choose hyperboloidal slices \\cite{Zenginoglu:2011jz,PanossoMacedo:2019npm,Ripley:2022ypi,gajic2021quasinormal}; see the conclusions for a discussion of this alternative in connection with our approach.}.\nWithout an inner product, it is not \nclear how to project onto quasinormal modes to study nonlinear\nmode mixing.\n\nThe main goal of this paper is to point out an unconventional bilinear form which may take\nthe place (for some purposes) of an inner product on quasinormal modes of Kerr. Before \nwe introduce this notion, we develop a general theory for conserved -- under time evolution --\nbilinear forms for Weyl scalars or metric perturbations. Similar to \\cite{carter1977killing}, the key idea is \nto start with a ``Klein-Gordon'' type current for Weyl scalars or metric perturbations and \nto apply symmetry operators to the entries of this bilinear expression. As we show, in Kerr spacetimes, \nsuch symmetry operators include, besides the obvious ones descending from the \nKilling symmetry, also an infinite tower of operators built from Carter's Killing tensor. \n(For the Weyl scalars, the symmetry operator of lowest differential order has two derivatives; \nfor metric perturbations, it has six derivatives). \nIn particular, using a combination of such operators we find an infinite set of new conserved, local, gauge invariant current associated with Carter's constant \\cite{Carter:1968ks} in Kerr.\\footnote{\n\\label{footnote1}\nFor an explanation of the relation with previous works \\cite{carter1977killing, carter1979generalized, grant2020class, grant2020conserved, andersson2015spin,aksteiner2019symmetries}, see section \\ref{sec:Symmetry}.}\n\nThe bilinear form of main interest for this paper is, however, not obtained from such differential symmetry operators but rather the symmetry operator associated with the discrete $t$--$\\phi$ reflection. We show that gravitational quasinormal modes\nwith different frequencies are orthogonal with respect to this\nbilinear form. For the reader interested in the main result, the bilinear form is presented explicitly for quasinormal modes in \\eqref{eq:mode-bilinear}. We show furthermore that the quasinormal mode excitation coefficients of a solution are given precisely by the projection of data onto the corresponding modes via the bilinear form.\n\n\nThe plan of this paper is as follows. In section \\ref{sec:Bilinear} we recall the standard recipe for \nconstructing conserved bilinear forms for partial differential operators. In section \\ref{sec:Symmetry}\nwe introduce symmetry operators (including symmetry operators related to the Killing tensor, see also footnote \\ref{footnote1}) \nto construct further conserved bilinear forms, and currents.\n In section \\ref{sec:bilinear_tphi} we construct the bilinear form $\\langle\\langle \\cdot , \\cdot \\rangle\\rangle$ \n using the $t$--$\\phi$ reflection symmetry, which gives orthogonality of quasinormal modes in section \\ref{sec:Ortho}.\nFinally, in section \\ref{sec:Lap transform} we explain the \nrelation with excitation coefficients. Some technical aspects of this paper are deferred to various appendices. \n\n\n\n\\section{Bilinear form -- basic construction}\n\\label{sec:Bilinear}\n\nConsider a partial differential operator $\\mathcal X$ acting on sections of some vector bundle, $E$, \nover a manifold $M$. We assume that $M$ is equipped with a volume form, \n$\\epsilon_{a_1 \\dots a_n}$; later we will always have a metric $g_{ab}$, so the volume form \nis chosen as the one compatible with the metric. Let $\\tilde E$ be the dual vector bundle, i.e., \neach fibre is given by the ${\\mathbb C}$-linear maps of the corresponding fiber of $E$. If $\\psi$ is a \nsection of $E$ and $\\tilde \\psi$ is a section of $\\tilde E$, we can pointwise form the scalar \n$\\tilde \\psi \\psi \\in {\\mathbb C}$. The formal adjoint is the unique differential operator $\\mathcal X^\\dagger$\ndefined by the formula \n\\begin{equation}\n(\\mathcal X^\\dagger \\tilde \\psi) \\psi - \\tilde \\psi \\mathcal X \\psi = \\nabla_a x^a[\\tilde \\psi, \\psi], \n\\end{equation}\nwhere $x^a[\\tilde \\psi, \\psi]$ is local, i.e., at any point built from finitely many \nderivatives of the fields at that point. The divergence operator on the right is defined by our \nvolume form and if it comes from a metric, as we assume from now, it \nis equal to the usual covariant derivative operator. Said differently, $\\mathcal X^\\dagger \\tilde \\psi$ is \nobtained by the usual ``partial integration'' procedure dropping surface terms as if the above \nequation were placed under an integral sign. Note that, by contrast to quantum mechanics, \n$\\dagger$ as defined above is ${\\mathbb C}$-linear, rather than anti-linear.\n\nNow let $(\\tilde \\psi, \\psi)$ be a pair of solutions to $\\mathcal X \\psi = 0 = \\mathcal X^\\dagger \\tilde \\psi$, and let \n$\\Sigma$ be a codimension 1 submanifold of $M$ (later to be chosen as a constant $t$ slice of Kerr). \nThen, by Gauss' theorem, if $\\tilde \\psi$, $\\psi$ have sufficient decay on $\\Sigma$ for the following integral to be suitably convergent (e.g., if they are compactly supported), \nthen the bilinear form \n\\begin{equation}\n\\label{Xdef}\nX[\\tilde \\psi, \\psi] := \\int_\\Sigma x^a [\\tilde \\psi, \\psi] \\, {\\rm d} \\Sigma_a \\equiv \\int_\\Sigma (\\star x) [\\tilde \\psi, \\psi]\n\\end{equation}\nis unchanged under local deformations of $\\Sigma$, and we say that it is ``conserved''.\n(Here $\\star$ denotes the Hodge dual.) \nAs a simple example, consider $\\mathcal X = \\nabla^a \\nabla_a - m^2$, the Klein-Gordon operator acting on real-valued functions $\\psi$, \nso $E = \\tilde E = {\\mathbb R}$ is the trivial bundle. Then $\\mathcal X^\\dagger = \\mathcal X$ and $x^a = - \\tilde \\psi \\nabla^a \\psi + \\psi \\nabla^a \\tilde \\psi$ is the Klein-Gordon (symplectic) current, \nwhich is of course conserved for any pair of solutions. \nThe bilinear form in this case is just the symplectic form for Klein-Gordon theory. It is anti-symmetric \nunder $\\tilde \\psi \\leftrightarrow \\psi$, but note that in the general case we cannot say that about the bilinear form since the bundles $E$\nand $\\tilde E$ cannot usually be identified in a natural way. \n\nAs a second example, let $\\mathcal{E}$ be the linearized Einstein operator on a Ricci-flat \u00b4spacetime. It acts on symmetric covariant rank-2\ntensors $h_{ab}$, so $E$ is equal to ${\\rm Sym}(T^*M \\otimes T^*M)$ in this case, and the dual bundle $\\tilde E$ corresponds to symmetric contravariant \nrank-2 tensors, ${\\rm Sym}(TM \\otimes TM)$. The formula is\n\\begin{align}\\label{eq:linearE}\n \\mathcal{E}_{ab}(h) \\equiv \\frac{1}{2}\\big[ &-\\nabla^c\\nabla_c h_{ab} - \\nabla_a\\nabla_bh + 2 \\nabla^c\\nabla_{(a} h_{b)c} \\nonumber\\\\\n & + g_{ab}(\\nabla^c \\nabla_c h - \\nabla^c\\nabla^d h_{cd}) \\big],\n\\end{align}\nand under the identification of $E$ with $\\tilde E$ (by using the metric $g^{ab}$ to raise indices), we have $\\mathcal{E}^\\dagger = \\mathcal{E}$.\nAs in the Klein-Gordon case, this last relation follows because the linearized \nEinstein equation arises from an action principle. By explicit calculation, the boundary term $w^a \\equiv x^a[\\tilde h, h]$\nis given by \\cite{iyer1994some}\n\\begin{equation}\n\\label{wadef}\nw^a = \np^{abcdef}\\left( h_{bc} \\nabla_d \\tilde h_{ef} - \\tilde h_{bc} \\nabla_d h_{ef}\\right),\n\\end{equation}\nwhere\n\\begin{align}\n p^{abcdef} = &g^{ae}g^{fb}g^{cd} - \\frac{1}{2}g^{ad}g^{be}g^{fc} - \\frac{1}{2}g^{ab}g^{cd}g^{df}\\nonumber\\\\ \n &- \\frac{1}{2}g^{bc}g^{ae}g^{fd} + \\frac{1}{2}g^{bc}g^{ad}g^{ef}.\n\\end{align}\nThe bilinear form \n\\begin{equation}\n\\label{Wdef}\nW[\\tilde h, h] = \\int_\\Sigma\np^{abcdef}\\left( h_{bc} \\nabla_d \\tilde h_{ef} - \\tilde h_{bc} \\nabla_d h_{ef}\\right) {\\rm d} \\Sigma_a ,\n\\end{equation}\nis the symplectic form of General Relativity \\cite{iyer1994some}.\n\nOur third, and most important, example concerns the Teukolsky operator(s) for the perturbed Weyl scalars of the Kerr spacetime $(M,g_{ab})$, \nto which we will restrict attention from now on. For this, we shall employ the GHP\nformalism~\\cite{Geroch:1973am,Bini:2002jx,Aksteiner:2010rh,Toth:2018ybm} in the following, and we now briefly review the essential portions of this formalism which \nsimplifies and also conceptualizes many calculations in the\nKerr (or more generally, Petrov type D) geometry. $l^a$ and $n^a$ are taken to be the repeated\nprincipal null directions which are completed to a null tetrad by defining a smooth pair of\ncomplex null rays $(m^a, \\bar m^a)$ that span the remaining\ndimensions. We choose the normalization $l_an^a=1$ and\n$m_a\\bar m^a=-1$, corresponding to the $-2$ signature. The metric\nthen takes the form\n\\begin{equation}\\label{eq:NP met}\ng_{ab} = 2l_{(a}n_{b)}-2m_{(a}\\bar m_{b)}.\n\\end{equation}\nThe basic idea is to contract any tensor field on $M$ into\nthe legs of the Newman-Penrose (NP) tetrad $(l^a, n^a, m^a, \\bar m^a)$\nin all possible ways\\footnote{We do not require tensor fields to be\n \\emph{fully} contracted with the tetrad, so in general we refer to\n NP \\emph{tensors}, not just scalars. In other words, there can\n remain tensor indices after contraction.} and to represent the\naction of the covariant derivative operator $\\nabla_a$ in terms of\nthese tetrad components, in a way that preserves a natural grading by\nspin and boost weights.\n\nFields $\\eta$ obtained by contracting with the tetrad are\nclassified according to their spin and boost weights as follows. Under\na local rotation that preserves the real null pair, the tetrad\ntransforms as $(l^a, n^a, e^{i\\Gamma} m^a, e^{-i\\Gamma} \\bar m^a)$,\nwhereas under a local boost that preserves the directions of the real\nnull pair, it transforms as\n$(\\Lambda l^a, \\Lambda^{-1}n^a, m^a, \\bar m^a)$, where $\\Lambda$,\n$\\Gamma$ are smooth real-valued functions. If we combine these\nfunctions into the complex function $\\lambda^2 = \\Lambda e^{i\\Gamma}$,\nthen $\\eta$ is said to possess (real) GHP weights $(p,q)$ if under\nthe above combined local rotation and boost of the tetrad, it\ntransforms as\n\\begin{equation}\\label{trafo}\n\\eta \\to \\lambda^p \\bar \\lambda^q \\eta. \n\\end{equation}\nWe write $\\eta \\ensuremath{\\circeq} (p,q)$ if this is the case. In the GHP\nformalism, only quantities with the same weight may be added, whereas\nweights behave additively under multiplication.\n\nFrom the mathematical viewpoint, the GHP formalism can be understood in terms of principal fibre bundles and\ntheir associated vector bundles, as follows. Consider the set of oriented null frames aligned with the given null directions. On each such frame, we may pointwise \nperform a boost\/rotation, which as we described can be combined into a nonzero complex number $\\lambda \\in {\\mathbb C}_\\times$. Thus, we have a multiplicative action of \n${\\mathbb C}_\\times$ on the set of frames which gives this set the structure of a principal $G$-bundle: A principal bundle is abstractly a bundle $P$ over $M$ such that a group $G$\ncan act by right multiplication $X \\to X \\cdot g$ in the fibre -- in our case $X$ is an NP frame aligned with the principal null direstions and $g \\leftrightarrow \\lambda$. Given a principal $G$-bundle and a representation $R$ of $G$ on some vector space $V$, there is a canonical construction of an ``associated'' vector bundle. The sections of this bundle correspond physically to quantities defined on $M$ that ``transform in the representation $R$''.\nMore precisely, the elements in this associated bundle are the equivalence classes of pairs $(X,v)$ where $X \\in P$ \nand $v \\in V$ where $(X,v)$ is declared to be equivalent to \n$(X \\cdot g, R(g)v)$. In the present example, $R_{p,q}(\\lambda)v = \\lambda^p \\bar \\lambda^q v$ and $V = {\\mathbb C}$, which corresponds precisely to the ``transformation law'' \\eqref{trafo}. The associated vector bundle is denoted in general by $P \\ltimes_R V$ and its fibres are isomorphic to $V$. In our case, we get 1-dimensional complex (``line'') bundles $L_{p,q}=P \\ltimes_{p,q} {\\mathbb C}$ over $M$ labelled by the GHP weights $(p,q)$. The number $s=\\frac{1}{2}(p-q)$ is commonly referred to as the spin. Of course, we could tensor $L_{p,q}$ with the usual tensor bundles $T^{(r,s)} M$ to host objects that have GHP weights and tensor indices at the same time such as $l^a$ or $R_{abcd} m^a m^d$.\n\nThe advantage of the above invariant viewpoint involving associated vector bundles is that we can naturally see what quantities are defined in a frame independent manner, which quantities can naturally be added, etc. This provides not only an extremely useful guiding principle in the -- usually very complicated -- calculations related to Kerr, but also means that one is always intrinsically dealing with objects that behave in a well-defined manner under a change of frame. To make the formalism really useful, one needs covariant derivative operators on the bundles $L_{p,q}$. These are given by\n\\begin{equation}\n\\Theta_a = \\nabla_a - \\tfrac{1}{2} (p-q) \\bar m^b \\nabla_a m_b -\\tfrac{1}{2} (p+q) n^b \\nabla_a l_b.\n\\end{equation}\nThe Teukolsky operators also feature the ``gravito-magnetic potential'' which is given by\n\\begin{equation}\n\\label{Bdef}\nB^a \\equiv -(\\rho n^a - \\tau \\bar m^a) \\ensuremath{\\circeq} (0,0), \n\\end{equation}\nwhere $\\rho, \\tau$ are related to spin-coefficients~\\cite{Geroch:1973am,Bini:2002jx,Aksteiner:2010rh,Toth:2018ybm}; see appendix \\ref{app:D}.\nThe Teukolsky operator acts on GHP-scalars of the same weight\\footnote{For the definition of $\\mathcal O$ and \n $\\mathcal{O}^\\dagger$ for general GHP weights see appendix \\ref{app:B}.} as the perturbed Weyl scalar $\\psi_0$, \ni.e., $(p,q) = (4,0)$ and is given by \n\\begin{equation}\n \\mathcal{O} = g^{ab}(\\Theta_a + 4 B_a)(\\Theta_b + 4 B_b) - 16 \\Psi_2 \n\\end{equation}\n with $\\Psi_2$ a background Weyl-scalar. So $E=L_{4,0}$ now. Since the dual vector bundle to $L_{p,q}$ is $L_{-p,-q}$, the adjoint \n Teukolsky operator $\\mathcal{O}^\\dagger$ acts on GHP scalars of weight $(-4,0)$. It is given by\n \\begin{equation}\\label{eq:Odagger}\n\\mathcal{O}^\\dagger = g^{ab}(\\Theta_a - 4 B_a)(\\Theta_b - 4 B_b) - 16 \\Psi_2 .\n\\end{equation} \n It follows that the boundary term $x^a[\\tilde \\Upsilon, \\Upsilon] \\equiv \\pi^a$ (with $\\tilde \\Upsilon \\ensuremath{\\circeq} (4,0), \\Upsilon \\ensuremath{\\circeq} (-4,0)$) \n is given in the case of the Teukolsky operator by \n \\begin{equation}\n \\label{pidef}\n \\pi^a = \\tilde \\Upsilon(\\Theta^a - 4B^a)\\Upsilon - \\Upsilon (\\Theta^a + 4 B^a) \\tilde \\Upsilon \n \\end{equation}\nWe denote the corresponding bilinear form -- formally similar to the Klein-Gordon inner product of a charged scalar field -- by \n\\begin{equation}\n\\label{Pidef}\n\\Pi[\\tilde \\Upsilon, \\Upsilon] = \\int_\\Sigma \\left[ \\tilde \\Upsilon(\\Theta^a - 4B^a)\\Upsilon - \\Upsilon (\\Theta^a + 4 B^a) \\tilde \\Upsilon \\right] {\\rm d} \\Sigma_a.\n\\end{equation}\n\nThe Teukolsky equation\/operator and the linearized Einstein equation\/operator are well-known to be related and this implies that the \nbilinear forms $W$ and $\\Pi$ as in \\eqref{Wdef} and \\eqref{Pidef} are related, too. \\cite{Prabhu:2018jvy} have shown that for $\\Upsilon$ a\nsmooth solution to $\\mathcal O^\\dagger \\Upsilon=0$ arising from compact\nsupport data and $h_{ab}$ a smooth solution to\n$\\mathcal E h_{ab} = 0$, an identity of the following form holds\n\\begin{equation}\\label{eq:intertwine_inf}\nw^a[h, \\mathcal S^\\dagger \\Upsilon] = - \\pi^a[\\mathcal T h, \\Upsilon] + \\nabla_b H^{ab}[\\Upsilon, h], \n\\end{equation}\nwhere $H^{ab}$ is a skew symmetric local tensor. Furthermore~\\cite{Aksteiner:2014thesis,Araneda:2016iwr}\n\\begin{subequations}\n \\begin{align}\n \\mathcal{S}(T) &= Z^{bcda} (\\Theta_a + 4 B_a) \\Theta_b T_{cd},\\\\\n \\mathcal{T}(h) &= - \\frac{1}{2}Z^{bcda} \\Theta_a \\Theta_b h_{cd},\n \\end{align}\n\\end{subequations}\nwhere $Z^{abcd} \\equiv Z^{ab}Z^{cd}$, and $Z^{ab} \\equiv 2l^{[a}m^{b]}$, are operators such that the Teukolsky-Wald identity holds:\n\\begin{equation}\\label{SEOT}\n\\mathcal S \\mathcal E = \\mathcal O \\mathcal T.\n\\end{equation}\nThis equation encodes that the action of $\\mathcal T h$ on a metric perturbation $h_{ab}$ (which equals the perturbed Weyl scalar $\\psi_0$)\ngives a solution to Teukolsky's equation $\\mathcal{O} \\psi_0 = 0$. Conversely, taking an adjoint of \\eqref{SEOT}, i.e., \n$\\mathcal{E} \\mathcal{S}^\\dagger = \\mathcal{T}^\\dagger \\mathcal{O}^\\dagger$, shows that any solution $\\mathcal{O}^\\dagger \\Upsilon = 0$ of \nGHP weight $(-4,0)$ (``Hertz potential'') is such that $h_{ab} = \\Re \\mathcal{S}^\\dagger_{ab} \\Upsilon$ is a solution to the linearized Einstein equations. \n\nRef.~\\cite{Prabhu:2018jvy} did not derive the explicit form for $H^{ab}$ but \nargued for the above equation \\eqref{eq:intertwine_inf} to hold on general grounds based on \\eqref{SEOT}. The main use of the above identity \n\\eqref{eq:intertwine_inf} is to \nrelate the corresponding bilinear forms $W[h, \\mathcal S^\\dagger \\Upsilon]$ and $\\Pi[\\mathcal T h, \\Upsilon]$ for a Cauchy surface $\\Sigma$\nof the exterior of Kerr. This identity is obtained by simply integrating the above identity over $\\Sigma$. If all \nfields are falling off rapidly at the horizon and spatial infinity, then the boundary term arising from $H^{ab}$ will not contribute; \nin other cases, $H^{ab}$ will contribute surface terms. Their computation is fairly long and non-trivial and therefore deferred to appendix \\ref{app:A}. \nIf $\\Sigma$ is a co-dimension one surface with boundary $\\partial \\Sigma$,\n$\\Upsilon$ is a\nsmooth solution to $\\mathcal O^\\dagger \\Upsilon=0$ and $h_{ab}$ a smooth solution to\n$\\mathcal E h_{ab} = 0$, then we have\n\\begin{equation}\\label{eq:intertwine}\n W[h, \\mathcal S^\\dagger \\Upsilon] = - \\Pi[\\mathcal T h, \\Upsilon] + B[h,\\Upsilon] \n\\end{equation}\nwhere $B = \\int_{\\partial \\Sigma} H^{ab} {\\rm d} \\Sigma_{ab}$. When $\\Sigma$ is \na slice of constant $t$ in Boyer-Lindquist coordinates, $\\partial \\Sigma$ would correspond to the bifurcation surface at $r=r_+$ and the sphere at $r=\\infty$.\nUsing this formula, the reader can readily transfer results on \nbilinear forms in this paper between the metric perturbation and Teukolsky variables. \n\n\\section{Bilinear forms from infinitesimal symmetry operators}\n\\label{sec:Symmetry}\n\nConsider again a general partial differential operator $\\mathcal X$ acting on sections of some vector bundle, $E$, \nover a manifold $M$. We have the corresponding conserved bilinear form $X[\\tilde \\psi, \\psi]$ defined \nby \\eqref{Xdef}. Now suppose $\\mathcal{C}$ is a partial differential operator acting on $E$ mapping \nsolutions to $\\mathcal X \\psi = 0$ to solutions -- this is equivalent to the statement that there is a partial differential operator $\\mathcal{D}$\nsuch that $\\mathcal X \\mathcal{C} = {\\mathcal D} \\mathcal X$. Such an operator is called a ``symmetry operator''. The symmetry operators form an algebra which is trivial for a generic operator $\\mathcal X$.\nIf we have a symmetry operator, then \n$X[\\tilde \\psi, \\mathcal{C} \\psi]$ is also a conserved \nbilinear form, i.e., invariant under local changes of the surface $\\Sigma$ in \\eqref{Xdef}, see e.g. \\cite{carter1977killing, carter1979generalized} for a similar observation.\n\nLet us apply this recipe to the linearized Einstein operator $\\mathcal{E}$ on the Kerr spacetime. The Kerr spacetime\nhas two Killing vector fields, $t^a, \\phi^a$ corresponding to asymptotic time translations and rotations. The Lie\nderivatives $\\mathcal{L}_t, \\mathcal{L}_\\phi$ evidently commute with $\\mathcal{E}$ and thus provide two conserved quadratic forms:\n\\begin{equation}\n\\label{canen}\nE[h] = W[h,\\mathcal{L}_t h], \\quad J[h] = W[h,\\mathcal{L}_\\phi h].\n\\end{equation}\nThey correspond to the canonical energy and canonical angular momentum of the perturbation $h_{ab}$ when $\\Sigma$ is a Cauchy surface\nstretching between the bifurcation surface and spatial infinity \\cite{hollands2013stability}. \n\nIf we want to repeat a similar construction for the Teukolsky operator $\\mathcal{O}$ and the corresponding bilinear form $\\Pi$ \nwe face the problem that the Lie-derivative \nin general is not well-defined on an arbitrary vector bundle (though it is on the usual bundles of tensors over $M$). \nIn the GHP formalism, the vector bundles $L_{q,p}$ in question \nare defined relative to an NP tetrad, and in such a case we can still give a definition of the Lie derivative along a Killing vector\nfield, though not an arbitrary vector field, as we now describe. The point is that if $g_{ab}$ has an isometry $\\varphi$ that\n preserves the globally defined null directions, then this \n constitutes an intrinsically\n defined action on GHP tensors $\\eta \\ensuremath{\\circeq} (p,q)$. More\n explicitly, if $\\varphi$ preserves the null directions, then it must\n be the case that it acts on a given null frame as\n $\\varphi_* l^a = \\Lambda l^a$, $\\varphi_* n^a = \\Lambda^{-1} n^a$,\n and $\\varphi_* m^a = e^{i\\Gamma} m^a$, for some real functions $\\Lambda$,\n $\\Gamma$ on $M$ that depend on the chosen frame and\n $\\varphi$. The action of $\\varphi$ on $\\eta$ is then invariantly\n defined since GHP tensors are functionals of the\n null tetrads giving rise to the prescribed pair of null\n directions. In the given null frame, this action amounts to\n $\\varphi^{\\text{GHP}}_*\\eta \\equiv \\lambda^{-p} \\bar \\lambda^{-q}\n \\varphi_*\\eta$, where $\\lambda^2 = \\Lambda e^{i\\Gamma}$ and\n $\\varphi_*$ is the standard pushforward on functions \n (or tensors). In particular, the\n tetrad vectors are invariant under $\\varphi_*^{\\text{GHP}}$.\n\n Infinitesimally, if $\\varphi_t$ is a 1-parameter group of transformations generated by\n a Killing field $\\chi^a$ with corresponding\n $\\lambda_t$, then the corresponding ``Lie'' transport of $\\eta \\ensuremath{\\circeq} (p,q)$ is given\n by~\\cite{edgar2000integration}\n \\begin{IEEEeqnarray}{rClCl}\\label{eq:GHPLie}\n \\text{\\L}_\\chi \\eta &=& \\lim_{t\\to0}\\frac{(\\varphi_{-t})^{\\text{GHP}}_\\ast \\eta - \\eta}{t} && \\nonumber\\\\\n &=& (\\mathcal{L}_\\chi - p w - q \\bar w) \\eta &\\ensuremath{\\circeq}& (p,q),\n \\end{IEEEeqnarray}\n in the given frame. Here, $\\mathcal{L}$ denotes the standard Lie derivative, and \n \\begin{align}\n w &= \\frac{d}{dt} \\log \\lambda_t \\bigg|_{t=0}\\\\\n &= \\frac{1}{2} \\left(n_a\\mathcal{L}_\\chi l^a - \\bar m_a \\mathcal{L}_\\chi m^a \\right).\n \\end{align}\n If we introduce the bivector \n $Y \\equiv n \\wedge l - \\bar m \\wedge m$ (for further details on the\n bivector calculus see, e.g.\n \\cite{fayos1990electromagnetic,Aksteiner:2014thesis}) and use\n the fact that $\\chi^a$ is a Killing field, so\n $\\nabla_{(a}\\chi_{b)} = 0$, then~\\eqref{eq:GHPLie} can be manipulated to obtain\n \\begin{equation}\\label{eq:GHPLie-simplified}\n \\text{\\L}_\\chi \\eta = \\left[ \\mathcal{L}^\\Theta_\\chi\n - \\frac{p}{4} Y^{ab}\\Theta_a\\chi_b\n - \\frac{q}{4} \\left( Y^{ab} \\Theta_a\\chi_b \\right)^\\ast \\right] \\eta,\n \\end{equation}\n where $\\mathcal{L}^\\Theta$ is the standard Lie derivative with $\\nabla_a$\n derivatives replaced by $\\Theta_a$ derivatives. In this notation, the GHP Lie derivative \n is also defined for GHP-tensors, i.e., \n sections in a bundle $L_{p,q}$ tensored with \n $TM$ or $T^*M$. In any case, the GHP Lie\n derivative defined here is manifestly GHP covariant, and it can be\n checked that it satisfies the Leibniz rule. The expression for \n $\\text{\\L}_\\chi$ \n in a chosen NP tetrad will depend on that choice. \n For the Kinnersley tetrad \\eqref{eq:Kintet}, $w=0$, but\n $w$ can be different from zero for other choices of the frame.\n\n With these definitions, it then follows that $\\text{\\L}_\\chi$ for $\\chi^a$ either $t^a$ or $\\phi^a$ \n commutes with the covariant derivative $\\Theta_a$ and annihlates $g_{ab}, n^a, l^a, m^a, \\bar m^a, B_a$. Therefore, \n $\\text{\\L}_\\chi$ also commutes with the Teukolsky operators, \n \\begin{equation}\n [\\text{\\L}_\\chi, \\mathcal{O}] = 0 = [\\text{\\L}_\\chi, \\mathcal{O}^\\dagger], \\quad \\chi^a = t^a , \\phi^a, \n \\end{equation}\n and it thus defines a symmetry operator. \n \n There exist other symmetry operators in the Kerr (and more generally, Petrov type D-) spacetimes\n related to the Killing tensor $K_{ab}$ that exists in those spacetimes. The construction of those operators for spin $s=0, \\tfrac{1}{2}$ in the Teukolsky equation goes back to \n \\cite{carter1977killing, carter1979generalized}; here we present the corresponding symmetry operator for arbitrary GHP-weights $(p,q)$. Similar operators have appeared also in \n \\cite{grant2020class, grant2020conserved}, eq. III.3, for spin $s=1, 2$, though not in the GHP covariant form presented here which makes manifest the relationship with the Killing tensor. This tensor \n is given by \n \\begin{equation}\n \\label{Kabdef}\n K^{ab} = - \\dfrac{1}{4} \\left( \\zeta - \\bar{\\zeta} \\right)^2 l^{(a} n^{b)} + \\dfrac{1}{4} \\left( \\zeta + \\bar{\\zeta} \\right)^2 m^{(a} \\bar{m}^{b)}\n \\end{equation} \n where we use the shorthand\n\\begin{equation}\n\\label{zetadef}\n\\zeta = - \\Psi^{-\\tfrac{1}{6}}_2 \\bar{\\Psi}^{-\\tfrac{1}{6}}_2 \\rho^{-\\tfrac{1}{2}} \\bar{\\rho}^{\\tfrac{1}{2}} \\circeq \\GHPw{0}{0}\n\\end{equation}\nwith \n$\\rho$ one of the spin coefficients in the GHP formalism. The desired symmetry operator $\\mathcal{K}$\nacting on GHP scalars of weights $(p,q)$ is defined as\n\\begin{align}\\label{eq:Koperatordef}\n\\mathcal{K} \\eta =&\\left( \\Theta_a + p B'_a + q \\bar{B}'_a \\right) K^{ab} \\left( \\Theta_b + p B'_b + q \\bar{B}'_b \\right) \\eta \\nonumber\\\\\n&+ 2 (p \\gamma + q \\bar{\\gamma}) \\text{\\L}_\\xi \\eta\n\\end{align}\nwhere \n\\begin{equation}\n\\label{xidef}\n\\xi_a = \\zeta \\left( B_a - B'_a \\right), \n\\end{equation}\nis proportional to a Killing vector field, and $\\gamma = ( \\zeta^2 - \\bar{\\zeta}^2 )\/(8 \\zeta)$. Here and in the following, a prime as in $B'_a$ means the GHP priming operation $n^a \\leftrightarrow l^a, m^a \\leftrightarrow \\bar m^a$.\nIn Boyer-Lindquist coordinates and the Kinnersly frame (see appendix \\ref{app:D}), $\\xi^a = M^{-1\/3} t^a$, $\\gamma = M^{- 1\/3} \\frac{- i a \\cos \\theta}{2(r - i a \\cos\\theta)}$ and $\\text{\\L}_\\xi \\eta = M^{-1\/3} \\partial_t \\eta$.\n$\\mathcal{K}$ is called a symmetry operator because one can show that \n\\begin{equation}\n\\label{commutator}\n[\\mathcal{K}, \\mathcal{O}] = 0 = [\\mathcal{K}, \\mathcal{O}^\\dagger] \n\\end{equation}\nwhen acting on GHP quantities of weight $(4,0)$ or $(-4,0)$, respectively. The proof of this statement is rather \nnontrivial and deferred to appendix \\ref{app:B}, where we also prove the commutation property for arbitrary $(p,q)$. It follows from the properties of the GHP Lie derivative that $[\\text{\\L}_\\chi, \\mathcal{K}]=0$ for any Killing vector field $\\chi^a$, so we have:\n\n\\begin{theorem}\n$\\text{\\L}_t, \\text{\\L}_\\phi, \\mathcal{K}$ generate a commutative, infinite-dimensional algebra of symmetry operators for Teukolsky's operator $\\mathcal O$ for any GHP weights $(p,q)$.\n\\end{theorem}\n\nHence, by the general scheme, if we have \nsolutions to $\\mathcal{O} \\tilde \\Upsilon = 0 = \\mathcal{O}^\\dagger \\Upsilon$,\nand symmetry operators $\\mathcal{A}, \\mathcal{B}$, then the bilinear \nform $\\Pi[\\mathcal{A}\\tilde \\Upsilon, \\mathcal{B} \\Upsilon]$, with $\\Pi$ as in \\eqref{Pidef}, is conserved, i.e. unchanged under local deformations of the Cauchy surface $\\Sigma$. We caution the reader that such bilinear forms can be trivial, i.e., be equivalent to forms that are conserved identically; see appendix \\ref{sec:trivial} for some discussion. \n \nIt is possible to derive symmetry operators also for the linearized Einstein tensor $\\mathcal{E}$ (and for the Maxwell equations) on Kerr or more generally, a Petrov type D spacetime. Let $n=0,1,2,\\dots$ and set\n\\begin{equation}\n\\label{Cndef}\n\\mathcal{C}_n = \\mathcal{S}^\\dagger \\mathcal{K}^n \\zeta^{2s} \\mathcal{T}',\n\\end{equation}\nas well as\n\\begin{equation}\n\\mathcal{D}_n = \\mathcal{T}^\\dagger \\mathcal{K}^n \\zeta^{2s} \\mathcal{S}',\n\\end{equation}\nwhere for spin-2 considered here we should take $s=2$, and where we use the GHP priming operation. \nThen\n\\begin{equation}\n\\begin{split}\n\\mathcal{E} \\mathcal{C}_n =& \\mathcal{E} \\mathcal{S}^\\dagger \\mathcal{K}^n \\zeta^{2s} \\mathcal{T}' \\\\\n=& \\mathcal{T}^\\dagger \\mathcal{O}^\\dagger \\mathcal{K}^n \\zeta^{2s} \\mathcal{T}'\\\\ \n=& \\mathcal{T}^\\dagger \\mathcal{K}^n \\mathcal{O}^\\dagger \\zeta^{2s} \\mathcal{T}' \\\\\n=& \\mathcal{T}^\\dagger \\mathcal{K}^n \\zeta^{2s} \\mathcal{O}' \\mathcal{T}'\\\\\n=& \\mathcal{T}^\\dagger \\mathcal{K}^n \\zeta^{2s} \\mathcal{S}' \\mathcal{E}\\\\\n=& \\mathcal{D}_n \\mathcal{E}\n\\end{split}\n\\end{equation}\nwhere we used twice the Teukolsky-Wald identity \\eqref{SEOT}, the commutation $[\\mathcal{O}^\\dagger, \\mathcal{K}]=0$, as well as the intertwining \nrelation $\\mathcal{O}^\\dagger \\zeta^{2s} = \n\\zeta^{2s} \\mathcal{O}'$.\nWhen acting on a perturbation $h_{ab}$, $\\mathcal{T}'(h)$ gives the perturbed Weyl scalar $\\psi_4$, which is gauge invariant. Therefore, we see that $\\mathcal{C}_n(h)=0$ for any gauge perturbation $h_{ab} = \\mathcal{L}_\\xi g_{ab}$. \n\nBy the results of appendix \\ref{app:B} another symmetry operator for $\\mathcal E$ would be $\\mathcal{C}_n = \\mathcal{S}^\\dagger \\mathcal{G}^n \\zeta^{2s} \\mathcal{T}'$, with $\\mathcal{D}_n = \\mathcal{T}^\\dagger \\mathcal{G}^{\\dagger n} \\zeta^{2s} \\mathcal{S}'$ (with similar proof, see appendix \\ref{app:B} for the definition of $\\mathcal G$), and further symmetry operators are obtained by the GHP prime- and overbar operations applied to these $\\mathcal{C}_n$'s. Finally, by putting $s=1$ in the above expressions, and defining $\\mathcal{T}, \\mathcal{S}$ so that the analog of the Teukolsky-Wald identity \\eqref{SEOT} holds for electromagnetic perturbations, where $({\\mathcal E}A)_a = \\nabla^b \\nabla_{[a} A_{b]}$, we get similar operators in the electromagnetic case.\n\nAs a consequence, in all cases, $\\mathcal{C}_n$ give symmetry operators for $\\mathcal{E}$ of order $4+2n$ for spin-2 and of order $2+2n$\nfor spin-1. Regarding our operator $\\mathcal{C}_0$ for spin-2, we remark that a very similar looking operator has been considered by \\cite{grant2020class}, Eq.~III.14. \nRegarding our operator $\\mathcal{C}_1$, \na similar looking operator has been considered in \n\\cite{grant2020class}, Eq.~III.47 and also in \n\\cite{aksteiner2019symmetries}, Thm.~16. However, closer inspection of the operator\\footnote{\\cite{grant2020class}, Eq. III.14 on the other hand is manifestly local.} in \\cite{grant2020class}, Eq. III.47 shows that it is non-local, while our operators are all local and also manifestly GHP covariant. The relation of our operators $\\mathcal{C}_1$ to the order 6 symmetry operator asserted in \\cite{aksteiner2019symmetries} is not completely clear to us and the same goes for our other operators $\\mathcal{C}_1'$, etc. For spin-1, symmetry operators of orders 2 and 4 have been discussed in \\cite{grant2020conserved,andersson2015spin}, and the comparison to ours is qualitatively similar.\n\nBy the general theory, for example ($n=0,1,2, \\dots$)\n \\begin{equation}\n\\chi_{(n)}[h] = W[\\overline{\\mathcal{C}_0 h}, \\mathcal{C}_n h]\n\\end{equation}\nwith $W$ as in \\eqref{Wdef} are conserved for all solutions $h_{ab}$ to the linearized Einstein equations, i.e.~unchanged under local deformations of the Cauchy surface $\\Sigma$. The corresponding conserved currents are \n \\begin{equation}\n \\label{jdef}\nj^a_{(n)} = w^a[\\overline{\\mathcal{C}_0 h}, \\mathcal{C}_n h]\n\\end{equation}\nwith $w^a$ as in \\eqref{wadef}. Note that each $j^a_{(n)}$ is \nlocal and gauge invariant from the properties of $\\mathcal{C}_n$. The concrete expressions of $j^a_{(n)}$ \nare very long and contain $2n+9$ derivatives of $h_{ab}$.\nFor the reason explained below, we call $j^a_{(n)}$ the ``Carter current(s)''.\n\nTo gain some insight into the meaning of the conserved quantities $\\chi_{(n)}$, we make a WKB (high frequency) analysis similar to \\cite{green2016superradiant}, see also \\cite{grant2020class}. If the momentum of the \nsharply collimated WKB wave packet $h_{ab}$ is $p_a$\nand its amplitudes defined with respect to a suitable basis of polarization tensors are $A_{+,\\times}$, the result is\n\\begin{equation}\n\\begin{split}\n\\label{chinint}\n\\chi_{(n)}[h] = & \\int_\\Sigma j^a_{(n)} {\\rm d} \\Sigma_a \\\\\n\\sim & \\, \\, -i (-1)^n \\int_\\Sigma p^a \n{\\rm Im}(A_+ \\bar A_\\times) \\times \\\\\n& \\qquad \\times Q(p)^{n+4} \\, {\\rm d} \\Sigma_a\n\\end{split}\n\\end{equation}\nwhere $K^{ab} p_a p_b = Q(p)$ denotes the Carter constant. See appendix \\ref{app:WKB} for more \ndetail on the derivation of this formula \nand on the \nprecise definitions of the WKB wave functions, polarizations, etc. \n\n\nWe can obviously form alternative conserved quantities by other combinations of the various symmetry operators of the linearized Einstein operator described above giving e.g., the GHP primed version of our Carter currents $j^{a \\prime}_{(n)}$. We note also that such currents could have alternatively been constructed from $\\pi^a$ \\eqref{pidef}, taking $\\Upsilon = \\zeta^4 \\psi_4$ and $\\tilde \\Upsilon = \\psi_0$ and acting on those with various symmetry operators for the Weyl scalars, as described above. \n\nWe finally remark that conserved currents for metric perturbations related to Carter's constant have also been considered in \\cite{grant2020class}, Eqs. IV.14-16. Eq. IV.14 is very similar to our $j_{(0)}^a$ but their currents Eqs. IV.15-16 are different from our Carter currents $j^a_{(n)}$ or their GHP primes because unlike ours, they are based on non-local currents requiring a mode decomposition of the solutions.\n\n\n \n\\section{Bilinear form from $t$--$\\phi$ reflection}\\label{sec:bilinear_tphi}\n\nIn the previous section, we combined the basic conserved bilinear form \\eqref{Pidef} with symmetry operators, which arise in particular \nfrom the Killing vector fields of Kerr. One naturally expects that a similar construction should be possible for the discrete isometry of\nKerr, namely the $t$--$\\phi$ reflection map $J: (t,\\phi) \\to (-t,-\\phi)$ where here and in the following we refer to Boyer-Lindquist coordinates.\nHowever, just as for Killing vectors, some care has to be taken when defining the action of $J$ on GHP scalars with nontrivial weights \n$(p,q)$. So we first turn to this issue.\n\nThe map $J$ swaps the null directions $l^a$ and $n^a$ and changes the\n orientation on the orthogonal complement of these null directions\n spanned by $m^a$, $\\bar m^a$. \n There must thus be $\\Lambda$, $\\Gamma$ depending on the null tetrad\n such that $J_* l^a = -\\Lambda n^a$, $J_* n^a = -\\Lambda^{-1} l^a$, and\n $J_* m^a = e^{i\\Gamma} \\bar m^a$, where we have defined $J$ to act on\n tensors by the push-forward. \n By analogy with the previous\n case of isometries which are continuously deformable to the identity, \n it is then natural to define for $\\eta \\ensuremath{\\circeq} (p,q)$ a GHP\n reflection\n \\begin{equation}\n \\label{Jdef}\n \\mathcal J \\eta \\equiv i^{p+q} \\lambda^{-p} \\bar \\lambda^{-q} \\eta \\circ J \\ensuremath{\\circeq} (-p,-q) \n \\end{equation}\n in the given frame. \n \n The operator $\\mathcal J$ is evidently a GHP priming operation combined with $t \\to -t, \\phi \\to -\\phi$, and\n is therefore easily seen to be GHP covariant (i.e.~defined intrinsically as a map from sections in $L_{p,q}$ to sections in $L_{-p,-q}$, irrespective of the chosen frame), \n but, by contrast to the ``pull-back'' arising from isometries continuously connected to the \n identity as considered above, it changes the GHP weights. In this sense it is similar to the CPT operator arising in quantum field theory. \n It is clear that $\\mathcal J^2 = 1$ and one can relatively easily show the ``anti-commutation'' relations \n $\\text{\\L}_{t} \\mathcal J = - \\mathcal J \\text{\\L}_{t}$, $\\text{\\L}_{\\varphi} \\mathcal J = - \\mathcal J \\text{\\L}_{\\varphi}$\n with the GHP Lie-derivative defined above.\n We also note an important intertwining property\nof the $t$--$\\phi$ reflection operator $\\mathcal J$ with the Teukolsky operator and its adjoint, namely,\n\\begin{align}\\label{eq:OJ}\n \\mathcal O \\Psi_2^{4\/3} \\mathcal J = \\Psi_2^{4\/3} \\mathcal J \\mathcal O^\\dagger,\n\\end{align}\nwhere we used basic properties of gravito-magnetic field $B_a$ and its GHP prime $B'_a$, as well as the relation\n\\begin{equation}\\label{eq:gradPsi}\n \\Theta_a \\Psi_2 = -3 (B_a + B'_a) \\Psi_2.\n\\end{equation}\n In the Kinnersley frame and Boyer-Lindquist coordinates (see appendix \\ref{app:D}), the $\\mathcal J$ operator corresponds to sending $t \\to -t, \\phi \\to -\\phi$\n and multiplication according to \\eqref{Jdef} by appropriate powers of $\\lambda, \\bar \\lambda$, where $\\lambda$ is given in this case \n explicitly by \n\\begin{equation}\\label{eq:boostParams}\n \\lambda \n = \\sqrt{2} (r-ia \\cos \\theta) \\Delta(r)^{-1\/2}.\n\\end{equation}\n\nWe are now in a position to define the bilinear form. For simplicity, we restrict at first to entries having compact support on the \nCauchy surface $\\Sigma$ in order to avoid any convergence problems.\n\n\\begin{definition}[Bilinear form for compact support]\n Let $\\Upsilon_1, \\Upsilon_2 \\ensuremath{\\circeq} (-4,0)$ be smooth GHP scalars\n of compact support on $\\Sigma$ in the kernel of\n $\\mathcal O^\\dagger$. Then we set\n \\begin{equation}\\label{bilinear}\n \\langle\\langle \\Upsilon_1, \\Upsilon_2 \\rangle\\rangle \\equiv \\Pi_\\Sigma[\\Psi_2^{4\/3} \\mathcal J \\Upsilon_1, \\Upsilon_2]\n \\end{equation}\n with $\\Pi$ as in \\eqref{Pidef}.\n\\end{definition}\n\n\\begin{lemma}\\label{lemma:compactsupport}\n Under the conditions of the definition, we have\n \\begin{enumerate}[label=(\\roman*), start=1]\n \\item $\\langle\\langle \\Upsilon_1, \\Upsilon_2 \\rangle\\rangle$ is ${\\mathbb C}$-linear in both entries.\n \\item\n $\\langle\\langle \\Upsilon_1, \\Upsilon_2 \\rangle\\rangle=\\langle\\langle \\Upsilon_2,\n \\Upsilon_1 \\rangle\\rangle$,\n \\item\n $\\langle\\langle \\text{\\L}_t\\Upsilon_1, \\Upsilon_2 \\rangle\\rangle=\\langle\\langle\n \\Upsilon_1, \\text{\\L}_t \\Upsilon_2 \\rangle\\rangle$ for $t^a$ the time\n translation Killing field, and\n \\item $\\langle\\langle \\Upsilon_1, \\Upsilon_2 \\rangle\\rangle$ is independent of\n the chosen Cauchy surface $\\Sigma$.\n \\end{enumerate}\n\\end{lemma}\nBefore we prove this lemma, we remark that, e.g. by \\eqref{eq:Kinnersley-bilinear}, the bilinear form may be viewed as defined on the initial data of the Teukolsky equation on \nthe Cauchy surface $\\Sigma$. On an initial data set $\\text{\\L}_t$ corresponds to the action of a suitably definined Hamiltonian operator $\\mathcal{H}$. \nThen item (iii) corresponds to the statement that \n\\begin{equation}\n \\langle\\langle \\Upsilon_1, \\mathcal{H} \\Upsilon_2 \\rangle\\rangle = \\langle\\langle \\mathcal{H} \\Upsilon_1, \\Upsilon_2 \\rangle\\rangle, \n\\end{equation}\ni.e.~to the fact that the Hamiltonian operator is symmetric with respect to our bilinear form. We refer the interested reader to appendix \\ref{sec:Lagrangian-Hamiltonian} for details \non the Hamiltonian formulation of the Teukolsky equation. \n\nWe also note that although we defined our bilinear form on $s=-2$\nGHP scalars (i.e., solutions to the adjoint Teukolsky equation), we\ncould also define a bilinear form on $s=+2$ solutions to the original\nTeukolsky equation. In this case, we set\n$\\langle\\langle\\tilde\\Upsilon_1, \\tilde\\Upsilon_2\\rangle\\rangle \\equiv\n\\Pi_\\Sigma[\\tilde\\Upsilon_1, \\Psi_2^{-4\/3} \\mathcal J\n\\tilde\\Upsilon_2]$. It can be shown that the $s=+2$ bilinear form\nsatisfies all the same properties as the $s=-2$ form. \n\\begin{proof}\n \\begin{enumerate}[label=(\\roman*),start=1]\n \\item This is obvious from the definition.\n \\item By explicit calculation, we have with $\\pi_{abc} = \\epsilon_{abcd} \\pi^d$ and $\\pi^a$ as in \\eqref{pidef},\n \\begin{widetext}\n \\begin{align}\n \\pi_{abc}(\\Psi_2^{4\/3}\\mathcal J \\Upsilon_1, \\Upsilon_2) &= \\epsilon_{dabc} \\left[ (\\Psi_2^{4\/3} \\mathcal J \\Upsilon_1) (\\Theta^d - 4 B^d) \\Upsilon_2 - \\Upsilon_2 (\\Theta^d + 4 B^d) (\\Psi_2^{4\/3} \\mathcal J \\Upsilon_1 )\\right] \\nonumber \\\\\n &= \\mathcal J \\epsilon_{dabc} \\left[ \\Psi_2^{4\/3} \\Upsilon_1 (\\Theta^d - 4 B^{\\prime d}) (\\mathcal J \\Upsilon_2 ) - (\\mathcal{J} \\Upsilon_2) (\\Theta^d + 4 B^{\\prime d}) (\\Psi_2^{4\/3} \\Upsilon_1) \\right] \\nonumber\\\\\n &= \\mathcal J \\epsilon_{dabc} \\left[ \\Upsilon_1 (\\Theta^d + 4 B^d) (\\Psi_2^{4\/3} \\mathcal J \\Upsilon_2) - (\\Psi_2^{4\/3} \\mathcal J \\Upsilon_2) (\\Theta^d - 4 B^d) \\Upsilon_1 \\right] \\nonumber\\\\\n &= - \\mathcal J \\pi_{abc}(\\Psi_2^{4\/3}\\mathcal J \\Upsilon_2, \\Upsilon_1),\n \\end{align}\n \\end{widetext}\n using $\\mathcal J^2 = 1$ and~\\eqref{eq:gradPsi}. Now integrate over $\\Sigma$. Since\n $\\mathcal J$ reverses the orientation of $\\Sigma$, the claim\n follows.\n \\item \n We first remark that, by Cartan's magic formula, we have that on solutions (where $\\pi = \\pi_{abc} {\\rm d} x^a \\wedge {\\rm d} x^b \\wedge {\\rm d} x^c$),\n \\begin{equation}\n \\mathcal{L}_t \\pi = {\\rm d} ( t \\cdot \\pi),\n \\end{equation}\n if ${\\rm d} \\pi=0$. Integrating over $\\Sigma$ and using Stokes's theorem,\n \\begin{equation}\\label{eq:pi-cartan}\n \\int_\\Sigma \\mathcal{L}_t \\pi = \\int_{\\partial \\Sigma} t \\cdot \\pi = 0.\n \\end{equation}\n as, for compact support data, the contribution on $\\partial \\Sigma$ evaluates to zero. \n In our case,\n $\\Upsilon_1 \\in \\ker \\mathcal{O}^\\dagger$, therefore\n $\\Psi_2^{4\/3} \\mathcal J \\Upsilon_1 \\in \\ker \\mathcal O$, thus\n $\\pi(\\Psi_2^{4\/3}\\mathcal J \\Upsilon_1, \\Upsilon_2)$ is indeed closed, ${\\rm d} \\pi=0$.\n On the other hand, we have, since background quantities are all\n GHP-Lie-derived by $t^a = M^{1\/3} \\xi^a$, and since\n $\\mathcal J \\text{\\L}_t = - \\text{\\L}_t \\mathcal J$, that\n \\begin{align}\\label{eq:lemmaiiib}\n & \\mathcal{L}_t \\pi(\\Psi_2^{4\/3}\\mathcal J \\Upsilon_1, \\Upsilon_2)\\nonumber \\\\\n &\\quad= \\pi( \\Psi_2^{4\/3} \\text{\\L}_t \\mathcal J \\Upsilon_1, \\Upsilon_2) + \\pi(\\Psi_2^{4\/3} \\mathcal J \\Upsilon_1, \\text{\\L}_t \\Upsilon_2) \\nonumber \\\\\n &\\quad= - \\pi(\\Psi_2^{4\/3} \\mathcal{J} \\text{\\L}_t \\Upsilon_1, \\Upsilon_2) + \\pi(\\Psi_2^{4\/3} \\mathcal J \\Upsilon_1, \\text{\\L}_t \\Upsilon_2).\n \\end{align}\n Inserting this into the left hand side of \\eqref{eq:pi-cartan} evaluated on the solutions $\\Psi_2^{4\/3}\\mathcal J \\Upsilon_1$ and $\\Upsilon_2$ immediately yields the claim.\n\n \\item Holds by Gauss's theorem because $\\pi$ is closed on solutions,\n and $\\Psi_2^{4\/3} \\mathcal J$ takes $\\ker \\mathcal O^\\dagger$ into\n $\\ker \\mathcal O$.\n \\end{enumerate}\n\\end{proof}\n\nWe end this section with an explicit expression of our bilinear form in Boyer-Lindquist coordinates and the Kinnersley frame:\n\\begin{widetext}\n\\begin{align}\\label{eq:Kinnersley-bilinear}\n \\langle\\langle \\Upsilon_1, \\Upsilon_2 \\rangle\\rangle \n = 4 M^{4\/3}\n \\int_\\Sigma {\\rm d} r \\, {\\rm d}\\theta {\\rm d}\\phi\\, \\frac{\\sin\\theta}{\\Delta^2} \\Bigg[\n & \n \\Upsilon_1\\Big|_{\\substack{t\\to-t \\\\ \\phi\\to-\\phi}} \\left( \\frac{\\Lambda}{\\Delta}\\partial_t + \\frac{2Mra}{\\Delta}\\partial_\\phi + 2 \\left[ -r - ia\\cos\\theta + \\frac{M}{\\Delta}(r^2 - a^2)\\right] \\right) \\Upsilon_2\n \\nonumber\\\\\n & \n + \\Upsilon_2 \\left[\\left( \\frac{\\Lambda}{\\Delta}\\partial_t + \\frac{2Mra}{\\Delta}\\partial_\\phi + 2 \\left[ -r - ia\\cos\\theta + \\frac{M}{\\Delta}(r^2 - a^2)\\right] \\right) \\Upsilon_1\\right]_{\\substack{t\\to-t \\\\ \\phi\\to-\\phi}}\n \\Bigg],\n \\end{align}\n\\end{widetext} \nwhere we refer to appendix \\ref{app:D} for the definitions of \n$\\Sigma, \\Delta$, and $\\Lambda$. \n\n\\section{Quasinormal mode orthogonality}\n\\label{sec:Ortho}\n\\subsection{Quasinormal modes}\nConsider modes of the form\n\\begin{equation}\\label{eq:modes}\n {}_s\\Upsilon_{\\ell m\\omega} = e^{-i\\omega t + i m \\phi} \\ensuremath{ {}_s R_{\\ell m \\omega }}(r) \\ensuremath{{}_s S_{\\ell m \\omega}}(\\theta),\n\\end{equation}\nwith $m \\in \\mathbb Z$ and $\\omega \\in \\mathbb C$, in the Kinnersley\nframe. This form leads to separation of the spin-$s$\nTeukolsky equation~\\cite{Teukolsky:1973ha}, $\\mathcal O \\Upsilon = 0$ (for any integer spin $s$), into an angular equation,\n\\begin{widetext}\n\\begin{align}\\label{eq:Sph eq}\n \\left[\\frac{1}{\\sin \\theta} \\frac{ {\\rm d}}{{\\rm d} \\theta}\\left(\\sin \\theta \\frac{{\\rm d} \\,}{{\\rm d} \\theta} \\right) \\right. \\left. + \\left( K - \\frac{m^2+s^2+2 m s \\cos \\theta}{\\sin^2 \\theta} - a^2 \\omega^2 \\sin^2 \\theta -2 a \\omega s \\cos \\theta \\right) \\right] \\ensuremath{{}_s S_{\\ell m \\omega}}(\\theta) = 0,\n\\end{align}\nand a radial equation,\n\\begin{align}\\label{eq:radial}\n \\left[ \\Delta^{-s} \\frac{{\\rm d}}{{\\rm d} r} \\left( \\Delta^{s+1} \\frac{{\\rm d}}{{\\rm d} r} \\right) \\right. \\left. + \\left( \\frac{H^2 - 2 i s (r-M)H}{\\Delta} + 4 i s \\omega r+2 a m \\omega - K +s(s+1) \\right) \\right] \\ensuremath{ {}_s R_{\\ell m \\omega }}(r) = 0,\n\\end{align}\n\\end{widetext}\nwith $H \\equiv (r^2+a^2)\\omega - a m$. Here $K$ is a separation\nconstant. Imposing regularity at the poles $\\theta=0,\\pi$, the angular\nequation leads to a discrete set of modes $\\ensuremath{{}_s S_{\\ell m \\omega}}$ and separation\nconstants $\\ensuremath{ {}_s K_{\\ell m \\omega}}$, both of which are indexed by\n$\\ell \\in \\mathbb Z^{\\ge \\max(|m|, |s|)}$. The functions\n$\\ensuremath{{}_s S_{\\ell m \\omega}}(\\theta)e^{im\\phi} $ are known as spin-weighted spheroidal\nharmonics~\\cite{Teukolsky:1973ha}. For $\\omega \\in \\mathbb R$, the\nangular problem reduces to a Sturm-Liouville eigenvalue problem. \nModes with the same $s$,\n$m$, and real $\\omega$, but different $\\ell$ are orthogonal, and\nwe normalize them such that\n\\begin{equation}\\label{eq:theta-orthogonality}\n \\int_0^\\pi {\\rm d}\\theta\\, \\sin\\theta \\, {}_sS_{\\ell m\\omega}(\\theta) {}_sS_{\\ell'm\\omega}(\\theta) = \\delta_{\\ell\\ell'}.\n\\end{equation}\nOrthogonality can be checked by verifying that the angular operator is\nsymmetric with respect to this product. \n\nTo discuss boundary conditions of the radial equation it is convenient to\nintroduce a ``tortoise'' coordinate ${\\rm d} r_*= (r^2+a^2)\/\\Delta {\\rm d} r$, see\n\\eqref{eq:rstar}. \nFor fixed $s,l,m,\\omega$ one considers the solutions\n$R^{\\rm in}$ and $R^{\\rm up}$ ``defined'' by the ``boundary conditions''\n\\begin{subequations}\\label{eq:R bcs}\n \\begin{align}\n &R^{\\rm in} \\sim \\frac{e^{-ikr_*}}{\\Delta^{s}}, \\qquad r_*\\to-\\infty,\\\\\n &R^{\\rm up} \\sim \\frac{e^{i\\omega r_*}}{r^{2s+1}}, \\qquad r_*\\to\\infty,\n \\end{align}\n\\end{subequations}\nwhere \n$k \\equiv \\omega-m\\Omega_H$, where $\\Omega_H$ is the angular frequency of the outer horizon\n$\\Omega_H=a\/(2Mr_+)$, and where the radii of the inner- and outer horizons (roots of $\\Delta$) are denoted by $r_\\pm$, respectively. \n\nThe conditions \\eqref{eq:R bcs} correspond physically to the\nabsence of incoming radiation from the past horizon and past null\ninfinity, respectively. As stated \\eqref{eq:R bcs} do not really pick out uniquely a solution in the case ${\\rm Im} \\omega<0$ because we may always add a multiple of the subdominant solution as $|r_*| \\to \\infty$ without affecting the asymptotic behavior. \nMore precisely, mode solutions may be obtained via series expansions \\cite{Leaver1986}, involving three-term recurrence relations for the coefficients.\n Selecting the so-called ``minimal solution'' \\cite{gautschi1967computational} of the recurrence relations ensures that the series represenation converges at the horizon (in) or infinity (up).\\footnote{This definition is satisfied by a radial solution of the form\n\\begin{equation}\n R(r) = e^{i\\omega r}(r-r_-)^{-1-s+i\\omega+i\\sigma_+}(r-r_+)^{-s-i\\sigma_+} f(r) ,\n\\end{equation}\nwhere $\\sigma_+=(\\omega r_+-am)\/(r_+-r_-)$ and $f(r)=\\sum_{n=0}^{\\infty}d_n\\left(\\frac{r-r_+}{r-r_-}\\right)^n$ with $d_n$ coefficients that are a minimal solution to a three-term recursion relation~\\cite{Leaver1985} so that the series is uniformly absolutely convergent as $r\\rightarrow\\infty$}\n Imposing both of these conditions simultaneously\n %\n\\footnote{The problem is made\ncomplicated, however, because $\\omega$ and $K$ appear in both the\nangular and radial equations, $\\omega$ nonlinearly. One must jointly\nsolve both equations to obtain a self-consistent solution of this\nnonlinear eigenvalue problem. Using Hamiltonian methods (see appendix \\ref{sec:Lagrangian-Hamiltonian})\none can recast this as the eigenvalue problem $\\mathcal{H} \\Upsilon = i\\omega \\Upsilon$,\ni.e., the problem is linear in $\\omega$, but the angular and radial\nproblems remain coupled.}\ngives rise to a discrete set of quasinormal modes $\\omega_n \\in \\mathbb{C}$, where $n=0,1,2,\\ldots$ are the so-called ``overtone'' numbers. We restrict to frequencies with ${\\rm Im}\\, \\omega \\le 0$, as modes growing exponentially in time are not in the specturm of Kerr \\cite{Whiting:1988vc}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[trim={0.cm 4.cm 0.cm 2.cm},clip,width=0.49\\linewidth]{contour_r.pdf}\n\\includegraphics[trim={0.cm 4.cm 0.cm 2.cm},clip,width=0.49\\linewidth]{contour_rstar.pdf}\n\\caption{{\\it Left:} Sketch of the complex $r$ contour $C_*$ defining the bilinear form on quasinormal modes. The contour cannot be pulled back to the real axis because the integrand crosses (an infinite number of) different sheets associated with the branch points $r_-$ and $r_+$. {\\it Right:} Same contour, but in the complex $r_*$ plane.\nNote that this contour cannot be pulled back to the real axis due to the presence of Stokes lines along which the integrand of the bilinear form would diverge.}\n\\label{fig:r_contour}\n\\end{figure*}\n\n\\subsection{Bilinear form on quasinormal modes}\n\nWe would now like to extend our definition of the bilinear form $\\langle\\langle \\cdot , \\cdot \\rangle\\rangle$, originally \nonly for compactly supported solutions\/data on the Cauchy surface $\\Sigma$, to quasinormal modes. \nThe immediate problem is that, according to the boundary conditions on the corresponding solutions \nto the radial equation, these blow up both at the horizon $r=r_+$ and infinity $r \\to \\infty$. \nIn this subsection, inspired by the work of~\\cite{LeungModes94}, we show that the Kerr\nbilinear form can be defined for quasinormal mode data by a suitable\ndeformation of the radial integration into the complex plane.\\footnote{\nIn the quantum mechanics literature, this method is known also as (exterior)\ncomplex scaling \\cite{Aguilar:1971ve}. \nComplex scaling and complex integration contours have already been used in the context of black hole quasinormal modes, see for instance~\\cite{Bony2007,Dyatlov:2011jd} and~\\cite{Glampedakis:2003dn,Leaver1986b}. } \n\nConsider the bilinear form acting on two quasinormal modes with\nquasinormal frequencies $\\omega_1$ and $\\omega_2$. The \nintegrand in the bilinear\nform~\\eqref{eq:mode-bilinear} goes as\n$\\sim e^{\\pm i(\\omega_1+\\omega_2)r_\\ast}$ as $r_\\ast \\to \\pm\\infty$,\nand therefore diverges exponentially for\n$\\Im(\\omega_1 + \\omega_2) < 0$, which is the case for all modes that decay\nin time. Therefore, we clearly see that the bilinear form as defined for compact support data~\\eqref{eq:Kinnersley-bilinear} is divergent.\n\nWe can obtain a finite bilinear form by analytic continuation in $r$.\nThe radial mode functions $R^{\\rm in\/up}(r)$ are analytic with branch points at $r=r_\\pm$ \\cite{Leaver1986}, and we take the branch cut as the wiggly line in Fig.~\\ref{fig:r_contour} going from $r_+$ to $r_-$. We take the branch cut for the tortoise coordinate\n\\eqref{eq:rstar} $r_*(r)$ to be identical, so that we can think of both the radial functions $R^{\\rm in\/up}$ and $r_*$ as \ndefined on the same multisheeted covering of the twice cut complex $r$-plane. The integrand of the bilinear form, given by the 3-form \n$\\pi_{abc} = \\epsilon_{abcd} \\pi^d$ [see \\eqref{pidef}] evaluated on two mode \nfunctions as in \\eqref{bilinear} or equivalently \\eqref{eq:Kinnersley-bilinear}, therefore has an analytic continuation on the multi-sheeted complex $r$-plane. \n\nIn \\eqref{bilinear} or equivalently \\eqref{eq:Kinnersley-bilinear}, we now define an integration contour going into \nthis complex $r$-plane as shown qualitatively \nin fig. \\ref{fig:r_contour}. In terms of $r_*(r)$, \nwhich is a function on the same multi-sheeted complex $r$-plane, the contour is defined in such a way that $0 < \\arg((\\omega_1 + \\omega_2) r_\\ast) < \\pi$\non the right limit, and\n$-\\pi < \\arg((\\omega_1 + \\omega_2) r_\\ast) < 0$ on the left, then as\n$|r_\\ast| \\to \\infty$, \nthe volume integral will converge exponentially with $|r_\\ast|$.\n\nTo achieve this for any $\\Im(\\omega_1 + \\omega_2) < 0$ me may take a snake\nshaped contour $\\lambda \\mapsto r_*(\\lambda,\\epsilon)$ of the radial\ncoordinate in the complex $r_*$ plane, with the properties\n\\begin{equation}\\label{eq:contour}\n \\begin{cases}\n r_*(\\lambda,\\epsilon)=\\lambda\n &\\text{for $\\lambda_1 < \\lambda < \\lambda_2$}\\\\\n \\arg r_*(\\lambda,\\epsilon) \\to +\\pi - \\epsilon & \\text{for $r_* \\to \\infty$}\\\\\n \\arg r_*(\\lambda,\\epsilon) \\to 0 + \\epsilon &\\text{for $r_* \\to -\\infty$,}\n \\end{cases}\n\\end{equation}\nwhere $\\lambda_1<0$, $\\lambda_2>0$ can in principle be chosen arbitrarily. We give a sketch of this contour, $C_*$, which corresponds \nto one in terms of $r$, in the right panel of Fig.~\\ref{fig:r_contour}. The corresponding 3-dimensional\n submanifold (depending on $\\epsilon > 0$ and on $t \\in {\\mathbb R}$) of the analytically continued Kerr manifold $M_{{\\mathbb C}}$ is denoted by\n$\\Sigma_{{\\mathbb C}} = \\{ (t,r_*(\\lambda,\\epsilon),\\theta,\\phi) \\mid \\lambda \\in {\\mathbb R} \\}$.\nIn practice, the angle $\\epsilon>0$ is chosen sufficiently small such that the integral in the following definition of the bilinear form converges, \n\\begin{align}\n \\langle\\langle \\Upsilon_1, \\Upsilon_2 \\rangle\\rangle &= \\Pi_{\\Sigma_{{\\mathbb C}}}[\\Psi_2^{4\/3}\\mathcal J \\Upsilon_1, \\Upsilon_2]. \\\\\n \\nonumber\n\\end{align}\nReplacing $\\Sigma$ with the contour $\\Sigma_{\\mathbb{C}}$ as described in section~\\ref{sec:bilinear_tphi}, \nthanks to the analyticity of the integrand and its fall off on $\\partial \\Sigma_{\\mathbb{C}}$, all properties of the bilinear form of of lemma \\ref{lemma:compactsupport} continue to hold on quasinormal modes. In particular, from item (iii) of lemma \\ref{lemma:compactsupport}, we get $(\\omega_1-\\omega_2) \\langle\\langle \\Upsilon_1 , \\Upsilon_2 \\rangle\\rangle=0$ for a pair of quasinormal modes with complex frequencies $\\omega_1, \\omega_2$. Furthermore, by (iv), the value of the bilinear form is independent of the precise choice of $t$, details of the complex integration contour such as the asymptotic angle $\\epsilon$ against the real half-axes and\/or $\\lambda_1, \\lambda_2$, as long as the integrand is exponentially decaying. \n\n\\begin{corollary}[Orthogonality of quasinormal modes]\n Let $\\Upsilon_1$ and $\\Upsilon_2$ be quasinormal modes for the $s=2$\n Teukolsky equation\n with frequencies $\\omega_1$ and $\\omega_2$. Then either\n $\\langle\\langle \\Upsilon_1, \\Upsilon_2 \\rangle\\rangle = 0$ or\n $\\omega_1 = \\omega_2$.\n\\end{corollary}\n\n\nOur bilinear form takes the following form on quasinormal mode solutions~\\eqref{eq:modes}. After plugging two $s=-2$ mode solutions in separated form\ninto~\\eqref{eq:Kinnersley-bilinear}, we can carry out the $\\phi$\nintegration to obtain \n\\begin{widetext}\n\\begin{eqnarray}\\label{eq:mode-bilinear}\n &&\\langle\\langle \\Upsilon_{\\ell_1m_1\\omega_1}, \\Upsilon_{\\ell_2m_2\\omega_2} \\rangle\\rangle \\nonumber\\\\&= &8\\pi M^{4\/3} \\delta_{m_1m_2} e^{-i(\\omega_2-\\omega_1)t}\n \\int_{C_*} {\\rm d} r_* \\int_0^\\pi {\\rm d}\\theta\\, \\frac{ (r^2+a^2)\\sin\\theta}{\\Delta} S_1(\\theta) S_2(\\theta) R_1(r) R_2(r) \n\n \\bigg( - \\frac{i\\Lambda}{\\Delta}(\\omega_1+\\omega_2) \\nonumber\\\\\n &&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac{2iMra}{\\Delta}(m_1+m_2) + 2 \\left[ -r - ia\\cos\\theta + \\frac{M}{\\Delta}(r^2 - a^2)\\right] \\bigg)\n\\end{eqnarray}\n\\end{widetext}\nwith $C_*$ the contour for the $r_*$-integration described above and the Kerr quantities $\\Delta, \\Sigma, \\Lambda$ as given in \nappendix \\ref{app:D}.\n\nThe integrands depend on $\\theta$ and $r$ in a nonfactorizable way, so\nthis expression is the best that can be achieved in general: for Kerr,\nthe orthogonality relation expressed by the previous corollary \n(vanishing of the above inner product for $\\omega_1 \\neq \\omega_2$) \nis fundamentally two-dimensional. This has\nto do with the fact that the orthogonality\nrelation~\\eqref{eq:theta-orthogonality} for spin-weighted spheroidal\nharmonics occurs between modes of different $\\ell$ but the \\emph{same}\n$m$ and $\\omega$; if $\\omega_1 \\ne \\omega_2$, then no such relation\nexists, and one cannot expect to be able to perform the $\\theta$\nintegration to obtain a $\\delta_{\\ell_1\\ell_2}$ factor. \n\nIn the $a\\to0$ Schwarzschild limit, however, the integral \\emph{does}\nfactorize: the $\\theta$ dependence of the integrand reduces to the\n$\\sin\\theta$ volume factor on the sphere, the spheroidal harmonics\nreduce to spherical harmonics (independent of $\\omega$), and the\n$\\theta$ integral is proportional to $\\delta_{\\ell_1\\ell_2}$. One is left\nwith a radial integration, which must vanish for\n$\\omega_1 \\ne \\omega_2$. \n\nAs we can see from the following figure \\ref{fig:ortho}, the contour integral in the bilinear form converges quite well, \nwhich is useful in practice when using it to extract excitation coefficients, as we describe in the next section. \nFurthermore, since orthogonality is an exact result for quasinormal modes, it can be used potentially as a benchmark check \nfor approximations. For example, we have considered approximations to quasinormal modes based on a matched asymptotic \nexpansion for near-extremal black holes, and have found that the orthogonality relation is typically satisfied to a very high accuracy. \n\\begin{figure}\n\\centering\n\\includegraphics[width=1.\\linewidth,trim={.cm .1cm .1cm .1cm},clip]{Ortho_Kerr_l_n.pdf}\n\\caption{Numerical check of the orthogonality between two Kerr quasinormal modes with the same $l=m=2$ and different $n=0$, 1 (upper panel) and modes with the same $n=0$, $m=2$ and different $l=2$, 3 (lower panel). We show the result of the numerical evaluation of the bilinear form~\\eqref{eq:mode-bilinear} along the most convergent contour (black points) $r_* \\to r_{*,\\rm lower\/upper}+ \\lambda e^{-i \\arg(\\omega_1+\\omega_2)+i\\theta}$, $\\theta=\\pi\/2$, integrating up to $\\lambda_{\\rm upper\/lower}$. We use the mode solutions provided by the Black Hole Perturbation Toolkit~\\cite{BHPToolkit}.\nFor the overtone orthogonality, we also show an exponential fit converging to zero as $\\lambda_{\\rm lower}\\to-\\infty$ (red line). Because of the presence of the branch cut, the lower integration limit sets the overall accuracy of the bilinear form. In this example, we set $M=1$, $a\\simeq0.7$, $r_{*,\\rm upper}=4$ and $r_{*,\\rm lower}=-6$.\n}\n\\label{fig:ortho}\n\\end{figure}\n\nWe remark that\nour ``norm'' on quasinormal modes \nhas some similarities with the ``norm'' of resonant state \nwave functions in quantum mechanics defined by~\\cite{Zeldovich:1961theory}. Rather than taking the integral of $|\\psi|^2$, the ``norm''\nused by \\cite{Zeldovich:1961theory} also involves $\\psi^2$, whereas our bilinear is complex linear in both arguments as opposed to an inner product (anti-linear in the first argument, complex linear in the second).\nOur regularization procedure differs from that proposed by~\\cite{Zeldovich:1961theory} but was rather inspired by the investigations \\cite{LeungModes94} in the context of leaky optical\none-dimensional cavities, and on Schwarzschild black holes in~\\cite{Ching:1993gt}.\nIn \\cite{leung1997twoa,leung1997twob} it was recognized that phase space was the natural setting for the bilinear form.\\footnote{A\nvariational method for computing quasinormal frequencies of ``dirty''\nSchwarzschild black holes was developed in~\\cite{Leung:1999rh,Leung:1999iq}.}\nIn fact, in several ways our work was inspired by some of these papers: we work\nwithin the Teukolsky formalism, we arrive at the bilinear form\nstarting from the symplectic form, and we recognize the fundamental\nimportance of the $t$--$\\phi$ reflection symmetry. \n\n\\medskip\n\n\\section{Excitation coefficients}\\label{sec:Lap transform} \n\nIf $\\langle\\langle \\cdot , \\cdot \\rangle\\rangle$ were an honest to God scalar product in a Hilbert space and $\\{ {}_s\\Upsilon_{\\ell mn} \\}$ an orthonormal basis, then an arbitrary \nwave function $\\Upsilon_s$ could evidently be expanded as\n\\begin{align}\n\\label{excited}\n\\Upsilon_s = \\sum_{\\ell mn} c_{\\ell mn} \\, {}_s\\Upsilon_{\\ell mn},\n\\end{align}\nwhere the excitation coefficients are\n\\begin{align}\nc_{\\ell mn} =\\frac{\\langle\\langle {}_s\\Upsilon_{\\ell mn} , \\Upsilon_s \\rangle\\rangle }{\\langle\\langle {}_s\\Upsilon_{\\ell mn} , {}_s\\Upsilon_{\\ell mn} \\rangle\\rangle }. \\label{eq:excitation coeff}\n\\end{align}\nHere $\\sum_{\\ell m n}$ denotes $\\sum_{\\ell = \\vert s\\vert}^\\infty\\sum_{m=-\\ell}^\\ell \\sum_{n=0}^\\infty$. In the present context, $\\langle\\langle \\cdot,\\cdot\\rangle\\rangle$ is of course \nonly a symmetric bilinear form on solutions to the spin $s$ Teukolsky equation (for the case of interest in this paper, $s=-2$). It is neither positive definite, nor is the set of quasi-normal modes, \nwhile being orthogonal, in any obvious mathematical sense a complete basis for a reasonable function space in as far as we can see. \n\nInspired by~\\cite{LeungModes94}, we will nevertheless show in this section \nthat for solutions $\\Upsilon_s$ to the adjoint Teukolsky equation with compact support on a Cauchy surface $\\Sigma$, \nthe above expansion can formally be ``derived'' in the Laplace transform formalism \\cite{Leaver1986b,Nollert:1999ji} for the retarded propagator if we deform the frequency \nintegration contours into the complex plane and collect only contributions from the quasinormal mode frequencies. Thus, \\eqref{eq:excitation coeff}, while not an exact \nequality, is expected to capture the transient behavior of the solution $\\Upsilon_s$.\n\n\\subsection{Laplace transform}\\label{eq:Laplace}\n\nThe Laplace transform $\\hat f(\\omega) = L f(t)$ of a function $f(t)$ is given by\n\\begin{equation}\\label{eq:Lap def}\n\\hat f(\\omega) = \\int_0^\\infty e^{i\\omega t} f(t) {\\rm d} t,\n\\end{equation}\n where $\\Im \\omega >0$. The Laplace transform is related to the Fourier transform $\\mathcal F \\!f = \\int_{-\\infty}^{\\infty} e^{i \\omega t} f(t) {\\rm d} t$ by sending $f(t) \\to f(t) \\theta (t)$, where $\\theta(t)$ is the Heaviside distribution. Sufficient conditions for the existence of $\\hat f(\\omega)$ are that the function $f(t)$ be Riemann integrable (continuous except on sets of measure zero) on every closed sub-interval of the path of integration and that it be of exponential order; i.e. at any $t$ one can find constants $a$ and $N$ such that $\\vert e^{-at} f(t) \\vert < N$. If the Laplace integral exists for some value of $\\omega=\\omega_0$, then it also exists for all $\\omega$ with $ \\Im \\omega>\\Im \\omega_0$. The lowermost $\\Im \\omega_0$ where convergence occurs is called the abscissa of convergence and the region above this line called the convergence region. The function $\\hat f(\\omega)$ is analytic in the convergence region. \n\nThe Laplace transform formalism is naturally adapted to the study of causal dynamics of linear second-order systems, as it incorporates the initial data into a source by taking time derivatives into field values at the initial time\n\\begin{align}\\label{eq:lap derivs}\nLf'(t) = &-i\\omega \\hat f(\\omega)-f(0), \\\\\nLf''(t) = &-\\omega^2 \\hat f(\\omega)+i\\omega f(0) - f'(0).\n\\end{align}\nThe Laplace transform $\\ensuremath{\\hat \\Upsilon_{s}}$ of the spin-$s$ master function $\\ensuremath{\\Upsilon_{s}}$ is given by\n\\begin{equation}\\label{eq:tilde U}\n\\ensuremath{\\hat \\Upsilon_{s}}(\\omega,r,\\theta,\\phi) = \\int_{0}^{\\infty} e^{i \\omega t} \\ensuremath{\\Upsilon_{s}}(t,r,\\theta,\\phi) {\\rm d} t.\n\\end{equation}\nThis decomposed into modes in the usual way\n\\begin{equation}\\label{eq:tilde Us mode}\n\\ensuremath{\\hat \\Upsilon_{s}} = \n\\sum_{\\ell m} \\ensuremath{{}_s S_{\\ell m \\omega}}(\\theta) \\, \\ensuremath{ {}_s R_{\\ell m \\omega }}(r) e^{i m \\phi}.\n\\end{equation}\nThe inverse transform is given by\n\\begin{equation}\\label{eq:inverseLap}\n\\Upsilon_s(t,r,\\theta,\\phi)= \\frac{1}{2\\pi}\\int_{-\\infty+ic}^{\\infty+ic} e^{-i\\omega t} \\ensuremath{\\hat \\Upsilon_{s}}(\\omega,r,\\theta,\\phi)\\, {\\rm d} \\omega,\n\\end{equation}\nwhere $c>0$ is chosen such that the integral contour lies within the convergence region.\n\nTo formulate the initial data problem within the mode decomposition, we take the Laplace transform of the Teukolsky master equation \nand substitute \\eqref{eq:tilde Us mode}. We then collect the terms in the master equation with transformed time derivatives to the right-hand side and project onto the angular mode function. \nThis yields a sourced equation for the radial function\n\\begin{equation}\\label{eq:radeq formal}\n\\mathcal{L} \\ensuremath{ {}_s R_{\\ell m \\omega }} = \\ensuremath{ {}_s I_{\\ell m \\omega}}.\n\\end{equation}\nHere, $\\mathcal{L}$ is given in \\eqref{eq:radial} and the source $\\ensuremath{ {}_s I_{\\ell m \\omega}}=\\ensuremath{ {}_s I_{\\ell m \\omega}}(r)$ is comprised of $(\\ell,m)$-projected initial data~\\cite{Campanelli:1997un}:\n\\begin{widetext}\n\\begin{align}\\label{eq:I source modes}\n \\ensuremath{ {}_s I_{\\ell m \\omega}}= \\int_0^{2\\pi}\\! \\! \\int_0^\\pi \\bigg[ \n\\frac{\\Lambda}{\\Delta} \\left( \\ensuremath{\\partial}_t \\ensuremath{\\Upsilon_{s}} - i \\omega \\ensuremath{\\Upsilon_{s}} \\right) -2s \\Big( \\frac{M(r^2-a^2)}{\\Delta} -r -ia\\cos\\theta \\Big) \\ensuremath{\\Upsilon_{s}} + \\frac{4 M ar}{\\Delta} \\ensuremath{\\partial}_\\phi \\ensuremath{\\Upsilon_{s}}\\bigg]_{t=0}\\ensuremath{{}_s S_{\\ell m \\omega}}(\\theta)\\, e^{-i m \\phi} \\sin \\theta\\, {\\rm d} \\theta \\, {\\rm d} \\phi\n\\end{align}\n\\end{widetext}\nwhich we take to be of compact support.\n Imposing the outgoing boundary conditions \\eqref{eq:R bcs} fixes the freedom of homogeneous solutions to \\eqref{eq:radeq formal} and therefore defines a radial Green's function\n${}_s g_{\\ell m\\omega}(r,r') = \\ensuremath{ {}_s R_{\\ell m \\omega }}^{\\rm in}(r_<)\\ensuremath{ {}_s R_{\\ell m \\omega }}^{\\rm up}(r_>)\/\\mathcal W $\nwhere $r_< \\,\\,(r_>)$ is the lesser (greater) of $r$ and $r'$. Here, for any two solutions of the radial equation at fixed\n$s, m, \\ell, \\omega$, the (``$\\Delta$-scaled'') Wronskian\n\\begin{equation}\\label{eq:Wronskian}\n\\mathcal W[R_1,R_2] = \\Delta^{1+s}\\left[ R_1 \\frac{{\\rm d} R_2}{{\\rm d} r} - R_2 \\frac{{\\rm d} R_1}{{\\rm d} r} \\right]\n\\end{equation}\nhas been defined which is independent of $r$. If $R_1$ and $R_2$ are linearly dependent, then\nthe Wronskian vanishes. Thus, if we take $R_1 \\to R^{\\text{in}}$,\n$R_2 \\to R^{\\text{up}}$, the Wronskian vanishes when $\\omega$ attains\na quasinormal frequency.\n\nThe quasinormal mode contribution to $\\Upsilon_s$ can be found by closing the contour of the Laplace integral in the lower-half complex $\\omega$ plane, and can be expressed \nas a discrete sum over the residues of the radial Green's function arising at the points in the complex frequency plane where the Wronskian vanishes:\n\\begin{align}\\label{eq:oops n}\n\\Upsilon_{s} =& -i \\sum_{n\\ell m} e^{-i\\omega_n t+im\\phi} {}_sS_{\\ell m n}(\\theta) \\nonumber\\\\\n&\\times \\int_{r_+}^{\\infty} \\frac{\n\\Rq{n}^{\\rm in}(r_<) \\Rq{n}^{\\rm up}(r_>)}{{\\rm d}\\mathcal W\/ {\\rm d}\\omega\\vert_{\\omega_n}} {}_s I_{\\ell m n}(r')\\Delta^s(r') {\\rm d} r',\n\\end{align}\nwhere ${}_s I_{\\ell m n}={}_s I_{\\ell m \\omega}\\vert_{\\omega=\\omega_n}$. In considering only the poles when closing the contour, we are effectively ignoring \nthe early-time ``direct'' contribution from the large-$\\omega$ arc and the late-time ``tail'' contribution resulting from the branch point at zero frequency.\nThus, the $=$ sign in the above equation is not actually justified and should be understood as meaning this approximation.\n\nOn a quasinormal mode,\n $R^{\\rm in}$ is a constant multiple of $R^{\\rm up}$, and either may be moved outside the radial integral to write the field as\n\\begin{equation}\\label{eq:oops nice}\n\\Upsilon_{s} = \\sum_{n \\ell m} c_{\\ell mn} \\,{}_s\\Upsilon_{lmn},\n\\end{equation}\nwhere we have isolated the familiar form of the excitation coefficient\n\\begin{align}\\label{eq:qnm exc}\nc_{n\\ell m} &= -\\frac{i}{{\\rm d} \\mathcal W\/{\\rm d}\\omega\\vert_{\\omega_n}}\\int_{r_+}^{\\infty}\n {}_s I_{\\ell m n}(r') \\, {}_s R_{\\ell mn}(r') \\Delta^s(r') \\, {\\rm d} r'.\n\\end{align}\n\n\\subsection{Equivalence between \n\\eqref{eq:qnm exc} and \\eqref{eq:excitation coeff}} \n\n To begin, for a given\nradial function $R$, and $\\ell, m, \\omega$, we can define a $s=-2$ GHP\nscalar $\\Upsilon_{\\ell m \\omega}$ in separated form~\\eqref{eq:modes}\nby appending a spin-weighted spheroidal harmonic and $e^{-i\\omega t}$\ntime-dependence. For the time being, we do not require $R$ to satisfy\nany equation. We have the following lemma relating the Wronskian to\n$t\\cdot\\pi$ integrated over the 2-sphere.\n\n\\begin{lemma}\\label{lemma:wronskian-boundary}\n Let $\\Upsilon_1, \\Upsilon_2 \\ensuremath{\\circeq} (-4,0)$ be two GHP scalars in\n separated form~\\eqref{eq:modes}, with the same $m, \\ell, \\omega$,\n where $S_1$, $S_2$ are normalized spin-weighted spheroidal harmonics\n solving the angular equation, but where $R_1, R_2$ are not\n necessarily solutions to the radial equation. Then\n \\begin{equation}\n 8\\pi M^{4\/3} \\mathcal{W}[R_1,R_2] = \\int_{S^2(t,r)} t \\cdot \\pi(\\Psi_2^{4\/3}\\mathcal J \\Upsilon_1, \\Upsilon_2),\n \\end{equation}\n where $S^2(t,r)$ is a sphere of constant $t$ and $r$, $\\pi_{abc} = \\epsilon_{abcd} \\pi^d$ and $\\pi^a$ as in \\eqref{pidef}.\n\\end{lemma}\n\\begin{proof}\nConsider the Cauchy surface $\\Sigma(t) = \\{t={\\rm const.}\\}$ in Boyer-Lindquist coordinates. The future directed normal \nto $\\Sigma$ and induced area element on $S^2(t,r)$ are given by, respectively\n\\begin{align}\n \\nu^a =& \\left( \\sqrt{\\frac{\\Lambda}{\\Delta\\Sigma}}, 0, 0, \\frac{2Mar}{\\sqrt{\\Delta\\Sigma\\Lambda}}\\right), \\\\\n {\\rm d} A =& \\sqrt{\\Sigma\\Lambda} \\sin\\theta \\, {\\rm d} \\theta {\\rm d} \\phi. \n\\end{align}\nFrom the first relation on can read off the lapse function $N$ of $t^a$ from $\\nu^a = (t^a - N^a)\/N$. \nThe action of the reflection reverses $\\nu^a$ and from this fact and the formula for $\\pi^a$, see \\eqref{pidef}, one can deduce that \n\\begin{widetext}\n \\begin{align}\\label{eq:int-tdotpi}\n \\int_{S^2(t,r)} t \\cdot \\pi(\\Psi_2^{4\/3}\\mathcal J \\Upsilon_1, \\Upsilon_2) = \\int_{S^2(t,r)} N \\Psi_2^{4\/3}\\left\\{ (\\mathcal J\\Upsilon_1) r^a(\\Theta_a - 4 B_a) \\Upsilon_2 - \\Upsilon_2 \\mathcal J [r^a (\\Theta_a - 4 B_a) \\Upsilon_1 ]\\right\\} {\\rm d} A\n \\end{align}\n\\end{widetext}\nwhere $r^a$ is the normal to $S^2(r,t)$ inside $\\Sigma(t)$.\nAn explicit calculation shows that in the Kinnersley frame,\n \\begin{equation}\n r^a(\\Theta_a - 4B_a)\\Upsilon = \\sqrt{\\frac{\\Delta}{\\Sigma}}\\partial_r\\Upsilon - 2\\frac{(r-M)}{\\sqrt{\\Delta\\Sigma}}\\Upsilon.\n \\end{equation}\n Using this, as well as expressions for $N$, ${\\rm d} A$, and\n $\\mathcal J$ [using~\\eqref{eq:boostParams}], we obtain\n \\begin{widetext}\n \\begin{equation}\n \\int_{S^2(t,r)} t \\cdot \\pi(\\Psi_2^{4\/3}\\mathcal J \\Upsilon_1, \\Upsilon_2)\n = \\frac{4M^{4\/3}}{\\Delta(r)} \\left( R_1\\frac{{\\rm d} R_2}{{\\rm d} r} - R_2 \\frac{{\\rm d} R_1}{{\\rm d} r}\\right) \\int_0^{\\pi}\\int_0^{2\\pi} \\ S_1(\\theta) S_2(\\theta) \\, \\sin\\theta {\\rm d} \\theta {\\rm d} \\phi .\n \\end{equation}\n\\end{widetext}\nFinally, perfoming the integration and using the normalization~\\eqref{eq:theta-orthogonality} for the angular\n functions, we obtain the result.\n\\end{proof}\n\nNext, we take $R_1$ and $R_2$ to be solutions ingoing at the horizon\nand outgoing at infinity. Considered as a function of $\\omega$, the\nWronskian vanishes at quasinormal frequencies $\\omega_n$, because at\nthese frequencies the two solutions become linearly dependent. The\nfirst derivative with respect to $\\omega$, however, is proportional to\nthe ``norm'' of the quasinormal mode.\n\\begin{lemma}\\label{lemma:Wronskian-derivative}\n Let $R_\\omega^{\\mathrm{in}}, R_{\\omega}^{\\mathrm{up}}$ be solutions to\n the radial equation for fixed $s=-2,\\ell,m$, and allowing $\\omega$\n to vary, that are ingoing at the horizon and outgoing at infinity,\n respectively, as in~\\eqref{eq:R bcs}. Construct\n $\\Upsilon^{\\mathrm{in}}_\\omega, \\Upsilon^{\\mathrm{up}}_\\omega \\ensuremath{\\circeq}\n (-4,0)$ as mode solutions based on the radial functions. Then the\n derivative of the Wronskian at a quasinormal frequency $\\omega_n$\n can be written\n \\begin{equation}\n \\left. \\frac{{\\rm d}}{{\\rm d} \\omega}\\mathcal{W}[R^{\\mathrm{in}}_\\omega, R^{\\mathrm{up}}_\\omega] \\right|_{\\omega = \\omega_n} = \\frac{-i}{8\\pi M^{4\/3}} \\langle\\langle \\Upsilon^{\\mathrm{in}}_{\\omega_n}, \\Upsilon^{\\mathrm{up}}_{\\omega_n} \\rangle\\rangle.\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Consider the current\n $\\pi\\left(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n},\n \\Upsilon^{\\text{up}}_\\omega\\right)$ evaluated at a generic\n frequency $\\omega$ and a quasinormal frequency $\\omega_n$. By\n Cartan's magic formula and the fact that $\\pi$ is closed on\n solutions,\n \\begin{align}\n &{\\rm d}\\left(t \\cdot \\pi\\left(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\Upsilon^{\\text{up}}_\\omega\\right)\\right) \\nonumber \\\\\n &\\quad = \\mathcal{L}_t \\pi\\left(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\Upsilon^{\\text{up}}_\\omega\\right) \n \\nonumber\\\\\n &\\quad = -i(\\omega - \\omega_n)\\pi\\left(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\Upsilon^{\\text{up}}_\\omega\\right) ,\n \\end{align}\n where in contrast to the previous lemma the right side does not\n vanish on account of the different frequencies. We integrate over a partial Cauchy surface $S$, and\n apply Stokes's theorem,\n \\begin{align}\\label{eq:balance}\n &\\int_{\\partial S} t \\cdot \\pi(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\Upsilon^{\\text{up}}_\\omega)\\nonumber\\\\\n &\\quad= -i (\\omega - \\omega_n) \\int_S \\pi(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\Upsilon^{\\text{up}}_\\omega).\n \\end{align}\n Next, we\n differentiate this equation with respect to $\\omega$ and take the\n limit $\\omega\\to \\omega_n$. On the right side, we trivially get\n \\begin{equation}\n \\left.\\frac{{\\rm d}}{{\\rm d}\\omega}\\right|_{\\omega=\\omega_n} \\text{r.h.s. of \\eqref{eq:balance}} \n = -i \\int_S \\pi(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\Upsilon^{\\text{up}}_{\\omega_n}).\n \\end{equation}\n The left side can be expressed as three terms, namely \n \\begin{align}\n &\\left.\\frac{{\\rm d}}{{\\rm d}\\omega}\\right|_{\\omega=\\omega_n} \\text{l.h.s. of \\eqref{eq:balance}} \\nonumber\\\\\n &\\quad= \\int_{\\partial S_+} t\\cdot\\pi\\left(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\left.\\frac{{\\rm d}}{{\\rm d}\\omega}\\right|_{\\omega=\\omega_n} \\Upsilon^{\\text{up}}_\\omega\\right) \\nonumber\\\\\n &\\qquad- \\left.\\frac{{\\rm d}}{{\\rm d}\\omega}\\right|_{\\omega=\\omega_n} \\int_{\\partial S_-} t\\cdot\\pi\\left(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_\\omega, \\Upsilon^{\\text{up}}_\\omega\\right) \\nonumber\\\\\n &\\qquad + \\int_{\\partial S_-} t\\cdot \\pi\\left( \\left.\\frac{{\\rm d}}{{\\rm d}\\omega}\\right|_{\\omega=\\omega_n} \\Psi_2^{4\/3} \\mathcal{J} \\Upsilon^{\\text{in}}_\\omega, \\Upsilon^{\\text{up}}_{\\omega_n} \\right).\n \\end{align}\n By lemma~\\ref{lemma:wronskian-boundary} we can write the second of\n these terms as the derivative of the Wronskian,\n \\begin{align}\n &\\left.\\frac{{\\rm d}}{{\\rm d}\\omega}\\right|_{\\omega=\\omega_n} \\int_{\\partial S_-} t\\cdot\\pi\\left(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_\\omega, \\Upsilon^{\\text{up}}_\\omega\\right) \\nonumber\\\\\n &\\quad= 8 \\pi M^{4\/3} \\left.\\frac{{\\rm d}}{{\\rm d}\\omega} \\mathcal W[R^{\\text{in}}_\\omega, R^{\\text{up}}_\\omega]\\right|_{\\omega = \\omega_n}.\n \\end{align}\n\n Summarizing our results so far, we have shown that\n \\begin{align}\\label{eq:balance2}\n &8 \\pi M^{4\/3} \\left.\\frac{{\\rm d}}{{\\rm d}\\omega} \\mathcal W[R^{\\text{in}}_\\omega, R^{\\text{up}}_\\omega]\\right|_{\\omega = \\omega_n} \\nonumber\\\\\n &\\quad = -i \\int_S \\pi(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\Upsilon^{\\text{up}}_{\\omega_n}) \\nonumber\\\\\n &\\qquad - \\int_{\\partial S_-} t\\cdot \\pi\\left( \\left.\\frac{{\\rm d}}{{\\rm d}\\omega}\\right|_{\\omega=\\omega_n} \\Psi_2^{4\/3} \\mathcal{J} \\Upsilon^{\\text{in}}_\\omega, \\Upsilon^{\\text{up}}_{\\omega_n} \\right)\n \\nonumber\\\\\n &\\qquad - \\int_{\\partial S_+} t\\cdot\\pi\\left(\\Psi_2^{4\/3} \\mathcal J \\Upsilon^{\\text{in}}_{\\omega_n}, \\left. \\frac{{\\rm d}}{{\\rm d}\\omega} \\right|_{\\omega=\\omega_n} \\Upsilon^{\\text{up}}_\\omega\\right).\n \\end{align}\nAs $S \\to \\Sigma_\\mathbb{C}$, the boundary integrals vanish exponentially, so the right hand side\nof~\\eqref{eq:balance2} reduces to $-i \\langle\\langle \\Upsilon^{\\text{in}}_{\\omega_n}, \\Upsilon^{\\text{up}}_{\\omega_n} \\rangle\\rangle$. \n\\end{proof}\n\nThe desired equivalence between \\eqref{eq:qnm exc} and \\eqref{eq:excitation coeff} can be seen immediately by\nsubstituting \\eqref{eq:I source modes} for ${}_s I_{\\ell m n}$, comparing with \\eqref{eq:Kinnersley-bilinear}, and applying lemma~\\ref{lemma:Wronskian-derivative}.\n\n\\section{Concluding remarks}\n\nWe end this paper with some potential applications and alternatives to our formalism. \n\nThe main motivation of this work is to provide some tools needed to study the black hole ringdown beyond linear order in perturbation theory. Higher orders are already needed to interpret high-precision numerical relativity simulations of binary mergers~\\cite{Mitman:2022qdl,Cheung:2022rbm}, and could be needed to analyse gravitational wave observations by future detectors. \nRoughly speaking, we wish to make an ansatz \nfor the solution of the non-linear system as a linear combination of quasinormal modes with \\emph{time dependent} excitation coefficients similar to \\eqref{excited}. The idea is that the bilinear form will help us writing down a dynamical system for these coefficients by analogy with wave equations on compact spaces, where the normal modes would be used instead to compute the overlap integrals required for terms in this dynamical system that are non-linear in the modes.\n\nAs extremality is approached, it is well known that a family of quasinormal modes becomes arbitrarily long-lived, with\n$\\Re \\omega \\approx m\\Omega_H$\n\\cite{PressTeukolsky1973,Detweiler1977,Leaver1985,Hod:2008zz,Yang:2012pj,Cook:2014cta}. With\na commensurate frequency spectrum and arbitrarily slow decay, these\nmodes have been conjectured to become \\emph{turbulent} as\n$a\\to M$~\\cite{Yang:2014tla}. By taking the extremal limit of the nonlinear excitation coefficients using the approach detailed above, we hope to establish (or rule out) the emergence of turbolent behavior. \n\nApplications along similar lines could include mode\nmixing in clouds of ultralight scalar fields that could form outside\nKerr black holes~\\cite{Arvanitaki:2010sy}. Here, once these clouds grow via\nthe superradiant instability, nonlinear interactions between the modes have been conjectured to give rise to a coherent\nemission of gravitational waves or a bosenova~\\cite{Yoshino:2013ofa}, see~\\cite{Baumann:2018vus,Baumann:2022pkl} for recent proposals based on heuristic methods. It would be interesting to see whether our methods could be used to conceptualize or shed more light on the theoretical basis of such proposals. \n\nIn \\cite{gajic2021quasinormal}, a different approach is taken to quasinormal modes. Their essential idea is to consider, instead of a time $t$\nCauchy surface intersecting the bifurcation surface and spatial infinity, a ``hyperboloidal'' slice intersecting the future event horizon and future null infinity. On such a slice, they define a certain space of ``almost analytic'' functions (a ``Gevrey space'') encoding somehow the ``boundary conditions'' \\eqref{eq:R bcs}. Their space is in fact a genuine Hilbert space, and the time evolution is represented on this space by a semigroup whose generator is essentially the Hamiltonian, $\\mathcal{H}$ (see appendix \\ref{sec:Lagrangian-Hamiltonian}). Their inner product is non-canonical -- and is not conserved -- and the generator of the semi-group is correspondingly not symmetric, as is also not physically expected due to the ``dissipative'' nature of quasinormal modes. Nevertheless, their analysis shows that quasinormal modes are genuine eigenfunctions $\\mathcal{H} \\Upsilon = i\\omega \\Upsilon$ in this space -- crucially, by contrast to their restriction to a constant $t$ surface, they do not blow up on the hyperboloidal slice as the horizon or scri are approached. \nWhile the quasinormal modes are not orthogonal in their inner product, the definition of our bilinear form with a hyperboloidal slice is also clearly possible and it would be interesting to see whether quasinormal modes, as defined in the \nsetting of \\cite{gajic2021quasinormal} (see also~\\cite{Zenginoglu:2011jz,PanossoMacedo:2019npm,Ripley:2022ypi}) are still orthogonal in this bilinear form, as we conjecture. This would provide an alternative to our regularization procedure involving complex contours. \n\nFinally, it will be interesting to explore the relation between our bilinear form and the adjoint-spheroidal functions introduced in Ref.~\\cite{London:2020uva}.\n\n\n\\medskip\n{\\bf Acknowledgements:} We thank M. Casals, E. Flanagan, D. Gajic and E. Flanagan for comments and discussions related to this work. SH thanks the Max-Planck Society for supporting the collaboration between MPI-MiS and Leipzig U., Grant Proj. Bez. M.FE.A.MATN0003. VT is grateful to the International Max Planck Research School, MPI-MiS for support through a studentship. This work makes use of the Black Hole Perturbation Toolkit.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction and Result}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain with $C^1-$boundary. The normal vectors $n(x), n(y)$ of two elements of the boundary, $x,y \\in \\partial \\Omega$,will point roughly in the same direction which is roughly orthogonal to $y-x$ if $x$ and $y$ are close. In regions of\nlarge curvature the normal vector changes quickly but convex domains whose boundary has regions with large curvature are `flatter' in\nother regions and it might all average out in the end. We prove a quantitative version of this notion.\n\n\\begin{thm} There exists $c_n > 0$ so that for any bounded $\\Omega \\subset \\mathbb{R}^n$ with $C^1-$boundary\n$$ \\int_{\\partial \\Omega \\times \\partial \\Omega} \\frac{\\left|\\left\\langle n(x), y - x \\right\\rangle \\left\\langle y - x, n(y) \\right\\rangle \\right| }{\\|x -y\\|^{n+1}}~d \\sigma(x) d\\sigma(y) \\geq c_n |\\partial \\Omega|$$\nwith equality if and only if the domain $\\Omega$ is convex.\n\\end{thm}\nIntegration is carried out with respect to the $(n-1)-$dimensional Hausdorff measure\nand the size of the boundary $ |\\partial \\Omega|$ is measured the same way. Somewhat to our surprise,\nwe were unable to find this statement in the literature. It can be interpreted as a global conservation law\nfor convex domains or as a geometric functional with an extremely large set of minimizers (all convex domains). The requirement of the boundary $\\partial \\Omega$ being $C^1$\ncan presumably be somewhat relaxed.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.8]\n\\draw[thick] (0,0) ellipse (2.5cm and 1cm);\n\\filldraw (-2,0.6) circle (0.06cm);\n\\filldraw (2.37,0.3) circle (0.06cm);\n\\draw (-2, 0.6) -- (2.37, 0.3);\n\\draw [thick, ->] (-2, 0.6) -- (-2.28, 1.05);\n\\draw [thick, ->] (2.37, 0.3) -- (2.8, 0.55);\n\\node at (-1.9, 0.3) {$x$};\n\\node at (2.15, 0.05) {$y$};\n\\node at (-2.9, 1) {$n(x)$};\n\\node at (3.4, 0.5) {$n(y)$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{If $x$ and $y$ are close, then $n(x)$ and $n(y)$ are nearly orthogonal to $x-y$ unless $x$ and $y$ are far apart.}\n\\end{figure}\n\n If $\\Omega$ is the unit ball in $\\mathbb{R}^n$, then\n$\\partial \\Omega = \\mathbb{S}^{n-1}$ and for any $x, y \\in \\mathbb{S}^{n-1}$, we have $n(x) = x$ and\n$ \\|x-y\\|^2 = 2 - 2\\left\\langle x, y \\right\\rangle$. This simplifies the expression since\n$$ \\left|\\left\\langle n(x), y - x \\right\\rangle \\left\\langle y - x, n(y) \\right\\rangle \\right| = (1 - \\left\\langle x,y \\right\\rangle)^2.$$\n Moreover, using rotational symmetry and\n $w = (1,0,0,\\dots,0)$ for the north pole,\n\\begin{align*}\n\\int_{\\mathbb{S}^{n-1} \\times \\mathbb{S}^{n-1}} \\frac{ (1 - \\left\\langle x, y \\right\\rangle)^2 }{\\|y - x\\|^{n+1}}~d \\sigma(x) d\\sigma(y) &=\n \\int_{\\mathbb{S}^{n-1} \\times \\mathbb{S}^{n-1}} \\frac{ (1 - \\left\\langle x, y \\right\\rangle)^2 }{(2-2\\left\\langle x, y\\right\\rangle)^{\\frac{n+1}{2}}}~d \\sigma(x) d\\sigma(y) \\\\\n &= \\frac{|\\mathbb{S}^{n-1}|}{2^{\\frac{n+1}{2}}} \\int_{\\mathbb{S}^{n-1} } (1-\\left\\langle x, w\\right\\rangle)^{-\\frac{n-3}{2}}~d \\sigma(x) \n\\end{align*}\nwhich implies\n\\begin{align*}\n c_n = \\frac{1}{2^{\\frac{n+1}{2}}} \\int_{\\mathbb{S}^{n-1} } (1-x_1)^{-\\frac{n-3}{2}}~d \\sigma(x)= \\frac{1}{2} \\int_{\\mathbb{S}^{n-1}} \\left| x_1 \\right| d\\sigma(x).\n \\end{align*}\nThe constant has a simple form in low dimensions where $c_2 = 2$ and $c_3 = \\pi$. \nOur proof can be best described as an application of Integral Geometry; we formulate and use a bilinear version of the Crofton formula. \nThe proof tells us a little bit more: if the domain $\\Omega$ is convex, stronger statements can be made.\n\\begin{corollary} For any convex, bounded $\\Omega \\subset \\mathbb{R}^n$ with $C^1-$boundary and all $x \\in \\partial \\Omega$\n$$ \\int_{\\partial \\Omega} \\frac{\\left|\\left\\langle n(x), y - x \\right\\rangle \\left\\langle y - x, n(y) \\right\\rangle \\right| }{\\|x -y\\|^{n+1}}~d\\sigma(y) = c_n.$$\nIf $x \\in \\Omega \\setminus \\partial \\Omega$ and $w \\in \\mathbb{S}^{n-1}$ is an arbitrary unit vector, then\n$$ \\int_{\\partial \\Omega} \\frac{\\left|\\left\\langle w, y - x \\right\\rangle \\left\\langle y - x, n(y) \\right\\rangle \\right| }{\\|x -y\\|^{n+1}}~d\\sigma(y) = 2 \\cdot c_n.$$\n\\end{corollary}\nWe note that $c_n$ is the exact same constant as above (which can be seen by integrating the first equation over $\\partial \\Omega$ with respect to $d \\sigma(x)$). Some of the conditions can presumably be relaxed a little. The Crofton formula is known to hold in a very general setting (see Santal\\'o \\cite{santa2}). It is an interesting question whether any of these results could be generalized to more abstract settings.\n\n\\section{Proof of the Theorem}\n The Crofton formula in $\\mathbb{R}^n$ (see, for example, Santal\\'o \\cite{santa}) states that for rectifiable $S$ of co-dimension 1 one has\n$$ |S| = \\alpha_n \\int_{L} n_{\\ell}(S) d\\mu(\\ell),$$\nwhere the integral runs over the space of all oriented lines in $\\mathbb{R}^n$ with respect to the kinematic measure $\\mu$ (which is invariant under all\nrigid motions of $\\mathbb{R}^n$) and $n_{\\ell}(S)$ is the number of times the line $\\ell$ intersects the surface $S$. The constant\n$\\alpha_n$ can be computed by picking $S = \\mathbb{S}^{n-1}$ but will not be needed for our argument.\n\n\n\\begin{lemma}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain with $C^1-$boundary. Almost all lines $\\ell$ (with respect to the kinematic measure) intersect the boundary $\\partial \\Omega$ either never or in exactly two points if and only if $\\Omega$ is convex. \\end{lemma}\n\\begin{proof}\nIf $\\Omega$ is convex, the result is immediate. Suppose now $\\Omega$ is not convex; then there exists a boundary point $x \\in \\partial \\Omega$\nsuch that the supporting hyperplane does not contain all of the domain on one side (note that because the boundary of $\\Omega$ is $C^1$, the\nsupporting hyperplane is unique).\nIn particular, there exists $y \\in \\Omega$ that is on the other side of the supporting hyperplane. The line $\\ell$ that goes through $x$ and $y$ satisfies\n$n_{\\ell}(\\partial \\Omega) \\geq 4$, moreover, this is stable under some perturbations of the line (and thus a set of kinematic measure larger than 0) because $\\partial \\Omega$ is $C^1$. \\end{proof}\n\n\\begin{proof}[Proof of the Theorem]\nWe first note for almost all lines $\\ell$ (with respect to the kinematic measure) the number of intersections $n_{\\ell}(\\partial \\Omega)$ is either 0 or at least 2: if a line enters the domain, it also has to exit the domain (lines that are tangential to the boundary are a set of measure 0). This implies\n$$ |\\partial \\Omega| = \\alpha_n \\int_{L} n_{\\ell}(\\partial \\Omega) ~d \\mu(\\ell) \\leq \\frac{\\alpha_n}{2} \\int_{L} n_{\\ell}(\\partial \\Omega)^2 ~d \\mu(\\ell)$$\nwith equality if and only if $\\Omega$ is convex.\nAt this point we pick a small $\\varepsilon > 0$ and decompose the boundary\n$ \\partial \\Omega = \\bigcup_i \\partial \\Omega_i$ into small disjoint regions that have diameter $\\leq \\varepsilon \\ll 1$ (and $\\varepsilon$ will later tend to 0).\nNaturally,\n$$ n_{\\ell}(\\partial \\Omega)^2 = \\left[ n_{\\ell}\\left( \\bigcup_i \\partial \\Omega_i\\right) \\right]^2 = \\left[ \\sum_i n_{\\ell}\\left( \\partial \\Omega_i\\right) \\right]^2= \\sum_{i,j} n_{\\ell}( \\partial \\Omega_i) n_{\\ell}(\\partial \\Omega_j)$$\nWe will now evaluate the integral over such a product. The diagonal terms $i=j$ behave\na little bit differently than the non-diagonal terms and we start with those. As $\\varepsilon \\rightarrow 0$, the fact that the \nboundary is $C^1$ implies that a `random' line will hit any such infinitesimal segment at most once and thus the Crofton formula implies\n\\begin{align*}\n \\frac{\\alpha_n}{2} \\int_{L} \\sum_i n_{\\ell}( \\partial \\Omega_i)^2 ~d \\mu(\\ell) &= (1+o(1)) \\frac{\\alpha_n}{2} \\int_{L} \\sum_i n_{\\ell}( \\partial \\Omega_i) ~d \\mu(\\ell)\\\\ \n &= (1+o(1)) \\frac{\\alpha_n}{2} \\int_{L} n_{\\ell}( \\partial \\Omega) ~d \\mu(\\ell) = (1+o(1)) \\frac{|\\partial \\Omega|}{2}\n \\end{align*}\n which is nicely behaved as $\\varepsilon \\rightarrow 0$ (the error could be made quantitative in terms of the modulus of continuity of the normal vector). It remains to analyze the off-diagonal terms.\n Let us assume that $\\partial \\Omega_x \\subset \\partial \\Omega$ is a small\nsegment centered around $x \\in \\partial \\Omega$ and $\\partial \\Omega_y \\subset \\partial \\Omega$ is a small\nsegment centered around $y \\in \\partial \\Omega$ and that both are scaled to have surface area $0 < \\varepsilon \\ll 1$. We can\nalso assume, because the surface is $C^1$ and we are allowed to take $\\varepsilon$ arbitrarily small, that they are approximately given by hyperplanes (and, as above, the error is a lower order term coming from curvature). \nThe quantity to be evaluated, \n$$ \\int_{L} n_{\\ell}( \\partial \\Omega_x) n_{\\ell}(\\partial \\Omega_y) d\\mu(\\ell), \\qquad \\mbox{can be seen in probabilistic terms}$$\nas the likelihood that a `random' line (random as induced by the kinematic measure $\\mu$) intersects\nboth $\\partial \\Omega_x$ and $\\partial \\Omega_y$. Appealing to the law of total probability\n$$ \\mathbb{P}\\left( n_{\\ell}( \\partial \\Omega_x) n_{\\ell}(\\partial \\Omega_y) = 1\\right) = \\mathbb{P}( n_{\\ell}(\\partial \\Omega_y) = 1 \\big| n_{\\ell}( \\partial \\Omega_x) =1) \\cdot \\mathbb{P}( n_{\\ell}( \\partial \\Omega_x) =1).$$\nThe last quantity is easy to evaluate: by Crofton's formula\n$$ \\mathbb{P}( n_{\\ell}( \\partial \\Omega_x) =1) = \\frac{1}{\\alpha_n} | \\partial \\Omega_x| = \\frac{\\varepsilon}{\\alpha_n}.$$\nIt remains to compute the second term: the likelihood of a `random' line hitting $\\partial \\Omega_y$ provided that it has already hit $\\partial \\Omega_x$. For this purpose, we first consider what we can say about random lines that have hit $\\partial \\Omega_x$. The distribution of $\\partial \\Omega_x \\cap \\ell$, provided it is not empty, is, to leading order, uniformly distributed over $\\partial \\Omega_x$ because $\\partial \\Omega_x$ is, to leading order, part of a hyperplane and the kinematic measure is translation-invariant. In contrast, the direction $\\phi$ of intersection (identified with unit vectors on $\\mathbb{S}^{n-1}$) is not uniformly distributed: the likelihood is proportional to the size of the projection of $\\Omega_x$ in direction of $\\phi$ which is proportional to the inner product of $\\phi$ with the normal vector $n(x)$. Hence the probability distribution of the direction of intersection $\\phi$ of lines conditioned on hitting $\\partial \\Omega_x$ is given by\n$$ \\Psi(\\phi) = \\frac{2 \\left\\langle n(x), \\phi \\right\\rangle } {\\int_{\\mathbb{S}^{n-1}} \\left|\\left\\langle w, n(x)\\right\\rangle \\right| d\\sigma(w) }, $$\nwhere the factor $2$ comes from the fact that each line creates two directions of intersections. \nThis allows us to perform a change of measure: we may assume that the lines are oriented uniformly at random provided that we later weigh the end result by $\\Psi$.\nIf the lines are oriented in all directions uniformly, then it is easy to see the likelihood of hitting $\\partial \\Omega_y$ provided one has already hit $\\partial \\Omega_x$: it is simply proportional to the size of the projection of $\\partial \\Omega_y$ onto the sphere of radius $\\|x-y\\|$ centered at $x$. The projection shrinks\nthe area by a factor of $ \\left| \\left\\langle n(y), (x-y)\/\\|x-y\\| \\right\\rangle \\right|$. The relative likelihood is then proprtional to\n$$P= \\Psi\\left( \\frac{x-y}{\\|x-y\\|}\\right) \\left| \\left\\langle n(y), \\frac{x-y}{\\|x-y\\|} \\right\\rangle \\right| \\frac{\\varepsilon}{\\|x-y\\|^{n-1}}.$$\nPlugging in the definition of $\\Psi$ this simplifies to\n$$ P = 2\\left( \\int_{\\mathbb{S}^{n-1}} \\left|\\left\\langle w, n(x)\\right\\rangle\\right| d\\sigma(w)\\right)^{-1} \\frac{\\left| \\left\\langle n(x), x-y\\right\\rangle \\left\\langle x-y, n(y) \\right\\rangle \\right|}{\\|x-y\\|^{n+1}} \\varepsilon,$$\nwhere we note that, by rotational symmetry of the sphere, the first integral is actually independent of the direction in which $n(x)$ is pointing.\nAltogether,\n\\begin{align*}\n |\\partial \\Omega|& = \\alpha_n \\int_{L} n_{\\ell}(\\partial \\Omega) ~d \\mu(\\ell) \\leq \\frac{\\alpha_n}{2} \\int_{L} n_{\\ell}(\\partial \\Omega)^2 ~d \\mu(\\ell) \\\\\n &= \\frac{\\alpha_n}{2} \\int_{L} \\sum_i n_{\\ell}(\\partial \\Omega_i) ~d \\mu(\\ell) + \\frac{\\alpha_n}{2} \\int_{L} \\sum_{i \\neq j} n_{\\ell}(\\partial \\Omega_i) n_{\\ell}(\\partial \\Omega_j) ~d \\mu(\\ell).\n\\end{align*}\nThe inequality is an equation if and only if $\\Omega$ is convex. As already discussed above, the first term tends to $|\\Omega|\/2$ as $\\varepsilon \\rightarrow 0$. Thus, for arbitrary $a \\in \\mathbb{S}^{n-1}$,\n\\begin{align*}\n\\frac{ |\\partial \\Omega|}{2} &\\leq \\lim_{\\varepsilon \\rightarrow 0} \\frac{\\alpha_n}{2} \\int_{L} \\sum_{i \\neq j} n_{\\ell}(\\partial \\Omega_i) n_{\\ell}(\\partial \\Omega_j) ~d \\mu(\\ell) \\\\\n&=\\left( \\int_{\\mathbb{S}^{n-1}} \\left|\\left\\langle w, a\\right\\rangle\\right| d\\sigma(w)\\right)^{-1} \\int_{\\partial \\Omega \\times \\partial \\Omega} \\frac{\\left|\\left\\langle n(x), y - x \\right\\rangle \\left\\langle y - x, n(y) \\right\\rangle \\right| }{\\|y - x\\|^{n+1}}~d \\sigma(x) d\\sigma(y).\n\\end{align*}\nThis establishes the inequality with constant\n$$ c_n =\\frac{1}{2} \\int_{\\mathbb{S}^{n-1}} \\left|\\left\\langle w, n\\right\\rangle\\right| d\\sigma(w) = \\frac{1}{2} \\int_{\\mathbb{S}^{n-1}} \\left| w_1 \\right| d\\sigma(w).$$\n\\end{proof}\n\n\n\\section{Proof of the Corollary}\n\\begin{proof} The proof of the Corollary is using the same computation as the proof of the Theorem in two additional settings leading to the two identities. Let $\\Omega$ be convex and let $x \\in \\partial \\Omega$. We start by considering an infinitesimal hyperplane segment $\\partial \\Omega_x$ centered around $x$. By convexity of $\\Omega$, almost all lines intersecting $\\partial \\Omega_x$ will intersect $\\partial \\Omega$ in exactly one other point. This implies, as the size of $\\partial \\Omega_x$ tends to 0, that\n$$ \\int_{L} n_{\\ell}( \\partial \\Omega_x) n_{\\ell}(\\partial \\Omega \\setminus \\partial \\Omega_x) d\\mu(\\ell) = (1+o(1)) \\cdot \\mu(\\partial \\Omega_x).$$\nAt the same time, by Crofton's formula, the likelihood of a line hitting $\\partial \\Omega_x$ is only a function of the surface area of $\\partial \\Omega_x$ and independent of everything else. Finally, using linearity, we can decompose $\\partial \\Omega \\setminus \\partial \\Omega_x$ into small hyperplane segments and use the computation above to deduce that\n$$ \\int_{\\partial \\Omega} \\frac{\\left|\\left\\langle n(x), y - x \\right\\rangle \\left\\langle y - x, n(y) \\right\\rangle \\right| }{\\|x -y\\|^{n+1}}~d\\sigma(y) = \\mbox{const}.$$\nIntegrating once more and applying the Theorem immediately implies that the constant has to be $c_n$. As for the second part, we can consider an infinitesimal hyperplane segment $H_x$ centered at $x \\in \\Omega \\setminus \\partial \\Omega$ with normal direction given by $w \\in \\mathbb{S}^{n-1}$. Every line hitting $H_x$ intersects $\\partial \\Omega$ in exactly two points and thus\n$$ \\int_{L} n_{\\ell}( H_x) n_{\\ell}(\\partial \\Omega) d\\mu(\\ell) = 2 \\cdot \\mu(H_x).$$\nBy Crofton's formula, the right-hand side does not depend on the shape or location of $H_x$ and is only a function of the surface area of the infinitesimal segment. As for the left-hand side, using the computation done in the proof of the Theorem shows\n $$ \\int_{L} n_{\\ell}( H_x) n_{\\ell}(\\partial \\Omega) d\\mu(\\ell) = (1+o(1)) \\int_{\\partial \\Omega} \\frac{\\left|\\left\\langle w, y - x \\right\\rangle \\left\\langle y - x, n(y) \\right\\rangle \\right| }{\\|x -y\\|^{n+1}}~d\\sigma(y)$$\nwhere the error term is with respect to the diameter of $H_x$ shrinking to 0.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\nAcquiring comprehensive data from human speech is a challenging task\nthat, however, is crucial for understanding and modelling speech\nproduction as well as developing speech signal processing algorithms.\nThe possible approaches can be divided into \\emph{direct} and\n\\emph{indirect methods}. Direct methods concern measurements carried\nout on test subjects either by audio recordings, acquisition of\npressure, flow velocity, or even electrical signals (such as takes\nplace in electroglottography), or using different methods of medical\nimaging during speech. Indirect methods concern simulations using\ncomputational models (such as described in \\cite{A-A-M-M-V:MLBVFVTOPI}\nand the references therein) or measurements from \\emph{physical\n models}\\footnote{Physical models are understood as artefacts or\n replicas of parts of the speech anatomy in the context of this\n article.}. Typically, computational and physical models are created\nand evaluated based on data that has first been acquired by direct\nmethods. The main advantage of indirect methods is the absence of the\nhuman component that leads to experimental restrictions and unwanted\nvariation in data quality.\n\nThe purpose of this article is to describe an experimental\narrangement, its validation, and some experiments on one type of\nphysical model for vowel production: \\emph{acoustic resonators}\ncorresponding to vocal tract (VT) configurations during prolonged\nvowel utterance. The anatomic geometry for such resonators has been\nimaged by Magnetic Resonance Imaging (MRI) with simultaneous speech\nrecordings as described in\n\\cite{A-A-H-J-K-K-L-M-M-P-S-V:LSDASMRIS,K-M-O:PPSRDMRI}. The MRI voxel\ndata has been processed to surface models as explained in\n\\cite{O-M:ASUAMRIVTGE} and then printed in ABS plastic by Rapid\nPrototyping as explained below in Section~\\ref{ProcessingSubSec}. In\nitself, the idea of using 3D printed VT models in speech research is\nby no means new: see, e.g.,\n\\cite{T-K-E-S-W:EEIFPSKGFTC,E-S-W:NIMARVTDP,T-M-K:AAVTDVPFDTDM}.\n\nJust creating physical models of the VT is not enough for model\nexperiments: also a suitable acoustic signal source is required with\ncustom instrumentation and software associated to it. As these\nexperiments involve a niche area in speech research, directly\napplicable commercial solutions do not exists and constructing a\ncustom measurement suite looks an attractive option. Thus, we propose\nan \\emph{acoustic glottal source} design shown in\nFig.~\\ref{TractrixHornFig} that resembles the loudspeaker-horn\nconstructions shown in \\cite[Fig.~1]{T-K-E-S-W:EEIFPSKGFTC},\n\\cite[Fig.~3]{E-S-W:NIMARVTDP}, \\cite[Fig.~2a]{Wolfe:AIS:2000} . \n\nAll such source\/horn constructions can\nbe regarded as variants of \\emph{compression drivers} used as high\nimpedance sources for horn loudspeakers. Unfortunately, most\ncommercially available compression drivers are designed for\nfrequencies over $500 \\, \\mathrm{Hz}$ whereas a construction based on\na loudspeaker unit can easily be scaled down to lower frequencies\nrequired in speech research. We point out that high quality acoustic\nmeasurements on VT physical models can be carried out using a\nmeasurement arrangement not based an impedance matching horn or a\ncompression driver of some other kind; see\n\\cite[Fig.~3]{T-M-K:AAVTDVPFDTDM} where the sound pressure is fed into\nthe model through the mouth opening, and the measurements are carried\nout using a microphone at the vocal folds position. However, excitation\nfrom the glottal position is desirable because the face and the\nexterior space acoustics are issues as well.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.21\\textwidth]{GS_pic_9.jpg}\\hspace{0.31cm}\n\\includegraphics[width=0.43\\textwidth]{GS_pic_6.jpg}\\hspace{0.31cm}\n\\includegraphics[width=0.29\\textwidth]{dummyload.jpg}\n\\end{center}\n \\caption{\\label{TractrixHornFig} Left: Measurement arrangement for\n the frequency response of vowel [\\textipa{\\textscripta}] from a 3D\n printed VT geometry. Middle: The tractrix horn and the loudspeaker\n unit assembly separated. Right: The dummy load used for\n calibration measurements as explained in\n Section~\\ref{CalibrationSec}. }\n\\end{figure}\n\n\nThe general principle of operation of the acoustic glottal source is\nfairly simple. The source consists of a loudspeaker unit and an\nimpedance matching horn as shown disassembled in\nFig.~\\ref{TractrixHornFig} (middle panel). The purpose of the horn is\nto concentrate the acoustic power from the low-impedance loudspeaker\nto an opening of diameter $6\\, \\mathrm{mm}$, the high-impedance output\nof the source. There is, however, a number of conflicting design\nobjectives that need be taken into account in a satisfactory way. For\nexample, the instrument should not be impractically large, and it\nshould be usable for acoustic measurements of physical models of human\nVTs in the frequency range of interest, specified as $80\n\\ldots 7350 \\, \\mathrm{Hz}$ in this article. To achieve these goals\nin a meaningful manner, we use a design methodology involving\n\\textrm{(i)} heuristic reasoning based on mathematical acoustics,\ntogether with \\textrm{(ii)} numerical acoustics modelling of the main\ncomponents and their interactions. Numerical modelling of all details\nis not necessary for a successful outcome. Optimising the source\nperformance using only the method of trial and error and extensive\nlaboratory measurements would be overly time consuming as well.\n\nThe design and construction process was incremental, and it consisted\nof the following steps that were repeated when necessary:\n\\begin{enumerate}\n\\item[(i)] Choice of the acoustic design and the main components,\n based on general principles of acoustics, horn design, and\n feasibility,\n\\item[(ii)] Finite element (FEM) based modelling of the horn acoustics\n to check overall validity of the approach, to detect and then\n correct the expected problems in construction,\n\\item[(iii)] the construction of the horn and the loudspeaker\n assembly together with the required instrumentation,\n\\item[(iv)] a cycle of measurements and modifications, such as\n placement of acoustically soft material and silicone sealings in\n various parts based on, e.g., the FEM modelling,\n\\item[(v)] development of MATLAB software for producing properly\n weighted measurement signals for sweep experiments that compensate\n most of the remaining nonidealities, and\n\\item[(vi)] development of MATLAB software for reproducing the\n Liljencrants--Fant (LF) glottal waveform excitation at the glottal\n position of the physical models.\n\\end{enumerate}\n\n\nFinally, the source is used for measuring the frequency responses of\nphysical models of VT during the utterance of Finnish vowels\n[\\textipa{\\textscripta, i, u}], obtained from a 26-year-old male (in\nfact, one of the authors of this article).\nThe measured amplitude frequency responses are compared with the\nspectral envelope data from vowel samples shown in\nFig.~\\ref{VTResponseFig}, recorded in anechoic chamber from the same\ntest subject. In addition to these responses, vowel signal is produced\nby acoustically exciting the physical models by a glottal pulse\nwaveform of LF type, reconstructed at the output of the source. The\nproduced signals for vowels [\\textipa{\\textscripta, i, u}] have good\naudible resolution from each other, yet they have the distinct\n``robotic'' sound quality that is typical of most synthetically\nproduced speech.\n\nResonant frequencies extracted from the measured frequency responses\nare used for development and validation of acoustic and phonation\nmodels such as the one introduced in \\cite{A-A-M-M-V:MLBVFVTOPI}. The\nsynthetic vowel signals are intended for benchmarking Glottal Inverse\nFiltering (GIF) algorithms as was done in\n\\cite{Alku:EstVoiceSrc:2006,Alku:IFReview:2011}. Large amounts of\nmeasurement data are required for these applications which imposes\nrequirements to the measurement arrangement.\n\nSo as to physical dimension of the measured signals, this article\nrestricts to sound pressure measurements using microphones. If\nacoustic impedances are to be measured instead, some form of acoustic\n(perturbation) velocity measurement need be carried out. The velocity\nmeasurement can be carried out, e.g., by hot wire anemometers\n\\cite{Kob:MMV:2002}, impedance heads consisting of several microphones\n\\cite{Wolfe:EEI:2013}, or even by a single microphone using a\nresistive calibration load coupled to a high impedance source\n\\cite{Singh:AIM:1978}; see \\cite[Table~1]{Wolfe:IPM:2006} for various\napproaches. In general, carrying out velocity measurements is much\nmore difficult and expensive that measuring just sound pressure.\nDetermining pressure-to-pressure -responses of VT physical models is,\nhowever, sufficient for the purposes of this article since\n(\\textrm{i}) resonant frequencies can be determined from pressures,\nand (\\textrm{ii}) the GIF algorithm can be configured to run on\npressure data.\n\n\\section{\\label{BackgroundSec} Background}\n\nWe review relevant aspects from mathematical acoustics, horn design,\nsignal processing, and MRI data acquisition.\n\n\\subsection{Acoustic equations for horns}\n\nAcoustic horns are impedance matching devices that can be described as\nsurfaces of revolution in a three-dimensional space. Thus, they are\ndefined by strictly nonnegative continuous functions $r = R(x)$ where\n$x \\in [0, \\ell]$, $\\ell > 0$ being the length of the horn, and $r$\ndenoting the radius of horn at $x$. The end $x = 0$ ($x = \\ell$) is\nthe \\emph{input end} (respectively, the \\emph{output end}) of the\nhorn. It is typical, though not necessary, that the function\n$R(\\cdot)$ is either increasing or decreasing.\n\n\nThere exists a wide literature on the design of acoustic (tractrix)\nhorns for loudspeakers; see, e.g.,\n\\cite{Dinsdale:1974,Edgar:1981,Delgado:2000,U-W-B:OVMAH}. As a\ngeneral rule, the matching impedance at an end of the horn is\ninversely proportional to the opening area. For uniform diameter\nwaveguides, the matching impedance coincides with the characteristic\nimpedance given by $Z_0 = \\rho c\/A_0$ where $A_0$ is the\nintersectional area. The constant $c$ denotes the speed of sound and\n$\\rho$ is the density of the medium.\n\nTo describe the acoustics of an air column in a cavity such as a horn,\nwe use two (partial) differential equations. The three dimensional\nacoustics is described by the lossless Helmholtz equation in terms of\nthe velocity potential\n\\begin{equation} \\label{HelmHoltzEq}\n \\lambda^2 \\phi_\\lambda = c^2 \\Delta \\phi_\\lambda \\text{ on } \\Omega \\quad\n \\text{ and } \\quad \n \\frac{\\partial \\phi_\\lambda}{\\partial \\nu}({\\bf r}) = 0 \\text{ on }\n \\partial \\Omega \\setminus \\Gamma_0\n\\end{equation}\nwhere the acoustic domain is denoted by $\\Omega \\subset {\\mathbb{R}}^3$ with\nboundary $\\partial \\Omega$. A part of the boundary, denoted by\n$\\Gamma_0$, is singled out as an interface to the exterior space. In\nhorn designs of Section~\\ref{HelmholtzCavitySec}, the interface\n$\\Gamma_0$ is the opening at the narrow output end of the horn. In\nSection~\\ref{CompValSec}, the symbol $\\Gamma_0$ denotes a spherical\ninterface around the mouth opening. For now, we use the Dirichlet\nboundary condition on $\\Gamma_0$\n\\begin{equation} \\label{DirichletBndry}\n \\phi_\\lambda({\\bf r}) = 0 \\text{ on } \\Gamma_0.\n\\end{equation}\nEqs.~\\eqref{HelmHoltzEq}-- \\eqref{DirichletBndry} have a countably\ninfinite number of solutions $(\\lambda_j, \\phi_j) = (\\lambda,\n\\phi_\\lambda) \\in \\mathbb{C} \\times H^1(\\Omega) \\setminus \\{ 0\\}$ for $j = 1,\n2, \\ldots$, and each of the solutions is associated to a\n\\emph{Helmholtz resonant frequency} $f_j$ of $\\Omega$ by $f_j =\n\\mathrm{Im}{\\lambda_j}\/2 \\pi$.\n\nIn addition to acoustic resonances, the acoustic transmission\nimpedance of the source is important. Because it is more practical to\ndeal with scalar impedances, we use the lossless Webster's resonance\nmodel for defining it, again in terms of Webster's velocity\npotential. It is given for any $s \\in \\mathbb{C}$ by\n\\begin{equation} \\label{WebsterModel}\n\\begin{aligned}\n s^2 \\psi_s & = \\frac{c^2}{A(x)} \\frac{\\partial}{\\partial x} \\left (A(x) \\frac{\\partial \\psi_s}{\\partial x} \\right ) \\text{ on } [0,\\ell], \\\\\n - A(0) \\frac{\\partial \\psi_s}{\\partial x}(0) & = \\hat i(s), \\text{ and } R_L A(\\ell)\n \\frac{\\partial \\psi_s}{\\partial x}(\\ell) = \\rho s \\phi_s(\\ell)\n\\end{aligned}\n\\end{equation}\nwhere $A(x) = \\pi R(x)^2$ is the intersectional area of the horn,\n$\\rho$ is the density of air, and $R_L \\geq 0$ is the termination\nresistance at the output end $x = \\ell$ \\footnote{Because the external\n termination resistance $R_L$ is the only loss term in\n Eq.~\\eqref{WebsterModel}, we call the model lossless.}. Again, the\nfrequencies and Laplace transform domain $s$ variables are related by\n$f = \\mathrm{Im} s\/ 2\\pi$. The function $\\hat i(s)$ is the Laplace\ntransform of the (perturbation) volume velocity used to drive the\nhorn, and the output is similarly given as the Laplace transform of\nthe sound pressure given by $\\hat p(s) = \\rho s \\phi_s(\\ell)$. Now,\nthe transmission impedance of the horn, terminated to the resistance\n$R_L > 0$, is given by\n\\begin{equation} \\label{TransmissionImpedance}\n Z_{R_L}(s) = \\hat p(s)\/\\hat i(s) \\text{ for all } s \\in \\mathbb{C}_+.\n\\end{equation}\nNote that when solving Eq.~\\eqref{WebsterModel} for a fixed $s$, we may\nby linearity choose $\\hat i(s) = 1$ when plainly $Z_{R_L}(s) = \\rho s\n\\phi_s(\\ell)$. Further, as an impedance of a passive system, the\ntransmission impedance satisfies the positive real condition\n\\begin{equation}\n \\mathop{Re} {Z_{R_L}(s)} \\geq 0 \\text{ for all } s \\in \\mathbb{C}^{+} := \\{ s\n \\in \\mathbb{C}: \\mathop{Re}{s} > 0 \\}.\n\\end{equation}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.35]{testTractrix.pdf} \\hspace{0.5cm}\n\\includegraphics[width=0.20\\textwidth]{tract_blend_3d.png} \\hspace{0.5cm}\n\\includegraphics[width=0.33\\textwidth]{tract_a_rend.png} \n\\end{center}\n \\caption{\\label{DesignOfSourceFig} Left: Wave propagation in a\n tractrix horn. The spherical wave front progressing along the\n centreline meets the horn surface at right angles. Middle: A 3D\n illustration of the impedance matching cavity within the\n source. Right: The geometry of the VT corresponding to vowel\n [\\textipa{\\textscripta}], equipped with a spherical boundary\n condition interface at the mouth opening.}\n\\end{figure}\n\n\n\\subsection{\\label{SupressionSubSec} Suppression of transversal modes in horns}\n\nBy transversal modes we refer to the resonant standing wave patterns\nin a horn where significant pressure variation is perpendicular to the\nhorn axis, as opposed to purely longitudinal modes. The purpose of\nthis section is to argue why transversal modes in horn geometries are\nundesirable from the point of view of the this article.\n\nAs a well-known special case, consider a wave\\-guide of length $\\ell$\nthat has a constant diameter, i.e., $A(x) = A_0$. Then the\ntransmission impedance given by\nEqs.~\\eqref{WebsterModel}--\\eqref{TransmissionImpedance} can be given\nthe explicit formula\n\\begin{equation} \\label{TransmissionLineImpedance}\n Z_{R_L}(s) = \\frac{Z_0 R_L}{Z_0 \\cosh{\\frac{s \\ell}{c}} + R_L \\sinh{\\frac{s \\ell}{c}}}\n\\end{equation}\nwhere $Z_0 := \\rho c\/A_0$ is called \\emph{characteristic\n impedance}. Because both $\\cosh$ and $\\sinh$ are entire functions,\nit is impossible to have $Z_{R_L}(s) = 0$ for any $s \\in \\mathbb{C}$. If the\ntermination resistance $R_L$ equals the characteristic impedance of\nthe wave\\-guide, the wave\\-guide becomes nonresonant, and we get the\npure delay $Z_{R_L}(s) = Z_0 e^{- s \\ell \/ c}$ of duration $T = \\ell \/\nc$ as expected.\n\nIt can be shown by analysing the Webster's model that the transmission\nimpedance $Z_{R_L}(s)$ given by Eq.~\\eqref{TransmissionImpedance} has\nno zeroes for $s \\in \\mathbb{C}$; i.e., it is an all-pole transmission impedance\nfor any finite value of termination resistance\n$R_L > 0$.\\footnote{This follows from Holmgren's uniqueness theorem\n for real analytic area functions $A(\\cdot)$.} The salient, desirable\nfeature of any all-pole impedance is that also the admittance\n$A_{R_L}(s) := Z_{R_L}(s)^{-1}$ is analytic and even\n$\\mathop{Re}{A_{R_L}(s)} > 0$ in $s \\in \\mathbb{C}^{+}$. This makes it easy to\nprecompensate the lack of flatness in the frequency response of\n$Z_{R_L}(s)$ by a causal, passive, rational filter whose transfer\nfunction approximates $A_{R_L}(s)$.\n\nOn the other hand, it has been shown in\n\\cite[Theorem~5.1]{L-M:PEEWEWP} that the time-dependent Webster's\nmodel describes accurately the transversal averages of a 3D wavefront\nin an acoustic wave\\-guide if the wavefront itself is constant on the\ntransversal sections of the wave\\-guide interior. Conversely,\nWebster's equation models only the longitudinal dynamics of the\nwave\\-guide acoustics by its very definition as can be understood\nfrom, e.g., \\cite{L-M:WECAD}. If the transversal modes in a\nwave\\-guide have been significantly excited, then Webster's equation\nbecomes a poor approximation, and all hopes of regarding the measured\ntransmission impedance of the wave\\-guide as an all-pole SISO system\nare lost. A more intuitive way of seeing why transversal acoustic\nmodes are expected to introduce zeroes to $Z_{R_L}(\\cdot)$ is by\nreasoning by analogy with Helmholtz resonators: the resonant side\nbranches of the wave\\-guide (eliciting transversal modes at desired\nfrequencies) can be used to eliminate frequencies from response.\n\nWe have now connected, via Webster's horn model, the appearance of\ntransversal modes in a horn to zeroes of the transmission impedance\n$Z_{R_L}(\\cdot)$. Because these zeroes are undesirable features in\ngood horn designs, we need to identify and suppress the transversal\nmodes as well as is feasible.\n\n\\subsection{Minimisation of transmission loss}\n\\label{TLMinSec}\n\nWhen a horn is excited from its input end, some of the excitation\nenergy is reflected back to the source with some delays. For horns of\nfinite length $\\ell$, there are two kinds of backward\nreflections. Firstly, the geometry of the horn may cause distributed\nbackward reflections over the the length of the horn. Secondly, there\nmay be backward reflections at the output end of the horn, depending\non the acoustic impedance seen by the horn at the termination point $x\n= \\ell$ in Eq.~\\eqref{WebsterModel}. We next consider only the\nbackward reflections of the first kind since only they can be affected\nby the horn design.\n\nBecause the acoustics of the horn described by\nEqs.~\\eqref{HelmHoltzEq}---\\eqref{WebsterModel} is internally\nlossless, minimising the TL amounts to minimising the backward\nreflections that take place inside the horn. This is a classical shape\noptimisation problem in designing acoustical horns, and modern\napproaches are based on numerical topology optimisation techniques as\npresented in, e.g., \\cite{U-W-B:OVMAH,Y-W-B:LOTSHMIFFDPAH} where also\nother design objectives (typical of loudspeaker horn design) are\ntypically taken into account.\n\nWe take another approach, and use analytic geometry and physical\nsimplifications of wave propagation for designing the function $r =\nR(x)$ on $[0, \\ell]$ following Paul~G.~A.~H.~Voigt who proposed a\nfamily of tractrix horns in his patent ``Improvements in Horns for\nAcoustic Instruments'' in 1926, see \\cite{PV:IHAI}. His invention was\nto use the surface of revolution of tractrix curve given by\n\\begin{equation} \\label{TractrixEq}\n x = a \\ln{\\frac{a + \\sqrt{a^2 - r^2}}{r}} - \\sqrt{a^2 - r^2}, \\quad r \\in [0, a]\n\\end{equation}\nwhere $a > 0$ is a parameter specifying the radius of the wide (input)\nend. Obviously, Eq.~\\eqref{TractrixEq} defines a decreasing function\n$x \\mapsto R(x) = r$ mapping $R:[0,\\infty) \\to (0,a]$ with $R(0) = a$\nand $\\lim_{x \\to \\infty}{R(x)} = 0$ which defines the \\emph{tractrix\n horn}. The required finite length $\\ell > 0$ of the horn is solved\nfrom $R(\\ell) = b$ where $0 < b < a$ is the required radius of the\n(narrow) output end.\n\nThe tractrix horn is known as the \\emph{pseudosphere} of constant\nnegative Gaussian curvature in differential geometry. That it acts as\na spherical wave horn is based on Huyghens principle and a geometric\nproperty of Eq.~\\eqref{TractrixEq}. More precisely, it can be seen\nfrom Fig.~\\ref{TractrixHornFig} (left panel) that a spherical wave\nfront of curvature radius $a$, propagating along the centreline of the\nhorn, meets the tractrix horn surfaces always in right\nangles. Disregarding, e.g., the viscosity effects in the boundary\nlayer at the horn surface, the right angle property is expected to\nproduce minimal backward reflections for spherical waves similarly as\na planar wavefront would behave in a constant diameter wave\\-guide far\naway from wave\\-guide walls.\n\n\\subsection{\\label{DeConvSec} Regularised deconvolution}\n\nA desired sound waveform target pattern will be reconstructed at the\nsource output by compensating the source dynamics in\nSection~\\ref{ImpulseSubSec}. Our approach is based on the idea of\n\\emph{constrained least squares filtering} used in digital image\nprocessing \\cite{Hunt:DLS:1972,Phillips:TNS:1962}.\n\nSuppose that a linear, time-invariant system has the real-valued\nimpulse response $h(t) = h_0(t) + h_e(t)$ that is expected to contain\nsome measurement error $h_e(t)$. When the input signal $u = u(t)$ is\nfed to the system, the measured output is obtained from\n\\begin{equation} \\label{ConvolutionEq}\ny(t) = (h_0*u)(t) + v(t) \\quad \\text{ with } \\quad v = h_e * u + w\n\\quad \\text{ for } \\quad t \\in [0, T].\n\\end{equation}\nAs usual, the convolution is defined by $(h_0*u)(t) = \\int_{-\\infty}^t\n{h_0(t - \\tau) u(\\tau) \\, d \\tau}$, and our task is to estimate $u$\nfrom Eq.~\\eqref{ConvolutionEq} given $y$ and some incomplete\ninformation about the output noise $v$. We assume $u, v \\in L^2(0,T)$\nand that $h_0$ is a continuous function. We define the noise level\nparameter by $\\epsilon = \\Vert v \\Vert_{L^2(0,T)} \/ \\Vert y \\Vert_{L^2(0,T)}$ and require that $0\n< \\epsilon < 1$ holds.\n\nUnfortunately, Eq.~\\eqref{ConvolutionEq} is not typically solvable for\nsmooth $y$ since the noise $v$ is not generally even continuous\nwhereas the convolution operator $h_0*$ is smoothing. Instead of\nsolving Eq.~\\eqref{ConvolutionEq}, we solve an estimate $\\v u$ for $u$\nfrom the regularised version of Eq.~\\eqref{ConvolutionEq}, given for\n$y \\in L^2(0,T)$ by\n\\begin{equation} \\label{RegularisedEq}\n \\begin{aligned}\n \\mathrm{Arg \\, min} & \\left ( \\kappa \\Vert \\v u \\Vert_{L^2(0,T)}^2 + \\Vert\n \\v u'' \\Vert_{L^2(0,T)}^2 \\right ) \\\\\n & \\text{ with the constraint } \\Vert\n y - h_0* \\v u \\Vert_{L^2(0,T)} = \\epsilon \\Vert y \\Vert_{L^2(0,T)}.\n \\end{aligned}\n\\end{equation}\nHere $T> 0$ is the sample length, $\\kappa > 0$ is a regularisation\nparameter, and $\\epsilon$ is the noise level introduced above in the\nview of $v$ in Eq.~\\eqref{ConvolutionEq}. Obviously, it is not\ngenerally possible to choose $\\epsilon = 0$ in\nEq.~\\eqref{RegularisedEq} without rendering $y = h_0* \\v u$\ninsolvable in $L^2(0,T)$.\n\nUsing Lagrange multipliers, the Lagrangian function takes the form\n\\begin{equation*}\n L_\\epsilon(\\v u, \\mu) = \\kappa \\Vert \\v u \\Vert_{L^2(0,T)}^2 \n + \\Vert \\v u'' \\Vert^2_{L^2(0,T)} \n -\\mu \\left (\\Vert y - h_0* \\v u \\Vert_{L^2(0,T)}^2 - \\epsilon^2 \\Vert y \\Vert^2_{L^2(0,T)} \\right ).\n\\end{equation*}\nUsing the variation $\\tilde{u}_\\eta = \\v u+\\eta w$ with $\\eta\n\\in {\\mathbb{R}}$, we get\n\\begin{equation*}\n\\begin{aligned}\n & \\frac{d}{d\\eta} L_\\epsilon (\\tilde{u}_{\\eta}, \\mu)\n \\bigg|_{\\eta=0} \\\\\n= & 2\\mathrm{Re} \\left( \\kappa \\langle w , \\v u \\rangle_{L^2(0,T)} + \\langle w'' , \\v u'' \\rangle_{L^2(0,T)}\n -\\mu \\langle h_0* w, y-h_0 *\\v u \\rangle_{L^2(0,T)}\n \\right) = 0\\,\n\\end{aligned}\n\\end{equation*}\nfor all test functions $w \\in \\mathcal{D}([0,T])$. Thus\n\\begin{equation*}\n \\kappa \\langle w, \\v u \\rangle + \\langle w'', \\v u'' \\rangle - \\mu \\langle h_0*w, y-h_0*\\v u \\rangle=0\n\\end{equation*}\nwhich, after partial integration and adjoining the convolution\noperator $h_0*$, gives\n\\begin{equation*}\n\\kappa \\v u + \\v u^{(4)} -\\mu (h_0*)^* \\left( y- h_0*\\v u \\right)\n = 0,\n\\end{equation*}\nleading to the normal equation\n\\begin{equation} \\label{NormalEq}\n\\v u = \\left[\\gamma \\left ( \\kappa + \\frac{d^4}{dt^4} \\right) +\n (h_0*)^*(h_0*) \\right]^{-1}(h_0*)^*y\n\\end{equation}\ntogether with the constraint $\\Vert y - h_0* \\v u \\Vert_{L^2(0,T)} =\n\\epsilon \\Vert y \\Vert_{L^2(0,T)}$ where $\\gamma= \\gamma(y,\\epsilon) \\in {\\mathbb{R}}$ satisfies $\\gamma\n= 1\/ \\mu$ (a constant independent of $t$). By a direct computation\nusing commutativity, we get for the residual\n\\begin{equation} \\label{ResidualEq}\n v_{\\kappa, \\mu} = y - h_0 * \\v u = \\left (\\kappa + \\frac{d^4}{dt^4}\n + \\mu (h_0*)^*(h_0*) \\right )^{-1} \\left (\\kappa y + y^{(4)} \\right ).\n\\end{equation}\nBecause $\\gamma, \\kappa > 0$, the inverses in\nEqs.~\\eqref{NormalEq}--\\eqref{ResidualEq} exist by positivity of the\noperators.\n\nSo, the possible noise components $v$ in Eq.~\\eqref{ConvolutionEq},\nconsistent with Eq.~\\eqref{NormalEq}, are the two parameter family $v\n= v_{\\kappa, \\mu}$ given in Eq.~\\eqref{ResidualEq} where $\\kappa, \\mu\n> 0$. For each $\\kappa$, we have\n\\begin{equation*}\n \\Vert v_{\\kappa, 0} \\Vert_{L^2(0,T)} \n = \\Vert y \\Vert_{L^2(0,T)} \n \\text{ and } \\lim_{\\mu \\to \\infty} { \\Vert v_{\\kappa, \\mu} \\Vert_{L^2(0,T)}} = 0.\n\\end{equation*}\nBy continuity and the inequality $0 < \\epsilon < 1$, there exists a\n$\\mu_0 = \\mu_0(\\epsilon, \\kappa)$ such that $\\Vert v_{\\kappa, \\mu_0}\n\\Vert_{L^2(0,T)} = \\epsilon \\Vert y \\Vert_{L^2(0,T)}$ as required. We\nconclude that $\\v u$ given by Eq.~\\eqref{NormalEq} with $\\gamma =\n1\/\\mu_0$ is a solution of the optimisation problem\n\\eqref{RegularisedEq}, and, hence, the regularised solution of\nEq.~\\eqref{ConvolutionEq} depending on parameters $\\epsilon, \\kappa >\n0$. In practice, the values of these regularising parameters must be\nchosen based on the original problem data $y$ and $v$.\n\nIn frequency plane, Eqs.~\\eqref{NormalEq}--\\eqref{ResidualEq} take the\nform\n\\begin{equation*}\n \\hat u(\\xi) = \\frac{\\overline{H(i \\xi)} \\hat y (\\xi)}\n {\\gamma \\left ( \\kappa + \\xi^4 \\right) + \\abs{H(i \\xi)}^2} \n\\end{equation*}\nwhere $H(s) = \\int_0^\\infty {e^{-st}h_0(t) \\, dt} $ is the transfer\nfunction corresponding to $h_0(t)$ and\n\\begin{equation}\\label{remainderEq}\n \\hat v_{\\kappa,\\mu}(\\xi) = G_{\\kappa,\\mu}(i \\xi) \\hat y (\\xi) \\quad \\text{ where } \\quad\n G_{\\kappa,\\mu}(s) = \\left (1 + \\frac{\\mu \\abs{H(s)}^2}{ \\kappa + s^4} \\right )^{-1}.\n\\end{equation}\nNote that $\\abs{G_{\\kappa,\\mu}(i \\xi)} < 1$, and the last equation\nindicates that the high frequency components of $y$ and\n$v_{\\kappa,\\mu}$ are essentially identical. By Parseval's identity,\nthe value of $\\gamma = 1\/\\mu_0$ is solved from $\\frac{1}{2\n \\pi}\\int_{-\\infty}^\\infty {\\abs{\\hat v_{\\kappa,\\mu}(\\xi)}^2\n \\, d \\xi } = \\epsilon^2 \\Vert y \\Vert^2_{L^2(0,T)} $.\n\n\n\n\\subsection{\\label{ProcessingSubSec} Processing of VT anatomic data and sound}\n\nThree-dimensional anatomic data of the VT is used for\ncomputational validations of the sound source as well as for carrying\nout measurements using physical models.\n\nVT anatomic geometries were obtained from a (then)\n26-year-old male (in fact, one of the authors of this article) using\n3D MRI during the utterance of Finnish vowels [\\textipa{\\textscripta,\n i, u}] as explained in \\cite{A-A-H-J-K-K-L-M-M-P-S-V:LSDASMRIS}. A\nspeech sample was recorded during the MRI, and it was processed for\nformant analysis by the algorithm described in \\cite{K-M-O:PPSRDMRI}.\nThe formant extraction for Section~\\ref{CompValSec} was carried out\nusing Praat \\cite{Praat:2016}. Three of the MR images\ncorresponding to Finnish quantal vowels [\\textipa{\\textscripta, i, u}]\nwere processed into 3D surface models (i.e., STL files) as explained\nin \\cite{O-M:ASUAMRIVTGE}. A spherical boundary condition interface\nwas attached at the mouth opening for the geometry corresponding\n[\\textipa{\\textscripta}] for producing the computational geometries\nshown in Fig.~\\ref{CoupledSystemRes}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.28\\textwidth]{a_print.jpg}\\hspace{0.2cm}\n\\includegraphics[width=0.277\\textwidth]{i_print.jpg}\\hspace{0.2cm}\n\\includegraphics[width=0.345\\textwidth]{u_print.jpg} \\hspace{0.2cm}\n\\end{center}\n \\caption{\\label{VTPrints} Physical VT models of\n articulation geometries corresponding to [\\textipa{\\textscripta,\n i, u}]. Adaptor sleeves have been glued to the glottis ends\n for coupling to the sound source.}\n\\end{figure}\n\nStratasys uPrint SE Plus 3D printer was used to produce physical\nmodels in ABS plastic from the STL files, shown in\nFig.~\\ref{VTPrints}. The printed models are in natural scale with\nwall thickness $2 \\, \\mathrm{mm}$, they extend from the glottal\nposition to the lips, and they were equipped with an adapter (visible\nin Fig.~\\ref{VTPrints}) for coupling them to a acoustic sound source\nshown in Fig.~\\ref{TractrixHornFig} (left panel).\n\n\\section{Design and construction}\n\nBased on the considerations of Section~\\ref{BackgroundSec}, we\nconclude that the following three design objectives are desirable for\nachieving a successful design:\n\\begin{enumerate}\n\\item[(i)] \\label{Req1} The transmission loss (henceforth, TL) from\n the the input to output should be as low as possible.\n\\item[(ii)] \\label{Req2} There should be no strong transversal\n resonant modes inside the impedance matching cavity of the device.\n\\item[(iii)] \\label{Req3} The frequency response\n $\\omega \\mapsto \\abs{Z_{R_L}(i \\omega)}$ of the transmission\n impedance should be as flat as possible for relevant termination\n resistances.\n\\end{enumerate}\nIt is difficult --- if not impossible --- to optimise all these\ncharacteristics in the same device. Fortunately, DSP techniques can\nbe used to cancel out some undesirable features, and instead of\nrequirement \\textrm{(iii)} it is more practical to pursue a more\nmodest goal:\n\\begin{enumerate}\n\\item[(iii')] \\label{Req3weaker} The frequency response $\\omega \\mapsto\n \\abs{Z_{R_L}(i \\omega)}$ should be such that its lack of flatness can be\n accurately precompensated by causal, rational filters.\n\\end{enumerate}\nWe next discuss each of these design objectives and their solutions in\nthe light of Section~\\ref{BackgroundSec}.\n\nThe tractrix horn geometry was chosen so as to minimise the TL as\nexplained in Section~\\ref{TLMinSec}. In the design proposed in this\narticle, we use $a = 50.0 \\, \\mathrm{mm}$, $b = 2.2 \\, \\mathrm{mm}$,\nand $\\ell = 153.0 \\, \\mathrm{mm}$ as nominal values in\nEq.~\\eqref{TractrixEq}. The physical size was decided based on reasons\nof practicality and the availability of suitable loudspeaker units.\n\n\nContrary to horn loudspeakers or gramophone horns having essentially\npoint sources at the narrow input end of the horn, the sound source is\nnow located at the wide end of the horn. Hence, it would be desirable\nto generate the acoustic field by a spherical surface source of\ncurvature radius $a$ whose centrepoint lies at the centre of the\nopening of the wider input end. This goal is impossible to precisely\nattain using commonly available loudspeaker units, but a reasonable\noutcome can be obtained just by placing the loudspeaker (with a\nconical diaphragm) at an optimal distance from the tractrix horn as\nshown in Fig.~\\ref{DesignOfSourceFig} (middle panel). This results in\na design where the \\emph{impedance matching cavity} of the source is a\nhorn as well, consisting of the tractrix horn that has been extended\nat its wide end by a cylinder of diameter $2 a = 100.0 \\, \\mathrm{mm}$\nand height $h = 20.0 \\, \\mathrm{mm}$. Thus, the total longitudinal\ndimension of the impedance matching cavity inside the sound source is\n$\\ell_{tot} = \\ell + h = 173.0 \\, \\mathrm{mm}$ as shown in\nFig.~\\ref{DesignOfSourceFig} (middle panel). This dimension\ncorresponds to the quarter wavelength resonant frequency at $f_{low} =\n1648 \\, \\mathrm{Hz}$, obtained by solving the eigenvalue problem\nEq.~\\eqref{HelmHoltzEq} by finite element method (FEM) shown in\nFig.~\\ref{HornSystemRes} (left panel). For frequencies much under\n$f_{low}$, the impedance matching cavity need not be considered as a\nwave\\-guide but just as a delay line.\n\nSince the geometry of impedance matching cavity has already been\nspecified, there is no \\emph{geometric} degrees of freedom left for\nimproving anything. Thus, it is unavoidable to relax design\nrequirement \\textrm{(iii)} in favour of the weaker requirement\n\\textrm{(iii')}. As discussed in Section~\\ref{SupressionSubSec},\nrequirement \\textrm{(iii')} can, however, be satisfactorily achieved\nif overly strong transversal modes of the impedance matching cavity\ncan be avoided, i.e., the design requirement \\textrm{(ii)} is\nsufficiently well met.\n\n\n\n\n\n\n\n\\subsection{\\label{HelmholtzCavitySec} Modal analysis of the impedance matching cavity}\n\nThe first step in treating transversal modes of the impedance matching\ncavity is to detect and classify them. Understanding the modal\nbehaviour helps the optimal placement of attenuating material. For\nthis purpose, the Helmholtz equation \\eqref{HelmHoltzEq} was solved by\nFEM in the geometry of the impedance matching cavity, producing\nresonances up to $8 \\, \\mathrm{kHz}$. Some of the modal pressure\ndistributions are shown in Fig.~\\ref{HornSystemRes}. As explained in\nSection~\\ref{CompValSec} below, also the acoustic resonances of the VT\ngeometry shown Fig.~\\ref{DesignOfSourceFig} (right panel) were\ncomputed in a similar manner, and their perturbations were evaluated\nwhen coupled to the impedance matching cavity as shown in\nFig.~\\ref{CoupledSystemRes}.\n\nThe triangulated surface mesh of the impedance matching cavity was\ncreated by generating a profile curve of the tractrix horn in MATLAB,\nfrom which a surface of revolution was created in Comsol where the\ncylindrical space and the loudspeaker profile were included.\nSimilarly, the surface mesh of the VT during phonation of the Finnish\nvowel \\textipa{[\\textscripta]} was extracted from MRI\ndata~\\cite{O-M:ASUAMRIVTGE}. This surface mesh was then attached to\nthe surface mesh of the spherical interface $\\Gamma{_0}$ shown in\nFig.~\\ref{DesignOfSourceFig} (right panel). For computations required\nin Section~\\ref{CompValSec}, the two surface meshes (i.e., the cavity\nand the VT) were joined together at the output end of the tractrix\nhorn and the glottis, respectively. Finally, tetrahedral volume meshes\nfor FEM computations were generated using GMSH~\\cite{gmsh} of all of\nthe three geometries with details given in Table~\\ref{MeshTable}.\n\n\\begin{table}[h]\n \\centerline{\n \\begin{tabular}{|l|c|c|c|c|}\n\\hline\n & tetrahedrons & d.o.f. \\\\\n\\hline\n\\textbf{Impedance matching cavity} & 71525 & 15246 \\\\\n\\textbf{Cavity joined with VT} & 175946 & 38020 \\\\\n\\textbf{VT} & 97847 & 21745 \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\label{MeshTable} The number of tetrahedrons of the three FEM\n meshes used for resonance computations in\n Sections~\\ref{HelmholtzCavitySec}~and~\\ref{CompValSec}. The degrees\n of freedom indicates the size of resulting system of linear\n equations for eigenvalue computations. }\n\\end{table}\n\nThe Helmholtz equation \\eqref{HelmHoltzEq} with the Dirichlet boundary\ncondition \\eqref{DirichletBndry} at the output interface $\\Gamma_0$ is\nsolved by FEM using piecewise linear elements. In this case, the\nproblem reduces to a linear eigenvalue problem whose lowest\neigenvalues give the resonant frequencies and modal pressure\ndistributions of interest. Some of these are shown in\nFig.~\\ref{HornSystemRes}.\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.22\\textwidth]{first_long.png}\\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{first_cylinder.png}\\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{first_uncertain.png} \\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{first_certain.png}\n\\end{center}\n \\caption{\\label{HornSystemRes} Pressure distributions of some\n resonance modes of the impedance matching cavity above the loudspeaker\n unit. The lowest mode is at $1648 \\, \\textrm{Hz}$, and it is\n purely longitudinal. The lowest transversal mode is at $1994 \\,\n \\textrm{Hz}$, and it is due to the cylindrical part joining the\n loudspeaker unit to the tractrix horn. At $4218 \\, \\textrm{Hz}$,\n a transversal mode appears where strong excitation exists between\n the cylindrical part and the horn. The lowest transversal mode\n that is solely due to the tractrix horn geometry is found at $5229\n \\, \\textrm{Hz}$.}\n\\end{figure}\n\n\n\n\n\nThe purely longitudinal acoustic modes were found at frequencies\n$1648 \\, \\mathrm{Hz}$, $2540 \\, \\mathrm{Hz}$, $3350 \\, \\mathrm{Hz}$,\n$3771 \\, \\mathrm{Hz}$, $4499 \\, \\mathrm{Hz}$, $5061 \\, \\mathrm{Hz}$,\n$5745 \\, \\mathrm{Hz}$, $6671 \\, \\mathrm{Hz}$, $7088 \\, \\mathrm{Hz}$,\n$7246 \\, \\mathrm{Hz}$, and $7737 \\, \\mathrm{Hz}$. All of these\nlongitudinal modes have multiplicity $1$. Transversal modes divide\ninto three classes: \\textrm{(i)} those where excitation is mainly in\nthe cylindrical part of the impedance matching cavity, \\textrm{(ii)}\nthose where the excitation is mainly in the tractrix horn, and\n\\textrm{(iii)} those where both parts of the impedance matching cavity\nare excited to equal extent. Resonances due to the cylindrical part\nappear at frequencies $1994 \\, \\mathrm{Hz}$, $3094 \\, \\mathrm{Hz}$,\n$4150 \\, \\mathrm{Hz}$, $6063 \\, \\mathrm{Hz}$, $6262 \\, \\mathrm{Hz}$,\n$6872 \\, \\mathrm{Hz}$, $6942 \\, \\mathrm{Hz}$, $7334 \\, \\mathrm{Hz}$,\nand $7865 \\, \\mathrm{Hz}$, and they all have multiplicity $2$ except\nthe resonance at $4150 \\, \\mathrm{Hz}$ that is simple. (Note that\nthere is a longitudinal resonance at $4150 \\, \\mathrm{Hz}$ as well.)\nThere are only four frequencies corresponding to the transversal modes\n(all with multiplicity $2$) in the tractrix horn: namely,\n$5229 \\, \\mathrm{Hz}$, $5697 \\, \\mathrm{Hz}$, $6764 \\, \\mathrm{Hz}$,\nand $6781 \\, \\mathrm{Hz}$. The peculiar mixed modes of the third kind\nwere observed at $4218 \\, \\mathrm{Hz} \\, (2)$,\nand $5200 \\, \\mathrm{Hz} \\, (3)$ where\nthe number in the parenthesis denotes the multiplicity.\n\nBased on these observations, the acoustic design of the impedance\nmatching cavity was deemed satisfactory as the transversal dynamics of\nthe tractrix horn shows up only above $5.2 \\, \\mathrm{kHz}$. The\nlower resonant frequencies of the wide end of the cavity are treated\nby placement of attenuating material as described in\nSection~\\ref{ConstructionDetailsSec}.\n\n\n\\subsection{\\label{ConstructionDetailsSec} Details of the construction}\n\n\nThe tractrix horn geometry was produced using the parametric Tractrix\nHorn Generator OpenSCAD script \\cite{TractrixGenerator:2014}. The horn\nwas 3D printed by Ultimaker Original in PLA plastic with wall thickness\nof $2 \\, \\mathrm{mm}$ and fill density of $100 $\\%. The inside surface\nof the print was coated by several layers of polyurethane lacquer,\nafter which it was polished. The horn was installed inside a cardboard\ntube, and the space between the horn and the tube was filled with\n$\\approx 1.2 \\, \\textrm{kg}$ of \\emph{plaster of Paris} in order to\nsuppress the resonant behaviour of the horn shell itself and to\nattenuate acoustic leakage through the horn walls.\n\nThe walls of the cylindrical part of the impedance matching cavity\nwere covered by felt in order to control the standing waves in the\ncylindrical part of the cavity. Acoustically soft material, i.e.,\npolyester fibre, was placed inside the source (partly including the\nvolume of the tractrix horn) by the method trial and improvement,\nbased on iterated frequency response measurements as explained in\nSection~\\ref{CompensationSubSec} and heuristic reasoning based on\nFig.~\\ref{HornSystemRes}. The main purpose of this work was to\nsuppress overly strong transversal modes shown in\nFig.~\\ref{HornSystemRes} in the impedance matching cavity shown in\nFig.~\\ref{DesignOfSourceFig} (middle panel). As a secondary\neffect, also the purely longitudinal modes got suppressed. Adding\nsound soft material resulted in the attenuation of unwanted resonances\nat the cost of high but tolerable increase in the TL of the source.\n\nThe loudspeaker unit of the source is contained in the hardwood box\nshown in Fig.~\\ref{TractrixHornFig}, and its wall thickness $40 \\,\n\\textrm{mm}$. The box is sealed air tight by applying silicone mass to\nall joints from inside in order to reduce acoustic leakage. Its exterior\ndimensions are $215 \\, \\textrm{mm} \\times 215 \\, \\textrm{mm} \\times\n145 \\, \\textrm{mm}$, and it fits tightly to the horn assembly\ndescribed above. The horn assembly and the space of the loudspeaker\nunit above the loudspeaker cone form the impedance matching cavity of\nthe source shown in Fig.~\\ref{DesignOfSourceFig}. There is another\nacoustic cavity under the loudspeaker unit whose dimension are $135.0\n\\, \\mathrm{mm} \\times 135.0 \\mathrm{mm} \\times 70.0 \\mathrm{mm}$. Also\nthis cavity was tightly filled with acoustically soft material to\nreduce resonances.\n\n\\subsection{Electronics and software for measurements}\n\nWe use a $4''$ two-way loudspeaker unit (of generic brand) whose\ndiameter determines the opening of the tractrix horn. Its nominal\nmaximum output power is $30 \\, \\mathrm{W \\, (RMS)}$ when coupled to a\n$4 \\, \\Omega$ source. The loudspeaker is driven by a power amplifier\nbased on TBA810S IC. There is a decouplable mA-meter in the\nloudspeaker circuit that is used for setting the output level of the\namplifier to a fixed reference value at $1 \\, \\mathrm{kHz}$ before\nmeasurements. The power amplifier is fed by one of the output channels\nof the sound interface ``Babyface'' by RME, connected to a laptop\ncomputer via USB interface.\n\nThe acoustic source contains an electret \\emph{reference microphone}\n(of generic brand, $\\oslash \\, 9 \\, \\mathrm{mm}$, biased at $5 \\,\n\\mathrm{V}$) at the output end of the horn. The reference microphone\nis embedded in the wave\\-guide wall, and there is an aperture of\n$\\oslash \\, 1 \\, \\mathrm{mm}$ in the wall through which the microphone\ndetects the sound pressure. The narrow aperture is required so as not\nto overdrive the microphone by the very high level of sound at the\noutput end of the horn, and it is positioned about $13.5 \\,\n\\mathrm{mm}$ below the position where vocal folds would be in the 3D\nprinted VT model (depending on the anatomy).\n\nThe measurements near the mouth position of 3D-printed VTs are carried\nout by a \\emph{signal microphone}. As a signal microphone, we use\neither a similar electret microphone unit as the reference microphone\nor Br\\\"uel \\& Kj\\ae{}ll measurement microphone model 4191 with the\ncapsule model 2669 (as shown in Fig.~\\ref{TractrixHornFig} (left\npanel)) and preamplifier Nexus 2691. The B\\&K unit has over $20 \\,\n\\mathrm{dB}$ lower noise floor compared to electret units which,\nhowever, has no significance when measuring, e.g., the resonant\nfrequencies of an acoustic load in a noisy environment. Measurements\nin the anechoic chamber yield much cleaner data when the B\\&K unit is\nused, and this is advisable when studying acoustic loads with higher\nTL and lower signal levels. Then, extra attention has to be paid to\nall other aspects of the experiments so as to achieve the full\npotential of the high-quality signal microphone.\n\nThe reference and the signal electret microphone units were picked\nfrom a set of $10$ units to ensure that their frequency responses\nwithin $80 \\, \\mathrm{Hz} \\ldots 8 \\ \\textrm{kHz}$ are practically\nidentical. It was observed that there are very little differences in\nthe frequency and phase response of any two such microphone\nunits. Furthermore, these microphones are practically\nindistinguishable from the Panasonic WM-62 units (with nominal\nsensitivity $-45 \\pm 4$ dB re $1$ V\/Pa at $1$ kHz) that were used in\nthe instrumentation for MRI\/speech data acquisition reported in\n\\cite{A-A-H-J-K-K-L-M-M-P-S-V:LSDASMRIS}.\n\nFinal results given in Section~\\ref{MeasSigSec} were measured using\nthe Br\\\"uel \\& Kj\\ae{}ll model 4191 at the mouth position. The results\nshown in \\cite[Fig.~5]{K-M-O:PPSRDMRI} were measured using the\nelectret unit matched with the similar reference microphone, embedded\nto the source at the glottal position. In this article, the electret\nmicrophone measurements at the mouth position were only used for\ncomparison purposes.\n\nBiases for both the electret microphones are produced by a custom\npreamplifier having two identical channels based on LM741 operational\namplifiers. The amplifier has nonadjustable $40 \\, \\mathrm{dB}$\nvoltage gain in its passband that is restricted to $40 \\, \\textrm{Hz}\n\\ldots 12 \\, \\textrm{kHz}$. Particular attention is paid to reducing\nthe ripple in the microphone bias as well as the cross-talk between\nthe channels. The input impedance $2.2 \\, \\mathrm{k \\Omega}$ of the\npreamplifier is a typical value of electret microphones,\nand the output is matched to $300 \\, \\Omega$ for the two input\nchannels of the Babyface unit.\n\nSignal waveforms and sweeps are produced numerically as explained in\nSections~\\ref{CalibrationSec} for all experiments. Frequency response\nequalisation and other kinds of time and frequency domain\nprecompensations are a part of this process. All computations are done\nin MATLAB (R2016b) running on Lenovo Thinkpad T440s, equipped with 3.3\nGHz Intel Core i7-4600U processor and Linux operating system. The\nexperiments are run using MATLAB scripts, and access from MATLAB to\nthe Babyface is arranged through Playrec (a MATLAB\nutility,~\\cite{Playrec}).\n\n\n\n\\subsection{Measurement arrangement}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{meas_setup.png}\\hspace{0.2cm}\\end{center}\n \\caption{\\label{SystemGraph} An illustration of the\n measurement system.}\n\\end{figure}\n\nAn outline of the measurement arrangement for sweeping a VT print is\nshown in Fig.~\\ref{SystemGraph}. Both the amplifiers, the digital\nanalogue converter (DAC), and the computer are located outside the\nanechoic chamber. The arrangement inside the anechoic chamber\ncontains two microphones: the reference unit at the glottal position\ninside the source, and the external microphone in front of the mouth\nopening. The position of the external microphone must be kept same in\nall measurements to have reproducibility.\n\nBecause of the quite high transmission loss of the VT print (in\nparticular, in VT configuration corresponding to [\\textipa{i}]) and\nthe relatively low sound pressure level produced by the source at the\nglottal position (compared to the sound pressure produced by human\nvocal folds), one may have to carry out measurements using an acoustic\nsignal level only about $20 \\ldots 30 \\, \\mathrm{dB}$ above the\nhearing threshold. The laboratory facilities require using\nwell-shielded coaxial microphone cables of length $10 \\, \\mathrm{m}$\nin order to prevent excessive hum. Another significant source of\ndisturbance is the acoustic leakage from the source directly to the\nexternal microphone. This leakage was be reduced by $\\approx 6 \\,\n\\mathrm{dB}$ by enclosing the sound source into a box made of\ninsulating material, and preventing sound conduction through\nstructures by placing the source on silicone cushions resting on a\nheavy stone block (not shown in\nFigs.~\\ref{TractrixHornFig}~and~\\ref{SystemGraph}).\n\n\n\\section{\\label{CompValSec} Computational validation using a VT load}\n\nWhen an acoustic load is coupled to a sound source containing an\nimpedance matching cavity, the measurements carried out using the\nsource necessarily concern the joint acoustics of the source and the\nload. Hence, precautions must be taken to ensure that the\ncharacteristics of the acoustic load truly are the main component in\nmeasurement results. In the case of the proposed design, the small\nintersectional area of the opening at the source output leads to high\nacoustic output impedance which is consistent with a reasonably good\nacoustic \\emph{current} source. Also, the narrow glottal position of\nthe VT helps in isolating the the two acoustic spaces from each other.\n\nWe proceed to evaluate this isolation by computing the Helmholtz\nresonance structures of the joint system shown in\nFigs.~\\ref{CoupledSystemRes} and compare them with \\textrm{(i)}\nformant frequencies measured from the same test subject during the MR\nimaging, and \\textrm{(ii)} Helmholtz resonances of the VT geometry\nshown in Fig.~\\ref{DesignOfSourceFig} (right panel). The VT part of\nboth the computational geometries is the same, and it corresponds to\nthe vowel [\\textipa{\\textscripta}]. The vowel [\\textipa{\\textscripta}]\nout of [\\textipa{\\textscripta, i, u}] was chosen because its three\nlowest formants are most evenly distributed in the voice band of\nnatural speech. \n\n\n\n\\begin{table}[h]\n \\centerline{\n \\begin{tabular}{|l|c|c|c|c|}\n\\hline\n & $F_1$ & $F_2$ & $F_3$ \\\\\n\\hline\n\\textbf{VT resonances} & $519$ & $1130$ & $2297$ \\\\\n\\textbf{VT + source resonances} & $594$ & $1136$ & $2290$ \\\\\n\\textbf{Formant frequencies} & $683$ & $1111$ & $2417$ \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\label{FormantCouplingTable} Vowel formants and Helmholtz\n resonances (in $\\textrm{Hz}$) of a VT during a production\n of [\\textipa{\\textscripta}]. In the first two row, only those\n resonances have been taken into account whose modal behaviour\n corresponds with the formants $F_1, F_2,$ and\n $F_3$.}\n\\end{table}\n\nIn numerical computations, the domain $\\Omega \\subset {\\mathbb{R}}^3$ for the\nHelmholtz equation~\\eqref{HelmHoltzEq} consists of the VT geometry of\n[\\textipa{\\textscripta}] either as such (leading to ``VT resonances''\nin Table~\\ref{FormantCouplingTable}) or manually joined to the\nimpedance matching cavity at the glottal position (leading to ``VT +\nsource resonances'' in Table~\\ref{FormantCouplingTable}). The FEM\nmeshes have been described in Section~\\ref{HelmholtzCavitySec}.\nThe acoustic modes and resonant frequencies have been computed from\nEq.~\\eqref{HelmHoltzEq}, and some of the resulting resonant\nfrequencies and modal pressure distributions are shown in\nFig.~\\ref{CoupledSystemRes}.\n\nIn contrast to Section~\\ref{HelmholtzCavitySec}, the symbol $\\Gamma_0$\nnow denotes the spherical mouth interface surface visible in\nFig.~\\ref{DesignOfSourceFig} (right panel), and instead of\nEq.~\\eqref{DirichletBndry} we use the boundary condition of Robin type\n\\begin{equation*}\n \\lambda \\phi_\\lambda({\\bf r}) + c\\frac{\\partial\n \\phi_\\lambda}{\\partial\\nu}({\\bf r}) = 0 \\text{ on } \\Gamma_0,\n\\end{equation*}\nmaking the interface absorbing. When computing VT resonances for\ncomparison values without the impedance matching cavity (the top row\nin Table~\\ref{FormantCouplingTable}), the interface at the glottal\nopening is considered as part of $\\Gamma_0$, too. The resulting\nquadratic eigenvalue problem was then solved by transforming it to a\nlarger, linear eigenvalue problem as explained in\n\\cite[Section~3]{Hannukainen:2007}.\nFor a similar kind of numerical experiment involving VT geometries but\nwithout a source, see \\cite{arnela:2013}.\n\n\nThe formant values given in Table~\\ref{FormantCouplingTable} have been\nextracted by Praat \\cite{Praat:2016} from post-processed speech\nrecordings during the acquisition of the MRI geometry as explained in\nSection~\\ref{ProcessingSubSec}. The extraction was carried out at $3.5\n\\, \\mathrm{s}$ from starting of the phonation, with duration $25 \\,\n\\textrm{ms}$.\n\nGiven in semitones, the discrepancies between the first two rows in\nTable~\\ref{FormantCouplingTable} are $-2.3$, $-0.1$, and\n$0.05$. Similarly, the discrepancies between the last two rows in\nTable~\\ref{FormantCouplingTable} are $-2.4$, $0.4$, and $-0.9$. The\nlargest discrepancy concerning the first formant $F_1$ is partly\nexplained by the challenges in formant extraction from the nonoptimal\nspeech sample pair of the MRI data used. In\n\\cite[Table~2]{A-A-H-J-K-K-L-M-M-P-S-V:LSDASMRIS}, the value for $F_1$\nfrom the same test subject was found to be $580 \\pm 23 \\, \\mathrm{Hz}$\nbased on averaging over ten speech samples during MRI and using a more\ncareful treatment for computing the spectral envelope, based on MATLAB\nfunction \\verb|arburg| .\n\nWe conclude that for Helmholtz resonances corresponding to $F_2$ and\n$F_3$ of the physical model of [\\textipa{\\textscripta}], the\nperturbation due to acoustic coupling with the impedance matching\ncavity are small fractions of the comparable natural variation in\nspoken vowels. So as to the lowest formant $F_1$, it seems that the\nimpedance matching cavity actually represents a better approximation\nof the true subglottal acoustics contribution than the mere absorbing\nboundary condition imposed at the glottis position of a VT\ngeometry. We further observe that the three lowest resonant modes of\nthe VT (corresponding to formants $F_1, F_2, F_3$) appear where the\nimpedance matching cavity remains in ``ground state''; see\nFig.~\\ref{CoupledSystemRes}. This supports the desirable property that\nthe narrowing of the horn at the vocal folds position effectively\nkeeps the impedance matching cavity of the source and the VT load only\nweakly coupled.\n\n\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics[width=0.22\\textwidth]{a2.png}\\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{a3.png} \\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{a4.png} \\vspace{0.5cm}\n\\includegraphics[width=0.22\\textwidth]{a5.png} \\\\ \\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{a6.png}\\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{a7.png} \\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{a8.png} \\hspace{0.2cm}\n\\includegraphics[width=0.22\\textwidth]{a9.png}\n\\end{center}\n \\caption{\\label{CoupledSystemRes} Pressure distributions of some\n resonance modes of the impedance matching cavity of the source\n coupled to a VT geometry of [\\textipa{\\textscripta}]. The modes\n corresponding to longitudinal VT resonances are at\n frequencies $593 \\, \\textrm{Hz}$, $1137 \\, \\textrm{Hz}$, and $2287\n \\, \\textrm{Hz}$, corresponding to formants $F_1, F_2,$ and $F_3$.\n The remaining pressure modes under $2465 \\, \\mathrm{Hz}$ are\n excitations of the impedance matching cavity of the source.}\n\\end{figure}\n\n\n\n\\section{Calibration measurements and source compensation}\n\\label{CalibrationSec}\n\n\\subsection{\\label{CompensationSubSec} Measurement and compensation of the frequency response}\n\nIn this section, we describe the production of an \\emph{exponential\n frequency sweep}\\footnote{Also known as the logarithmic chirp.} with\nuniform sound pressure at the glottal position. The defining property\nof such sweeps is that each increase in frequency by a semitone takes\nan equal amount of time. In this work, the frequency interval of such\nsweeps is $80 \\ldots 7350 \\, \\mathrm{Hz}$ with duration of $10 \\,\n\\mathrm{s}$. All measurements leading to curves in\nFigs.~\\ref{EnvelopeResidualFig}--\\ref{LissajousFig} were carried out\nusing the \\emph{dummy load} shown in Fig.~\\ref{TractrixHornFig} (right\npanel) as the standardised acoustic reference load.\n\nIf one plainly introduces a constant voltage amplitude exponential\nsinusoidal sweep to the loudspeaker unit, the sound pressure at the\nsource output (as seen by the adjacent reference microphone) will vary\nover $20 \\, \\mathrm{dB}$ over the frequency range of the sweep as\nshown in Fig.~\\ref{EnvelopeResidualFig} (left panel). The key\nadvantage in producing a \\emph{constant amplitude sound pressure} at\nthe source output is that excessive external noise contamination of\nmeasured signals can be avoided on frequencies where the output power\nwould be low. Standardising the sound pressure at the output of the\nsource also makes the source acoustics less visible in the\nmeasurements of the load. This reduces the perturbation effect at\n$F_1$ that was computationally observed in Section~\\ref{CompValSec}.\n\nAn essentially flat sound pressure output shown in\nFig.~\\ref{EnvelopeResidualFig} (right panel) can be obtained from the\nsource by applying the frequency dependent amplitude weight\n$\\mathbf{w}$ shown in Fig.~\\ref{EnvelopeResidualFig} (middle panel) to\nthe voltage input to the loudspeaker unit. As is to be expected, both\nthe weighted and unweighted voltage sweeps have almost identical phase\nbehaviours as can be seen in Fig.~\\ref{LissajousFig} (left panel). In\ncontrast, the voltage sweep and the resulting sound pressure at the\nreference microphone are out of phase in a very complicated frequency\ndependent manner; see Fig.~\\ref{LissajousFig} (middle panel). Such\nphase behaviour cannot be explained by the relatively sparsely located\nacoustic resonances of the impedance matching cavity.\n\nAn iterative process requiring several sweep measurements was devised\nto obtain the weight shown in Fig.~\\ref{EnvelopeResidualFig} (middle\npanel), and it is outlined below as\nAlgorithm~\\ref{SweepAlgorithm}. Various parameters in the algorithm\nwere tuned by trial and error so as to produce convergence to a\nsatisfactory compensation weight. During the iteration, different\nversions of the measured sweeps have to be temporally aligned with\neach other. The required synchronisation is carried out by detecting a\n$1 \\, \\mathrm{kHz}$ cue of length $1 \\, \\mathrm{s}$, positioned before\nthe beginning of each sweep. This is necessary because there are\nwildly variable latency times in the DAC\/software combination used for\nthe measurements.\n\n\\begin{figure}[t]\n \\includegraphics[width=0.32\\textwidth]{orig_freq_res-crop.pdf}\n \\includegraphics[width=0.32\\textwidth]{compensation_weights.pdf}\n \\includegraphics[width=0.32\\textwidth]{final_freqres-crop.pdf}\n \\caption{\\label{EnvelopeResidualFig} Left panel: The pressure signal\n envelope of the measurement system at the glottal position when a\n constant amplitude exponential voltage sweep was used as the\n loudspeaker input. The first longitudinal resonance of the\n impedance matching cavity appears at $1648 \\, \\mathrm{Hz}$. The\n source was terminated to the dummy load shown in\n Fig.~\\ref{TractrixHornFig} (right panel). Middle panel: The\n inverse weights that are applied to the constant amplitude\n exponential sweep in order to get the output in the next panel.\n Right panel: The envelope of the weighted exponential sweep at the\n glottal position where the weight has been produced by\n Algorithm~\\ref{algorithm}. The produced sound pressure sweep at\n the source output has residual amplitude dynamics of approximately\n $0.5 \\,\\mathrm{dB}$.}\n\\end{figure}\n\n\\begin{algorithm}\n\\caption{Computation of the equalisation weight $\\mathbf{w}$}\\label{algorithm}\n\\begin{algorithmic}[1]\n\\Procedure{CalibrateCompensation}{n,t}\n\\State $\\mathbf{w}\\gets [1,1,\\ldots,1]$\n\\For {$k\\gets 0\\ldots N$}\n\\State $\\mathbf{x} \\gets w\\cdot $ExponentialChirp(t)\n\\State $\\mathbf{y} \\gets $Play $ (\\mathbf{x}) $\n\\State $H \\gets $ComputeEnvelope$ (\\mathbf{y})$\n\\State $d \\gets $Dynamics $(H)$\n\\State $r \\gets $Regularization $(d)$\n\\State $ \\mathbf{w} \\gets \\frac{1}{\\abs{H}+r}\\cdot \\mathbf{w}$\n\\EndFor\n\\State\\Return $\\mathbf{w}$\n\\EndProcedure\n\\end{algorithmic} \\label{SweepAlgorithm}\nWe consider the calibration successful if the measured dynamics at the\nfinal iteration stage is below $1 \\, \\mathrm{dB}$.\n\\end{algorithm}\n\n\nThe system comprising the power amplifier, the loudspeaker and the\nacoustic load is somewhat nonlinear which becomes evident in wide\nfrequency ranges and high amplitude variations. Even though the curves\nin Fig.~\\ref{EnvelopeResidualFig} (left and middle panels) are\nobviously related, they do not sum up to a constant that would be\nindependent of the frequency. Not even the dynamical ranges of these\ncurves coincide as would happen in a linear and time-invariant\nsetting. In spite of nonlinearity, it is possible to use of a very\nslowly increasing sweep to produce an accurate voltage gain from the\noutput of DAC to the output of reference microphone preamplifier over\na very wide range of frequency. One example of such voltage gain\nfunction is shown in Fig.~\\ref{EnvelopeResidualFig} (left panel) but\nits inverse is not a good candidate for the compensation weight.\n\n\n\\begin{figure}[hb]\n \\centering\n \\includegraphics[width=0.3\\textwidth]{lissajous_original_to_compensated_3000hz.pdf} \\hspace{0.2cm}\n \\includegraphics[width=0.3\\textwidth]{lissajous_1540hz_orig_sweep_to_compensated_nose.pdf} \\hspace{0.2cm} \n \\includegraphics[width=0.33\\textwidth]{impulse_response_dummy.pdf}\n\\caption{\\label{LissajousFig} Left: Lissajous plot of the original,\n unweighted voltage sweep against the sweep near $3 \\, \\mathrm{kHz}$\n weighted by $\\mathbf{w}$ produced by\n Algorithm~\\ref{SweepAlgorithm}. Middle: Lissajous plot of the\n unweighted voltage sweep against the corresponding output as\n recorded by the reference microphone near $1540 \\mathrm{Hz}$ where\n the phase difference varies around $\\pi\/2$. Right: The measured\n impulse response of from the voltage input to reference microphone\n output. In both the measurements, the source was terminated to the\n dummy load shown in Fig.~\\ref{TractrixHornFig} (right panel).}\n\\end{figure}\n\n\n\\noindent The results of sweep measurement from physical models of VT\nare given in Section~\\ref{SweepMeasSubSec}.\n\n\n\n\\subsection{Compensation of the source response for reference tracking}\n\\label{ImpulseSubSec}\n\nAnother important goal is to be able to reconstruct a desired waveform\nas the sound pressure output of the source as observed by the\nreference microphone. In the context of speech, a good candidate for a\ntarget waveform is the Liljencrants--Fant (LF) waveform \\cite{Fant:1985}\ndescribing the the flow through vibrating vocal folds; see\nFig.~\\ref{ReconstructedWaveformFig} (top row, left panel).\n\nBecause there is an acoustic transmission delay of $\\approx 0.5 \\,\n\\mathrm{ms}$ in the impedance matching cavity in addition to various,\nmuch larger latencies in the DAC\/computer instrumentation and software,\na simple feedback-based PID control strategy is not feasible for\nsolving any trajectory tracking problem. Instead, a \\emph{feedforward\n control solution} is required where the response of the acoustic\nsource and the electronic instrumentation is cancelled out by\n\\emph{regularised deconvolution}, so as to obtain an input waveform\nthat produces the desired output. For this, we use the version of\nconstrained least squares filtering whose mathematical treatment in\nsignal processing context is given in Section~\\ref{DeConvSec}.\n\nThe regularised deconvolution requires estimating the impulse response\nof the whole measurement system that corresponds to the convolution\nkernel $h_0$ in Eq.~\\eqref{ConvolutionEq}. This response is estimated\nusing the sinusoidal sweep excitation described in\n\\cite{Muller:TFM:2001}, and the result of the measurement can be seen\nin~Fig.~\\ref{LissajousFig} (right panel). Because the deconvolution\ncontains regularisation parameters $\\gamma$ and $\\kappa$, it tolerates\nsome noise always present in the estimated impulse response.\n\nLet us proceed to describe how the mathematical treatment given in\nSection~\\ref{DeConvSec} can be turned into a workable signal\nprocessing algorithm in discrete time. All signals (including the\nestimated impulse response corresponding to kernel $h_0$) are\ndiscretised at the sampling rate $44 100 \\, \\mathrm{Hz}$ used in all\nsignal measurements. We denote the sample number of a discretised\nsignal, say, $x[n]$ by $N = 44 100 \\, \\mathrm{Hz} \\cdot T$ where $T$\nis the temporal length of the original (continuous) signal $x(t)$, $t\n\\in [0, T]$, and sampling is carried out by setting, e.g., \n\\begin{equation*}\n x[n] = \\frac{1}{T_s} \\int_{(n-1) T_s}^{n T_s}{ x(t) \\, dt} \n \\quad \\text{ where } \\quad 1 \\leq n \\leq N \\quad\n \\text{ and } \\quad T_s = \\mathrm{s}\/44 \\, 100 .\n\\end{equation*}\nThe measured (discrete) impulse response $h_0[n]$ is extended to match\nthe signal length $N$ by padding it with zeroes, if necessary.\n\nIn discrete time, the regularised deconvolution given in\nEqs.~\\eqref{NormalEq}--\\eqref{ResidualEq} takes the matrix\/vector form\n\\begin{equation} \\label{DiscretisedRegConv}\n\\begin{aligned}\n \\v{\\mathbf{u}} & = \\left(\\gamma \\left (\\kappa I + R^{T}R \\right ) + H^{T}H \\right)^{-1}H^{T} \\mathbf{y} \\quad \\text{ and } \\\\\n \\mathbf{v}_{\\gamma, \\kappa} & =\n\\left(\\kappa I + R^{T}R + \\gamma^{-1} H^{T}H \\right)^{-1} \\left ( \\kappa I + R^{T}R \\right ) \\mathbf{y}.\n\\end{aligned}\n\\end{equation}\nThe components of the $N \\times 1$ column vectors $\\v{\\mathbf{u}},\n\\mathbf{y}, \\mathbf{v}_{\\gamma, \\kappa}$ are plainly the discretised\nvalues $\\v{u}[n], y[n], {v}_{\\kappa, \\mu}[n]$ for $n = 1, \\ldots , N$\nof signals $\\v{u}, y, {v}_{\\kappa, \\mu}$, respectively, given in\nEqs.~\\eqref{NormalEq}--\\eqref{ResidualEq} where $\\mu = \\gamma^{-1}$.\nThe second order difference $N \\times N$ matrix $R$ is the symmetric\nmatrix whose top row is $\\left [ 2,-1,0,\\ldots,0 , -1 \\right ]$,\nmaking it circulant. The nonsymmetric $N \\times N$ matrix $H = \\left [\n h_{j,k} \\right ]$ is constructed by setting $h_{jk}=h_0[(N + j -\n k)\\,\\mathrm{mod} \\,N+1]$ for $1 \\leq j, k \\leq N$. Because all of\nthe matrices $R = R^{T}, H, H^{T}$ are now circulant, so is the\nsymmetric matrix $\\gamma \\left (\\kappa I + R^{T}R \\right ) + H^{T}H$\nin Eq.~\\eqref{DiscretisedRegConv}. Hence, the matrix\/vector products\nin Eq.~\\eqref{DiscretisedRegConv} can be understood as circular\ndiscrete convolutions that can be implemented in $N\\log(N)$ time using\nthe Fast Fourier Transform (FFT). This leads to very efficient\nsolution for $ \\v{\\mathbf{u}}$ given $\\mathbf{y}$ even for long\nsignals.\n\nDefining the transfer functions $\\widehat R(z)$, $\\widehat H(z)$ and\nthe transforms $\\widehat{y}(z), \\widehat{v}_{\\gamma, \\kappa} (z)$ for\n$z = e^{i \\theta}$ as\n\\begin{equation*}\n\\begin{aligned}\n & \\widehat R(z) = - z^{-N} - z^{-1} + 2 - z - z^{N}, \\quad \n \\widehat H(z) = \\sum_{n = 0}^N {h_{n 0} z^{n}} + \\sum_{n = -N}^{-1} {h_{0 n} z^{n}}, \\\\\n & \\widehat{y}(z) = \\sum_{n = 1}^N {y[n] z^{n}}, \\text{ and } \n \\widehat{v}_{\\gamma, \\kappa} (z) = \\sum_{n = 1}^N {{v}_{\\gamma, \\kappa}[n] z^{n}},\n\\end{aligned}\n\\end{equation*}\nwe observe that the latter of Eqs.~\\eqref{DiscretisedRegConv} takes\nthe form of Discrete Fourier Transform (DFT)\n\\begin{equation} \\label{DFTTransferFunctionEq}\n \\frac{\\widehat{v}_{\\gamma, \\kappa} (z_k) }{\\widehat{y}(z_k)} = \\frac{\\kappa + \\abs{\\widehat R(z_k)}^2 }\n {\\kappa + \\abs{\\widehat R(z_k)}^2 + \\gamma^{-1} \\abs{\\widehat H(z_k)}^2 },\n\\end{equation}\nrealised in MATLAB code, where $z_k = e^{2 \\pi k\/N}$ and $k = 1,\n\\ldots , N$ enumerates the discrete frequencies. By Parseval's\nidentity, we interpret the residual equation \\eqref{remainderEq} in\ndiscretised form as\n\\begin{equation*}\n\\sum_{k = 1}^N {\\abs{\\widehat{v}_{\\gamma, \\kappa} (z_k)}^2} = \\epsilon^2 \\sum_{k = 1}^N {\\abs{\\widehat{y} (z_k)}^2}\n\\end{equation*}\nwhich, together with Eq.~\\eqref{DFTTransferFunctionEq}, gives an\nequation from which $\\gamma = \\gamma(\\epsilon, \\kappa)$ can be solved for each $0 < \\epsilon < 1$\nand $\\kappa \\geq 0$. This is done using MATLAB's \\texttt{fminbnd}\nfunction to ensure that $\\gamma>0$. The values for $\\epsilon, \\kappa$\nare chosen based on the experiments.\n\n\n\n\\section{\\label{MeasSigSec} Results}\n\nTwo kinds of measurements on 3D printed VT physical models were\ncarried out. Firstly, the measurement of the magnitude frequency\nresponse to determine spectral characteristics (such as the lowest\nresonant frequencies) of the VT geometry. Secondly, the classical LF\nsignal was fed into the VT physical model to simulate vowel acoustics\nin a spectrally correct manner.\n\n\\subsection{\\label{SweepMeasSubSec} Sweep measurements}\n\nThe power spectral density is obtained from VT physical models by the\nsweep measurements. The sweep is constructed as described in\nSection~\\ref{CompensationSubSec} to obtain a constant sound pressure\nat the output of the source when terminated to the dummy load. The\nsignal from the measurement microphone at the mouth position of the\nphysical model is then transformed to an amplitude envelope (similar\napproach can be found in~\\cite[Fig. 2]{Wolfe:EMS:2016}) by an envelope\ndetector (i.e., computing a moving average of the nonnegative signal\namplitude). Finally, this output envelope is divided by the similar\nenvelope from the reference microphone at the source output. The\nresulting amplitude envelopes are shown in the top curves of\nFig.~\\ref{VTResponseFig}, and the resonance data is given in\nTable~\\ref{sweepFormantTable}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.3\\textwidth]{a_sweep_anec.pdf}\n \\includegraphics[width=0.3\\textwidth]{i_sweep_anec.pdf}\n \\includegraphics[width=0.3\\textwidth]{u_sweep_anec.pdf}\n \\caption{\\label{VTResponseFig} The measured frequency amplitude\n response of physical models of VT anatomies corresponding to\n vowels [\\textipa{\\textscripta, i, u}]. The spectral maxima\n extending to $7350\\,\\mathrm{Hz}$ were selected so that two peaks\n had to be at least $100\\,\\mathrm{Hz}$ apart from another with at\n least $4\\,\\mathrm{dB}$ peak prominence have been marked with\n circles. The lower curves are power spectral envelopes extracted\n from the vowel utterances of the same test subject, recorded in\n the anechoic chamber. }\n\\end{figure}\n\n\n\\begin{table}[h]\n \\centerline{\n \\begin{tabular}{|l|c|c|c|c|c|c|}\n\\hline\n & $P_1$ & $P_2$ & $P_3$ &$P_4$ &$P_5$&$P_6$ \\\\\n\\hline\n\\textbf{[a]} & 635 * & 1104 * & 2364 * & 3167 & 4038& X \\\\\n\\textbf{[i]} & 316 * & 658 & 984 & 2104 * & 2957 * & 5740 \\\\\n\\textbf{[u]} & 386 * &819 * &2132 * &3206 &4732&5736 \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\label{sweepFormantTable} Peak frequency positions from sweep\n measurements on 3D printed VT physical models. The peaks\n corresponding to the three lowest formants $F_1, F_2$, and $F_3$\n are denoted by an asterisk.}\n\\end{table}\n\n\n\n\\subsection{\\label{GlottalPulseReconSec} Glottal pulse reconstruction}\n\nThe second goal is to reconstruct acoustically reasonable pressure\nwaveforms at the source output as observed by the reference\nmicrophone. For reproducing nonsinusoidal target signals, a general\nmethod described in Sections~\\ref{DeConvSec}~and~\\ref{ImpulseSubSec}\nis used to track them. We use the LF waveform shown in\nFig.~\\ref{fig:direct} (left panel) as the target signal since it\nmodels the action of the vocal folds during phonation. The regularised\nconvolution is successful in producing the desired tracking as can be\nseen in Fig.~\\ref{fig:deconv} (right panel). For results shown in\nFig.~\\ref{ReconstructedWaveformFig}, the impulse response and all\nsignals have been measured with the source terminated to the vowel\ngeometry [\\textipa{\\textscripta}].\n\n\n\n\\section{Discussion}\n\nAfter many design cycles for improvements, the proposed acoustic\nglottal source appears well suited for its intended use. We now\nproceed to discuss remaining shortcomings and possible improvements\nfor the design and algorithms.\n\nThe three most serious shortcomings in the final design are\n\\textrm{(i)} high TL in the impedance matching cavity due to\nattenuation by polyester fibre, \\textrm{(ii)} acoustic leakage through\nthe source chassis, and \\textrm{(iii)} the usable low frequency limit\nat $\\approx 80 \\, \\mathrm{Hz}$. Since the proposed design is scalable,\nthe latter two deficiencies are easiest treated by increasing the\nphysical dimensions, chassis wall thickness, and, hence, the mass of\nthe source. Using $6''$ or even $8''$ loudspeaker unit with lower bass\nresonant frequencies could be considered, equipped with separate\nconcentric tweeters for producing the higher frequencies. Overly\nincreasing the size of the source makes it, however, impractical for\ndemonstration purposes.\n\n\\begin{figure}[t]\n \\begin{center}\n \\begin{subfigure}[b]{0.485\\textwidth}\n \\centering\n \\includegraphics[width=0.49\\textwidth]{LF_target.pdf}\n \\includegraphics[width=0.49\\textwidth]{h_star_x.pdf}\n \\caption{The LF waveform input to the measurement system (left)\n and the corresponding pressure output at the glottal position\n (right).}\n \\label{fig:direct}\n \\end{subfigure}\n \\hspace{0.05cm}\n \\begin{subfigure}[b]{0.485\\textwidth}\n \\centering\n \\includegraphics[width=0.49\\textwidth]{input.pdf}\n \\includegraphics[width=0.485\\textwidth]{deconv_x.pdf}\n \\caption{The input waveform produced by regularised\n deconvolution (left) and the corresponding output,\n replicating the LF waveform (right).}\n \\label{fig:deconv}\n \\end{subfigure}\n \\caption{Output waveform reconstruction at $180 \\, \\mathrm{Hz}$\n using measured impulse response and regularised\n deconvolution.}\\label{ReconstructedWaveformFig}\n \\end{center}\n\\end{figure}\n\n\nTransversal resonances were checked by adding polyester fibre to the\nwide parts of the impedance matching cavity which results in a marked\nincrease in TL of the source. Considering the amplitude response\ndynamics of $\\approx 35 \\, \\mathrm{dB}$ of the source shown in\nFig.~\\ref{EnvelopeResidualFig}, the output volume remains relatively\nlow in uniform amplitude sweeps that are produced as explained in\nSection~\\ref{CompensationSubSec}. Even though the VT physical models\nhave additional TL of order $20 \\ldots 40 \\, \\mathrm{dB}$ depending on\nthe vowel and test subject, it is possible to carry out frequency\nresponse of formant position measurements without an anechoic chamber\nor a high quality measurement microphone at the mouth position, and\nthe results are quite satisfactory; see\n\\cite[Fig.~5]{K-M-O:PPSRDMRI}. To obtain the high quality frequency\nresponse data or carry out waveform reconstructions presented in\nSection~\\ref{MeasSigSec}, one has to do the utmost to reduce acoustic\nleakage, hum, and noise level, including using of the Br\\\"uel \\&\nKj\\ae{}ll measurement microphone in the anechoic chamber. Then\nsecondary error components emerge as can be observed, e.g., as\nroughness between the formant peaks in Fig.~\\ref{VTResponseFig}. We\npoint out that the quality of the microphone used at the mouth opening\ndoes not affect the measured frequencies of the formant\npeaks. However, the microphone position or the paraboloid concentrator\nshown in \\cite[Fig.~4]{K-M-O:PPSRDMRI} does have a small yet\nobservable effect, in particular, on the lowest resonance frequency of\nthe physical model.\n\nAn attractive way of getting a louder sound source is to use\n\\emph{Smith slits} \\cite{Smith:1953,Dodd:2009} for checking the transversal\nresonances within the wide part of the impedance matching cavity. The\nrequired design work is best carried out using computational design\noptimisation methods introduced in\n\\cite{U-W-B:OVMAH,Y-W-B:LOTSHMIFFDPAH}.\n\nThis article does not concern impedance measurements rather than\nresponse between two acoustic pressures. For impedance measurements,\nthe perturbation velocity should be measured at the output of the\nsource for which a number of approaches, based on microphones, have\nbeen proposed \\cite{Wolfe:AIS:2000, Wolfe:IPM:2006,Wolfe:EEI:2013}. In\nthe current design, hot wire anemometry at the reference microphone\nposition would be most suitable; see \\cite{Pratt:MAI:1977,\n Kob:MMV:2002}. Even the smallest Microflown unit (see\n\\cite{Eerden:EWN:1998,Bree:TMN:1996,Bree:TMF:1997}) commercially\navailable, placed in the middle of the source output channel of\ndiameter $6\\, \\mathrm{mm}$, would cause severe back reflections.\n\nWe have used two different response compensation techniques in\nSection~\\ref{CalibrationSec}: weighting for sweeps and regularised\ndeconvolution for more complicated signals. Using deconvolution for\nproducing sweeps tens of seconds long is not a practical since the\ndimension of Eqs.~\\eqref{DiscretisedRegConv} would be too high. As\nopposed to weighted sweeps, regularised deconvolution takes into\naccount the phase response of the full measurement system. The\ndeconvolution is a linear operation whereas the measurement system\nshows signs of amplitude nonlinearity in\nFig.~\\ref{EnvelopeResidualFig}. This is one of the reasons why\ntracking more challenging targets than the LF waveform (e.g., the ramp\nsignal) will not give as good an outcome. The compensation weight\nreconstruction method in Section~\\ref{CompensationSubSec} does not\nrely on linearity at all, and its performance can be improved by\nincreasing the sweep length.\n\nOne of the challenging secondary objectives is to design dummy loads\nof \\emph{reasonable physical size} for the source that would present a\nconstant resistive load over a wide range of frequencies. The dummy\nload shown in Fig.~\\ref{TractrixHornFig} (right panel) consists of a\ntractrix horn tightly filled with polyester fibre, and it has the\nproperty of not being resonant to an observable degree. Two\nparticularly inspiring examples on the construction of resistive\nacoustic loads are given in \\cite{Wolfe:AIS:2000} ($42 \\,\\mathrm{m}$\nof insulated steel pipe of inner diam. $7.8 \\, \\mathrm{mm}$) and\n\\cite{Wolfe:IPM:2006} ($97 \\,\\mathrm{m}$ of straight PVC pipe of inner\ndiam. $15 \\, \\mathrm{mm}$). The practical challenges in such\napproaches are considerable.\n\nWe conclude by discussing the numerical efficiency of the discretised\ndeconvolution proposed in Section~\\ref{ImpulseSubSec}. In order to\nobtain an $N \\log{N}$ algorithm, the $N \\times N$ matrices $R$ and $H$\nwere forced to be circulant. Another way to proceed is allowing $R$ to\nbe the usual tridiagonal, symmetric, second order difference matrix, \nand $H$ to be the upper triangular matrix obtained from the impulse\nresponse, both noncirculant Toeplitz matrices. Then the symmetric\nmatrix $\\gamma \\left (\\kappa I + R^{T}R \\right ) + H^{T}H$ in\nEq.~\\eqref{DiscretisedRegConv} is a slightly perturbed Toeplitz\nmatrix, and the required (approximate) solution of the linear system\ncan be carried out by Toeplitz-preconditioned Conjugate Gradients at\nsuperlinear convergence speed; see, e.g., \\cite{JM:PIT}. Again, an $N\n\\log{N}$ algorithm is obtained if the matrix\/vector products are\nimplemented by FFT.\n\n\\section{Conclusions}\n\nA sound source was proposed for acoustic measurements of vocal tract\nphysical models, produced by Fast Prototyping methods from Magnetic\nResonance Images. The source design requires only commonly available\ncomponents and instruments, and it can be scaled to different\nfrequency ranges. Heuristic and numerical methods were used to\nunderstand and to optimise the source design and performance. Two\nkinds of algorithms were proposed for compensating the source\nnonoptimality: (\\textrm{i}) an iterative process for producing uniform\namplitude sound pressure sweeps, and (\\textrm{ii}) a method based on\nregularised deconvolution for replicating target sound pressure\nwaveforms at the source output. The sound source together with the\ntwo compensation algorithms, written in MATLAB code, was deemed\nsuccessful based on measurements on the vocal tract geometry\ncorresponding to vowel [\\textipa{\\textscripta}] of a male speaker.\n\n\\section*{Acknowledgments}\n\n\n The authors wish to thank for consultation and facilities\n Dept. Signal Processing and Acoustics, Aalto University\n (Prof.~P.~Alku, Lab.~Eng.~I.~Huhtakallio, M.~Sc.~M.~Airaksinen) and\n Digital Design Laboratory, Aalto University (M.~Arch.~A.~Mohite).\n\n The authors have received financial support from Instrumentarium\n Science Foundation, Magnus Ehrnrooth Foundation, Niilo Helander\n Foundation, and Vilho, Yrj\\\"o and Kalle V\\\"ais\\\"al\\\"a Foundation.\n\n\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}