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+{"text":"\\section{Introduction}\n\nDue to launch in late 2015, LISA Pathfinder \\cite{Armano2015} is a European Space Agency (ESA) precursor mission for a space-based gravitational wave observatory \\cite{Amaro2012}. At the heart of the experiment are two cubic gold-platinum test masses, free-falling under gravity and each enclosed within their own capacitive inertial sensor. The gold-coated test masses act as mirrors for a laser interferometer used to measure changes in their separation with pico-metre accuracy. The spacecraft follows the motion of the test masses along the axis of their separation using micro-Newton thrusters while electrostatic forces are applied to control the motion in other degrees of freedom. To demonstrate the feasibility of detecting gravitational waves, the test masses need to maintain pure geodesic motion achieved by the elimination of all non-gravitational forces to a level below $6\\times10^{-14}\\,\\textup{NHz}^{-1\/2}$ in a frequency band between 1 and 10\\,mHz.\n\nAlthough the test masses are enclosed within the satellite there will be an inevitable build up of electric charge on the test masses due to incident ionising radiation from space \\cite{Jafry1996}. A charged test mass will experience electrostatic forces through interaction with electric fields within the inertial sensor \\cite{Weber2012}, as well as Lorentz forces via coupling with interplanetary magnetic fields \\cite{Sumner2000}. Left unchecked, these forces can limit the performance of the instrument. They also introduce operational constraints and artefacts in the data \\cite{Shaul2005} which make it necessary to control the  charge. While missions such as CHAMP \\cite{Reigber2002}, GRACE \\cite{Tapley2004},  GOCE \\cite{Rummel2011} and MICROSCOPE \\cite{Touboul2001} rely on a wire connection to ground to avoid charge build-up, the force noise goals of LISA Pathfinder necessitate a non-contact method to avoid disturbing the test masses.  The use of photoelectric emission from UV photons is such a method and was first demonstrated on the Gravity Probe B mission \\cite{Buchman1995}.\n\nLISA Pathfinder includes a charge management subsystem (CMS) \\cite{Sumner2009} exploiting the photoelectric effect by using $253.7\\,\\textup{nm}$ light from low-pressure mercury discharge lamps to transfer charge between the gold-coated surfaces of the test mass and the surrounding housing and electrodes of the sensor. In its most simplistic form this means illuminating the housing and electrodes to add negative charge to the test mass or illuminating the test mass itself in order to remove negative charge. \n\nThe capacitive inertial sensors of a future ESA gravitational wave observatory will be based on a similar design to those aboard LISA Pathfinder and as such will also require a charge management system \\cite{Vitale2014} as will any other mission using isolated proof-masses, such as STEP \\cite{Sumner2007}. Since the design and production of the LISA Pathfinder CMS, several types of deep ultraviolet light-emitting diodes (UV LEDs) have become commercially available with peak wavelengths below $260\\,\\textup{nm}$. These devices offer many advantages that make them promising candidates to replace mercury lamps as a CMS light source. However, unlike mercury lamps which have space heritage prior to the LISA Pathfinder mission \\cite{Buchman1995, Adams1987}, such devices have seldom been used in space and never in an extended mission.\n\nThe aim of this work is to compare three types of commercially available UV LEDs and determine their suitability for use in space via a series of rigorous tests. A particular emphasis is placed on the unique and strict requirements needed to be fulfilled by any potential CMS light source.\n\n\n\n\\section{LISA Pathfinder Test Mass Discharging}\n\nBefore considering what will be required of an improved light source for a generic CMS it is important to summarise the performance and functionality of the LISA Pathfinder CMS. The LISA Pathfinder CMS consists of three hardware parts, the main one being the UV Light Unit (ULU). It contains six programmable, customised Pen-Ray low-pressure mercury discharge lamps\\footnote{http:\\textbackslash\\textbackslash uvp.com\\textbackslash penraylightsources.html}, three for each inertial sensor, as well as the electronics required for their operation. Each lamp has an integrated heater to allow operation at low temperatures as well as an optics barrel to collect, filter and focus the emitted light into a fibre. A band-pass filter transmits the mercury line at $253.7\\,\\textup{nm}$ line responsible for photoemission while blocking light at $184.4\\,\\textup{nm}$, which is harmful to subsequent elements of the optical path and longer wavelength lines and continuum emission to avoid injecting unnecessary light into the sensor.\n\nThe other two hardware items are the Fibre Optic Harness (FOH) and Inertial Sensor UV Kit (ISUK). The FOH routes the light from the ULU in the outer compartment of the spacecraft towards each inertial sensor at the centre and consists of a series of custom made fibre optics with separate chains for each lamp. The UV light is finally injected into the sensors inside their vacuum enclosures via the ISUKs which are custom-made ultra-high vacuum fibre feed-throughs. There are three ISUKs for each sensor, two pointing at the housing and one at the test mass. Ideally, any future light source would be compatible with the LISA Pathfinder FOH and ISUK, allowing the same designs and materials to be reused.\n\nThe maximum UV power entering the sensor varies depending on the optical chain but is typically of order 100\\,nW with a dynamic range of $\\approx200$ achieved using a pulse-width modulation technique at kHz frequencies. Test mass discharging can be performed in two different modes: in fast discharge mode, the test mass is allowed charge up over several days until it reaches a level where charge-related disturbances become problematic, typically $\\approx10^{7}$ elementary charges (e). The sensor is then illuminated at a relatively high UV power level to reduce the charge below $10^{5}$\\,e over the course of 10 to 20 minutes. In continuous discharging mode a lower UV power level is used such that the discharging rate cancels the environmental charging rate, predicted to be $\\approx10\\,\\textup{to}\\,100\\,\\textup{es}^{-1}$ \\cite{Araujo2005, Wass2005}, and the test-mass charge is maintained at a level below $10^5$\\,e.\n\nEach lamp requires its own high-voltage electronics to produce the $\\approx600\\,\\textup{V}$ needed to initiate discharge in the mercury vapour, falling to around half this level during operation. The typical power consumption of a mercury lamp at full power is $\\approx4\\,\\textup{W}$. When initially switched on, the rise-time of the UV output is temperature dependent where the time for the output to rise from 5\\% to 95\\%  of its maximum value ranges from $\\approx50\\,\\textup{s}$ at $20\\,^{\\circ}\\textup{C}$ to $\\approx15\\,\\textup{s}$ at $40\\,^{\\circ}\\textup{C}$. The vapour pressure of mercury varies with temperature and therefore UV power emitted by a lamp at a particular operational setting is also temperature-dependent. Related to the vapour pressure, at temperatures below $15\\,^{\\circ}\\textup{C}$ it is necessary to pre-warm lamps prior to operation. While lamp output will degrade during the LISA Pathfinder mission lifetimes are predicted to be over $1000\\,\\textup{hours}$ of continuous use at high output power.\n\n\n\\subsection{Complications}\n\nThere are three complications to the simple concept of discharging presented so far. The first is the absorption of UV light by unintended surfaces. For example, although the light may be directed at one surface, say the test mass, light is inevitably reflected leading to absorption on the sensor housing or electrodes opposite, generating a photocurrent acting against the intended direction of discharge. At $253.7\\,\\textup{nm}$, the reflectivity of the gold coated surfaces is 36\\% at normal incidence but increases rapidly for incident angles shallower than $45^{\\circ}$ \\cite{Johnson1972}. In the inertial sensor,  most light is incident at between 45 and 70$^{\\circ}$ relative to the surface normal and as such, significant amounts of light will be absorbed by secondary surfaces.\n\nA second complication arises due to unavoidable contamination of the gold surfaces inside the inertial sensor. The work function of pure gold, deposited in vacuum and measured \\textit{in situ} is $5.2\\,\\textup{eV}$ \\cite{Huber1966}. Upon exposure to air, adsorbates reduce the effective work function to as low as $4.2\\,\\textup{eV}$ \\cite{Saville1995}, with water and hydro-carbons having particular influence. This surface contamination persists even when the sample is placed in vacuum and after modest baking \\cite{Hechenblaikner2012}. Given that the energy of the usable photons from mercury lamps is $4.89\\,\\textup{eV}$, it is only through surface adsorbates that discharge can occur at all. However, relying on inherently uncertain surface properties leads to nominally identical sensor surfaces having significantly different photoelectric properties. Studies have found that the quantum yield, the number of emitted photoelectrons per absorbed photon, of gold can vary from $10^{-6}$ to $10^{-4}$ \\cite{Schulte2009}. When combined with the distribution of absorbed light due to reflections, in the most extreme case a significant asymmetry in yield can prevent discharging in one direction no matter which surface is initially illuminated.\n\nThe final complication is the presence of local electric fields within the inertial sensor. Both alternating and direct current (AC and DC) voltages can be applied to the 18 separate electrodes surrounding the test mass to enable the test mass position sensing and actuation \\cite{Weber2007}. The voltage used to bias the test mass for capacitive sensing is a $100\\,\\textup{kHz}$ sine wave with a nominal $5.4\\,\\textup{V}$ amplitude applied to six of the electrodes in the sensor. This so called injection bias results in a test mass potential with respect to the grounded electrode housing of $\\pm0.6\\,\\textup{V}$. Sinusoidal actuation voltages are also applied to the remaining 12 position-sensing electrodes at a range of audio frequencies with amplitudes of up to $7\\,\\textup{V}$. The potential difference between test mass and sensor therefore varies greatly depending on location. Crucially, the energy of the emitted photoelectrons is less than $1\\,\\textup{eV}$ meaning that photocurrents are strongly influenced by these electric fields, to the point where they can be completely suppressed. Although avoided as it introduces electrostatic forces in the measurement bandwidth, DC voltages may also be applied to individual electrodes to aid discharging.\n\n\n\n\\section{UV LEDs}\n\nUsing UV LEDs as a CMS light source has several obvious advantages over the mercury lamps used for LISA Pathfinder. They offer significant mass and volume savings not only because of their smaller size but also because of a reduction in the complexity of the associated electronics and optics. Electrical power consumption is significantly reduced and, depending on how they are operated, they offer a higher range of UV output power and longer lifetimes than mercury lamps.\n\nUV LEDs are available that can produce light with a wavelength $<254\\,\\textup{nm}$ and therefore photons of higher energy than the mercury spectral line. Moving away from the work function of contaminated gold not only should the quantum yield increase but at a photon energy $>5.2\\,\\textup{eV}$ it is possible to liberate electrons from pure gold. This opens the possibility of reducing or removing surface contamination, for example by aggressive baking under vacuum or ion sputtering, without the fear of preventing discharging. Without the unpredictable photoelectric properties of a contaminated surface, the risk of bipolar discharging not being possible would be removed. A parallel study has involved measuring the quantum yield from a number of prepared surfaces and this will be reported separately \\cite{Wass2015}.\n\nBeing semiconductor devices, it is possible to pulse UV LEDs at high frequencies and synchronise them with the AC voltages present in the inertial sensor \\cite{Sun2006}. Switching the device on only when the electric fields in the region under illumination enhance the desired direction of discharge elegantly mitigates the problem of asymmetric photoelectric properties as the photo-current would be able to flow in one direction while being partially or completely suppressed in the opposite, undesirable direction. This method also provides redundancy as shifting the phase of the pulses by $180^{\\circ}$ would reverse the direction of net photocurrent and allow a single point of UV injection to discharge both positively and negatively. Furthermore, the discharge rate could be tuned by adjusting the phase of the UV pulses with respect to the synchronising voltage, increasing the dynamic range of the discharge system. Finally, synchronous discharging would be more efficient compared to a DC scheme where for half of the injection bias cycle the voltages act against the desired direction.\n\nWith the potential advantages clear, work has already been carried out by other groups to demonstrate that UV LEDs are suitable for space applications. These have mainly focused on one particular type of device with a peak wavelength at approximately $255\\,\\textup{nm}$ supplied by Sensor Electronic Technology (SET) \\cite{SETi2015}. These tests have yielded promising results suggesting that this type of device has impressive lifetimes, can be pulsed at high frequencies and are radiation hard \\cite{Sun2006, Sun2009}. However, pulsed discharging has only been demonstrated at frequencies of $1\\,\\textup{kHz}$ and $10\\,\\textup{kHz}$ with $50\\%$ duty cycles. The obvious electric field with which to to synchronise in the inertial sensor would be the $100\\,\\textup{kHz}$ injection voltage, with duty cycles $<50\\%$ resulting in pulses of $<5\\,\\mu\\textup{s}$ duration. In addition, previous work has not focussed on dynamic range or presented a systematic study on a number of devices.\n\nMore recently, shorter wavelength devices from the same supplier have come to market which nominally peak at $240\\,\\textup{nm}$. This shorter wavelength could offer additional potential benefits for discharging. More recently still, devices which peak at $250\\,\\textup{nm}$ and supplied by Crystal IS (CIS) have appeared \\cite{CIS2015}. For space applications a second source supplier can be crucial.\n\nBoth types of device (SET and CIS) consist of a structure of AlGaN layers grown on a substrate via patented chemical vapour deposition processes. Broadly speaking the ratio of aluminium and gallium within the $Al_{x}Ga_{1-x}N$ alloy determines the peak wavelength of the device with increased aluminium content leading to lower wavelengths, \\cite{Hirayama2005}. For the SET devices sapphire substrates are used while CIS use aluminium nitride, which they claim produces fewer lattice defects, leading to superior lifetimes and higher output powers. Both suppliers offer devices surface mounted or pre-packaged with integrated optics and according to their data sheets offer UV output powers of $\\approx100\\,\\mu\\textup{W}$ at electrical powers of $<100\\,\\textup{mW}$. Both also have wide operating temperature ranges of $-30\\,^{\\circ}\\textup{C}$ to $+50\\,^{\\circ}\\textup{C}$.\n\nDespite the promise of UV LEDs as a future CMS light source several unanswered questions remain. Unlike mercury lamps, which have a space heritage pre-dating the LISA Pathfinder mission, UV LEDs have very limited short-term space exposure. Before a future CMS can be designed as a whole a precise understanding of the properties of its light source is required. Survivability needs to be demonstrated under environmental stresses such as thermal vacuum cycling, radiation exposure, vibration and shock.  Furthermore, a detailed understanding of the spectral properties of a device under a variety of operational scenarios, including any changes with operational temperature, drive current, pulse width, age or after irradiation is vital as it directly determines the yield and energy distribution of the photoelectrons emitted from a surface. Both the electrical properties and UV output of the devices under consideration are known to be strongly temperature dependent and this needs to be fully understood. \n\nTests needs to be carried out against a clear requirement specification; the outcome of a recent focussed ESA technology study has been used here to define the range and extent for this work \\cite{Sumner2015}. Testing was carried out on a range of commercially available UV LEDs that aimed to quantify and compare a variety of device characteristics including their general electrical and UV output properties, spectral stability, pulsed performance, temperature dependence as well as thermal vacuum, radiation and vibration survivability. It was equally important to test the underlying technologies such that if and when even shorter wavelength devices are released in future critical issues can be identified.\n\n\n\n\\section{Test Devices}\n\nIn total, nine devices of three different types were tested. Two types of device were supplied by Sensor Electronic Technology (SET) with part references UVTOP-255-TO39-BL and UVTOP-240-TO39-HS while the other was supplied by Crystal IS (CIS) with a part reference OPTAN250J. The SET devices of the same type were produced from the same wafer, while the CIS devices were selected from a bin of devices with similar output properties as tested by the manufacturer. All the devices were packaged in TO-39 cans and identified by their nominal peak wavelength. The two SET device types had nominal peak wavelengths of $255\\,\\textup{nm}$ and $240\\,\\textup{nm}$ while the CIS device was $250\\,\\textup{nm}$. The SET-255 devices were sold as `research grade' and had an integrated ball lens focusing their output to a spot approximately $2\\,\\textup{mm}$ in diameter at a distance of about $20\\,\\textup{mm}$. The SET-240 devices had a hemispherical lens creating a beam with a typical $6^\\circ$ spread, while the CIS-250 devices also had an integrated ball lens with a slightly shorter focal length than that of the SET device. According to the manufacturer the CIS-250 devices had received a $48\\,\\textup{hour}$ \"burn-in\" at $100\\,\\textup{mA}$ prior to delivery which was not the case for the SET devices. All the devices remained completely untouched and in secure storage prior to testing.\n\nBefore each device was tested it underwent an initial inspection where it was weighed, photographed and checked visually. It then had flying leads attached and was assigned a unique label and stored in an individual Electrostatic Discharge (ESD) bag. Three of the nine devices are shown in Figure \\ref{fig:DevicePhotos}.\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}[c]{0.35\\textwidth}\n\\includegraphics[width=1.0\\textwidth]{DSC_1780_fixed.pdf}\n\\end{minipage}\n\\hfil\n\\begin{minipage}[c]{0.40\\textwidth}\n\\caption[Photograph of devices with flying leads attached.]{\\label{fig:DevicePhotos} From left to right the devices are SET-255-01, SET-240-01 and CIS-250-01. Each device has soldered flying leads attached, with the joint insulated with Kapton tape. Note that each device has a different type of lens while the scale on the left is in mm.}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Testing}\n\nTesting was carried out from June to October 2014 and other than radiation and vibration, which were performed at external facilities, all tests were carried out with custom-made test equipment developed and built at Imperial College. All the work described here was done under ESA contract and included a high level of documentation and traceability with formal test procedures and task sheets. Data processing and analysis were carried out with the LISA Technology Package Data Analysis (LTPDA) toolbox \\cite{LTPDA2015}.\n\nUnless otherwise stated, the tests were performed with the device under test (DUT) fixed in a copper mount within a light sealed enclosure. The mount temperature was controlled using a thermoelectric system with two thermistors embedded within the mount (Peltier: DA-014-12-02, Controller: PR-59, both supplied by Laird Technologies). During low temperature testing the enclosure was flooded with nitrogen to prevent condensation and\/or ice forming. A calibrated Hamamatsu S1337-1010BQ UV photodiode with a $1\\,\\textup{cm}^2$ sensitive area could be fixed opposite the DUT, in the same mount, capturing the total UV output of the device. Alternatively, with the photodiode removed, the output from the DUT could be injected into a $1\\,\\textup{mm}$ diameter, UV transparent fibre. Within the enclosure, the fibre itself was mounted on a 3-axis translation stage to position the fibre tip for optimal UV acceptance. Depending on the DUT, it was possible to couple $10\\%$ to $20\\%$ of the total light emitted into the fibre. The fibre was routed out of the enclosure through a rubberised seal and interfaced with either a spectrometer or a Hamamatsu H6780-06 photomultiplier tube (PMT).\n\nSpectral measurements were made using a customized Princeton Instruments Model VM-502 spectrometer with slit widths chosen to give a resolution of $\\approx0.3\\,\\textup{nm}$. The spectrometer diffraction grating was positioned with a stepper motor and its calibration was checked on a weekly basis against the well-defined position of spectral lines from an unfiltered mercury lamp source. When used in conjunction with a broadband deuterium lamp the spectrometer could also be operated as a monochromator. This made it possible to use a calibrated power meter (Sensor: 918D-UV-OD3R, Meter: 841-P-USB, both supplied by Newport) to cross-calibrate the absolute response as a function of wavelength of the photodiodes and PMT used during testing with a systematic uncertainty of $\\pm2\\%$. The temperature dependence of both the response and dark current for the photodiodes was measured over a temperature range from $-10\\,^{\\circ}\\textup{C}$ to $+60\\,^{\\circ}\\textup{C}$. Over this range the response was found to vary by less than $0.1\\%$, the limit of the measurement. The dark current was found to increase exponentially with temperature measuring $0.01\\,\\textup{nA}$ at $-10\\,^{\\circ}\\textup{C}$ and $1\\,\\textup{nA}$ at $+60\\,^{\\circ}\\textup{C}$. Although this was small compared to the typical $\\mu\\textup{A}$ photodiode signals measured it was nonetheless subtracted during runs where a dark reading could be made.\n\nGreat care was taken throughout testing to comply with handling and usage recommendations for each device given on their data sheet and with ESA European Cooperation for Space Standardisation (ECSS) guidelines \\cite{ECSS2009}. Precautions for handling ESD-sensitive devices were taken and UV LEDs were only handled wearing clean nitrile gloves, while also taking care to avoid mechanical shocks. Appropriate heat sinking was also used when soldering devices. A $10\\,\\textup{mA}$ maximum average drive current was chosen for the devices, well within datasheet recommendations. At this level, UV output powers at the end of a fibre were predicted to be $>10\\,\\mu\\textup{W}$, several times higher than that delivered by the LISA Pathfinder CMS and consistent with the requirement specification \\cite{Sumner2015}. A Keithley 2602A source meter was used to supply the devices with DC current. To drive the devices in a pulsed mode, a Tektronix AFG 3101 signal generator was used to trigger a custom-made set of electronics producing a variable current-limited pulse amplitude. A Tektronix AM503 current probe and an Agilent Infiniium MS09254A oscilloscope were used to study the pulsed behaviour.\n\nThe mainly automated laboratory-based tests were carried out in a sequential order taking a week per device. Once all  devices had been characterised in the laboratory, radiation testing was performed followed by vibration and shock, again in accordance with \\cite{Sumner2015}. At several points in the test campaign a reference test was performed on each device. This consisted of a small subset of performance tests aimed at giving a baseline against which any potential changes in characteristics could be monitored. Reference measurements was performed on each device before and after the laboratory based tests, after the radiation test and after the vibration test.\n\n\n\n\\section{Results}\n\nThe laboratory based tests were carried out on each device individually and are collated here to allow comparison.\n\n\\subsection{Initial Reference Characteristics}\n\nThe reference measurement consisted of three parts; a current-voltage (IV) scan performed while simultaneously measuring the total UV output of the DUT, a spectral scan while the DUT was driven at $1\\,\\textup{mA}$ DC, and a waveform measurement while the DUT was pulsed at $100\\,\\textup{kHz}$, $10\\%$ duty cycle with a $10\\,\\textup{mA}$ drive current amplitude. The IV scan was done between $0\\,\\textup{mA}$ and $10\\,\\textup{mA}$, in steps of $0.1\\,\\textup{mA}$, dwelling at each setting for $1\\,\\textup{second.}$ All measurements were made with the DUT sealed in the air filled temperature controlled enclosure, held at $20\\,^{\\circ}\\textup{C}$.\n\n\\begin{figure}[h]\n\\begin{minipage}[t]{0.5\\textwidth}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{UVcurves.pdf}\n\\includegraphics[width=1.0\\textwidth]{IVcurves.pdf}\n\\end{minipage}\n\\begin{minipage}[t]{0.5\\textwidth}\n\\includegraphics[width=1.0\\textwidth]{Spectra.pdf}\n\\caption[Initial device characteristics.]{\\label{fig:DeviceChar} Clockwise from bottom left: IV scans, simultaneously measured total UV output power and spectral scans. Note the unexpected jump at $2\\,\\textup{mA}$ during the SET-240-01 UV measurement. The cause of this anomaly is not clear but no such jump appears in the IV curve and the behaviour was not seen again in any other test.}\n\\end{minipage}\n\\end{figure}\n\nReferring to Figure \\ref{fig:DeviceChar}, all the initial IV scans show broad agreement with device data sheets and the three device types fall within a similar range, approximately $5\\,\\textup{V}$ to $7.5\\,\\textup{V}$ at $10\\,\\textup{mA}$. Note that the nine curves are approximately grouped by device type but there is a spread within each of these groups, the SET-255s showing the least variation. The simultaneously measured total UV output shows seven out of the nine devices had a maximum output at $10\\,\\textup{mA}$ of between $100\\,\\mu\\textup{W}$ and $120\\,\\mu\\textup{W}$. However, there are two devices that lie outside this range at $78\\,\\mu\\textup{W}$ and $145\\,\\mu\\textup{W}$. While keeping in mind the small sample size, the SET-240s produce the highest output while the CIS-250s have the lowest, at least at drive currents above $5\\,\\textup{mA}$. At the lowest current setting tested ($0.1\\,\\textup{mA}$) the devices emitted between $15\\,\\textup{nW}$ and $80\\,\\textup{nW}$. This equates to dynamic ranges of between $10^{3}$ and $10^{4}$, even in this simple operational scenario. It is also worth noting that the output from the CIS-250 devices appears slightly more linear with drive current than for the SET devices. The typical electrical power consumptions are in the tens of mW range leading to conversion efficiencies of $0.1\\%$ to $0.2\\%$. \n\nFigure \\ref{fig:DeviceChar} also shows the measured spectrum of each device, normalised to have equal areas. The positions of the mercury $253.7\\,\\textup{nm}$ spectral line and the wavelength equivalent to the work function of pure gold are indicated for comparison. The spectra produced by SET devices of a particular type are almost indistinguishable from one another. While this is not unexpected as all devices were manufactured from the same wafer, it demonstrates the reproducibility of the measurement given that each scan was taken weeks apart. The CIS-250 devices show noticeable differences with CIS-250-02 standing out in particular with an unexpected secondary peak at $290\\,\\textup{nm}$ which accounts for approximately $10\\%$ of the light emitted. \n\nA quantitative analysis was performed on each spectrum in order to extract both the peak wavelength and full width at half maximum (FWHM). The peak position was obtained by fitting a Gaussian function to the 50 points around the central maximum while ignoring the tails and the FWHM by finding the point at which the measured spectrum crossed (up\/down) half its maximum value, interpolating between the two consecutive points where this occurs on either side. The results are summarised in Table \\ref{tab:DeviceInitialSpecTable}.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{ | c | c | c | c | c |}\n\\hline\n\\multirow{2}{*}{Device}   &  Peak Wavelength                                       &  FWHM                                       &  \\multicolumn{2}{|c|}{Fraction Below}                       \\\\\n                                          &  (nm)                                                            &  (nm)                                          &  $253.7\\,\\textup{nm} $  &  $237.4\\,\\textup{nm}$   \\\\\n\\hline\n  SET-255-01                     &  $258.34 \\pm 0.06$                                     &  $10.88 \\pm 0.02$                     &  $0.13$                            &  $0.00$                            \\\\\n  SET-255-02                     &  $258.63 \\pm 0.06$                                     &  $11.01 \\pm 0.01$                     &  $0.11$                            &  $0.00$                            \\\\\n  SET-255-03                     &  $258.53 \\pm 0.06$                                     &  $10.96 \\pm 0.03$                     &  $0.12$                            &  $0.00$                            \\\\\n  SET-240-01                     &  $247.20 \\pm 0.06$                                     &  $10.11 \\pm 0.04$                     &  $0.86$                            &  $0.01$                            \\\\\n  SET-240-02                     &  $247.16 \\pm 0.06$                                     &  $10.07 \\pm 0.12$                     &  $0.85$                            &  $0.01$                            \\\\\n  SET-240-03                     &  $247.15 \\pm 0.06$                                     &  $10.21 \\pm 0.16$                     &  $0.86$                            &  $0.01$                            \\\\\n  CIS-250-01                     &  $252.52 \\pm 0.06$                                     &  $10.71 \\pm 0.07$                     &  $0.53$                            &  $0.00$                            \\\\\n  CIS-250-02                     &  $252.48 \\pm 0.06$                                     &  $10.31 \\pm 0.16$                     &  $0.45$                            &  $0.00$                            \\\\\n  CIS-250-03                     &  $252.20 \\pm 0.06$                                     &  $9.58   \\pm 0.11$                     &  $0.57$                            &  $0.00$                            \\\\\n\\hline\n\\end{tabular}\n\\caption[Device's initial spectral properties.]{\\label{tab:DeviceInitialSpecTable} Initial spectral properties, measured at $20\\,^{\\circ}\\textup{C}$ while driven at $1\\,\\textup{mA}$ DC. Also shown is the fraction of each spectrum below the mercury $253.7\\,\\textup{nm}$ spectral line and the wavelength equivalent of the pure gold work function ($237.4\\,\\textup{nm}$). Note the error  in the peak wavelength is dominated by the uncertainty in the spectrometer calibration.}\n\\end{center}\n\\end{table}\n\nAll three device types have peak wavelengths higher than their nominal values of $255\\,\\textup{nm}$, $240\\,\\textup{nm}$ and $250\\,\\textup{nm}$, though within the limits specified by the suppliers, up to $+7\\,\\textup{nm}$ for the SET devices and $+2.5\\,\\textup{nm}$ for CIS. While all three device types produce significant amounts of light at a wavelength shortwards of the $253.7\\,\\textup{nm}$ used in the LISA Pathfinder CMS, only the SET-240s produce light at an energy greater than the work function of pure gold, albeit only 1\\% of their total.\n\n\n\\subsection{Spectral Stability}\n\nEach device spectrum was remeasured under a variety of operational scenarios, including various temperatures ($-10\\,^{\\circ}\\textup{C}$, $+20\\,^{\\circ}\\textup{C}$ and $+40\\,^{\\circ}\\textup{C}$, driven at $1\\,\\textup{mA}$ DC), DC drive currents ($1\\,\\textup{mA}$, $5\\,\\textup{mA}$ and $10\\,\\textup{mA}$, held at $20\\,^{\\circ}\\textup{C}$) and pulsed duty cycle (5\\%, 25\\% or 50\\%, at $100\\,\\textup{kHz}$, $20\\,\\textup{mA}$ amplitude). Including the reference measurements carried out before and after laboratory testing, ten measured spectra for each device were available for comparison and to check spectral stability under different operational conditions. Each spectrum was first quantified by extracting the spectral peak and FWHM as described previously. These data were then plotted for each device against the operational condition of interest, for example temperature, allowing the observation of any significant trends in the data. \n\nWithin the uncertainty of the measurement, the spectra (peak position and FWHM) of all devices was found to be stable with DC drive current and pulsed duty cycle, the one exception being CIS-250-02. It was found that with increasing drive current, whether DC or pulsed, the amplitude of the secondary peak diminished relative to the primary. At $10\\,\\textup{mA}$ it made up $<1\\%$ of the total emitted light compared to $10\\%$ at $1\\,\\textup{mA}$. However, the main peak showed no measurable variation in terms of peak position or FWHM. \n\nTemperature was found to affect the FWHM of all devices as well as the peak position of the lower wavelength devices. In both cases the relationship was linear over the temperature ranged studied and two typical examples are shown in Figure \\ref{fig:LinSpecProp}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.25\\textheight]{FWHM_SET-255-03.pdf}\n\\hfil\n\\includegraphics[height=0.25\\textheight]{Peak_SET-240-02.pdf}\n\\caption[Linear spectral properties.]{\\label{fig:LinSpecProp} Left: Change in FWHM with temperature for SET-255-03. Right: Change in spectral peak position with temperature for SET-240-02.}\n\\end{figure}\n\nLinear fits were performed on each set of FWHM and spectral peak data with the results summarized in Table \\ref{tab:DeviceLinFitSpec}. The three device types produced consistent results within each group. For the SET-255 and SET-240 devices a $1.5\\,\\textup{nm}$ reduction in FWHM was observed over the $50\\,^{\\circ}\\textup{C}$ temperature range studied, while for the CIS-250 devices it was $1.0\\,\\textup{nm}$. While no effect was observed for the SET-255 devices within the uncertainty of the measurement ($\\approx0.002\\,\\textup{nm}\/^{\\circ}\\textup{C}$), the other two types also saw a shift in their spectral peak position of approximately $0.75\\,\\textup{nm}$ over the same temperature range.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{ | c | c | c | c | c | }\n\\hline\n\\multirow{3}{*}{Device}  &  \\multicolumn{2}{|c|}{FWHM Linear Fit}                                                &  \\multicolumn{2}{|c|}{Spectral Peak Linear Fit}                                            \\\\\n                                         &  Gradient                                                             &  Intercept                &  Gradient                                                             &  Intercept                        \\\\\n                                         &  ($\\textup{nm}\/^{\\circ}\\textup{C}$)                &  (nm)                        &  ($\\textup{nm}\/^{\\circ}\\textup{C}$)                &  (nm)                                \\\\\n\\hline\n  SET-255-01                     &  $0.031 \\pm 0.003$                                            &  $10.30 \\pm 0.07$  &  $-$                                                                      &  $-$                                  \\\\\n  SET-255-02                     &  $0.024 \\pm 0.004$                                            &  $10.5   \\pm 0.1  $  &  $-$                                                                      &  $-$                                  \\\\\n  SET-255-03                     &  $0.028 \\pm 0.002$                                            &  $10.34 \\pm 0.04$  &  $-$                                                                      &  $-$                                  \\\\\n  SET-240-01                     &  $0.030 \\pm 0.001$                                            &  $9.50   \\pm 0.02$  &  $0.016 \\pm 0.002$                                             &  $246.87 \\pm 0.05$        \\\\\n  SET-240-02                     &  $0.030 \\pm 0.002$                                            &  $9.53   \\pm 0.04$  &  $0.017 \\pm 0.002$                                             &  $246.78 \\pm 0.04$        \\\\\n  SET-240-03                     &  $0.029 \\pm 0.005$                                            &  $9.7     \\pm 0.1  $  &  $0.017 \\pm 0.002$                                             &  $246.80 \\pm 0.04$        \\\\\n  CIS-250-01                     &  $0.017 \\pm 0.004$                                            &  $10.2   \\pm 0.1  $  &  $0.009 \\pm 0.002$                                             &  $252.29 \\pm 0.06$        \\\\\n  CIS-250-02                     &  $0.015 \\pm 0.004$                                            &  $9.9     \\pm 0.1  $  &  $0.011 \\pm 0.002$                                             &  $252.2   \\pm 0.1  $        \\\\\n  CIS-250-03                     &  $0.022 \\pm 0.002$                                            &  $9.3     \\pm 0.1  $  &  $0.013 \\pm 0.001$                                             &  $251.90 \\pm 0.03$        \\\\\n\\hline\n\\end{tabular}\n\\caption[Linear fits of spectral properties with temperature.]{\\label{tab:DeviceLinFitSpec} A summary of the linear fits found for each device describing the change in their spectral properties with temperature. Note that no significant temperature dependence was found for the position of the SET-255 spectral peaks.}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Pulsed Capabilities}\n\nPulsed capabilities were tested with the DUT mounted in the light sealed, temperature controlled enclosure. The UV output was routed to either the PMT or alternatively the calibrated power meter which allowed the average UV output to be measured. Each device was driven at a range of frequencies ($100\\,\\textup{Hz}$, $1\\,\\textup{kHz}$, $100\\,\\textup{kHz}$ and $1\\,\\textup{MHz}$ all at a $50\\%$ duty cycle, $20\\,\\textup{mA}$ amplitude), duty cycles ($25\\%$, $10\\%$ and $5\\%$ all at $100\\,\\textup{kHz}$, $20\\,\\textup{mA}$ amplitude) and current amplitudes ($10\\,\\textup{mA}$, $5\\,\\textup{mA}$ and $1\\,\\textup{mA}$ all at $100\\,\\textup{kHz}$, $25\\%$ duty cycle). A full set of measurements were made at $-10\\,^{\\circ}\\textup{C}$, $+20\\,^{\\circ}\\textup{C}$ and $+40\\,^{\\circ}\\textup{C}$ giving a total of 30 individual readings for each device. Example traces are shown in Figure \\ref{fig:PulsedProp}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{SET-255-01_T20_F100k_D50_A20_N1_cropped.pdf}\n\\includegraphics[width=0.49\\textwidth]{SET-255-01_T20_F100k_D50_A20_N5_cropped.pdf}\n\\caption[Pulsed properties.]{\\label{fig:PulsedProp} SET-255-01 pulsed at $100\\,\\textup{kHz}$, $50\\%$ duty cycle at $20\\,\\textup{mA}$ drive current amplitude. The measurement was made at $20\\,^{\\circ}\\textup{C}$ with a single pulse shown on the left and consecutive pulses on the right. Going from the top down, the first signal (green) is the voltage across the device, the second (yellow) is the initial pulse generator signal, the third (blue) is the current through the device (converted to a voltage at $10\\,\\textup{mV\/mA}$) and the fourth (red) signal shows the amplified response of the PMT, where the rise\/fall times are limited by the amplifier.}\n\\end{figure}\n\nThe oscilloscope was used to measure the $10\\%$ to $90\\%$ rise time of the voltage across the DUT and it was found to scale linearly with the inverse of the drive current amplitude for all three device types. \nFor the SET-255 devices the average rise time was $1.63\\pm0.07\\,\\mu\\textup{s}$ at $1\\,\\textup{mA}$ falling to $0.110\\pm0.004\\,\\mu\\textup{s}$ at $20\\,\\textup{mA}$ with the SET-240 devices showing similar behaviour with an average of $1.49\\pm0.09\\,\\mu \\textup{s}$ at $1\\,\\textup{mA}$ and $0.109\\pm0.004\\,\\mu \\textup{s}$ at $20\\,\\textup{mA}$. However, the CIS-250 devices were about four times slower over the current range studied with average rise times of $6.46\\pm0.05\\,\\mu \\textup{s}$ at $1\\,\\textup{mA}$ and $0.323\\pm0.003\\,\\mu \\textup{s}$ at $20\\,\\textup{mA}$. \n\nAll three device types were able to be pulsed to greater than $100\\,\\textup{kHz}$ and due to their shorter rise times the SET devices up to $1\\,\\textup{MHz}$ with a $50\\%$ duty cycle. It was also found that temperature had no measurable effect on any of the device modulation properties and at each temperature the average UV output power scaled linearly with pulse duration. As expected the average UV output at a particular drive current amplitude did vary with temperature as discussed Section \\ref{TV}.\n\n\n\\subsection{Thermal Vacuum Performance}\\label{TV}\n\nThermal testing was carried out with a single device at a time mounted in a vacuum chamber at a pressure of $<10^{-5}\\,\\textup{mbar}$. The DUT was fixed in a copper mount with temperature control provided by a custom, two-stage Peltier system. The same type of Hamamatsu large area photodiode used previously was mounted opposite to the DUT to allow \\textit{in situ} measurements of the total UV output. The first part of the test was designed to quantify the relationship between UV output and temperature, at a range of DC drive currents. The temperature was raised from $-10\\,^{\\circ}\\textup{C}$ to $+40\\,^{\\circ}\\textup{C}$ in steps of $10\\,^{\\circ}\\textup{C}$, with this sequence repeated twice. Once the temperature had stabilised at each setting, an IV scan from $0\\,\\textup{mA}$ to $10\\,\\textup{mA}$ in steps of $0.1\\,\\textup{mA}$ was taken. In addition, the DUT was driven for 120 seconds in turn at $0.1\\,\\textup{mA}$, $0.5\\,\\textup{mA}$, $1\\,\\textup{mA}$, $5\\,\\textup{mA}$ and $10\\,\\textup{mA}$. The test was completely automated and lasted approximately 8 hours per device. The data were split by temperature and the average UV output was calculated from 40 seconds of data at each drive current setting. Data where the output was still settling were ignored. All nine devices were tested in this way with a selection of results shown in Figure \\ref{fig:OutputVsTemp}.\n\n\\begin{figure}[h]\n\\begin{minipage}[t]{0.5\\textwidth}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{OutputVsTemp_SET-240-03.pdf}\n\\includegraphics[width=1.0\\textwidth]{OutputVsTemp_SET-255-02.pdf}\n\\end{minipage}\n\\begin{minipage}[t]{0.5\\textwidth}\n\\includegraphics[width=1.0\\textwidth]{OutputVsTemp_CIS-250-01.pdf}\n\\caption[UV output with varying temperature.]{\\label{fig:OutputVsTemp} Variation in UV output between $-10\\,^{\\circ}\\textup{C}$ and $+40\\,^{\\circ}\\textup{C}$, at five different current settings. Note two separate readings were made at each current setting and each measurement has been normalized to the one made at $20\\,^{\\circ}\\textup{C}$. Clockwise form bottom left: SET-255-02, SET-240-03 and CIS-250-01.}\n\\end{minipage}\n\\end{figure}\n\nAs expected the UV output of all devices was found to have a significant temperature dependence which varied depending on the DC drive current. The behaviour was quantitatively similar for devices of a particular type but, as can be seen in Figure \\ref{fig:OutputVsTemp}, differences were observed between types. The SET-255 devices were the most sensitive to temperature with a linear relationship seen which changed with current setting, though this begins to become non-linear at $0.1\\,\\textup{mA}$. Qualitatively similar results were seen with the CIS-250 devices but for the SET-240 devices the linear behaviour only held at $10\\,\\textup{mA}$ and $5\\,\\textup{mA}$. Considering just the $10\\,\\textup{mA}$ data where all devices demonstrated linear behaviour, the SET-255 devices showed a $1.5\\,\\%\/^{\\circ}\\textup{C}$ change compared to $20\\,^{\\circ}\\textup{C}$, the SET-240 devices $1.0\\,\\%\/^{\\circ}\\textup{C}$ and the CIS-250 $0.5\\,\\%\/^{\\circ}\\textup{C}$.\n\nThe second part of the thermal vacuum test was non-operational survival cycling carried out in the same system but with the DUT turned off. The temperature was cycled ten times between $-40\\,^{\\circ}\\textup{C}$ and $+60\\,^{\\circ}\\textup{C}$ with a $1\\,\\textup{hour}$ dwell at each extreme. The pressure was $<10^{-5}\\,\\textup{mbar}$ throughout and all nine devices survived without any sign of degradation.\n\n\n\\subsection{Repeated Reference Characteristics}\n\nFollowing the laboratory-based testing, the reference measurements described previously were repeated. Within the measurement uncertainty, no device showed any change in spectral properties or pulsed behaviour. However, some changes were seen in both IV curves and UV output. The results are shown in Figure \\ref{fig:ChangePropLab}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{PostLabChangeIV.pdf}\n\\includegraphics[width=0.49\\textwidth]{PostLabChangeUV.pdf}\n\\caption[Changes in device properties during laboratory tests.]{\\label{fig:ChangePropLab} Left: The change in voltage required to drive a particular current. Right: The change in UV output at a particular drive current. Note the SET-240-01 data has been truncated below $2.2\\,\\textup{mA}$ to remove the data including the anomaly observed during the initial measurement. The SET-240-02 data rises steeply below $0.5\\,\\textup{mA}$ to approximately $+150\\%$.}\n\\end{figure}\n\nConsidering the IV properties first: CIS-250 devices showed no measurable changes; SET-255 devices experienced a reduction of $0.5\\%$ to $0.25\\%$ in drive voltage at all currents while the change for the SET-240 was larger and more complicated, though qualitatively similar for the three devices of this type. SET-240-02 stands out at lower currents as it transitions to an increase in required drive voltage.\n\nTurning to the UV output, the CIS-250 devices again showed no significant change, at least at drive currents above $1\\,\\textup{mA}$ while the SET-255 devices experienced a reduction of $5\\%$ to $15\\%$ depending on the drive current. Two of the SET-240 devices showed similar behaviour but again SET-240-02 stands out. Below $2\\,\\textup{mA}$ its UV output appears to have increased significantly by up to $+150\\%$ at drive currents below $0.5\\,\\textup{mA}$. Generally all devices showed more complex behaviour at lower drive currents, which was perhaps to be expected given the lower UV output. Recall that for all devices the total UV output power is around $100\\,\\mu\\textup{W}$ at $10\\,\\textup{mA}$ and around $50\\,\\textup{nW}$ at $0.1\\,\\textup{mA}$. What is reassuring though is the repeatability of all the CIS-250 measurements as they provide confidence that the changes observed with the other devices are real.\n\n\n\\subsection{Radiation}\n\nFollowing the laboratory-based tests the devices underwent radiation testing at the Cobalt-60 facility of the European Space Research and Technology Centre (ESTEC). A Co-60 source produced a diverging gamma-ray beam with photon energies of $1.17\\,\\textup{MeV}$ and $1.33\\,\\textup{MeV}$. The DUT dose rate was varied by adjusting the distance from the source. The total ionising dose and dose rate (water equivalent) was measured by a calibrated dosimeter provided by the facility. The test requirements called for a total dose of $30\\,\\textup{kRad}$ and this was delivered in three separate runs over the course of two days. The dose rates were adjusted to fit the test schedule and facility operating hours. For each of the device types one was irradiated while driven at $1\\,\\textup{mA}$ DC (01 devices), one was irradiated while off (02 devices) and one acted as a reference that was driven at $1\\,\\textup{mA}$ DC but was not irradiated (03 devices).\n\nBoth the irradiated and reference devices were mounted within perspex holders with the flying leads of driven devices connected to a custom made printed circuit board (PCB), Figure \\ref{fig:RadPhotos}. The devices that were driven were at a fixed $1\\,\\textup{mA}$ drive current, while the corresponding drive voltage was measured at $1\/32\\,\\textup{Hz}$ sampling frequency by a data acquisition system provided by the facility. No active temperature control was used for the devices though the room temperature was controlled and measured to be between $23\\,^{\\circ}\\textup{C}$ and $24\\,^{\\circ}\\textup{C}$ throughout. There was no real-time monitoring of the UV output but IV scans as described previously were carried out in between irradiation runs.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.22\\textheight]{IMG_1405_fixed.pdf}\n\\includegraphics[height=0.22\\textheight]{IMG_1419_fixed.pdf}\n\\caption[Radiation testing.]{\\label{fig:RadPhotos} Left: The six irradiated devices. Note that the leads of the unpowered devices are not connected to the board but have their lead tips covered with Kapton tape to avoid accidental electrical shorting. Right: The devices in place (to the right held in clamp stand) prior to irradiation.}\n\\end{figure}\n\nThe first run lasted 16 hours 45 minutes and delivered an absorbed dose of $13.26\\,\\textup{kRad}$, the second run lasted 3 hours 55 minutes and delivered an absorbed dose of $12.62\\,\\textup{kRad}$ and the third run lasted 16 hours 7 minutes and delivered an absorbed dose of $12.80\\,\\textup{kRad}$. The total absorbed dose in terms of water was $38.68\\,\\textup{kRad}$. \n\nSoon after the test began it became clear from the drive voltage monitoring that SET-255-01 was behaving erratically, as can be seen in Figure \\ref{fig:ChangePropRad}. This behaviour continued during the intermediate IV scans with the output of the device output changing by $\\approx10\\%$, although the device never completely failed. A subsequent inspection of data recorded upon arrival at the facility revealed that the drive voltage of the device was unstable prior to irradiation. This was in contrast to data recorded before transportation that showed no unusual behaviour. Thus, it would appear that the device could have been damaged either during transport or possibly during integration in the test rig and the subsequent behaviour was not caused by irradiation. More will be said about this in Section 5.7.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.22\\textheight]{RadiationRefVolt.pdf}\n\\includegraphics[height=0.22\\textheight]{PostRadChangeUV.pdf}\n\\caption[Changes in device properties during radiation tests.]{\\label{fig:ChangePropRad} Left: The monitoring voltages for the six devices that were driven at $1\\,\\textup{mA}$ during the test, three of which were also irradiated. SET-255-01 exhibited unstable behaviour during the first run and dropped by $1.75\\,\\textup{V}$ at the start of the second. Right: The change in UV output with drive current, as measured before and after the radiation test. Below $0.5\\,\\textup{mA}$, CIS-250-01 and CIS-250-02 rise to $12\\%$ and $29\\%$ respectively.}\n\\end{figure}\n\nFollowing the test a full set of reference measurements were made at Imperial College, except for SET-255-01 which was too unstable. Within the uncertainty of the measurements, no changes were observed in either the spectral or modulation properties for any device and the IV scans showed no change in the drive voltage. Five out of the remaining eight devices also showed no significant  change in their UV output, though as can be seen in Figure \\ref{fig:ChangePropRad}, three did. The device SET-255-02 which was irradiated but not driven showed a $\\approx10\\%$ reduction in UV output at all current settings. No such reduction was observed for the device SET-255-03 which was driven at $1\\,\\textup{mA}$ but not irradiated which suggests that the radiation exposure caused the degradation. The two SET-240 devices which were driven at $1\\,\\textup{mA}$ showed a $3\\%$ to $4\\%$ reduction in UV output at all current settings. However, device SET-240-02 which was irradiated but not driven showed no such reduction suggesting that ageing effects from the device being driven caused the degradation and not the radiation exposure. At high drive currents none of the CIS-250 devices showed any significant change though below $1\\,\\textup{mA}$ the output of all three increased.\n\n\n\\subsection{Vibration and Shock}\n\nVibration and shock testing was performed at the Airbus Defence and Space facility in Stevenage, UK. The device displaying erratic behaviour was replaced by a nominally identical one (SET-255-04) and all nine were soldered onto a custom made PCB according to ECSS including staking using Scotch-Weld 2216A\/B (as can be seen in Figure \\ref{fig:VibPhotos}) and strain relief \\cite{ECSS2009}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.22\\textheight]{IMG_1582_fixed.pdf}\n\\includegraphics[height=0.22\\textheight]{IMG_1603_fixed.pdf}\n\\caption[Vibration testing.]{\\label{fig:VibPhotos} Left: The nine devices soldered and staked to the test board, the SET-255 devices on the left, CIS-255 devices in the middle and the SET-240 devices on the right. Right: The test board mounted in the y-axis configuration with control and monitoring accelerometers visible. For the test in the x-axis the board was rotated $90^{\\circ}$ and for the z-axis it was re-mounted such that the board was parallel to the base plate.}\n\\end{figure}\n\nEach axis was tested in turn by performing a quasi-static (In Plane: $\\pm20\\,\\textup{g}$ at $35\\,\\textup{Hz}$ for 2 seconds. Out of Plane: $\\pm30\\,\\textup{g}$ at $30\\,\\textup{Hz}$ for 2 seconds.) followed by a sine, random and shock test at the levels shown in Figure \\ref{fig:VibLevels}, defined in \\cite{Sumner2015}. In between each test a low level sine sweep was performed at $0.2\\,\\textup{g}$, from $5\\,\\textup{Hz}$ to $2\\,\\textup{kHz}$ at $2\\,\\textup{oct\/min}$ to look for any changes.\n\nSurprisingly, all three of the SET-255 devices suffered a complete failure during the first x-axis run each producing an open circuit during an inter-axis electrical turn on test. The three SET-240 and CIS-250 devices showed no change and successfully completed all tests. Upon return to Imperial College, all nine devices were carefully removed from the board and had their flying leads re-soldered. The reference tests were repeated and showed no significant change in either the spectral or modulation properties for any device. The IV scans showed no change in the voltage or UV output for any of the CIS-250 devices but some differences in the SET-240 properties. The drive voltage of all three devices had changed $\\pm1\\,\\%$ over the $0\\,\\textup{mA}$ to $10\\,\\textup{mA}$ current range and the UV output of devices SET-240-02 and SET-240-03 had fallen by $\\approx9\\%$ and $\\approx6\\%$ respectively. It seems unlikely that the vibration test itself caused the changes but their are several other possibilities. For example, some degradation may have occurred due to a lack of heat sinking or temperature control while the devices were driven at $2\\,\\textup{mA}$ and $10\\,\\textup{mA}$ during the inter-axis electrical checks. Alternatively, the cleaning and re-soldering between reference tests could have led to measurable changes.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.32\\textwidth]{VibSine_Xaxis.pdf}\n\\hfil\n\\includegraphics[width=0.32\\textwidth]{VibRand_Xaxis.pdf}\n\\hfil\n\\includegraphics[width=0.32\\textwidth]{VibSRS_Xaxis.pdf}\n\\caption[Vibration levels.]{\\label{fig:VibLevels} From left to right are the x-axis sine vibration at $11$\\,mm 0--peak 20\\,g, random $17.7$\\,g rms and shock SRS test data. The tests were also carried out at the same levels in y and z.}\n\\end{figure}\n\nA visual inspection of the three failed SET-255 devices was performed with the aid of a microscope. No external damage was visible so an attempt was made to assess the inside of each device through the lens. With careful alignment this was indeed possible and revealed that the thin internal wire joining the chip to the device's legs had broken in both SET-255-02 and SET-255-04. The failure mode for SET-255-03 was less obvious but due to the difficult viewing conditions it remains possible that a more subtle break could have gone undetected. This points to insufficient internal stress relief and may also be contributing to the erratic behaviour of SET-255-01.\n\n\n\n\\section{Conclusions}\n\nThe tests described here have shown that UV LEDs can offer superior performance in almost every way when compared to the mercury lamps employed in the LISA Pathfinder CMS. All three device types would be capable of producing more light than the mercury lamp system at wavelengths less than $254\\,\\textup{nm}$, with the SET-240 also emitting $\\approx1\\%$ at an energy greater than the work function of pure gold. Even in a simple DC drive scenario they would also offer a considerable improvement in dynamic range, as over the drive current range studied ($0.1\\,\\textup{mA}$ to $10\\,\\textup{mA}$) dynamic ranges of order $10^{4}$ were observed. \n\nAlthough the electrical properties of the devices were consistent with data sheet values, all three types had measured peak wavelengths at the upper limit of their quoted values. The spectra of the light emitted by the UV LEDs were shown to be very stable with DC drive current, pulsed duty cycle, irradiation and age. Temperature was shown to have a small effect on spectral FWHM and peak position but given the spacecraft will be very thermally stable by necessity, this is unlikely to be an issue.\n\nIt was also shown that all three device types can be pulsed to at least $100\\,\\textup{kHz}$ with a $50\\%$ duty cycle, allowing them to be synchronised with the injection bias that will be present in the inertial sensor. As outlined, the ability to synchronise with the injection bias offers several additional advantages including mitigating the risk of asymmetric surface properties, increasing dynamic range and improving the efficiency of discharging. Pulsed performance was found to be stable with temperature but the rise times of all devices varied with the inverse of the drive current amplitude. With respect to rise times, both SET devices were similar but the CIS devices were found to be around four times slower over the current range studied.\n\nAll nine devices survived thermal vacuum cycling and were shown to operate over a temperature range of at least $-10\\,^{\\circ}\\textup{C}$ to $+40\\,^{\\circ}\\textup{C}$. As expected, the UV output of all device types was significantly temperature dependant and this relationship was also found to vary with DC drive current. Additionally, all devices were found to be radiation hard, though a possible $\\approx10\\%$ reduction in UV output was observed in SET-255-02. During this test SET-255-01 began to behave erratically but there is evidence to suggest the behaviour was not caused by the irradiation, though it may have exasperated the problem.\n\nBoth the SET-240 and CIS-250 devices survived vibration and shock testing in all three axes. However, unexpectedly all three of the SET-255 devices suffered a complete failure after the first x-axis test. Further investigation revealed that at least two of the three devices failed due to the breakage of a thin wire bond within the TO-39 package. As all three device types appear to have similar internal mechanical structure it is not clear why only the SET-255 devices would fail in this way. It is possible that the devices were just part of a bad batch or maybe the issue relates to the integrated ball lens, which was the largest of the three types studied. On a positive note the failure was not due to the underlying UV chip technology so a change in internal design could mitigate any risk.\n\nThe UV output of both types of SET device gradually degraded over the course of the tests, despite the fact usage was fairly minimal and always within data sheet recommendations. As measured at the end of the radiation tests the UV output of the SET-255 devices had fallen by an average of $\\approx15\\%$ at drive currents above $1\\,\\textup{mA}$. The SET-240 devices were similar with an average fall of $\\approx13\\%$ at drive currents above $1\\,\\textup{mA}$. This is in contrast to the CIS-250 devices where the measured change was $<1\\%$ at drive currents above $1\\,\\textup{mA}$. Interestingly, all nine devices showed significant changes in UV output below approximately $1\\,\\textup{mA}$ where increases and decreases of tens of percent were observed for devices of each type.\n\nEach device accumulated an average total usage of $30\\,\\textup{mA hours}$ during testing, with the devices that were driven during the radiation test (devices 01 and 03) gaining an additional $37\\,\\textup{mA hours}$. This can be considered a low-level of usage as according to the manufacturers, the CIS-250 devices receive a burn-in of $4800\\,\\textup{mA hours}$ ($48\\,\\textup{hours}$ at $100\\,\\textup{mA}$) prior to delivery. This burn-in may go some way to explaining the apparent output stability of the CIS-250 devices, at least at higher DC currents. Nevertheless the results do add support to the manufacturers claim that the technology used in their devices offer superior lifetimes.\n\nWith a few caveats, it has been demonstrated that the two underlying technologies behind the three devices studied are suitable for use in space. When compared directly there are several pros and cons for all three device types tested, with ultimately there being a trade off between shorter wavelengths, faster rise times and reliability. The study has also highlighted that thermal management and monitoring will be an important design consideration in the final charge management system and that a cautious approach should be employed with regards to vibration. Initial results from ongoing device lifetime testing have also been positive and will be reported separately \\cite{Hollington2015}.\n\n\n\n\\section*{Acknowledgements}\n\nThe authors would like to thank the whole LTPDA software development team \\cite{LTPDA2015} as well as Michele Muschitiello at ESTEC and Jaime Fensome at Airbus Stevenage for their assistance during the radiation and vibration tests respectively. Our thanks also go to Shahid Hanif at Imperial College for designing and building the pulsed drive electronics. We also acknowledge the financial support of the European Space Agency under contract C4000103768 for this work.\n\n\n\n\\section*{References}\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Action to Lowest Order\\label{SecLowest}}\nIn this section, we calculate and explicitly solve the equations of motion to lowest order in the two lengthscale expansion. First, however, we write out the complete, rescaled action showing explicitly the dependence on $\\epsilon$.  Inserting the decomposition \\eqref{Ricci2expansion} of the Ricci scalar and the rescaled metric \\eqref{scaledmetric} into the action \\eqref{CompleteAction} [following the prescription of Eq. \\eqref{epsilonaction}], we obtain\n\\begin{align}\nS_\\epsilon \\left[ \\tensor[^n]{\\hat{\\gamma}}{_{ab}}, \\n\\Phi, \\n\\chi, \\n\\phi \\right] ={} &\n\\sum_{n = 0}^N \\int_{\\rn} d^5 \\lrn x \\Bigg[ \\sqrt{-\\n{\\hat{\\gamma}}} \\frac{e^{2 \\, \\n\\chi }}{2 \\kapfs \\, \\n\\Phi } \\bigg( - \\frac{1}{4}\\n[^{ab}]{\\hat{\\gamma}} \\n[_{bc,y}]{\\hat{\\gamma}} \\n[^{cd}]{\\hat{\\gamma}} \\n[_{da,y}]{\\hat{\\gamma}} - 5 (\\n[_{,y}]\\chi)^2 - 4 \\n[_{,yy}]\\chi \\nonumber\n\\\\\n{}& + 4 \\frac{\\n[_{,y}]\\Phi}{\\n\\Phi} \\n[_{,y}]\\chi - 2 \\kapfs \\, \\n[^2]\\Phi \\lrn \\Lambda \\bigg) + \\n\\lambda (x^a, y) \\left(\\sqrt{-\\n{\\hat{\\gamma}}} - 1\\right) \\Bigg] \\nonumber\n\\\\\n{}& {} + \\sum_{n = 0}^{N-1} \\int_{\\bn} d^4 \\lrn w e^{2 \\, \\n\\chi(n)} \\sqrt{-\\n{\\hat{\\gamma}}} \\left[ \\frac{2}{\\kapfs} \\left(\\left.\\frac{\\n[_{,y}]\\chi}{\\n\\Phi}\\right|_{\\lrn y = n} - \\left.\\frac{\\np[_{,y}]\\chi}{\\np\\Phi}\\right|_{\\tensor*{y}{_{n+1}} = n} \\right) - \\lrn \\sigma \\right] \\nonumber\n\\\\\n{}& + \\sum_{n = 0}^N \\epsilon^2 \\int_{\\rn} d^5 \\lrn x \\sqrt{-\\n{\\hat{\\gamma}}} \\frac{e^{\\n \\chi}}{2 \\kapfs} \\bigg(\\n \\Phi \\n[^{(4)}]R - 3 \\n \\Phi \\nabla^a \\nabla_a \\n\\chi  - \\frac{3}{2} \\n \\Phi (\\nabla^a \\n\\chi) (\\nabla_a \\n\\chi) \\nonumber\n\\\\\n{}& - 2 \\nabla^a \\nabla_a \\n\\Phi - 2 (\\nabla^a \\n\\chi) (\\nabla_a \\n\\Phi) \\bigg) + \\sum_{n = 0}^{N-1} \\epsilon^2 \\, \\n[_m]S [\\n[_{ab}]h, \\n \\phi] \\label{scaledepsilonaction}\n\\end{align}\nwhere we include the Lagrange multiplier terms \\eqref{Lagrangemults} discussed in Appendix \\ref{AppExactEOMS}, and where the factor of $\\epsilon^2$ in front of the matter action comes from the process described in the previous section. This form explains the choice of the $\\epsilon^4$ factor in Eq. \\eqref{epsilonaction}, and shows the decomposition into $O(1)$ and $O(\\epsilon^2)$ terms, as claimed in Eq. \\eqref{completeepsilonaction}.\n\nFrom the form of Eq. \\eqref{scaledepsilonaction}, we see that we can neglect the last two lines in the limit $\\epsilon \\rightarrow 0$. We can obtain a more precise characterization of the domain of validity of this low energy approximation by estimating the ratio between the terms dropped and the terms retained. As an example, consider the first term on the $4^{\\mathrm{th}}$ line and the first term on the first line. Their ratio is (dropping the `$n$' labels)\n\\begin{align}\n\\left[e^\\chi \\Phi R^{(4)}\\right] \\left[ \\frac{e^{2\\chi}}{\\Phi} {\\hat{\\gamma}}^{ab} {\\hat{\\gamma}}_{bc,y} {\\hat{\\gamma}}^{cd} {\\hat{\\gamma}}_{da,y} \\right]^{-1} \\sim {}& \\left[ e^{\\chi} \\Phi R^{(4)} \\right] \\left[ \\frac{e^{2 \\chi}}{\\Phi \\tilde{y}^2} \\right]^{-1} \\nonumber\n\\\\\n\\sim {}& \\left[ e^{-\\chi} R^{(4)} \\right] \\left[ \\Phi^2 \\tilde{y}^2 \\right]\n\\end{align}\nwhere $\\tilde{y}$ is the coordinate lengthscale over which $\\hat{\\gamma}_{ab}$ varies. We recognize the first factor as essentially the Ricci scalar of the induced metric $e^\\chi \\hat{\\gamma}_{ab}$, which is of order ${\\cal L}_{c}^{-2}$. We recognize the second factor as the square of the physical lengthscale in the $y$ direction over which $\\hat{\\gamma}$ varies, which is always $\\sim {\\cal L}^2$ (see the explicit solution \\eqref{metricansatz} below). Thus, the ratio is $({\\cal L}\/{\\cal L}_c)^2$, confirming the identification of the low energy regime as ${\\cal L} \\ll {\\cal L}_c$.\n\n\n\\subsection{Varying the Action}\nIn the action \\eqref{scaledepsilonaction} at zeroth order in $\\epsilon$, we have three fields to vary (in $N$ regions): $\\n\\Phi (x^c, y)$, $\\n\\chi (x^c, y)$, and $\\n[_{ab}]{\\hat{\\gamma}} (x^c, y)$. There is a subtlety in the variation however. The constraint that $\\det \\left(\\n[_{ab}]{\\hat{\\gamma}}\\right) = -1$ must be imposed either at the level of the equations of motion, or by a Lagrange multiplier. The Lagrange multiplier is explicitly included in Eq. \\eqref{scaledepsilonaction}. Further details are provided in Appendix \\ref{AppExactEOMS}.\n\nWe begin by varying with respect to $\\n\\Phi$. From this variation, we find a single equation of motion in each region,\n\\begin{align}\n\\frac{1}{4} \\n[^{ab}]{\\hat{\\gamma}} \\; \\n[_{bc, y}]{\\hat{\\gamma}} \\n[^{cd}]{\\hat{\\gamma}} \\; \\n[_{da, y}]{\\hat{\\gamma}} - 3 \\; \\n[_{, y}^2]\\chi - 2 \\kapfs \\n[^2]\\Phi \\lrn{\\Lambda} = 0 . \\label{eqmotion2}\n\\end{align}\n\n\nNext, we vary with respect to $\\n[_{ab}]{\\hat{\\gamma}}$. Note that in varying the action, we obtain boundary terms from neighboring regions from the relationship \\eqref{firstIsraelgamma}. The variation produces an equation of motion in each bulk region,\n\\begin{align}\n\\n[_{ad, yy}]{\\hat{\\gamma}} = \\n[_{ab,y}]{\\hat{\\gamma}} \\; \\n[^{bc}]{\\hat{\\gamma}} \\; \\n[_{cd, y}]{\\hat{\\gamma}} - \\n[_{ad, y}]{\\hat{\\gamma}} \\left( 2 \\; \\n[_{,y}]{\\chi} - \\frac{\\n[_{,y}]\\Phi}{\\n\\Phi}\\right). \\label{eqYuck}\n\\end{align}\n(If using Lagrange multipliers, this equation results after the Lagrange multiplier is eliminated by tracing the equation using \\n[^{ab}]{\\hat{\\gamma}} and back substituting). Note that tracing over the indices and using Eq. \\eqref{gammamotion} leads to Eq. \\eqref{gammaswaps} as expected. We also find a boundary condition to be satisfied at each brane,\n\\begin{align}\n\\frac{1}{\\n\\Phi} \\n[_{ab, y}]{\\hat{\\gamma}} (y_{n} = n) ={}& \\frac{1}{\\np\\Phi} \\np[_{ab, y}]{\\hat{\\gamma}} (y_{n+1} = n). \\label{boundarygamma}\n\\end{align}\n\n\nFinally, we vary with respect to $\\n \\chi$. Here, we once again have boundary terms arising from integrating bulk terms by parts in neighboring regions.\nThere is an equation of motion in each bulk region,\n\\begin{equation}\n\\frac{1}{12} \\n[^{ab}]{\\hat{\\gamma}} \\; \\n[_{bc, y}]{\\hat{\\gamma}} \\n[^{cd}]{\\hat{\\gamma}} \\; \\n[_{da, y}]{\\hat{\\gamma}} + \\n[_{, y}^2]\\chi + \\n[_{, yy}]\\chi - \\frac{\\n[_{,y}]\\Phi}{\\n\\Phi} \\n[_{,y}]\\chi + \\frac{2}{3} \\kapfs \\n[^2]\\Phi \\lrn{\\Lambda} = 0. \\label{eqmotion1}\n\\end{equation}\nWe also find a boundary condition at each brane,\n\\begin{equation}\n\\left.\\frac{\\n[_{,y}]\\chi}{\\n\\Phi}\\right|_{\\lrn y = n} - \\left.\\frac{\\np[_{,y}]\\chi}{\\np\\Phi}\\right|_{\\lrnp y = n} = \\frac{2}{3} \\kapfs \\lrn{\\sigma}. \\label{jumpconditions}\n\\end{equation}\n\n\\subsection{Solving the Equations of Motion\\label{eqmotions}}\nWe have three equations of motion for each bulk region, as well as numerous boundary conditions for the fields at the branes [Eqs. \\eqref{metricjunction}, \\eqref{firstIsraelgamma}, \\eqref{eqmotion2}, \\eqref{eqYuck}, \\eqref{boundarygamma}, \\eqref{eqmotion1}, and \\eqref{jumpconditions}]. Note that these equations all describe the dynamics along a fibre of constant $x^a$ which doesn't couple to any other fibres, and so solving these equations of motion consists of solving the dynamics of the extra dimension of the model.\n\nWe begin by solving Eq. \\eqref{eqYuck}. It is convenient to solve this equation in matrix notation. Let\n\\begin{align}\n[{\\hat{\\gamma}}_{ab}] = \\boldsymbol{\\hat{\\gamma}}\n\\end{align}\nwhere we suppress indices $n$. Then in matrix notation, Eq. \\eqref{eqYuck} is\n\\begin{align}\n\\boldsymbol{\\ddot{\\hat{\\gamma}}} = \\boldsymbol{\\dot{\\hat{\\gamma}}} \\ \\boldsymbol{\\hat{\\gamma}}^{-1} \\boldsymbol{\\dot{\\hat{\\gamma}}} - \\boldsymbol{\\dot{\\hat{\\gamma}}} \\left( 2 \\chi_{,y} - \\frac{\\Phi_{,y}}{\\Phi}\\right),\n\\end{align}\nwhere dots denote derivatives with respect to $y$. It is straightforward to check that a solution to this differential equation in region $n$ is\n\\begin{align}\n\\boldsymbol{\\hat{\\gamma}} (x^a, y) ={}& \\mathbf{A}(x^a) \\exp\\left( \\mathbf{B}(x^a) \\int^y_{n-1} \\Phi(x^a, y^\\prime) e^{-2\\chi(x^a, y^\\prime)} dy^\\prime \\right). \\label{gammasoln}\n\\end{align}\nwhere $\\mathbf{A}$ and $\\mathbf{B}$ are arbitrary $4 \\times 4$ real matrix functions of $x^a$. The lower limit on the integral is chosen so that the boundary conditions may be matched at the previous brane (obviously, care must be taken in the first region). The expression \\eqref{gammasoln} has the correct number of integration constants to satisfy arbitrary boundary conditions. From our knowledge of $\\hat{\\gamma}_{ab}$, $\\mathbf{A}$ must be a symmetric matrix with determinant $-1$. The exponential has unit determinant, and so $\\mathbf{B}$ must be traceless. The condition that $\\boldsymbol{\\hat{\\gamma}}$ is symmetric implies that $\\mathbf{B}^T = \\mathbf{A} \\ \\mathbf{B} \\ \\mathbf{A}^{-1}$. The quantity which appears in Eqs. \\eqref{eqmotion2} and \\eqref{eqmotion1} is\n\\begin{align}\n\\n[^{ab}]{\\hat{\\gamma}} \\; \\n[_{bc, y}]{\\hat{\\gamma}} \\n[^{cd}]{\\hat{\\gamma}} \\; \\n[_{da, y}]{\\hat{\\gamma}} ={}& \\n[^{ab}]{\\gamma} \\; \\n[_{ab, yy}]{\\gamma} \\nonumber\n\\\\\n={}& \\mathrm{Tr} \\left( \\mathbf{B}^2 (x^a) \\right) \\Phi^2 e^{-4 \\chi}.\n\\end{align}\nWe define\n\\begin{align}\nb(x^a) = \\frac{1}{12} \\mathrm{Tr} \\left( \\mathbf{B}^2 (x^a) \\right)\n\\end{align}\nwhere the factor of 12 has been chosen for later convenience.\nFrom combining Eq. \\eqref{boundarygamma} with Eqs. \\eqref{metricjunction} and \\eqref{firstIsraelgamma}, we see that $\\mathbf{B}$ (and thus $b(x^a)$) is independent of region, while $\\mathbf{A}$ will change with each region according to Eq. \\eqref{firstIsraelgamma}.\n\nFrom Eq. \\eqref{eqmotion2}, we find\n\\begin{align}\n\\n[_{,y}]\\chi ={}& \\pm \\sqrt{b \\n[^2]\\Phi \\exp(-4 \\n\\chi) - \\frac{2}{3} \\kapfs \\n[^2]\\Phi \\lrn \\Lambda} \\nonumber\n\\\\\n={}& P_n \\n\\Phi \\sqrt{b \\exp(-4 \\n\\chi)- \\frac{2}{3} \\kapfs \\lrn \\Lambda} \\label{eqchi}\n\\end{align}\nwhere $P_n$ is either $\\pm 1$ and is constant in each bulk region. Differentiating Eq. \\eqref{eqchi} gives\n\\begin{align}\n\\n[_{,yy}]\\chi ={}& \\frac{\\n[_y]\\Phi}{\\Phi} \\chi_{,y} - 2 b \\, \\n[^2]\\Phi e^{-4 \\n\\chi}.\n\\end{align}\nThe same result is obtained by substituting Eq. \\eqref{eqmotion2} into Eq. \\eqref{eqmotion1}, and so we see that these equations of motion are degenerate. This leaves only one equation of motion (Eq. \\eqref{eqchi}) and one boundary condition (Eq. \\eqref{jumpconditions}) to satisfy.\n\n\\subsection{Classes of Solutions\\label{SecClasses}}\nIf $\\mathbf{B}(x^a) \\equiv \\mathbf{0}$, then the induced metric on all the branes are related to one another by conformal transformations, and a four-dimensional effective action is easily calculated. On the other hand, when $\\mathbf{B}(x^a) \\neq \\mathbf{0}$, the induced metrics on each brane are not simply related conformally, but through the equations \\eqref{firstIsraelgamma} and \\eqref{gammasoln}. If solutions with $\\mathbf{B}(x^a) \\neq \\mathbf{0}$ were to exist, the four-dimensional effective theory would contain more than one massless tensor degree of freedom; i.e., it would constitute a multigravity theory (see Damour and Kogan \\cite{Damour2002}). No such degrees of freedom have been seen in any linearized analyses\\footnote{In addition, it can be shown that in orbifolded models, there are no solutions with $\\mathbf{B}(x^a) \\neq \\mathbf{0}$; see Section \\ref{secRSI}.}. It is important to note that this is not a Kaluza-Klein mode. We believe that solutions with $\\mathbf{B}(x^a) \\neq \\mathbf{0}$ are ruled out due to divergences at $y \\rightarrow \\pm \\infty$, leading to a lack of global hyperbolicity in the spacetime, although we have been unable to prove this rigorously. We will restrict attention to the case $\\mathbf{B}(x^a) = \\mathbf{0}$ for the remainder of this paper.\n\n\\subsection{General Solutions at Leading Order\\label{BraneTensionDerivation}}\nWith $\\mathbf{B}(x^a) \\equiv \\mathbf{0}$, the field $\\n[_{ab}]{\\hat{\\gamma}}$ becomes independent of $y$ [see Eq. \\eqref{gammasoln}], and also independent of $n$ by Eq. \\eqref{firstIsraelgamma}. This means that we can drop the index $n$ from $x^a_n$, $w_n$, and $\\n[_{ab}]{\\hat{\\gamma}}$ without causing confusion. With $b(x^a) = 0$, the remaining equation of motion and boundary condition simplify somewhat. Equation \\eqref{eqchi} becomes\n\\begin{align}\n\\n[_{,y}]\\chi ={}& P_n \\n\\Phi \\sqrt{- \\frac{2}{3} \\kapfs \\lrn \\Lambda}, \\label{neweqmotion}\n\\end{align}\nwhich implies that $\\lrn \\Lambda < 0$, and so the bulk regions must be slices of anti de-Sitter space. Define\n\\begin{align}\nk_n = \\sqrt{\\frac{-\\kapfs \\lrn \\Lambda}{6}}. \\label{kndef}\n\\end{align}\nWe can use Eq. \\eqref{neweqmotion} for $\\n\\chi$ in Eq. \\eqref{jumpconditions} to obtain\n\\begin{equation}\n\\lrn{k} \\lrn{P} - \\lrnp{k} \\lrnp{P} = \\frac{1}{3} \\kapfs \\lrn \\sigma. \\label{branetunings}\n\\end{equation}\nThese relations are the well-known ``brane tunings'', which determine the branes tensions required in order to avoid a cosmological constant on the branes \\cite{Randall1999}.\n\nWe may integrate Eq. \\eqref{neweqmotion} to find\n\\begin{align}\n\\n\\chi(x^a, y) =\n\\begin{cases}\n2 k_0 P_0 \\int_0^y \\n\\Phi(x^a, y^\\prime) dy^\\prime + f(x^a) & n = 0,\n\\\\\n\\nm\\chi(x^a, n-1) + 2 \\lrn{k} \\lrn{P} \\int_{n-1}^y \\n\\Phi(x^a, y^\\prime) dy^\\prime & n > 0\n\\end{cases} \\label{chidef}\n\\end{align}\nwhere $f(x^a)$ is an arbitrary function. Note that the field $\\n\\chi$ is related to the distance from the previous brane to $y$ along a geodesic normal to the branes, made dimensionless by the appropriate lengthscale in the bulk. In particular, $\\chi$ describes the number of $e$-foldings the warp factor in the metric provides between two points in the five-dimensional spacetime. Assuming that $\\Phi$ is not divergent, if $\\exp(\\n\\chi(y))$ approaches zero or $\\infty$ anywhere, it can only occur as $y \\rightarrow \\pm \\infty$. We will restrict attention to the cases\n\\begin{align}\nP_0 = +1, \\ \\ \\mathrm{and} \\ \\ P_N = -1. \\label{boundaryp}\n\\end{align}\nWhen these signs fail to hold, then the warp factor increases monotonically as one goes to infinity, and it seems likely that the spacetime cannot be globally hyperbolic. We exclude cases where $\\exp(\\n\\chi(y)) \\rightarrow 0$ at finite $y$ by restricting ourselves to topologically connected spacetimes \\cite{Flanagan2001a, Karch2001}.\n\n\\section{The Action to Second Order\\label{SecSecond}}\nIn this section, we investigate the action to second order in $\\epsilon$. By integrating out the previously determined high energy dynamics, we find the four-dimensional effective action.\n\n\\subsection{Acquiring the Four-Dimensional Effective Action\\label{SecSecond1}}\nUsing the metric \\eqref{metricansatz} in Eqs. \\eqref{CompleteAction} and \\eqref{scaledaction}, we can calculate the second order contribution to the action $S_2$. The result is\n\\begin{align}\n S_2 \\left[ \\tensor[^n]{\\hat{\\gamma}}{_{ab}}, \\n\\chi, \\n\\phi \\right] = \\sum_{n = 0}^N \\int_{\\rn} d^5 & \\lrn x \\sqrt{-\\hat{\\gamma}} \\frac{e^{\\n\\chi}}{4 \\kapfs \\lrn k \\lrn P} \\Big[ \\n[_{,y}]\\chi R^{(4)} - 3 \\; \\n[_{,y}]\\chi \\nabla^2 \\n\\chi - \\frac{3}{2} \\n[_{,y}]\\chi (\\nabla^a \\n\\chi) (\\nabla_a \\n\\chi) \\nonumber\n\\\\\n & \\qquad {}- 2 \\nabla^2 \\n[_{,y}]\\chi - 2 (\\nabla^a \\n\\chi) (\\nabla_a \\n[_{,y}]\\chi) \\Big] + \\sum_{n = 0}^{N-1} \\n[_m]S \\left[ e^{\\n\\chi (x^a, n)} \\hat{\\gamma}_{ab}, \\n\\phi \\right]. \\label{s2action}\n\\end{align}\nNote that covariant derivatives as written here are associated with the metric $\\hat{\\gamma}_{ab}$, as is the four-dimensional Ricci scalar $R^{(4)}$.\n\nTo obtain the effective four-dimensional action, we integrate over $y$ in the five-dimensional action \\eqref{s2action}, as the dynamics of this dimension have already been solved. The term involving the Ricci scalar can be integrated straightforwardly, as $R^{(4)}$ has no $y$ dependence, but the other terms require more manipulation. We can combine the last four terms in the five-dimensional integral in the following way:\n\\begin{align}\n -3 e^{\\n\\chi} \\; \\n[_{,y}]\\chi \\nabla^2 \\n\\chi & - \\frac{3}{2} e^{\\n\\chi} \\; \\n[_{,y}]\\chi (\\nabla^a \\n\\chi) (\\nabla_a \\n\\chi) - 2 e^{\\n\\chi} \\nabla^2 \\n[_{,y}]\\chi - 2 e^{\\n\\chi} (\\nabla^a \\n\\chi) (\\nabla_a \\n[_{,y}]\\chi) \\nonumber\n\\\\\n ={}& \\frac{3}{2} \\frac{\\partial}{\\partial y} \\left( e^{\\n\\chi} (\\nabla^a \\n\\chi) (\\nabla_a \\n\\chi) \\right) - \\nabla^a \\left( 3 e^{\\n\\chi} \\; \\n[_{,y}]\\chi \\nabla_a \\n\\chi + 2 e^{\\n\\chi} \\nabla_a \\n[_{,y}]\\chi \\right)\n\\end{align}\nThe covariant derivative commutes with the integration over the fifth dimension in the action, and thus gives rise to a boundary term at $x^a \\rightarrow \\infty$, which we discard. We obtain\n\\begin{align}\n S_2 \\left[ \\tensor[^n]{\\hat{\\gamma}}{_{ab}}, \\n\\chi, \\n\\phi \\right] ={}& \\sum_{n = 0}^N \\int_{\\rn} d^5 \\lrn x \\sqrt{-\\hat{\\gamma}} \\frac{1}{4 \\kapfs \\lrn{k} \\lrn{P}} \\frac{\\partial}{\\partial y} \\left\\{ e^{\\n\\chi} R^{(4)} + \\frac{3}{2} e^{\\n\\chi} (\\nabla^a \\n\\chi) (\\nabla_a \\n\\chi) \\right\\} \\nonumber\n\\\\\n &{} + \\sum_{n = 0}^{N-1} \\n[_m]S \\left[ e^{\\n\\chi (x^a, n)} \\tensor{\\hat{\\gamma}}{_{ab}}, \\n\\phi \\right].\n\\end{align}\nIntegrating over $y$, we find boundary terms at each brane and at $y = \\pm \\infty$.\nWe note that the integral converges in the first and last regions because of the choices $P_0 = +1$ and $P_N = -1$, and so the terms evaluated at $\\pm \\infty$ vanish. We can rearrange the remaining terms into a sum over the branes.\n\\begin{align}\n S_2 \\left[ \\tensor[^n]{\\hat{\\gamma}}{_{ab}}, \\n\\chi, \\n\\phi \\right] ={}& \\sum_{n = 0}^{N-1} \\int d^4 x \\sqrt{-\\hat{\\gamma}} \\frac{1}{4 \\kapfs} \\left( \\frac{1}{\\lrn{k} \\lrn{P}} - \\frac{1}{\\lrnp k \\lrnp P} \\right) \\left[ e^{\\n\\chi} R^{(4)} + \\frac{3}{2} e^{\\n\\chi} (\\nabla^a \\n\\chi) (\\nabla_a \\n\\chi) \\right]_{y=n} \\nonumber\n\\\\\n & {} + \\sum_{n = 0}^{N-1} \\n[_m]S \\left[ e^{\\n\\chi (x^a, n)} \\tensor{\\hat{\\gamma}}{_{ab}}, \\n\\phi \\right]. \\label{ActionBeforeRedefs}\n\\end{align}\n\n\\subsection{Field Redefinitions\\label{FieldRedefs}}\nWe now have a four-dimensional Ricci scalar and a number of scalar fields, $\\n\\chi(x^a, n)$, whose values depend on the distance from the first brane to the $(n+1)^{\\mathrm{th}}$ brane. We modify our previous notational conventions as follows. We redefine the fields $\\n\\chi$ and $\\hat{\\gamma}_{ab}$ via\n\\begin{align}\n\\hat{\\gamma}_{ab} \\rightarrow {}& e^{f(x^a)} \\hat{\\gamma}_{ab}(x^a) \\\\\n\\n\\chi(x^a, y) \\rightarrow {}& \\n\\chi(x^a,y) - f(x^a)\n\\end{align}\nwhere $f(x^a)$ is defined in Eq. \\eqref{chidef}. Thus, we no longer impose the constraint that $\\det (\\hat{\\gamma}) = -1$; instead we have $\\det (\\hat{\\gamma}) = - \\exp (4 f)$. With the relaxation of this constraint, a Lagrange multiplier term in the action to enforce the constraint is no longer necessary. As $\\chi$ will only be evaluated on the branes from now on, we further depart from our previous conventions, and write $\\chi_n(x^a)$ to represent $\\n\\chi(x^a, n)$. Our new conventions also enforce $\\chi_0 = 0$, which means that the metric $\\hat{\\gamma}_{ab}$ is the Jordan frame metric of the first brane, ${\\cal B}_0$. Note that the five-dimensional metric is unchanged by these notational changes.\n\nFor convenience, we define the quantities\n\\begin{align}\nA_n ={}& \\left| \\frac{1}{k_n P_n} - \\frac{1}{k_{n+1} P_{n+1}} \\right|, \\label{DefofA}\n\\\\\nB_n ={}& \\frac{A_n}{A_0}, \\label{DefofB}\n\\\\\n\\epsilon_n ={}& \\mathrm{sgn} \\left( \\frac{1}{k_n P_n} - \\frac{1}{k_{n+1} P_{n+1}} \\right), \\label{epsilondef}\n\\end{align}\nfor $0 \\le n \\le N-1$. It is useful to note that $\\epsilon_n$ can be written as\n\\begin{align}\n\\epsilon_n\n={}& - \\mathrm{sgn} \\left( \\sigma_n P_n P_{n+1} \\right) \\label{epsilonuseful}\n\\end{align}\nby using Eq. \\eqref{branetunings}. We define an effective four-dimensional gravitational constant $\\kappa_4^2$ that is measured by observers on the first brane, ${\\cal B}_0$, by\n\\begin{equation}\n\\frac{1}{2 \\kappa_4^2} = \\frac{1}{4 \\kappa_5^2} A_0. \\label{4dNewtonConst}\n\\end{equation}\nFinally, we define fields $\\psi_n$ by\n\\begin{align}\n\\psi_n ={}& \\sqrt{B_n e^{\\chi_n}} \\label{psidef}\n\\end{align}\nfor $1 \\le n \\le N-1$. The domain of $\\psi_n$ is the positive reals. Rewriting our action in terms of these new variables (and suppressing the subscript `2' from now on), we obtain\n\\begin{align}\n S \\left[\\hat{\\gamma}_{ab}, \\lrn \\psi, \\n\\phi\\right] ={}& \\int d^4 x \\sqrt{-\\hat{\\gamma}} \\frac{\\epsilon_0}{2 \\kappa_4^2} \\left[ R^{(4)} \\left( 1 + \\sum_{n = 1}^{N-1} \\epsilon_0 \\epsilon_n \\psi_n^2 \\right) + 6 \\sum_{n = 1}^{N-1} \\epsilon_0 \\epsilon_n (\\nabla^a \\psi_n) (\\nabla_a \\psi_n) \\right] \\nonumber\n\\\\\n & {} + \\tensor[^0]S{_m}[\\tensor{\\hat{\\gamma}}{_{ab}}, \\tensor[^0]{\\phi}{}] + \\sum_{n = 1}^{N-1} \\n[_m]S \\left[ \\frac{\\psi_n^2}{B_n} \\hat{\\gamma}_{ab}, \\n\\phi \\right]. \\label{firstjordanaction}\n\\end{align}\nThis is the four-dimensional effective action in the Jordan conformal frame of the first brane, ${\\cal B}_0$. Note that the target space metric, determined by the kinetic energy term for the scalar fields, is flat, and the target space manifold is a subset of the quadrant of $\\mathbb{R}^{N-1}$ in which all the coordinates $\\lrn \\psi$ are positive, bearing in mind that each $\\lrn \\psi$ will be bounded either above or below by their definition \\eqref{psidef} and Eq. \\eqref{chidef}.\n\n\\subsection{Transforming to the Einstein Conformal Frame\\label{Einsteintransform}}\nThe Einstein conformal frame metric for the action \\eqref{firstjordanaction} is $g_{ab} = \\hat{\\gamma}_{ab} |\\Theta|$, where $\\Theta$ is given by\n\\begin{align}\n\\Theta = 1 + \\sum_{n = 1}^{N-1} \\epsilon_0 \\epsilon_n \\psi_n^2. \\label{thetadef}\n\\end{align}\nThe four-dimensional effective action becomes\n\\begin{align}\n S \\left[g_{ab}, \\lrn \\psi, \\n\\phi\\right] ={}& \\int d^4 x \\sqrt{- g} \\, \\frac{\\epsilon_0 \\, \\mathrm{sgn}(\\Theta)}{2 \\kappa_4^2} \\left[ \\tilde{R}^{(4)}[g] - \\frac{3}{2 \\Theta^{2}} (\\tilde{\\nabla}^a \\Theta)(\\tilde{\\nabla}_a \\Theta) + 6 \\sum_{n = 1}^{N-1} \\frac{\\epsilon_0 \\epsilon_n}{\\Theta} (\\tilde{\\nabla}^a \\psi_n) (\\tilde{\\nabla}_a \\psi_n) \\right] \\nonumber\n\\\\\n & {} + \\tensor[^0]S{_m}\\left[\\frac{1}{|\\Theta|}\\tensor{g}{_{ab}}, \\tensor[^0]{\\phi}{}\\right] + \\sum_{n = 1}^{N-1} \\n[_m]S \\left[ \\frac{\\psi_n^2}{B_n | \\Theta |} g_{ab}, \\n\\phi \\right], \\label{effectiveaction}\n\\end{align}\nwhere tildes refer to the metric $g_{ab}$. Note that the kinetic energy terms in this action \\eqref{effectiveaction} have apparent divergences at $\\Theta = 0$. However, for any given set of signs $\\epsilon_n$ (which correspond to a choice of model), it can be shown that $|\\Theta|$ is bounded away from zero. This arises because of the way each $\\lrn\\psi$ is bounded either above or below.\n\n\n\\section{Analysis of the Action\\label{secanalysis}}\nIn this section, we analyze the four-dimensional effective action \\eqref{effectiveaction} in a variety of cases. We begin with the cases of one and two branes, which serve to highlight some features of the model in the general case. In these special cases, our results reduce to previously known results. We then analyze the general situation.\n\n\\subsection{One Brane Case}\nIn the one brane case, the effective action simplifies greatly.\n\\begin{align}\n S [g_{ab}, \\tensor[^0]{\\phi}{}] ={}& \\int d^4 x \\sqrt{- g} \\frac{\\epsilon_0}{2 \\kappa_4^2} \\tilde{R}^{(4)}[g] + \\tensor[^0]{S}{_m} \\left[ g_{ab}, \\tensor[^0]{\\phi}{} \\right].\n\\end{align}\nThe four-dimensional effective action is just general relativity ($\\epsilon_0 = +1$ if the brane has positive tension). This corresponds to the RS-II model \\cite{Randall1999a}.\n\n\\subsection{Two Brane Case}\nHere the parameter of importance is $\\epsilon_0 \\epsilon_1$, which from Eqs. \\eqref{branetunings}, \\eqref{boundaryp} and \\eqref{epsilondef} is given by\n\\begin{align}\n\\epsilon_0 \\epsilon_1\n={}& - \\mathrm{sgn} \\left( \\sigma_0 \\sigma_1 \\right).\n\\end{align}\nWith $P_0$ and $P_2$ predetermined, it is possible for one brane tension to be negative, but not both. Therefore $\\epsilon_0 \\epsilon_1$ is positive if there is a negative tension brane, and is negative if both branes have positive tension. Using the definition \\eqref{thetadef} of $\\Theta$, the action \\eqref{effectiveaction} becomes\n\\begin{align}\n S [g_{ab}, \\psi_1, \\tensor[^0]{\\phi}{}, \\tensor[^1]{\\phi}{}] ={}& \\int d^4 x \\sqrt{- g} \\; \\frac{\\epsilon_0 \\, \\mathrm{sgn} (1+\\epsilon_0 \\epsilon_1 \\psi_1^2)}{2 \\kappa_4^2} \\left[ \\tilde{R}^{(4)}[g] + 6 \\frac{\\epsilon_0 \\epsilon_1}{(1 + \\epsilon_0 \\epsilon_1 \\psi_1^2)^2} (\\tilde{\\nabla}^a \\psi_1) (\\tilde{\\nabla}_a \\psi_1) \\right] \\nonumber\n\\\\\n & {} + \\tensor[^0]{S}{_m} \\left[ \\frac{1}{|1 + \\epsilon_0 \\epsilon_1 \\psi_1^2|} g_{ab}, \\tensor[^0]{\\phi}{} \\right] + \\tensor[^1]{S}{_m} \\left[ \\frac{\\psi_1^2}{B_1 |1 + \\epsilon_0 \\epsilon_1 \\psi_1^2|} g_{ab}, \\tensor[^1]{\\phi}{} \\right]. \\label{twobraneaction}\n\\end{align}\n\n\\subsubsection{Positive Brane Tensions}\nWhen both branes have positive tension, $\\epsilon_0 \\epsilon_1 = -1$. Which of $\\epsilon_0$ and $\\epsilon_1$ is negative depends on the sign of $\\Theta$. Combining Eqs. \\eqref{thetadef} and \\eqref{psidef},\n\\begin{align}\n\\Theta = 1 - B_1 e^{\\chi_1}. \\label{psi1ex}\n\\end{align}\nFrom Eqs. \\eqref{boundaryp} and \\eqref{epsilonuseful}, we see that\n\\begin{align}\n\\epsilon_0 = - \\epsilon_1 = - \\mathrm{sgn} (P_1).\n\\end{align}\nCombining this with Eq. \\eqref{chidef} and recalling that $\\chi_0 = 0$, we see that the exponential function in Eq. \\eqref{psi1ex} is greater than unity for $P_1 = +1$, and less than unity for $P_1 = -1$. If $P_1 = +1$, then the brane tensions [Eq. \\eqref{branetunings}] require that $k_0 > k_1$, and we see that $B_1 > 1$, giving $\\Theta < 0$ for $\\epsilon_0 = -1$, $\\epsilon_1 = +1$. If $P_1 = -1$, then the brane tensions dictate that $k_0 < k_1$. Thus, in this case, $B_1 < 1$, and so $\\Theta > 0$ for $\\epsilon_0 = +1$, $\\epsilon_1 = -1$.\n\nAssuming that $0 < \\psi_1 < 1$ ($\\Theta > 0$, $P_1 = -1$, $\\epsilon_0 = +1$), we define\n\\begin{align}\n\\varphi ={}& \\E \\tanh^{-1} (\\psi_1)\n\\end{align}\nwhere\n\\begin{align}\n\\E ={}& \\frac{\\sqrt{6}}{\\kappa_4}.\n\\end{align}\nThe domain of $\\varphi$ is $0$ to $\\infty$.\nThe action \\eqref{twobraneaction} then becomes\n\\begin{align}\n S [g_{ab}, \\psi_1, \\tensor[^0]{\\phi}{}, \\tensor[^1]{\\phi}{}] ={}& \\int d^4 x \\sqrt{- g} \\left[ \\frac{1}{2 \\kappa_4^2} \\tilde{R}^{(4)}[g] - \\frac{1}{2} (\\tilde{\\nabla}^a \\varphi) (\\tilde{\\nabla}_a \\varphi) \\right] \\nonumber\n\\\\\n & {} + \\tensor[^0]{S}{_m} \\left[ \\cosh^2 \\left(\\frac{\\varphi}{\\E} \\right) g_{ab}, \\tensor[^0]{\\phi}{} \\right] + \\tensor[^1]{S}{_m} \\left[ \\frac{1}{B_1} \\sinh^2 \\left(\\frac{\\varphi}{\\E} \\right) g_{ab}, \\tensor[^1]{\\phi}{} \\right]. \\label{PosBranes1}\n\\end{align}\n\nRequiring that the branes do not intersect or overlap gives\n\\begin{align}\n0 < \\psi_1 < \\sqrt{B_1} ={}& \\sqrt{\\frac{1 - k_1\/k_2}{1 + k_1\/k_0}}. \\label{intersectcondition}\n\\end{align}\nNote that $k_1 < k_2$ to satisfy Eq. \\eqref{branetunings}, and that $\\sqrt{B_1} < 1$ (responsible for $\\Theta > 0$). Thus, Eq. \\eqref{intersectcondition} is a more stringent constraint than $0 < \\psi_1 < 1$.\n\nIn the situation where $\\psi_1 > 1$ ($\\Theta < 0$, $P_1 = +1$, $\\epsilon_1 = +1$), we define\n\\begin{align}\n\\varphi ={}& \\E \\tanh^{-1} \\left(\\frac{1}{\\psi_1}\\right).\n\\end{align}\nThe domain of $\\varphi$ is from $0$ to $\\infty$.\nThe action \\eqref{twobraneaction} then becomes\n\\begin{align}\n S [g_{ab}, \\psi_1, \\tensor[^0]{\\phi}{}, \\tensor[^1]{\\phi}{}] ={}& \\int d^4 x \\sqrt{- g} \\left[ \\frac{1}{2 \\kappa_4^2} \\tilde{R}^{(4)}[g] - \\frac{1}{2} (\\tilde{\\nabla}^a \\varphi) (\\tilde{\\nabla}_a \\varphi) \\right] \\nonumber\n\\\\\n & {} + \\tensor[^0]{S}{_m} \\left[ \\sinh^2 \\left(\\frac{\\varphi}{\\E} \\right) g_{ab}, \\tensor[^0]{\\phi}{} \\right] + \\tensor[^1]{S}{_m} \\left[ \\frac{1}{B_1} \\cosh^2 \\left(\\frac{\\varphi}{\\E} \\right) g_{ab}, \\tensor[^1]{\\phi}{} \\right], \\label{PosBranes2}\n\\end{align}\nwhich coincides with the previous action \\eqref{PosBranes1} if we swap the actions $\\tensor[^0]{S}{_m}$ and $\\tensor[^1]{S}{_m}$ and rescale units in each matter action by factors of $B_1^{\\pm 1\/2}$.\n\nThe constraint on the radion field we impose to ensure that the branes do not overlap in this case is\n\\begin{align}\n\\psi_1 > \\sqrt{B_1} ={}& \\sqrt{\\frac{1 + k_1\/k_2}{1 - k_1\/k_0}} > 1,\n\\end{align}\nwhere $k_1 < k_0$ from the brane tunings (Eq. \\eqref{branetunings}).\n\nThe two cases $P_1 = -1$, described by the action \\eqref{PosBranes1}, and $P_1 = +1$, described by the action \\eqref{PosBranes2}, lie on opposite sides of $\\Theta = 0$. Each set of values $\\epsilon_0, \\ldots, \\epsilon_{N-1}$ gives different theories, and each theory comes with its own field space.\n\nThe actions \\eqref{PosBranes1} and \\eqref{PosBranes2} coincide with formulae in the literature for the action for the RS-I model, up to a rescaling of units \\cite{Randall1999, Chiba2000, Goldberger2000} (also, c.f. Eq. \\eqref{RSIendaction}). They describe a Brans-Dicke like scalar-tensor theory of gravity, with matter on each brane having a different coupling strength to the scalar component.\n\n\\subsubsection{One Negative Brane Tension}\nIf $\\epsilon_0 \\epsilon_1 = 1$ then $\\Theta > 0$ always, and by requiring the conditions \\eqref{boundaryp}, both $\\epsilon_0$ and $\\epsilon_1$ must be positive. We define\n\\begin{align}\n\\varphi ={}& \\E \\tan^{-1} (\\psi_1),\n\\end{align}\nwhere the domain of $\\varphi$ is $0$ to $(\\pi\/2) \\E$.\nThe action \\eqref{twobraneaction} becomes\n\\begin{align}\n S [g_{ab}, \\psi_1, \\tensor[^0]{\\phi}{}, \\tensor[^1]{\\phi}{}] ={}& \\int d^4 x \\sqrt{- g} \\left[ \\frac{1}{2 \\kappa_4^2} \\tilde{R}^{(4)}[g] + \\frac{1}{2} (\\tilde{\\nabla}^a \\varphi) (\\tilde{\\nabla}_a \\varphi) \\right] \\nonumber\n\\\\\n & {} + \\tensor[^0]{S}{_m} \\left[ \\cos^2 \\left(\\frac{\\varphi}{\\E} \\right) g_{ab}, \\tensor[^0]{\\phi}{} \\right] + \\tensor[^1]{S}{_m} \\left[ \\frac{1}{B_1} \\sin^2 \\left(\\frac{\\varphi}{\\E} \\right) g_{ab}, \\tensor[^1]{\\phi}{} \\right].\n\\end{align}\nNote that $\\varphi$ is a ghost field, which gives rise to the usual instability associated with a negative tension brane.\n\n\\subsection{General Case of N branes\\label{GeneralCase}}\nIn the general case of $N$ branes with $N>2$, we have $N-1$ scalar fields $\\psi_1, \\ldots, \\psi_{N-1}$. It will be useful to regard the $(N-1)$-dimensional space of field configurations as an $N-1$-dimensional hypersurface in an $N$-dimensional space with coordinates $\\psi_1, \\ldots, \\psi_{N}$, where\n\\begin{align}\n\\psi_N \\equiv \\sqrt{-\\epsilon_0 \\epsilon_N \\Theta} = \\sqrt{ \\left| 1 + \\sum_{n = 1}^{N-1} \\epsilon_0 \\epsilon_n \\psi_n^2 \\right| } \\label{psiNdef}\n\\end{align}\nby Eq. \\eqref{thetadef}. We also define $\\epsilon_N = - \\epsilon_0 \\mathrm{sgn}(\\Theta)$.\nThe action \\eqref{twobraneaction} becomes\n\\begin{align}\n S [g_{ab}, \\psi_n, \\n\\phi] ={}& \\int d^4 x \\sqrt{- g} \\; \\epsilon_0 \\, \\mathrm{sgn} (\\Theta) \\left[ \\frac{1}{2 \\kappa_4^2} \\tilde{R}^{(4)}[g] + \\frac{1}{2} \\E^2 \\sum_{n = 1}^N \\frac{\\epsilon_0 \\epsilon_n}{\\Theta} (\\tilde{\\nabla}^a \\psi_n) (\\tilde{\\nabla}_a \\psi_n) \\right] \\nonumber\n \\\\\n {}& + \\tensor[^0]S{_m}\\left[\\frac{1}{|\\Theta|}\\tensor{g}{_{ab}}, \\tensor[^0]{\\phi}{}\\right] + \\sum_{n = 1}^{N-1} \\n[_m]S \\left[ \\frac{\\psi_n^2}{B_n |\\Theta|} g_{ab}, \\n\\phi \\right],\n\\end{align}\nwhere it is understood that the action is constrained by Eq. \\eqref{psiNdef} (this constraint may be enforced by a Lagrange multiplier if desired).\n\nThe N-dimensional field space metric ($d\\Sigma^2$) is now conformally flat (and indeed, is a form of the metric on $N$-dimensional AdS space with an unusual signature). The physical field space metric is obtained by computing the induced $(N-1)$-dimensional metric on the hypersurface \\eqref{psiNdef} from the $N$-dimensional metric. Of the $N-1$ fields $\\psi_1, \\ldots, \\psi_{N-1}$, let $P$ with $0 \\le P \\le N-1$ be the number of fields with $\\epsilon_0 \\epsilon_n$ positive, and let $M = N-1-P$ be the number of fields with $\\epsilon_0 \\epsilon_n$ negative. We now relabel the fields based on which have positive kinetic coefficient, and which have negative kinetic coefficient. Call the fields $p_i$, $1 \\le i \\le P$, and $m_j$, $1 \\le j \\le M$. Following the convention in Eq. \\eqref{generalEinsteinframe} for the metric on the $N$-dimensional target space of the scalar fields, we can write it as\n\\begin{align}\n\\frac{\\Theta}{\\E^2} d\\Sigma^2 ={}& - \\sum_{n=1}^{N} \\epsilon_0 \\epsilon_n d\\psi_n^2\n\\\\\n={}& - \\sum_{i=1}^P dp_i^2 + \\sum_{j=1}^M dm_j^2 + \\mathrm{sgn}({\\Theta}) d\\psi_N^2.\n\\end{align}\nAs the metric on the positive (negative) coordinates is Euclidean, we can transform them into spherical polar coordinates. Define new coordinates $\\zeta, \\theta_1, \\ldots, \\theta_{P-1}$ and $\\eta, \\lambda_1, \\ldots, \\lambda_{M-1}$, such that\n\\begin{subequations}\n\\label{littlefieldeqns}\n\\begin{align}\n(p_1, \\ldots, p_P) = {}& \\zeta \\left(\\cos(\\theta_1), \\sin(\\theta_1) \\cos(\\theta_2), \\ldots, \\sin(\\theta_1) \\sin(\\theta_2) \\cdots \\sin(\\theta_{P-1})\\right),\n\\\\\n(m_1, \\ldots, m_N) = {}& \\eta \\left(\\cos(\\lambda_1), \\sin(\\lambda_1) \\cos(\\lambda_2), \\ldots, \\sin(\\lambda_1) \\sin(\\lambda_2) \\cdots \\sin(\\lambda_{M-1})\\right).\n\\end{align}\n\\end{subequations}\nIn these coordinates, the field space metric is\n\\begin{equation}\n\\frac{\\Theta}{\\E^2} d\\Sigma^2 = - d\\zeta^2 - \\zeta^2 d\\Omega_p^2 + d\\eta^2 + \\eta^2 d\\Omega_m^2 + \\mathrm{sgn}({\\Theta}) d\\psi_N^2,\n\\end{equation}\nwhere $d\\Omega_p^2 = d\\theta_1^2 + \\sin^2(\\theta_1) d\\theta_2^2 + \\ldots$ is the metric on the unit $(P-1)$-sphere, and similarly for $d\\Omega_m^2$. All of the angular fields ($\\theta_i$ and $\\lambda_j$) have a domain of $0$ to $\\pi\/2$, as each $\\psi_n$ is positive. We will refer to this region as the ``positive quadrant'' of the maximally extended fieldspace (where the domains of the angular fields are from $0$ to $\\pi$ or $2 \\pi$ as per usual). The following relationships hold:\n\\begin{align}\n\\zeta^2 ={}& \\sum_{i=1}^P p_i^2,\n\\\\\n\\eta^2 ={}& \\sum_{j=1}^M m_j^2,\n\\\\\n\\Theta ={}& 1 + \\zeta^2 - \\eta^2. \\label{newthetadef}\n\\end{align}\nThe domain of $\\eta$ and $\\zeta$ is from $0$ to $\\infty$.\n\nWe now compute the physical field space metric on the $(N-1)$-dimensional target space ($d\\sigma^2$) using the constraint \\eqref{psiNdef}. We obtain\n\\begin{align}\nd \\psi_N^2 ={}& \\frac{1}{|1 + \\zeta^2 - \\eta^2|} (\\zeta d\\zeta - \\eta d \\eta)^2,\n\\\\\nd\\sigma^2\n={}& \\frac{\\E^2}{1 + \\zeta^2 - \\eta^2} \\left[ - d\\zeta^2 \\left( \\frac{1 - \\eta^2}{1 + \\zeta^2 - \\eta^2} \\right) - \\zeta^2 d\\Omega_p^2 + d\\eta^2 \\left( \\frac{1 + \\zeta^2}{1 + \\zeta^2 - \\eta^2} \\right) + \\eta^2 d\\Omega_m^2 - \\frac{2 \\eta \\zeta}{1 + \\zeta^2 - \\eta^2} d\\eta d\\zeta \\right]. \\label{FullFieldMetric}\n\\end{align}\n\nTo summarize, our final result for the four dimensional action is a massless multiscalar-tensor theory in a nonlinear sigma model, of the form\n\\begin{align}\n S[g_{ab}, \\Phi^A, \\n\\phi] ={}& \\int d^4 x \\sqrt{-g} \\left\\{ \\frac{1}{2 \\kappa_4^2} R [g_{ab}] - \\frac{1}{2} \\gamma_{AB}(\\Phi^C) \\nabla_a \\Phi^A \\nabla_b \\Phi^B g^{ab} \\right\\} \\nonumber\n\\\\\n &{} + \\sum_{n=0}^{N-1} \\n[_m]S \\left[ e^{2 \\alpha_n (\\Phi^C)} g_{ab}, \\n\\phi \\right],\n\\end{align}\nwhere we assume that $\\epsilon_0 \\mathrm{sgn}(\\Theta) = +1$, which is required for a physical theory, given our conventional choices. The scalar fields $\\Phi^A$ are\n\\begin{align}\n\\left\\{ \\Phi^A \\right\\} = \\left\\{ \\zeta, \\theta_1, \\ldots, \\theta_{P-1}, \\eta, \\lambda_1, \\ldots, \\lambda_{M-1} \\right\\}.\n\\end{align}\nThe field space metric $\\gamma_{AB}(\\Phi^C)$ is given by $d \\sigma^2$ [Eq. \\eqref{FullFieldMetric}], and the brane coupling functions $\\alpha_n (\\Phi^C)$ by\n\\begin{subequations}\n\\begin{align}\ne^{2 \\alpha_0} = {}& \\frac{1}{|1 + \\zeta^2 - \\eta^2|},\n\\\\\ne^{2 \\alpha_n} = {}& \\frac{1}{|1 + \\zeta^2 - \\eta^2|} \\frac{\\psi_n^2}{B_n}, \\ 1 \\le n \\le N-1,\n\\end{align}\n\\end{subequations}\nwhere $B_n$ is given by Eq. \\eqref{DefofB}, and $\\psi_n$ is defined by the relevant expression in Eq. \\eqref{littlefieldeqns}.\n\n\\section{Five-Dimensional Ricci Scalars and Exact Equations of Motion\\label{AppExactEOMS}}\nHere we present the dimensionally reduced Ricci scalar and the exact equations of motion for the action \\eqref{CompleteAction}. We include the order at which terms appear in terms of our scaling parameter, $\\epsilon$.\n\n\\subsection{Dimensional Reduction of the Ricci Scalar}\nThe constraint $\\det \\hat{\\gamma} = -1$ may be enforced either at the level of the equations of motion, or by using a Lagrange multiplier.\n\nIf the constraint $\\det \\hat{\\gamma} = -1$ is being enforced at the level of the equations of motion, then it is simplest to compute the equations of motion using the metric \\eqref{firstmetric}, and then perform a conformal transformation on the quantities in the equations of motion. In this metric, the five-dimensional Ricci scalar is given by\n\\begin{align}\n\\n[^{(5)}]R ={}& \\epsilon^2 \\left(\\n[^{(4)}]R - \\frac{2 \\nabla^a \\nabla_a \\n\\Phi}{\\n\\Phi}\\right) - \\frac{\\n[^{ab}]\\gamma \\n[_{ab,yy}]\\gamma}{\\n[^2]\\Phi} + \\n[^{ab}]\\gamma \\n[_{ab,y}]\\gamma \\frac{\\n[_{,y}]\\Phi}{\\n[^3]\\Phi} \\nonumber\n\\\\\n{}& - \\frac{1}{4 \\n[^2]\\Phi} \\left( \\n[^{ab}]\\gamma \\n[_{ab,y}]\\gamma \\right)^2 + \\frac{3}{4 \\n[^2]\\Phi} \\n[^{ab}]\\gamma \\n[_{ac,y}]\\gamma \\n[^{cd}]\\gamma \\n[_{db,y}]\\gamma , \\label{Ricci1expansion}\n\\end{align}\nwhere covariant derivatives and the four-dimensional Ricci scalar are those associated with $\\n[_{ab}]\\gamma$.\n\nFor the constraint $\\det \\hat{\\gamma} = -1$ to be enforced at the level of the action, a Lagrange multiplier term must be added to the action\n\\begin{align}\n\\Delta S = \\sum_{n=0}^{N} \\int_\\rn d^5 x_n \\n\\lambda (x^a, y) \\left(\\sqrt{-\\n{\\hat{\\gamma}}} - 1\\right), \\label{Lagrangemults}\n\\end{align}\nwhere $\\n\\lambda (x^a, y)$ are the Lagrange multiplier fields. Using the metric \\eqref{scaledmetric}, the five-dimensional Ricci scalar is given by\n\\begin{align}\n\\n[^{(5)}]R ={}& \\epsilon^2 e^{-\\n\\chi} \\left(\\n[^{(4)}]R - 3 \\nabla^a \\nabla_a \\n\\chi  - \\frac{3}{2} (\\nabla^a \\n\\chi) (\\nabla_a \\n\\chi) - \\frac{2 \\nabla^a \\nabla_a \\n\\Phi}{\\n\\Phi} - \\frac{2 (\\nabla^a \\n\\chi) (\\nabla_a \\n\\Phi)}{\\n\\Phi} \\right) \\nonumber\n\\\\\n{}& + \\frac{1}{\\n\\Phi^2} \\left( -\\frac{1}{4}\\n[^{ab}]{\\hat{\\gamma}} \\n[_{ac,y}]{\\hat{\\gamma}} \\n[^{cd}]{\\hat{\\gamma}} \\n[_{db,y}]{\\hat{\\gamma}} - 5 (\\n[_{,y}]\\chi)^2 - 4 \\n[_{,yy}]\\chi + 4 \\frac{\\n[_{,y}]\\Phi}{\\n\\Phi} \\n[_{,y}]\\chi \\right) , \\label{Ricci2expansion}\n\\end{align}\nwhere covariant derivatives and the four-dimensional Ricci scalar are those associated with $\\n[_{ab}]{\\hat{\\gamma}}$. To obtain this form, we use the following two formulae which may be derived from the fact that $\\det (\\n[_{ab}]{\\hat{\\gamma}}) = -1$:\n\\begin{align}\n\\n[^{ab}]{\\hat{\\gamma}} \\; \\n[_{ab, y}]{\\hat{\\gamma}} ={}& 0, \\label{gammamotion}\n\\\\\n\\n[^{ab}]{\\hat{\\gamma}} \\; \\n[_{ab, yy}]{\\hat{\\gamma}} ={}& \\n[^{ab}]{\\hat{\\gamma}} \\; \\n[_{bc, y}]{\\hat{\\gamma}} \\n[^{cd}]{\\hat{\\gamma}} \\; \\n[_{da, y}]{\\hat{\\gamma}}. \\label{gammaswaps}\n\\end{align}\nThe complete action (with $\\epsilon$ scaling and Lagrange multipliers) is given by Eq. \\eqref{scaledepsilonaction}.\n\n\\subsection{Varying the Action}\nWe use $\\n[_{ab}]{\\hat{\\gamma}}$ to compute covariant derivatives, the four-dimensional Ricci scalar ($\\n[^{(4)}]{R}$) and the four-dimensional Einstein tensor ($\\n[^{(4)}_{ab}]{G}$). Indices will also be raised and lowered using this metric.\n\nVarying the action \\eqref{CompleteAction} with respect to $\\n\\Phi$, we find the bulk equation of motion\n\\begin{align}\n\\epsilon^2 e^{-\\n\\chi} \\left( \\n[^{(4)}]{R} - \\frac{3}{2} ({\\nabla}^a \\n\\chi) ({\\nabla}_a \\n\\chi) - 3 {\\nabla}^a{\\nabla}_a \\n\\chi \\right) - \\frac{3}{\\n[^2]{\\Phi}} \\n[^2_{,y}]\\chi + \\frac{1}{4 \\n[^2]{\\Phi}} \\n[^{ab}]{\\hat{\\gamma}}\\n[_{bc,y}]{\\hat{\\gamma}}\\n[^{cd}]{\\hat{\\gamma}}\\n[_{da,y}]{\\hat{\\gamma}} - 2 \\kapfs \\lrn \\Lambda = 0. \\label{exacteom1}\n\\end{align}\nFrom combining the variations with respect to $\\n[_{ab}]{\\hat{\\gamma}}$ and $\\n\\chi$ (after eliminating the Lagrange multiplier by tracing over the $\\n[_{ab}]{\\hat{\\gamma}}$ equation of motion, or enforcing $\\det \\hat{\\gamma} = -1$ on the equations of motion), we obtain a traceless tensor equation of motion in the bulk\n\\begin{align}\n{}& \\frac{1}{2} \\n[^2]{\\Phi} \\epsilon^2 e^{- \\n\\chi} \\left( 4 \\n[^{(4)}_{ab}]{G} + \\n[_{ab}]{\\hat{\\gamma}} \\n[^{(4)}]{R} + 2 ({\\nabla}_a \\n\\chi) ({\\nabla}_b \\n\\chi) - \\frac{1}{2} \\n[_{ab}]{\\hat{\\gamma}} ({\\nabla}^c \\n\\chi) ({\\nabla}_c \\n\\chi) - 4 {\\nabla}_a{\\nabla}_b \\n\\chi + \\n[_{ab}]{\\hat{\\gamma}} {\\nabla}^c{\\nabla}_c \\n\\chi \\right) \\nonumber\n\\\\\n{}& + \\frac{3}{2} \\n{\\Phi} \\epsilon^2 e^{- \\n\\chi} \\left( - 4 {\\nabla}_a {\\nabla}_b \\n{\\Phi} + 4 ({\\nabla}_{(a} \\n\\Phi) ({\\nabla}_{b)} \\n\\chi) + \\n[_{ab}]{\\hat{\\gamma}} {\\nabla}^c {\\nabla}_c \\n{\\Phi} - \\n[_{ab}]{\\hat{\\gamma}} ({\\nabla}_c \\n{\\Phi}) ({\\nabla}^c \\n{\\chi}) \\right) \\nonumber\n\\\\\n{}& - \\n[_{ab,yy}]{\\hat{\\gamma}} + \\frac{\\n[_{,y}]{\\Phi}}{\\n{\\Phi}} \\n[_{ab,y}]{\\hat{\\gamma}} - 2 \\n[_{,y}]{\\chi} \\n[_{ab,y}]{\\hat{\\gamma}} + \\n[_{ac,y}]{\\hat{\\gamma}}\\n[^{cd}]{\\hat{\\gamma}}\\n[_{db,y}]{\\hat{\\gamma}} = 0,\n\\end{align}\nand a scalar equation of motion in the bulk\n\\begin{align}\n{}& \\frac{1}{2} \\n[^2]{\\Phi} \\epsilon^2 e^{-\\n\\chi} \\left( - \\n[^{(4)}]{R}  + \\frac{3}{2} ({\\nabla}^c \\n\\chi) ({\\nabla}_c \\n\\chi) + 3 {\\nabla}^c{\\nabla}_c \\n\\chi + \\frac{5}{\\n\\Phi} {\\nabla}^c {\\nabla}_c \\n{\\Phi} + \\frac{5}{\\n\\Phi} ({\\nabla}_c \\n\\Phi) ({\\nabla}^c \\n\\chi) \\right) \\nonumber\n\\\\\n{}& + \\frac{1}{4} \\n[^{ab}]{\\hat{\\gamma}}\\n[_{ab,yy}]{\\hat{\\gamma}} + 3 \\n[_{,yy}]{\\chi} + 3 (\\n[_{,y}]{\\chi})^2 - 3 \\frac{\\n[_{,y}]{\\Phi}}{\\n{\\Phi}} \\n[_{,y}]{\\chi} + 2 \\n[^2]{\\Phi} \\kapfs \\lrn \\Lambda = 0.\n\\end{align}\nThese variations also give rise to the boundary conditions on the branes\n\\begin{align}\n\\frac{1}{\\n\\Phi} \\n[_{ab, y}]{\\hat{\\gamma}} - \\frac{1}{\\np\\Phi} \\np[_{ab, y}]{\\hat{\\gamma}} = 2 \\kapfs \\epsilon^2 e^{-\\n\\chi} \\left( \\n[_{ab}]{T} - \\n[_{ab}]{\\hat{\\gamma}} \\frac{1}{4} \\, \\n[^{cd}]{\\hat{\\gamma}} \\n[_{cd}]{T} \\right),\n\\end{align}\nand\n\\begin{align}\n- \\frac{3 \\n[_{,y}]{\\chi}}{\\n\\Phi} + \\frac{3 \\np[_{,y}]{\\chi}}{\\np\\Phi} + 2 \\kapfs \\lrn{\\sigma} = \\frac{1}{2} \\kapfs \\epsilon^2 e^{-\\n\\chi} \\, \\n[^{ab}]{\\hat{\\gamma}} \\n[_{ab}]{T}. \\label{branetuningsepsilon}\n\\end{align}\nHere, the four-dimensional stress energy tensors on the branes ($\\n[_{ab}]{T}$) are defined by\n\\begin{align}\n\\n[_m]{S} [\\n[_{ab}]{h} + \\delta \\n[_{ab}]{h}, \\n\\phi] = \\n[_m]{S} [\\n[_{ab}]{h}, \\n\\phi]\n- \\frac{1}{2} \\int d^4 w_n \\sqrt{-\\n{h}} \\n[_{ab}]{T} \\delta \\n[^{ab}]{h}.\n\\end{align}\n\nNote that every factor of $\\epsilon^2$ is accompanied by a factor of $\\exp({-\\n\\chi})$. Also note that the $O(1)$ terms in these equations are exactly our equations of motion \\eqref{eqmotion2} to \\eqref{jumpconditions}.\n\n\\section{Introduction and Summary}\nThe past ten years has seen a flurry of activity  related to extra-dimensional models. Motivated largely by string theory and M-theory, extra-dimensional models have displayed the potential to provide natural solutions to issues such as the hierarchy and cosmological constant problems, as well as providing interesting models for dark matter. Braneworld models, motivated by the works of Arkani-Hamed et al. \\cite{ADD1998} and Randall and Sundrum \\cite{Randall1999, Randall1999a}, have become a very active field of research, with many papers investigating extensions to the basic ideas (see, eg, \\cite{Brax2003, Rubakov2001, Maartens2004} and citations therein).\n\nSeveral different approximation and computational methods have been used to extract physical predictions from extra-dimensional models. In particular, many models have an effective four-dimensional regime at low energies, where the radius of curvature of spacetime measured by four-dimensional observers is much larger than a certain microphysical lengthscale. We review some of the computational methods that have been used to obtain a four-dimensional description of five-dimensional braneworld models, in order to place our results in context.\n\nOne method is to linearize the higher dimensional equations of motion about simple background solutions, then specialize to the long-lengthscale limit in order to obtain the linearized four-dimensional effective theory (which roughly corresponds to discarding the Kaluza-Klein modes). This method was used by Garriga and Tanaka \\cite{Garriga2000} in their analysis of the RS-I model \\cite{Randall1999}, who showed that linearized Einstein gravity is recovered on one of the branes in a particular regime. Further analyses to quadratic order and analyses on other backgrounds have also been performed; see, for example, Refs. \\cite{Kudoh2001, Kudoh2002, Carena2005, Callin2005}. Linearized analyses have many advantages: they are quick and simple, and serve to identify all of the dynamical degrees of freedom in the theory, particularly the Kaluza-Klein modes. However, the linearized method is inherently limited and cannot describe strong field phenomena such as cosmology and black holes.\n\nA second method is to project the five-dimensional equations of motion onto a brane; see, for example, Ref. \\cite{Shiromizu2000}. This ``covariant curvature'' formalism fully incorporates the nonlinearities of the theory. However, the projected description includes nonlocal terms, and the truncation to a low energy effective theory is nontrivial, except in cases with high degrees of symmetry.\n\nIn order to overcome some of these shortcomings, Kanno and Soda \\cite{Kanno2002a, Kanno2002, Soda2003} suggested a perturbation expansion of the covariant curvature formalism known as the ``gradient expansion method'', which involves expanding the theory in powers of the ratio between a microphysical scale and the four-dimensional curvature lengthscale. This approach allows a low-energy description of the model to be found, while retaining the nonlinearities of the theory. This method has been particularly successful in investigating the cosmology of braneworld models \\cite{Kanno2002a, Soda2003, Zen2005, Arroja2008}, and has the benefit of providing an explicit calculation of the five-dimensional metric, but is algebraically complex and requires assumptions on the form of the metric.\n\nAn alternative approach to obtaining a four-dimensional effective action, discussed by Wiseman \\cite{Wiseman2002}, focusses on the radion mode of the RS-I model. Treating the radion mode as a deflection of the branes, the approach uses a derivative expansion to calculate its nonlinear behavior. Although this method nicely captures the nonlinearities of the theory, it is highly nontrivial, and ``guesses'' the four-dimensional effective action, based on the first order equations of motion the method finds.\n\nA final method involves making an ansatz for the form of the five-dimensional metric in terms of four-dimensional fields, and integrating over the fifth dimension to obtain a four-dimensional action. Examples of this method in the literature include Refs. \\cite{Arroja2008, Chiba2000, Goldberger2000, Cotta-Ramusino2004, Kanno2005a}. The benefits of this method are the automatic truncation of the massive Kaluza-Klein modes, and the computational efficiency in dealing strictly at the level of the action. The main drawback is that the five-dimensional metric ansatz must be found (or guessed) using another method.\n\nFour-dimensional effective descriptions typically contain moduli fields (radion modes) which describe the distances between branes. Often, such modes appear as massless scalar fields which couple to gravity in a Brans-Dicke like manner (see, eg, Ref. \\cite{Chiba2000}). This occurs in the RS-I model of two branes in a compactified bulk with orbifold symmetry, for example. In this model, the radion mode must be stabilized by some mechanism (for example, by using a bulk scalar field as in the Goldberger-Wise mechanism \\cite{Goldberger1999a}), or else the theory is ruled out for observers on the TeV brane (see, eg, Refs. \\cite{Garriga2000, Bean2008a}). In theories including multiple branes, one expects several radion modes which may have nontrivial couplings to one another and to the four-dimensional metric at the nonlinear level.\n\nOne common extension of the RS models is to consider models with more than one or two branes. A variety of papers have considered three-brane models, usually on an orbifold (see, eg, \\cite{Zen2005, Cotta-Ramusino2004, Kogan2000, Kogan2001, Kogan2002}). Some special cases have been considered for arbitrary $N$-brane models, mostly to investigate their cosmological properties \\cite{Kogan2001, Flanagan2001a}. A few papers comment that their methods should extend to arbitrary $N$-brane situations (eg, \\cite{Damour2002}), but little analysis has actually been performed in this regard.\n\nIn this paper, we present a new method to obtain a four-dimensional effective theory from an $N$-brane model in five dimensions. We assume that matter is confined to branes with the only bulk field being gravity, and we do not invoke mechanisms to stabilize the radion modes. The method utilizes a two-lengthscale expansion to find solutions to the five-dimensional equations of motion in a low energy regime. We do not require assumptions about the form of the metric, or the existence of Gaussian normal coordinates. The method is computationally efficient, and does not require the explicit use of the five-dimensional Einstein equations or the Israel junction conditions. Instead, one always works at the level of the action. Furthermore, our method is very general and can be applied to various models. The method has similarities to the gradient expansion method (see especially \\cite{Kanno2005a}), but is computationally much simpler, and can deal with multiple branes in a straightforward manner. A particular strength of the method is that it performs a rigorous treatment of all radion modes, and automatically truncates massive modes. We present a brief example of the method for the case of the RS-I model \\cite{Randall1999}, before illustrating the method in detail for the case of $N$ four-dimensional branes in an uncompactified extra dimension, deriving the four-dimensional effective action for a general configuration.\n\n\\section{Separation of Lengthscales\\label{SecLengthscale}}\nWe now describe the approximation method, based on a two-lengthscale expansion, which we use to obtain a four-dimensional description of the system. We begin by defining the appropriate lengthscales, and then detail how the theory simplifies in the regime where the ratio of lengthscales is small.\n\n\\subsection{Two Lengthscales}\nThere are three groups of parameters in our model: the five-dimensional gravitational scale $\\kappa_5^{2}$, the brane tensions $\\{\\sigma_n\\}$, and the bulk cosmological constants $\\{\\Lambda_n\\}$. We assume that all parameters in a group are of the same order of magnitude, and so will just consider typical parameters $\\sigma$ and $\\Lambda$. Working with units in which $c=1$, the dimensionality of these parameters in terms of mass units $M$ and length units $L$ are $[\\kapfs] = {L^2}\/{M}$, $[\\sigma] = {M}\/{L^3}$, and $[\\Lambda] = {M}\/{L^4}$.\n\nWe assume that the dimensionless parameter $\\sigma^2 \\kapfs\/\\Lambda$ is roughly of order unity; this will be enforced by the brane tuning conditions we derive below [see Eq. \\eqref{branetunings}]. Eliminating \\kapfs, we can then define a lengthscale by\n\\begin{align}\n{\\cal L} = {\\sigma}\/{\\Lambda} \\label{defL}\n\\end{align}\nand a mass scale by\n\\begin{align}\n{\\cal M} = {\\sigma^4}\/{\\Lambda^3}. \\label{defM}\n\\end{align}\n\nFor a given configuration, we also define a \\textit{four-dimensional curvature lengthscale} ${\\cal L}_c(y)$ on each slice of constant $y$, as follows. We take the minimum of the transverse lengthscale over which the induced metric varies, and the transverse lengthscale over which the metric coefficient $\\Phi$ varies. In other words,\n\\begin{align}\n{\\cal L}_c (y) \\sim \\min \\left\\{ \\left|\\tensor*{R}{^{(4)}_{\\hat{a} \\hat{b} \\hat{c} \\hat{d}}}\\right|^{-1\/2}, \\left|\\tensor*{\\nabla}{_{\\hat{a}}} \\tensor*{R}{^{(4)}_{\\hat{b} \\hat{c} \\hat{d} \\hat{e}}}\\right|^{-1\/3}, \\ldots, \\frac{\\left|\\Phi\\right|}{\\left|\\tensor*{\\nabla}{_{\\hat{a}}} \\Phi\\right|}, \\frac{\\left|\\Phi\\right|^{1\/2}}{\\left|\\tensor*{\\nabla}{_{\\hat{a}}} \\tensor*{\\nabla}{_{\\hat{b}}} \\Phi\\right|^{1\/2}}, \\ldots \\right\\}\n\\end{align}\nwhere $\\hat{a}, \\hat{b}, \\ldots$ denotes an orthonormal basis of the induced metric, $R_{\\hat{a} \\hat{b} \\hat{c} \\hat{d}}$ is the Riemann tensor of the induced metric, and dots denote similar terms with more derivatives.\n\nThus, for a given configuration, we have two natural lengthscales: the microphysical lengthscale ${\\cal L} = \\sigma \/ \\Lambda$ (the same for all configurations), and the macrophysical curvature lengthscale ${\\cal L}_c$ (where the $c$ is intended to denote ``curvature'').\n\n\\subsection{Separating the Lengthscales}\nWe now evaluate the action \\eqref{CompleteAction} in the low energy regime ${\\cal L}_c \\gg {\\cal L}$, in which the theory admits a four-dimensional description. We will find that there is a leading order term of order $\\sim {\\cal ML}$, and a subleading term of order $\\sim {\\cal ML} ({\\cal L}\/{\\cal L}_c)^2$. Our strategy will be to separate the contributions to the action at each order, minimize the leading order piece of the action, and then substitute the general solutions obtained from that minimization into the subleading piece of the action. The result will be a four-dimensional action which gives the effective description of the system in the low energy regime.\n\nWe write the action \\eqref{CompleteAction} as a sum $S = S_g + S_m$ of a gravitational part $S_g$ and a matter part $S_m$, where the matter part is the last term in Eq. \\eqref{CompleteAction} and the gravitational part comprises the remaining terms.\n\nWe first discuss the expansion of the gravitational action $S_g$, which is a functional of a bulk metric $\\tensor{g}{_{\\alpha \\beta}}$ and brane embedding functions $\\tensor[^n]{x}{^\\Gamma}$. We define a mapping $T_\\epsilon$ which acts on these variables\n\\begin{align}\nT_\\epsilon : (\\tensor{g}{_{\\alpha \\beta}}, \\tensor[^n]{x}{^\\Gamma}) \\rightarrow (\\tensor*{g}{_{\\alpha \\beta}^\\epsilon}, \\tensor*[^n]{x}{^\\Gamma_\\epsilon}),\n\\end{align}\nwhere $\\epsilon > 0$ is a dimensionless parameter, as follows: (i) We specialize to our chosen gauge, (ii) replace the metric \\eqref{metric} with the rescaled version\n\\begin{equation}\n\\tensor*{ds}{^2_\\epsilon} = \\frac{1}{\\epsilon^2} e^{\\chi(x^c, y)} \\tensor{\\hat{\\gamma}}{_{a b}} (x^c, y) dx^{a} dx^{b} + \\Phi^{2} (x^c, y) dy^2 \\label{scaledmetric},\n\\end{equation}\nwhere indices indicating regions have been suppressed, and (iii) leave the embedding functions in our chosen gauge unaltered. We may think of $\\epsilon$ as a parameter which tunes the ratio of the microphysical lengthscale to the macrophysical lengthscale. As $\\epsilon$ is decreased, lengthscales on the brane are inflated, and so ${\\cal L}_c$ increases. Thus, as $\\epsilon$ decreases, so does the ratio ${\\cal L} \/ {\\cal L}_c$. In particular, we have\n\\begin{align}\n\\left(\\frac{{\\cal L}}{{\\cal L}_c}\\right)_\\epsilon = \\epsilon \\frac{{\\cal L}}{{\\cal L}_c}.\n\\end{align}\nIt is important to note that this $\\epsilon$ scaling does not map solutions to solutions, but just provides a means of keeping track of the dependence on the various lengthscales.\n\nWe can construct a one-parameter family of action functionals by using these rescaled metrics in our original action \\eqref{CompleteAction}\\footnote{The factor of $\\epsilon^4$ in Eq. \\eqref{epsilonaction} is inserted for convenience, so that Eq. \\eqref{scaledaction} contains terms of $O(1)$ and $O(\\epsilon^2)$. This is explicitly shown in Section \\ref{SecLowest}.}:\n\\begin{equation}\n\\tensor{S}{_{g, \\epsilon}} \\left[\\tensor*{g}{_{\\alpha \\beta}}, \\n[^\\Gamma]x \\right] \\equiv \\epsilon^4 S_{g} \\left[\\tensor*{g}{_{\\alpha \\beta}^\\epsilon}, \\n[^\\Gamma_\\epsilon]x \\right]. \\label{epsilonaction}\n\\end{equation}\nWe can expand this action in powers of $\\epsilon$ by\n\\begin{equation}\n\\tensor{S}{_{g, \\epsilon}} \\left[\\tensor*{g}{_{\\alpha \\beta}}, \\n[^\\Gamma]x \\right] = \\tensor{S}{_{g, 0}}\\left[\\tensor*{g}{_{\\alpha \\beta}}\\right] + \\epsilon^2 \\tensor{S}{_{g, 2}}\\left[\\tensor*{g}{_{\\alpha \\beta}}\\right], \\label{scaledaction}\n\\end{equation}\nwhere on the right hand side we omit the dependence on the embedding functions since we have used the gauge freedom to fix those. The expansion \\eqref{scaledaction} truncates after 2 terms; there are no higher order terms in $\\epsilon$. Note that there is no $O(\\epsilon)$ term, as when the action \\eqref{epsilonaction} is evaluated, terms of $O(\\epsilon^2)$ arise from contractions in the Ricci scalar using $g^{ab}$ ($O(1)$ terms arise from $g^{yy}$ contractions). Terms of order $O(\\epsilon)$ would arise from contractions using $g^{ay}$, but as these components of the metric have been gauge-fixed to zero, they are not present. This can be seen explicitly in the expansion of the Ricci scalar \\eqref{Ricci2expansion}. As we tune $\\epsilon \\rightarrow 0$, we move further into the low energy regime, and so we identify the zeroth order term as the dominant contribution to the action, and the second order term as the subleading term. This provides the separation of lengthscales we desire.\n\n\nLet us now turn to the matter contribution to the action, $S_m$. We expect the matter action to contribute at $O(\\epsilon^2)$, the same order as the subleading gravitational term. To see this, note that the brane tensions scales as $\\sigma \\sim {\\cal M}\/{\\cal L}^3$, where the scales ${\\cal M}$ and ${\\cal L}$ were defined in Eqs. \\eqref{defL} and \\eqref{defM}. The matter action will be roughly $S_m \\sim \\int \\rho \\; d^4 x$, where $\\rho$ is a 4-dimensional energy density. The four-dimensional Newton constant $\\kappa_4^2 = 8 \\pi G$ is of order $\\kappa_4^2 \\sim {\\cal L} \/ {\\cal M}$ by dimensional analysis [c.f. Eq. \\eqref{4dNewtonConst} below], and so $\\rho$ will be of order\n\\begin{align}\n\\rho \\sim \\frac{1}{\\kappa_4^2 {\\cal L}_c^2} \\sim \\frac{{\\cal M}}{{\\cal L} {\\cal L}_c^2}.\n\\end{align}\nTaking the ratio $\\rho \/ \\sigma$ now gives\n\\begin{align}\n\\frac{\\rho}{\\sigma} \\sim \\frac{{\\cal M} \/ {\\cal L} {\\cal L}_c^2}{{\\cal M} \/ {\\cal L}^3} \\sim \\frac{{\\cal L}^2}{{\\cal L}_c^2} \\propto \\epsilon^2. \\label{rhoscaling}\n\\end{align}\nFormally, the scaling \\eqref{rhoscaling} can be achieved by replacing the matter action $S_m$ with a rescaled action $S_{m, \\epsilon}$ given by (i) multiplying by $\\epsilon^4$ as in Eq. \\eqref{epsilonaction}, (ii) rescaling all fields and dimensional constants with dimensions $(\\mathrm{mass})^r (\\mathrm{length})^s$ by factors of $\\epsilon^{-(r+s)}$. The expansion of the full action is then\n\\begin{align}\nS_\\epsilon = S_{g, \\epsilon} + S_{m, \\epsilon} = {}& S_{g, 0} + \\epsilon^2 \\left[ S_{g, 2} + S_m \\right] \\nonumber\n\\\\\n= {}& S_0 + \\epsilon^2 S_2. \\label{completeepsilonaction}\n\\end{align}\nIt can be seen that given brane tensions tuned to the bulk cosmological constants, $\\sigma^2 \\sim \\Lambda\/\\kapfs$, we require that the matter density on a brane should be small, so as not to spoil the tuning. This also yields $\\rho \\ll \\sigma$, which roughly corresponds to the separation of lengthscales condition ${\\cal L} \\ll {\\cal L}_c$.\n\nWe perform this $\\epsilon$ scaling separately in each bulk region of the model. The contribution to the action from each region will separate into zeroth and second order terms.\n\n\\subsection{The Low Energy Regime\\label{secdiscuss}}\nNow that the contributions to each order have been identified, we can minimize the leading order term in the action, $\\tensor{S}{_0}$. Once general solutions to the equations of motion have been found, we can use these solutions in the second order term in the action. Thus, we solve for the high energy (short lengthscale) dynamics first, and use the solution to this as a background solution for the low energy (long lengthscale) dynamics. At this point, we may let $\\epsilon \\rightarrow 1$, and rely on the ratio $({\\cal L}\/{\\cal L}_c)^2$ being sufficiently small to provide the separation of lengthscales.\n\nThe effect of this separation of lengthscales is to enforce a decoupling of the high energy dynamics from the low energy dynamics. We will see below that the equation of motion for the high energy dynamics contains $y$ derivatives, but no $x^a$ derivatives. The theory at this order thus reduces to a set of uncoupled theories, one along each fiber $x^a = \\mathrm{const}$ in the bulk. These theories are coupled together at $O(\\epsilon^2)$, and thus in the regime of interest, the coupling is minimal. After solving the high energy dynamics along these fibres, a four-dimensional effective description of the system remains.\n\nThe low energy regime, in which the theory admits a four-dimensional description, is the regime\n\\begin{align}\n{\\cal L}_c \\gg {\\cal L}. \\label{comparisonofls}\n\\end{align}\nThis regime is also frequently characterized in the literature by the condition\n\\begin{align}\n\\rho \\ll \\sigma, \\label{sigmarho}\n\\end{align}\nwhere $\\rho$ is the mass density on a brane and $\\sigma$ is a brane tension [c.f. Eq. \\eqref{rhoscaling} above]. One can interpret the condition \\eqref{sigmarho} as saying that the mass density on the brane must be sufficiently small that the brane-tuning conditions [Eq. \\eqref{branetunings} below which enforces $\\sigma^2 \\sim \\Lambda \/ \\kapfs$] are not appreciably modified. However, the condition \\eqref{sigmarho} is less general than the condition \\eqref{comparisonofls}, and although necessary, is actually insufficient. First, as discussed above, \\eqref{sigmarho} only applies to branes, whereas \\eqref{comparisonofls} applies at each value of $y$, including away from the branes. Second, even when the density on a given brane vanishes, four-dimensional gravitational waves on that brane can give rise to radii of curvature ${\\cal L}_c$ that are comparable to ${\\cal L}$. In this case, the separation of lengthscales will not apply and the four-dimensional effective theory will not be valid, despite the fact that the condition \\eqref{sigmarho} is satisfied. Curvature associated with the metric coefficient $\\Phi$ can also yield similar results.\n\nFinally, we discuss a subtlety in our definition of the ``low energy regime''. As noted in the previous paragraph, ${\\cal L}_c$ varies with position in the five-dimensional universe. Our separation of lengthscales will break down when the induced metric on any slice of constant $y$ has a radius of curvature ${\\cal L}_c$ comparable to that of the microphysical lengthscale ${\\cal L}$; it is insufficient to require that ${\\cal L} \\gg {\\cal L}$ on each brane. When this happens, the terms of order $\\epsilon^2$ will couple strongly to the $O(1)$ terms, and our approximate solutions for the five-dimensional metric will no longer be valid. This will generically occur at sufficiently large distances from the branes, as $\\exp(\\n\\chi)$ typically grows exponentially small away from the branes, and ${\\cal L}_c^{-2} \\propto \\exp(-\\n\\chi) R^{(4)}$. Despite this breakdown, the contribution to the action from these regimes is exponentially suppressed by the warp factor, and thus provides only a small deviation from the effective theory. It is unlikely that the warp factor can grow without bound after encountering this regime while maintaining a globally hyperbolic spacetime.\n\n\\section{Discussion and Conclusions\\label{endsection}}\nWe have presented a new method to calculate the four-dimensional effective action for five-dimensional models involving $N$ non-intersecting branes in the low energy regime. Although we have only illustrated an application of the method to an uncompactified extra dimension, it is generally applicable, and is expected to work for circularly compactified and orbifolded models also.\n\n\\subsection{Domain of Validity of the Four-Dimensional Description\\label{HiEnergy}}\nWe begin our discussion of the domain of validity of the four-dimensional description given by Eq. \\eqref{effectiveaction} by recapping the method of computation discussed in Section \\ref{SecLengthscale}. Starting from the five-dimensional action $S$, we define a rescaled action $S_\\epsilon$ which has the expansion\n\\begin{equation}\nS_\\epsilon = \\tensor{S}{_0} + \\epsilon^2 \\tensor{S}{_2}. \\label{recapscaledaction}\n\\end{equation}\nIn Section \\ref{SecLowest} we found the most general solution of $\\delta S_0 = 0$, and substituting that solution into $S_2$, gave the four-dimensional action functional of Section \\ref{Einsteintransform}\\footnote{The action $S_0$ for the solution is zero, assuming the brane tunings \\eqref{branetunings}.}.\n\nThe basis of our approximation method is the smallness of the bulk radius of curvature $1\/k_n$ compared to the radius of curvature ${\\cal L}_c$ of the four-dimensional metric $e^{\\chi} \\hat{\\gamma}_{ab}$. However, although this approximation is valid on all the branes, it inevitably breaks down as $y \\rightarrow \\pm \\infty$, far from the branes, as ${\\cal L}_c \\rightarrow 0$, as discussed in Section \\ref{secdiscuss}. It is worth noting that in the special case where all of the induced metrics on the branes are flat and there are no matter fields, the metric ansatz (with $\\Phi = \\mathrm{const}$) is an exact solution to the five-dimensional Einstein equations, and this breakdown does not occur.\n\nOne might expect contributions from the regime far from the branes to invalidate our four-dimensional effective description. However, we expect that the contribution to the action far from the brane will negligibly change the calculation, as in the region in which we expect large departures from the derived metric, the warp factor exponentially suppresses any contributions.\n\nIt is possible for our two-lengthscale expansion to break down not only asymptotically, but also in between branes. A number of models (eg, \\cite{Carena2005, Flanagan2001a, Damour2002, Karch2001} to cite but a few) discuss ``bounce'' behavior in the warp factor, where it decreases and increases again in between branes, as with a $\\cosh^2$ dependence. Typically, this behavior appears when the metric $\\hat{\\gamma}$ is a curved FRW metric. It is a limitation of our method that this bounce is not evident in our solutions, as it explicitly requires coupling between the $O(1)$ and $O(\\epsilon^2)$ components (in particular, the four-dimensional Ricci scalar). Thus, this behavior is excluded by the underlying assumptions of our method, as near the turning point of these bounces, the separation of lengthscales has broken down. We note, however, that $\\cosh^2$ behavior is likely to be forbidden in the first or last ($y \\rightarrow \\pm \\infty$) regions by global hyperbolicity. It is also possible to produce $\\sinh^2$ behavior in the warp factor. In between branes, this can lead to topologically disconnected regions of spacetime as discussed in \\cite{Flanagan2001a}, which we have excluded by assumption. In the first or last regions, correctly accounting for this behavior requires that the integration over the fifth dimension be truncated. However, the contributions to our effective action from integrating beyond these regions is again exponentially suppressed and negligible. In the regime in which the separation of lengthscales is valid, our solutions are in agreement with models displaying these types of behavior.\n\nFor black holes, the solution given by our effective action is subject to the Gregory-Laflamme instability \\cite{Gregory1993} and the final outcome is uncertain (see \\cite{Chamblin2000} and citations thereof). The five-dimensional stability of solutions for which the induced metric on the branes is not nearly flat (eg, neutron stars) is an interesting open question. We conjecture that all the solutions without horizons are stable and are reasonably described by our four-dimensional effective action.\n\nWe may also consider the regime in which ${\\cal L}_c \\ll {\\cal L}$, such as will occur a long way away from the branes. In this limit, the physical description would change from being that of decoupled fibres to that of decoupled four-dimensional hypersurfaces [one should solve the $O(\\epsilon^2)$ contribution to the action first, and substitute that into the $O(1)$ contribution to the action]. This approach may yield a matched asymptotic expansion approach to obtaining a solution far from the branes. Our method may therefore be useful for investigating the regime between Minkowski space on a brane and a black hole on a brane.\n\nIt is important to note that our method does not yield the leading order five-dimensional metric. This can be seen from the fact that our four dimensional action depends only on the fields $\\n\\chi$ evaluated on the branes, and the values of these fields between the branes are not determined. However, knowledge of the leading order five-dimensional metric is, rather surprisingly, \\emph{not} a prerequisite for correctly capturing the leading order four-dimensional dynamics. Most other methods rely on knowledge of the five-dimensional behavior of the metric to calculate the effective four-dimensional equations of motion, and our method is somewhat unique in this regard.\n\nOur method of computation correctly captures the leading order dynamics of the system. However, there will be higher order corrections, suppressed by powers of $\\epsilon^2$. In particular, the fields $\\n\\chi$ and $\\n\\Phi$ can be expanded as\n\\begin{subequations}\n\\label{fieldexpansions}\n\\begin{align}\n\\n\\chi = {}& \\n[^{(0)}]{\\chi} + \\epsilon^2 \\n[^{(2)}]{\\chi} + O(\\epsilon^4),\n\\\\\n\\n\\Phi = {}& \\n[^{(0)}]{\\Phi} + \\epsilon^2 \\n[^{(2)}]{\\Phi} + O(\\epsilon^4).\n\\end{align}\n\\end{subequations}\nThroughout this paper, we have dealt only with the fields $\\n[^{(0)}]{\\chi}$ and $\\n[^{(0)}]{\\Phi}$. The necessity for the higher order terms can be seen from the exact, five-dimensional equations of motion, which are derived in Appendix \\ref{AppExactEOMS}. For example, the exact Israel junction conditions are given by Eq. \\eqref{branetuningsepsilon}. If we substitute the expansions \\eqref{fieldexpansions} into Eq. \\eqref{branetuningsepsilon}, and use \\eqref{jumpconditions} [with \\n\\chi\\ and \\n\\Phi\\ replaced by $\\n[^{(0)}]{\\chi}$ and $\\n[^{(0)}]{\\Phi}$] together with the brane tuning conditions \\eqref{branetunings}, we find that the higher order corrections $\\n[^{(2)}]{\\chi}$ and $\\n[^{(2)}]{\\Phi}$ are related to the matter stress energy tensors on the brane. Our results confirm the suggestion of Kanno and Soda that these higher order corrections do not affect the four-dimensional effective action to leading order \\cite{Kanno2005a}.\n\n\\subsection{Models Which Violate the Brane Tension Tunings\\label{BraneTensions}}\nIf a brane's tension is adjusted so as to violate the tuning condition \\eqref{branetunings}, then it is possible to view the situation as having either detuned brane tensions or detuned bulk cosmological constants. For accounting purposes, it is simpler to think of the bulk cosmological constants as being detuned. When this occurs, the exact equations of motion in the bulk \\eqref{exacteom1} imply that a nonzero Ricci curvature is induced to compensate for the detuning. Exact solutions have been calculated in highly symmetric cases, see for example Ref. \\cite{Karch2001}. In general, the exact nature of the perceived detuning is nontrivial, as the bulk cosmological constants on either side of the offending brane(s) can appear detuned by different amounts to compensate.\n\nIf the deviation from the brane tuning conditions is small [$\\Delta \\sigma \/ \\sigma^T = O(\\epsilon^2)$], then we can approximate the contribution to the four-dimensional effective action as\n\\begin{align}\n\\Delta S = - \\sum_{n=0}^{N-1} \\int d^4 x \\sqrt{-\\n{h}} (\\sigma_n - \\sigma_n^T),\n\\end{align}\nwhere $\\sigma_n^T$ is the tuned value for the $n^{\\mathrm{th}}$ brane, given by \\eqref{branetunings}. This approximation is of the same order as the other approximations we have made in our method. The net result is then an effective cosmological constant on each brane, given by\n\\begin{align}\n\\Lambda^{(4)}_n = \\sigma_n - \\sigma_n^T,\n\\end{align}\nwhich vanishes when the brane tensions are tuned. [Note: This differs from the result given in the literature for the RS-II model, see for example Ref. \\cite{Maartens2004}, but the difference is $O(\\epsilon^4)$].\n\nIf the detuning of a brane's tension from its tuned value should become too large [$O(1)$ rather than $O(\\epsilon^2)$], then the curvature induced by the four-dimensional effective cosmological constant can cause the radius of curvature on a slice of constant $y$ close to the branes to violate the approximations used in our method, which implies that our four-dimensional effective action will not be a good description of a system in this regime.\n\n\n\\subsection{Multigravity\\label{Discussions}}\nTheories with more than one independent dynamical tensor field are called multigravity theories; see the general discussion in Damour and Kogan \\cite{Damour2002}. The models in this paper may exhibit two forms of multigravity, although we have ignored one of them entirely.\n\nThe first form of multigravity is the possible existence of a second tensor field, given by the matrix $\\mathbf{B}(x^a)$ in Eq. \\eqref{gammasoln}. We argued in Section \\ref{SecClasses} that this form of multigravity is likely forbidden.\n\nThe second form of multigravity arises from the the fact that outside of the low energy regime, the models will contain Kaluza-Klein graviton modes. These modes will have masses that are formally of order ${\\cal L}^{-1}$, but may be much lighter due to exponential suppression factors, and so be phenomenologically important (so-called ``ultra-light modes'')\\cite{Kogan2000, Kogan2001}. Our method of analysis automatically excludes all massive fields (formally, we take $\\epsilon$ sufficiently small to overcome any large exponential factors), so we have neglected all graviton Kaluza-Klein modes. It is likely that some of these modes are in fact ultralight in our model, as in the analyses of Damour and Kogan \\cite{Kogan2000, Kogan2001, Damour2002}.\n\n\\subsection{Conclusions}\nThe method we have developed is a useful tool for investigating the four-dimensional, low energy behavior of five-dimensional braneworld models. We have illustrated the method for a system of $N$ branes in an uncompactified extra dimension. We intend to apply the method developed here to orbifolded models. We also plan to investigate experimental constraints on the model illustrated here, especially solar system constraints. Finally, we will investigate dark matter models obtained by placing matter fields on various branes.\n\n\\begin{acknowledgements}\nWe would like to thank S.-H. Henry Tye and Ira Wasserman for helpful discussions. This research was supported in part by NSF grants 0757735 and 0555216, and NASA grant NNX08AH27G.\n\\end{acknowledgements}\n\n\\subsection{Applicable Models\\label{SecModel}}\nIn this section, we introduce the parameters, metrics, and coordinate systems used to define the braneworld models which we apply our method to. The most basic model assumes that the extra dimension is infinite and not compactified, but the generalization to circularly compactified and orbifolded systems is straightforward, and is described in Section \\ref{secRSI}.\n\nWe consider a system of $N$ four-dimensional branes in a five-dimensional universe with one temporal dimension, with coordinates $x^\\Gamma = (x^0, \\ldots, x^4)$. We denote the bulk metric by $\\tensor{g}{_{\\Gamma \\Sigma}}(x^\\Theta)$, and the associated five-dimensional Ricci scalar by $R^{(5)}$. For simplicity, we assume that there are no physical singularities in the spacetime.\n\nThe $N$ branes are labeled by an index $n = 0, 1, \\ldots, N-1$, so that adjacent branes are labeled by successive values of $n$. We assume that the branes are non-intersecting. Denote the $n^{\\mathrm{th}}$ brane by \\bn. On \\bn, we introduce a coordinate system $\\tensor*{w}{_n^a} = (\\tensor*{w}{_n^0}, \\ldots, \\tensor*{w}{_n^3})$. The location of the branes in the five-dimensional spacetime is described by $N$ embedding functions $\\tensor[^n]{x}{^\\Gamma}(\\tensor*{w}{_n^a})$. From these embedding functions, we can calculate the induced metric \\n[_{ab}]{h} on \\bn,\n\\begin{align}\n\\n[_{ab}]{h}(\\tensor*{w}{_n^c}) = \\left.\\frac{\\partial \\, \\tensor[^n]{x}{^\\Gamma}}{\\partial \\tensor*{w}{_n^a}} \\frac{\\partial \\, \\tensor[^n]{x}{^\\Sigma}}{\\partial \\tensor*{w}{_n^b}} \\tensor{g}{_{\\Gamma \\Sigma}} \\left[\\tensor[^n]{x}{^\\Theta} \\right]\\right|_{\\tensor*{w}{_n^c}}. \\label{Embedding}\n\\end{align}\nWe associate a non-zero brane tension $\\sigma_n$ with each brane \\bn, and we also take there to be matter fields $\\n{\\phi}(\\tensor*{w}{_n^a})$ which live on \\bn, with their own matter action $\\n[_m]{S}{}[\\n[_{ab}]{h}, \\n{\\phi}]$.\n\nIn between each brane there exists a bulk region of spacetime, which we will denote ${\\cal{R}}_0, \\ldots , {\\cal{R}}_N$, with \\rn\\ lying between branes $n-1$ and $n$. The first (last) bulk region will describe the region between the first (last) brane and spatial infinity in the bulk. In each bulk region \\rn\\ we allow for a bulk cosmological constant \\lrn{\\Lambda} (see Ref. \\cite{Flanagan2001a} for a possible microphysical origin for such piecewise constant cosmological constants).\n\nFinally, the action for the model is\n\\begin{align}\nS \\left[ \\tensor{g}{_{\\Gamma \\Sigma}}, \\tensor[^n]{x}{^\\Gamma}, \\n\\phi \\right] ={}& \\int d^5 x \\sqrt{-g} \\left(\\frac{R^{(5)}}{2 \\kapfs} - \\Lambda(x^\\Gamma) \\right) - \\sum_{n = 0}^{N-1} \\lrn{\\sigma} \\int_{\\bn} d^4 \\lrn{w} \\sqrt{-\\n{h}} + \\sum_{n = 0}^{N-1} \\n[_m]{S}[\\n[_{ab}]{h}, \\n{\\phi}], \\label{generalaction}\n\\end{align}\nwhere \\kapfs\\ is the five-dimensional Newton's constant, and $\\Lambda(x^\\Gamma)$ takes the value \\lrn{\\Lambda} in \\rn.\n\n\\subsection{Overview of the Method and Results\\label{SecResults}}\nOur method works in five steps.\n\n\\textbf{Step 1: Gauge specialize.} From the general action (Eq. \\eqref{generalaction} in the model we discuss here), we perform a gauge transformation to specialize the metric to the straight gauge \\cite{Carena2005}, illustrated in Figure \\ref{FigSetup}.\n\n\\textbf{Step 2: Separate lengthscales in the action.} There are two characteristic lengthscales in the model. The first, which we call the microphysical lengthscale, is the lengthscale associated with the bulk cosmological constants, which is typically assumed to be on the order of the micron scale or smaller. The second lengthscale is the four-dimensional radius of curvature felt on the branes. When the ratio of the microphysical lengthscale to the four-dimensional radius of curvature is small (the low energy limit), the dynamics of the extra dimension effectively decouples from the four-dimensional dynamics, leading to a four-dimensional effective theory. We introduce a small parameter to tune this ratio, and use this parameter to perform a two lengthscale expansion of the action.\n\n\\textbf{Step 3: Solve equations of motion.} The equations of motion at zeroth order in this small parameter are calculated and explicitly solved. As expected in this type of model, all of the bulk cosmological constants must be negative, and at this order, the brane tensions are required to be tuned to a specific value\\footnote{We consider small deviations from this value in Section \\ref{BraneTensions}.} \\cite{Randall1999} in order to avoid an effective cosmological constant on the branes. The solution to the zeroth order equations of motion provides a background metric solution, which is perturbed at the next order in our small parameter (the metric is an exact solution if the four-dimensional space is flat).\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width = 0.7\\textwidth]{Figure1.pdf}\n    \\caption{An illustration of the model a) before and b) after gauge fixing. The bulk cosmological constants, brane tensions, and metrics are labeled.} \\label{FigSetup}\n\\end{figure}\n\n\\textbf{Step 4: Integrate five-dimensional dynamics.} The five-dimensional dynamics of the theory are integrated out by substituting the metric ansatz into the action, and integrating over the extra dimension. The action to zeroth order in the small parameter is minimized by the solution, leaving only the four-dimensional terms in the action.\n\n\\textbf{Step 5: Redefine fields.} The final step is to redefine fields to cast the four-dimensional effective action in the form of a four-dimensional multiscalar-tensor theory in a nonlinear sigma model. In the Einstein conformal frame, the general form of the four-dimensional effective action is given by\n\\begin{align}\n S[g_{ab}, \\Phi^A, \\n\\phi] ={}& \\int d^4 x \\sqrt{-g} \\left\\{ \\frac{1}{2 \\kappa_4^2} R [g_{ab}] - \\frac{1}{2} \\gamma_{AB}(\\Phi^C) \\nabla_a \\Phi^A \\nabla_b \\Phi^B g^{ab} \\right\\} + \\sum_{n=0}^{N-1} \\n[_m]S \\left[ e^{2 \\alpha_n \\left(\\Phi^C\\right)} g_{ab}, \\n\\phi \\right] \\label{generalEinsteinframe}\n\\end{align}\nwhere $\\Phi^A$, $1 \\le A \\le N-1$, are massless scalar fields (radion modes) which encode the interbrane distances. Also, $\\kappa_4^2 \\ (= 8 \\pi G_N)$ is the effective four-dimensional Newton's constant, which is a function of $\\kappa_5^2$ and the bulk cosmological constants. Finally, $\\gamma_{AB}(\\Phi^C)$ is the field space metric of the nonlinear sigma model, and $\\alpha_n (\\Phi^C)$ are the brane coupling functions. The functional form of both of these depends on the specifics of the model.\n\nOne of the features of the method used here is that five-dimensional gravitational perturbations which give rise to massive four-dimensional fields are automatically truncated. The mass scales of these fields are typically of order $\\hbar\/{\\cal L}$, where ${\\cal L}$ is the microphysical lengthscale of the theory. However, Damour and Kogan \\cite{Kogan2000, Kogan2001, Damour2002} have shown that it is possible to have graviton Kaluza-Klein modes where masses are of order $\\hbar\/{\\cal L} \\exp (-l\/{\\cal L})$, where $l$ is an inter-brane separation. Because of the exponential factor, these second graviton modes can be ultralight and observationally relevant. Although the models we consider are likely to contain such ultralight graviton modes, our method excludes their possible contributions to a four-dimensional effective theory.\n\nOur method has similarities to the gradient expansion method of Kanno and Soda \\cite{Kanno2002a, Kanno2002, Soda2003}. Our small expansion parameter coincides with theirs, and the zeroth order solutions from both methods agree in cases where both methods are applicable. However, beyond this point, the methods diverge. Our method Taylor expands the action, but not the metric as in the covariant curvature formalism. Although higher order corrections to the metric do exist, they are intrinsically five-dimensional interactions which are unnecessary for the construction of a four-dimensional effective theory, and their contributions to the effective theory are exponentially suppressed within the low energy regime. Furthermore, our method arrives at a four-dimensional effective action, rather than working only at the level of the equations of motion. This provides for computational efficiency and a more intuitive understanding of the final result.\n\nThis paper is structured as follows. We begin by briefly illustrating our method in the context of the RS-I model in Section \\ref{secRSI}. We then move on to detailing our method in the context of an uncompactified $N$-brane model with an infinite extra dimension. In Section \\ref{SecCoordinates}, we specialize our coordinate systems to a convenient gauge, and in Section \\ref{SecLengthscale} we discuss our two-lengthscale expansion in detail. In sections \\ref{SecLowest} and \\ref{SecSecond}, we solve the high energy dynamics to obtain the background metric ansatz, and derive the four-dimensional effective action of the model. We then specialize to one-brane and two-brane cases, before deriving the general $N$-brane action (Section \\ref{secanalysis}). Finally, we discuss the nature of the approximations employed, before summarizing and concluding in Section \\ref{endsection}.\n\n\n\\section{Application of the Method to the Randall Sundrum Model\\label{secRSI}}\nTo briefly illustrate an application of our method, we apply it to the well known case of the Randall Sundrum (RS-I) model with a general background. The derivation of results in this section follows the details on the uncompactified model treated in the remainder of this paper closely.\n\nMany papers have used a metric ansatz for the RS-I model (e.g. \\cite{Randall1999, Chiba2000}), guessing at the form of the five-dimensional metric, and using this to compute the four-dimensional effective action. Such metrics are typically of the form\n\\begin{align}\nds^2 = e^{\\chi (x^c, y)} \\gamma_{ab}(x^c) dx^a dx^b + \\left(\\frac{\\chi_{,y}(x^c, y)}{2k}\\right)^2 dy^2\n\\end{align}\nwhere $k = \\sqrt{-\\kapfs \\Lambda\/6}$. Rather than guessing at the form of the five-dimensional metric, our method derives a five-dimensional metric solution, from which the four-dimensional action is calculated.\n\nThe RS-I model contains two branes on a circular orbifold. We consider the circle of circumference $2L$, with the branes at $y=0$ and $y=L$, with $-L < y < L$. We will let the $y=0$ brane be the Planck brane, and the $y=L$ brane be the TeV brane. Points $y$ and $-y$ are identified. To write this in the language of regions used above, we treat the regions $-L < y < 0$ and $0 < y < L$ as two distinct regions, but identify fields by $\\phi(-y) = \\phi(y)$.\n\nWe now follow the computational steps outlined in Section \\ref{SecResults}.\n\n\\textbf{Step 1.} Write the action in the straight gauge \\cite{Carena2005}. In this gauge, the general metric is given by\n\\begin{align}\nds^2 = e^{\\chi (x^c, y)} \\gamma_{ab}(x^c, y) dx^a dx^b + \\Phi^2(x^c, y) dy^2\n\\end{align}\nwhere $\\det \\gamma = -1$, and we take $\\Phi$ to be positive. The action is given by\n\\begin{align}\nS ={}& \\int d^4 x \\left( \\int_{0^+}^{L^-}dy + \\int_{-L^+}^{0^-}dy \\right) \\sqrt{-g} \\left(\\frac{R^{(5)}}{2 \\kapfs} - \\Lambda \\right) - \\sigma_0 \\int_{{\\cal{B}}_0} d^4 x \\sqrt{-\\tensor[^0]{h}{}} - \\sigma_L \\int_{{\\cal{B}}_L} d^4 x \\sqrt{-\\tensor[^L]{h}{}} \\nonumber\n\\\\\n& + \\frac{1}{\\kapfs} \\int_{{\\cal{B}}_0} d^4 x \\sqrt{-\\tensor[^0]{h}{}} \\left(\\tensor[^0]{K}{^+} + \\tensor[^0]{K}{^-}\\right) + \\frac{1}{\\kapfs} \\int_{{\\cal{B}}_L} d^4 x \\sqrt{-\\tensor[^L]{h}{}} \\left(\\tensor[^L]{K}{^+} + \\tensor[^L]{K}{^-} \\right) + \\tensor[^0]S{_m} \\left[ \\tensor[^0]h{_{ab}}, \\tensor[^0]\\phi{} \\right] + \\tensor[^L]S{_m} \\left[ \\tensor[^L]h{_{ab}}, \\tensor[^L]\\phi{} \\right]. \\label{Rsaction}\n\\end{align}\nThe indices $0$ and $L$ refer to the Planck and TeV branes, respectively. $h_{ab}$ is the four-dimensional induced metric on a brane, and $\\sigma$ is the brane tension. $K^+$ and $K^-$ are the extrinsic curvature tensors on either side of the branes, and $S_m$ is the matter action on each brane.\n\n\\textbf{Step 2.} Now, expand the action \\eqref{Rsaction} to lowest order in the two-lengthscale expansion detailed in Section \\ref{SecLengthscale}. The action to lowest order in this model is given by\n\\begin{align}\nS ={} & \\int d^4 x \\left( \\int_{0^+}^{L^-}dy + \\int_{-L^+}^{0^-}dy \\right) \\sqrt{-\\gamma} \\frac{e^{2\\chi}}{2 \\kapfs \\Phi} \\left( - \\frac{1}{4} \\gamma^{ab} \\gamma_{bc,y} \\gamma^{cd} \\gamma_{da,y} - 5 \\left(\\chi_{,y}\\right)^2 - 4 \\chi_{,yy} + 4 \\frac{\\Phi_{,y}}{\\Phi} \\chi_{,y} - 2 \\kapfs \\Phi^2 \\Lambda \\right) \\nonumber\n\\\\\n&+ \\int d^4 x \\left( \\int_{0^+}^{L^-}dy + \\int_{-L^+}^{0^-}dy \\right) \\lambda (x^a, y) \\left(\\sqrt{-\\gamma} - 1 \\right) + \\int_{{\\cal{B}}_0} d^4 x \\sqrt{-\\gamma} e^{2 \\chi(0)} \\left[ \\frac{2}{\\kapfs} \\left( \\left. \\frac{\\chi_{,y}}{\\Phi} \\right|_{y=0^-} - \\left. \\frac{\\chi_{,y}}{\\Phi} \\right|_{y=0^+}\\right) - \\sigma_0 \\right] \\nonumber\n\\\\\n&\n+ \\int_{{\\cal{B}}_L} d^4 x \\sqrt{-\\gamma} e^{2 \\chi(L)} \\left[ \\frac{2}{\\kapfs} \\left( \\left. \\frac{\\chi_{,y}}{\\Phi} \\right|_{y=L^-} - \\left. \\frac{\\chi_{,y}}{\\Phi} \\right|_{y=-L^+}\\right) - \\sigma_L \\right]\n\\label{rslowaction}\n\\end{align}\nHere, $\\chi(0)$ denotes $\\chi(x^a, 0)$, and similarly for $\\chi(L)$. The second line in this action includes a Lagrange multiplier ($\\lambda$) to enforce the condition $\\det \\gamma = -1$.\n\n\\textbf{Step 3.} Varying the action \\eqref{rslowaction} with respect to the three fields $\\chi$, $\\gamma$ and $\\Phi$, the following equations of motion are obtained.\n\\begin{align}\n0 ={}& \\frac{1}{4} \\gamma^{ab} \\gamma_{bc, y} \\gamma^{cd} \\gamma_{da, y} - 3 \\chi_{, y}^2 - 2 \\kapfs \\Phi^2 \\Lambda\n\\\\\n\\gamma_{ad, yy} ={}& \\gamma_{ab,y} \\gamma^{bc} \\gamma_{cd, y} - \\gamma_{ad, y} \\left( 2 \\chi_{,y} - \\frac{\\Phi_{,y}}{\\Phi}\\right) \\label{rssecondeq}\n\\\\\n0 ={}& \\frac{1}{12} \\gamma^{ab} \\gamma_{bc, y} \\gamma^{cd} \\gamma_{da, y} + \\chi_{, y}^2 + \\chi_{, yy} - \\frac{\\Phi_{,y}}{\\Phi} \\chi_{,y} + \\frac{2}{3} \\kapfs \\Phi^2 \\Lambda\n\\end{align}\nThe following boundary conditions at the branes are also obtained.\n\\begin{align}\n\\gamma_{ab,y}(y=0, L) ={}& 0 \\label{rsfirstboundary}\n\\\\\n\\chi_{,y}(y=0^+) ={}& - \\frac{1}{3} \\kapfs \\sigma_0 \\Phi \\label{rssecondboundary}\n\\\\\n\\chi_{,y}(y=L^-) ={}& \\frac{1}{3} \\kapfs \\sigma_L \\Phi \\label{rsthirdboundary}\n\\end{align}\n\nWe now solve the equations of motion. The solution to \\eqref{rssecondeq} is given by (in matrix notation)\n\\begin{align}\n\\boldsymbol{\\gamma} (x^a, y) ={}& \\mathbf{A}(x^a) \\exp\\left( \\mathbf{B}(x^a) \\int^y_{0} \\Phi(x^a, y^\\prime) e^{-2\\chi(x^a, y^\\prime)} dy^\\prime \\right)\n\\end{align}\nwhere $\\mathbf{A}$ and $\\mathbf{B}$ are arbitrary $4 \\times 4$ real matrix functions of $x^a$, subject to the constraint that $\\boldsymbol{\\gamma}$ is a metric. This can be combined with \\eqref{rsfirstboundary} to yield $\\mathbf{B} = \\mathbf{0}$, and so $\\gamma$ is a function of $x^a$ only. The only remaining equation of motion is then $\\chi_{, y}^2 = - 2 \\kapfs \\Phi^2 \\Lambda \/3$. Defining $k = \\sqrt{-\\kapfs \\Lambda\/6}$, this gives $\\chi_{,y} = \\pm 2 k \\Phi$. Choose the negative solution, so that the brane at $y=0$ corresponds to the Planck brane. The other boundary conditions \\eqref{rssecondboundary} and \\eqref{rsthirdboundary} yield\n\\begin{align}\n\\sigma_0 = \\frac{6k}{\\kapfs} \\ \\ \\ \\ \\mathrm{and} \\ \\ \\ \\ \\sigma_L = - \\frac{6k}{\\kapfs}\n\\end{align}\nwhich are the well known brane tuning conditions. Combining these solutions, the metric solution is then\n\\begin{align}\nds^2 = e^{\\chi (x^c, y)} \\gamma_{ab}(x^c) dx^a dx^b + \\left(-\\frac{\\chi_{,y}(x^c, y)}{2k}\\right)^2 dy^2.\n\\end{align}\n\n\\textbf{Step 4.} We now have the zeroth-order metric solution, which has solved the five-dimensional dynamics. The next step is to use this metric in the original action and integrate over the fifth dimension (c.f. \\cite{Chiba2000}). The zeroth order part of the action integrates to exactly zero, while the remainder of the action (the original second order terms) yields the following four-dimensional effective action.\n\\begin{align}\nS ={}& \\int d^4 x \\frac{\\sqrt{-\\gamma}}{2k \\kapfs} \\left[ \\left(1 - e^{\\chi(L)}\\right) R^{(4)} - \\frac{3}{2} e^{\\chi(L)} (\\nabla^a \\chi(L))(\\nabla_a \\chi(L)) \\right] + \\tensor[^0]S{_m} \\left[ \\gamma_{ab}, \\tensor[^0]\\phi{} \\right] + \\tensor[^L]S{_m} \\left[ e^{\\chi(L)} \\gamma_{ab}, \\tensor[^L]\\phi{} \\right]\n\\end{align}\nThe constraint $\\det \\gamma = -1$ has been relaxed, instead choosing $\\chi(0) = 0$.\n\n\\textbf{Step 5.} Transforming to the Einstein frame, let $g_{ab} = \\left(1 - e^{\\chi(L)}\\right) \\gamma_{ab}$, and define $\\exp(\\chi(x^a, L)\/2) = \\tanh \\left(\\kappa_4 \\varphi(x^a)\/\\sqrt{6}\\right)$. Let $\\kappa_4^2 = k \\kapfs$ be the four-dimensional gravitational scale. The action in the Einstein frame is then given by\n\\begin{align}\nS ={}& \\int d^4 x \\sqrt{-g} \\left[ \\frac{R^{(4)}}{2 \\kappa_4^2} - \\frac{1}{2} (\\nabla^a \\varphi)(\\nabla_a \\varphi) \\right] + \\tensor[^0]S{_m} \\left[ \\cosh^2 \\left( \\frac{\\kappa_4 \\varphi}{\\sqrt{6}} \\right) g_{ab}, \\tensor[^0]\\phi{} \\right] + \\tensor[^L]S{_m} \\left[ \\sinh^2 \\left( \\frac{\\kappa_4 \\varphi}{\\sqrt{6}} \\right) g_{ab}, \\tensor[^L]\\phi{} \\right]. \\label{RSIendaction}\n\\end{align}\nThis action corresponds to the four-dimensional effective action arrived at by other means, such as the covariant curvature formalism \\cite{Kanno2002, Chiba2000}.\\\\\n\nFor the rest of this paper, we confine our discussions to uncompactified $N$-brane models.\n\n\\section{The Five-Dimensional Action in a Convenient Gauge\\label{SecCoordinates}}\nIn this section, we take the action \\eqref{generalaction} and make coordinate choices to simplify the expression. We also separate out contributions due to discontinuities in the connection across branes. We specialize the coordinate system to that of the straight gauge \\cite{Carena2005}, and give the action corresponding to Eq. \\eqref{generalaction} in this gauge. Again, while the details presented here are specific to an uncompactified extra dimension, they generalize straightforwardly to the other situations described previously.\n\nIn general, the five-dimensional Ricci scalar can have distributional components at the branes, as the metric will have a discontinuous first derivative due to the brane tensions. It is convenient to separate these distributional components from the continuous parts. It is further convenient to use separate bulk coordinates $\\tensor*{x}{_n^\\Gamma}$ in each bulk region \\rn, rather than using a single global coordinate system. We will therefore have a bulk metric in each region \\rn, rather than one global metric. We note that the $n^{\\mathrm{th}}$ brane will then have two embedding functions: $\\tensor*[^n]{x}{_n^\\Gamma} (\\tensor*{w}{_n^a})$ in the coordinates $\\tensor*{x}{_n^\\Gamma}$ of \\rn, and $\\tensor*[^n]{x}{_{n+1}^\\Gamma}(\\tensor*{w}{_n^a})$ in the coordinates $\\tensor*{x}{_{n+1}^\\Gamma}$ of ${\\cal{R}}_{n+1}$.\n\nCombining these modifications, we can write Eq. \\eqref{generalaction} as\n\\begin{align}\nS \\left[ \\tensor{g}{_{\\Gamma \\Sigma}}, \\tensor[^n]{x}{^\\Gamma}, \\n\\phi \\right] ={}& \\sum_{n = 0}^N \\int_{\\rn} d^5 \\lrn{x} \\sqrt{-\\n g}\n\\left(\\frac{\\n[^{(5)}]R}{2 \\kapfs} - \\lrn \\Lambda \\right)+ \\sum_{n = 0}^{N-1} \\frac{1}{\\kapfs} \\int_{\\bn} d^4 \\lrn{w} \\sqrt{-\\n h} \\left(\\n[^+]K + \\n[^-]K \\right) \\nonumber\n\\\\\n& {}- \\sum_{n = 0}^{N-1} \\lrn{\\sigma}\\int_{\\bn} d^4 \\lrn{w} \\sqrt{-\\n h} + \\sum_{n = 0}^{N-1} \\n[_m]S[\\n[_{ab}]{h}, \\n\\phi] \\label{coordinateAction}\n\\end{align}\nwhere $\\n[^+]{K}$ is the trace of the extrinsic curvature tensor of the $n^{\\mathrm{th}}$ brane in the bulk region ${\\cal{R}}_{n+1}$, and $\\n[^-]{K}$ is the trace of the extrinsic curvature tensor of the $n^{\\mathrm{th}}$ brane in the bulk region \\rn, where the normals are always defined to be pointing away from the bulk region and towards the brane [see Eqs. \\eqref{normal1} and \\eqref{normal2} below]. These terms are just the usual Gibbons-Hawking terms \\cite{Gibbons1977}.\n\nA note on our conventions. Many functions, coordinates and parameters will be indexed by some index $n$ in this paper. For coordinates and parameters, the index will always be in the lower right, e.g, \\lrn{x}. For functions, the index will be in the upper left, e.g, \\n[_{\\alpha \\beta}]{g}. We use capital Greek letters ($\\Gamma, \\Sigma, \\Theta$) to index five-dimensional tensors in arbitrary coordinate systems. When we specialize our coordinate system, we will use lowercase Greek letters ($\\alpha, \\beta, \\gamma$) to index five-dimensional tensors. We use Roman letters $(a, b, c)$ for four-dimensional tensors. The metric $g$ refers to a five-dimensional metric, while the metric $h$ refers to a four-dimensional metric. Finally, we use the $(-++++)$ metric signature.\n\n\\subsection{Specializing the Coordinate System}\nWe begin by specializing the coordinate systems in each bulk region. Denote the coordinates by $\\tensor*{x}{_n^\\Gamma} = (\\tensor*{x}{_n^a}, \\lrn{y})$, where $a$ indicates one temporal and three spatial dimensions. Without loss of generality, we can choose the coordinates such that the branes bounding the region are located at fixed \\lrn{y}. Next, choose the \\lrn{y} coordinates such that the branes are located at $\\lrn{y} = n-1$ and $\\lrn{y} = n$. In other words, in the brane embedding functions $\\tensor*[^n]{x}{^\\Gamma_n}(\\tensor*{w}{_n^a})$,\n\\begin{align}\n\\tensor[^{n-1}]{y}{_n}(\\tensor*{w}{_{n-1}^a}) ={}& n-1,\n\\\\\n\\tensor[^{n}]{y}{_n}(\\tensor*{w}{_{n}^a}) ={}& n.\n\\end{align}\nIn this way, the first brane will be located at $\\tensor{y}{_0}  = \\tensor{y}{_1} = 0$, and the last brane located at $\\tensor{y}{_{N-1}} = \\tensor{y}{_N} = N-1$. The $n^{\\mathrm{th}}$ bulk region \\rn\\ then extends from $\\tensor{y}{_n} = n-1$ to $\\tensor{y}{_n} = n$, with the exceptions of the first and last bulk regions, which extend away from the branes to $\\mp \\infty$ respectively.\n\nNext, we use some of the available gauge freedom to remove off-diagonal elements of the metrics. Carena et al. \\cite{Carena2005} have shown that it is always possible to find a coordinate transformation in \\rn\\ of the form $\\tensor*{x}{_n^a} \\rightarrow \\tensor*{b}{_n^a} (\\tensor*{x}{_n^c}, \\lrn{y})$ to make $\\n[_{ya}]{g} = 0$ while simultaneously maintaining that the branes be located at $\\tensor{y}{_n} = n-1$ and $\\tensor{y}{_n} = n$. After such a transformation, the metric in \\rn\\ can be written as\n\\begin{align}\n\\n[^2]{ds} ={}& \\n[_{ab}]{\\gamma}(\\tensor*{x}{_n^c}, \\lrn{y}) \\tensor*{dx}{_n^a} \\tensor*{dx}{_n^b} + \\n[^2]{\\Phi}(\\tensor*{x}{_n^c}, \\lrn{y}) \\tensor*{dy}{_n^2} \\label{firstmetric}\n\\end{align}\nwhere the sign of $\\n[_{yy}]{g}$ is known from the signature of the metric. We choose the sign of $\\n\\Phi$ to be positive.\n\nThe brane positions are now hyperplanes located at $y_n = \\mathrm{integer}$. It is obvious that only coordinate transformations for which $y \\rightarrow g(y)$ (with no $x^a$ dependence) can preserve this form for the hyperplanes. With this condition, only coordinate transformations for which $x^a \\rightarrow f^a(x^b)$ will preserve the form of the metric. Thus, the remaining gauge freedom lies in coordinate transformations of the form $x^a \\rightarrow f^a(x^b)$ and $y \\rightarrow g(y)$ such that the positions of the branes are preserved.\n\nFor later simplicity, we choose the following parameterization of the four-dimensional metric $\\n[_{ab}]\\gamma$. In each bulk region, let\n\\begin{equation}\n\\n[_{a b}]\\gamma (\\tensor*{x}{^c_n}, y_n) = e^{\\n\\chi (\\tensor*{x}{^c_n},\ny_n)} \\, \\n[_{a b}]{\\hat{\\gamma}} (\\tensor*{x}{^c_n}, y_n)\n\\end{equation}\nsuch that the determinant of $\\n[_{a b}]{\\hat{\\gamma}}$ is constrained to be $-1$. The function $\\exp(\\n\\chi)$ is sometimes called the warp factor. The metric in \\rn\\ is then\n\\begin{equation}\n\\n[^2]{ds} = e^{\\n\\chi (\\tensor*{x}{^c_n}, y_n)} \\,\\n[_{a b}]{\\hat{\\gamma}} (\\tensor*{x}{^c_n}, y_n) dx_n^a dx_n^b + \\n[^2]{\\Phi} (\\tensor*{x}{^c_n}, y_n) \\tensor*{dy}{_n^2}. \\label{metric}\n\\end{equation}\n\n\\subsection{Embedding Functions, Coordinate Systems on the Branes, and Induced Metrics}\nWe now specialize the coordinate system $\\tensor*{w}{_n^a}$ on the $n^{\\mathrm{th}}$ brane \\bn. We choose the coordinate system on ${\\cal{B}}_0$ to coincide with the first four coordinates of the bulk coordinate system of ${\\cal{R}}_0$, evaluated on the brane. Thus,\n\\begin{subequations}\n\\begin{align}\n\\tensor*[^0]{x}{_0^\\Gamma}(\\tensor*{w}{_0^a}) ={}& (\\tensor*[^0]{x}{_0^a}(\\tensor*{w}{_0^a}), \\tensor*[^0]{y}{_0}(\\tensor*{w}{_0^a}))\n\\\\\n={}& (\\tensor*{w}{_0^a}, 0).\n\\end{align}\nNow, transform the coordinates in the second bulk region by transforming $\\tensor*{x}{_1^a}$ such that\n\\begin{equation}\n\\tensor*[^0]{x}{_1^\\Gamma}(\\tensor*{w}{_0^a}) = (\\tensor*{w}{^a_0}, 0).\n\\end{equation}\nSuch a transformation only requires a mapping of the form $\\tensor*{x}{_1^a} \\rightarrow f^a(\\tensor*{x}{_1^b})$, and so the locations of the branes are preserved. Next, choose a coordinate system $\\tensor*{w}{_1^a}$ on ${\\cal{B}}_1$ such that\n\\begin{equation}\n\\tensor*[^1]{x}{_1^\\Gamma}(\\tensor*{w}{_1^a}) = (\\tensor*{w}{^a_1}, 1)\n\\end{equation}\nand continue this process until all branes and bulk regions have related coordinate systems. The coordinate systems we acquire have the property that for a point ${\\cal P}$ on \\bn, we have\n\\begin{equation}\n\\tensor*[^n]{x}{_{n}^\\Gamma} ({\\cal P}) = \\tensor*[^n]{x}{_{n+1}^\\Gamma} ({\\cal P}). \\label{CoordCondition}\n\\end{equation}\n\\end{subequations}\nNote that while the condition \\eqref{CoordCondition} implies that the coordinate patches can be joined continuously from one region to another in a straightforward manner, they need not form a global coordinate system as they may not join smoothly across the branes.\n\n\nFrom the embedding functions in these coordinate systems we can calculate the induced metric on the branes, using Eq. \\eqref{Embedding}. As each brane is adjacent to two bulk regions, there will be two induced metrics, one from each bulk region. For \\bn, the induced metric from \\rn\\ is\n\\begin{align}\n\\n[_{ab}^-]{h}(\\tensor*{w}{_n^c}) = e^{\\n\\chi (\\tensor*{w}{^c_n}, n)} \\,\\n[_{a b}]{\\hat{\\gamma}} (\\tensor*{w}{^c_n}, n)\n\\end{align}\nwhile the induced metric from ${\\cal{R}}_{n+1}$ is\n\\begin{align}\n\\n[_{ab}^+]{h}(\\tensor*{w}{_n^c}) = e^{\\np\\chi (\\tensor*{w}{^c_{n}}, n)} \\,\\np[_{a b}]{\\hat{\\gamma}} (\\tensor*{w}{^c_{n}}, n)\n\\end{align}\nWe will restrict attention to configurations where the two induced metrics coincide (as would be enforced by the first Israel junction condition \\cite{Israel1966}). We then have\n\\begin{align}\n\\n[_{ab}]{h}(\\tensor*{w}{_n^c}) ={}& \\n[_{ab}^-]{h}(\\tensor*{w}{_n^c}) = \\n[_{ab}^+]{h}(\\tensor*{w}{_n^c})\n\\\\\n\\n[_{ab}]{h}(\\tensor*{w}{_n^c}) ={}& e^{\\n\\chi (\\tensor*{w}{^c_n}, n)} \\,\\n[_{a b}]{\\hat{\\gamma}} (\\tensor*{w}{^c_n}, n) = e^{\\np\\chi (\\tensor*{w}{^c_n}, n)} \\,\\np[_{a b}]{\\hat{\\gamma}} (\\tensor*{w}{^c_n}, n). \\label{firstIsrael}\n\\end{align}\nTaking the determinant of this expression and using the fact that the determinants of $\\hat{\\gamma}_{a b}$ are constrained to be $-1$, we find that\n\\begin{equation}\n\\n\\chi (\\tensor*{w}{^c_n}, n)= \\np\\chi (\\tensor*{w}{^c_n}, n). \\label{metricjunction}\n\\end{equation}\nThen by equation \\eqref{firstIsrael}, it follows that\n\\begin{equation}\n\\n[_{a b}]{\\hat{\\gamma}} (\\tensor*{w}{^c_n}, n) = \\np[_{a b}]{\\hat{\\gamma}} (\\tensor*{w}{^c_n}, n).\n\\label{firstIsraelgamma}\n\\end{equation}\n\n\\subsection{The Action}\nNow that we have specialized the coordinate systems for every region and brane in our model, we can rewrite our action \\eqref{coordinateAction} in terms of these coordinates.\n\nWe can evaluate the extrinsic curvature tensor terms as follows. Each brane has two normal vectors, one each from the two adjacent bulk regions. We define the normal vectors $\\n[^\\pm]{\\vec{n}}$ at \\bn\\ to be the inward pointing normals from ${\\cal{R}}_{n+1}$ and \\rn. Since the branes are at fixed values of the coordinates \\lrn{y}, this gives\n\\begin{equation}\n\\n[^-]{\\vec{n}}(\\tensor*{w}{_n^a}) = \\frac{1}{\\n\\Phi (\\tensor*{w}{_n^a}, n)} \\partial_{\\lrn y} \\label{normal1}\n\\end{equation}\nas the normal vector from \\rn\\, and\n\\begin{equation}\n\\n[^+]{\\vec{n}}(\\tensor*{w}{_n^a}) = -\\frac{1}{\\np\\Phi (\\tensor*{w}{_n^a}, n)} \\partial_{\\tensor*{y}{_{n+1}}} \\label{normal2}\n\\end{equation}\nas the normal vector from ${\\cal{R}}_{n+1}$. The vector $\\n[^-]{\\vec{n}}$ points to the right of bulk region $n$ towards brane $n$, while $\\n[^+]{\\vec{n}}$ points to the left of region $n+1$ towards brane $n$.\n\nFor the extrinsic curvature tensors, we have by definition\n\\begin{align}\n\\n[_{ab}^-]K(\\tensor*{w}{_n^c}) ={}& \\left.\n\\frac{\\partial (\\tensor[^n]{x}{^\\alpha})}{\\partial \\tensor*{w}{_n^a}}\n\\frac{\\partial (\\tensor[^n]{x}{^\\beta})}{\\partial \\tensor*{w}{_n^b}}\n\\tensor{\\nabla}{_\\beta} \\n[_\\alpha^{-}]n \\right|_{\\tensor*{x}{^c_n} = \\tensor*{w}{_n^c}, \\lrn{y} = n} ,\n\\\\\n\\n[_{ab}^+]K(\\tensor*{w}{_n^c}) ={}& \\left.\n\\frac{\\partial (\\tensor[^{n+1}]{x}{^\\alpha})}{\\partial \\tensor*{w}{_n^a}}\n\\frac{\\partial (\\tensor[^{n+1}]{x}{^\\beta})}{\\partial \\tensor*{w}{_n^b}}\n\\tensor{\\nabla}{_\\beta} \\n[_\\alpha^{+}]n \\right|_{\\tensor*{x}{^c_{n+1}} = \\tensor*{w}{_n^c}, \\tensor{y}{_{n+1}} = n}.\n\\end{align}\nEvaluating these using the explicit form of the normals, we have\n\\begin{align}\n\\n[_{ab}^-]K(\\tensor*{w}{_n^c}) ={}& \\frac{1}{2} \\frac{1}{\\n \\Phi} \\left( \\n[_{,y}]{\\chi} e^{\\n\\chi} \\,\\n[_{ab}]{\\hat{\\gamma}} + e^{\\n\\chi} \\,\\n[_{ab,y}]{\\hat{\\gamma}} \\right) (\\tensor*{w}{_n^c}, n) ,\n\\\\\n\\n[_{ab}^+]K(\\tensor*{w}{_n^c}) ={}& -\\frac{1}{2} \\frac{1}{\\np \\Phi} \\left( \\np[_{,y}]{\\chi} e^{\\np\\chi} \\,\\np[_{ab}]{\\hat{\\gamma}} + e^{\\np\\chi} \\,\\np[_{ab,y}]{\\hat{\\gamma}} \\right) (\\tensor*{w}{_n^c}, n).\n\\end{align}\nTo take the trace of the extrinsic curvature tensor, we contract with the inverse induced metric\n\\begin{equation}\n\\n[^{ab}]h = e^{-\\n \\chi} \\,\\n[^{ab}]{\\hat{\\gamma}} = e^{-\\np \\chi} \\,\\np[^{ab}]{\\hat{\\gamma}}.\n\\end{equation}\nWe find\n\\begin{align}\n\\n[^+]K (\\tensor*{w}{_n^c})\n={}& \\left.-\\frac{2 \\;\\np[_{,y}]{\\chi}}{\\np \\Phi}\\right|_{\\tensor*{w}{_n^c}, n}, \\label{ExtrinsicP}\n\\\\\n\\n[^-]K (\\tensor*{w}{_n^c}) ={}& \\left.\\frac{2 \\;\\n[_{,y}]\\chi}{\\n\\Phi}\\right|_{\\tensor*{w}{_n^c}, n}. \\label{ExtrinsicM}\n\\end{align}\nIn deriving these equations, we used the fact that $\\n[^{ab}]{\\hat{\\gamma}} \\; \\n[_{ab, y}]{\\hat{\\gamma}} = 0$, which follows from $\\det (\\n[_{ab}]{\\hat{\\gamma}}) = -1$.\n\n\nFrom Eq. \\eqref{metric}, the determinant of the five-dimensional metric can be written as\n\\begin{align}\n\\sqrt{-\\n g} = \\n \\Phi e^{2 \\, \\n \\chi} \\sqrt{-\\n{\\hat{\\gamma}}}. \\label{metricexpansion}\n\\end{align}\nWe do not substitute $\\sqrt{-\\n{\\hat{\\gamma}}} = 1$ at this stage; instead we choose to enforce this at the level of the action by a Lagrange multiplier (see Appendix \\ref{AppExactEOMS}). Using Eqs. \\eqref{ExtrinsicP}, \\eqref{ExtrinsicM}, and \\eqref{metricexpansion}, the action \\eqref{coordinateAction} can be written as\n\\begin{align}\nS \\left[ \\tensor[^n]{\\hat{\\gamma}}{_{ab}}, \\n\\Phi, \\n\\chi, \\n\\phi \\right] ={} & \\sum_{n = 0}^N \\int_{\\rn} d^5 \\lrn x \\n \\Phi e^{2 \\, \\n \\chi} \\sqrt{-\\n{\\hat{\\gamma}}}\n\\left(\\frac{\\n[^{(5)}]R}{2 \\kapfs} - \\lrn \\Lambda\\right) \\nonumber\n\\\\\n& {} + \\sum_{n = 0}^{N-1} \\frac{2}{\\kapfs} \\int_{\\bn} d^4 \\lrn w e^{2 \\, \\n \\chi(n)} \\sqrt{-\\n{\\hat{\\gamma}}} \\left(\\left.\\frac{\\n[_{,y}]\\chi}{\\n\\Phi}\\right|_{\\lrn y = n} - \\left.\\frac{\\np[_{,y}]\\chi}{\\np\\Phi}\\right|_{\\tensor*{y}{_{n+1}} = n} \\right) \\nonumber\n\\\\\n& - \\sum_{n = 0}^{N-1} \\lrn \\sigma \\int_{\\bn} d^4 \\lrn w e^{2 \\, \\n \\chi(n)} \\sqrt{-\\n{\\hat{\\gamma}}} + \\sum_{n = 0}^{N-1} \\n[_m]S [\\n[_{ab}]h, \\n \\phi]. \\label{CompleteAction}\n\\end{align}\n\n\\subsection{Summary}\nWe summarize our results so far. We have $N$ branes, each with a brane tension which has been carefully adjusted, according to Eq. \\eqref{branetunings}. The branes divide our system into $N+1$ regions. Our coordinates are $x^\\alpha$, describing four-dimensional space, and $y$, describing the extra dimension.\n\nWe expanded the action in terms of our $\\epsilon$ scaling parameter to separate the high and low energy contributions. Specializing to a low energy regime, we solved for the high energy dynamics, arriving at the metric for each region of our system:\n\\begin{align}\n\\n[^2]{ds} = e^{\\n \\chi(x^c, y)} \\hat{\\gamma}_{ab} (x^c) dx^a dx^b + \\frac{\\n[^2_{,y}]\\chi (x^c, y)}{4 k_n^2} dy^2, \\label{metricansatz}\n\\end{align}\nwith $\\n\\chi$ given by Eq. \\eqref{chidef}, where $\\n\\Phi(x^a, y)$ can be chosen freely. The parameters $k_n$ are determined by the bulk cosmological constants and the five-dimensional Newton's constant, by Eq. \\eqref{kndef}. The derivative $\\chi_{,y}$ has fixed sign $P_n = \\pm 1$ in each region, although the derivative may approach zero as $y \\rightarrow \\pm \\infty$.\n\nAs an aside, when the metric in each region is in the form \\eqref{metricansatz}, the zeroth order action $S_0 [g_{ab}]$ [Eq. \\eqref{scaledaction}] evaluates to exactly zero. This can be seen by substituting the metric \\eqref{metricansatz} into the action and explicitly evaluating the integral over the $y$ dimension. All of the integrals become total derivatives whose boundary terms exactly cancel the boundary terms present in the action at this order.\n\nThe background metric ansatz \\eqref{metricansatz} is essentially the same as the zeroth order metric calculated by Kanno and Soda \\cite{Kanno2002}, taking $\\Phi^2(x^a, y) = \\exp(2 \\phi (y, x))$ in their notation. However, from here, we proceed without their assumption that $\\phi(y, x) = \\phi(x)$. The ``na\\\"{\\i}ve'' ansatz and the CGR ansatz of Chiba \\cite{Chiba2000} are also in the form of our metric \\eqref{metricansatz}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe many similarities shared by mesoscopic devices based on quantum dots\n(QD) and atomic or molecular systems justify experiments that can test\nat mesoscopic level effects encountered earlier in molecular physics.\n An important illustration of the above idea is the recent\nexperiment carried out by Kobayashi {\\it et al.} \\cite{F1,F2}\nthat revealed the counterpart of the Fano effect from atomic physics \\cite{Fano}\nin a hybrid mesoscopic system, namely an Aharonov-Bohm  interferometer\nwith an embedded dot. The Fano effect is a resonance produced by\nthe quantum interference between a discrete state embedded in a continuous spectrum and\nthe continuous spectrum. In Ref. \\cite{F1}\nthe discrete level is provided by the resonant transport through the QD,\nand the continuous background comes from the free arm of the ring.\n\nThe mesoscopic Fano effect leads to a series of\nasymmetric conductance peaks appearing as the gate voltage pushes the\ndot levels across the Fermi energy of the leads. The experiment of\nKobayashi {\\it et al.} presented even more interesting features\n: i) The orientation of the Fano line shape changes periodically with the magnetic flux.\n ii) Between two Fano line shapes the conductance exhibits clear AB oscillations\nproving the transport coherency. iii) The phase of the oscillations changes\nby $\\pi$ across a resonance.\n\nThe Fano effect in AB rings with quantum dots was predicted theoretically\nby Bu{\\l}ka {\\it et al.} \\cite{BS} The conductance of a ring with a single level\nQD was computed in the Keldysh formalism by Hofstetter {\\it et al.}\n\\cite{HKS} and the interplay between the Fano and Kondo effects has\nbeen discussed.\nLater on the Fano interference was discussed in the context of phase measurements\nin Aharonov-Bohm interferometers \\cite{AEY,AEI,EAIL}.\nThe calculations were done for a triangular tight-binding interferometer for\nwhich the scattering matrix can be computed exactly. Like the quantum dot, the reference\narm of the ring was restricted to a single site.\nA similar model was considered by Ueda {\\it et al.} \\cite{Ueda}\nKim and Hershfield \\cite{KH} utilized a more complex description of the quantum dot\nfor studying the role of the sign in the tunneling matrix on the transmittance zeros and phase.\n\nIn the present work we attempt a theoretical description of the\nFano interference for a two-dimensional\ndot and a much larger ring (in Ref. \\cite{F1} the dot accommodates up to 80\nelectrons while the ratio between the area of the ring and\nthe dot area is around 56). There are two reasons for this generalization.\nFirst, one can describe the effects of the magnetic field on the dot levels and\nsecondly, the investigation of finite size effects on the Fano interference observation\n is possible. Another aim of our study is to discuss the tuning of the\n Fano resonance by varying the magnetic field.\n The formalism we used to calculate the conductance has been described in\nRef. \\cite{MTAT} and is summarized in Section II. It relies on the\nLandauer-B\\\"{uttiker} formula and is similar to the scattering approach developed\nby Hackenbroich and Weidenm\\\"{u}ller. \\cite{HW2,Ha}\n\nSince the Coulomb repulsion plays an important role in QD systems it is natural\nto question about correlations effects. In the first experiment of\nKobayashi {\\it et al.} \\cite{F1} the Kondo effect was not detected but in a\nrecent work \\cite{F3} a Fano-Kondo anti-resonance was observed in the case of a\nquantum wire with a side-coupled QD, provided the lead-dot coupling is strong.\nThis issue was discussed theoretically in \\cite{MSU,OAY}.\nFor weak coupling the correlation between the localized spins in the dot\nand the incident ones from the leads is unlikely and one recovers\nthe Fano line shapes.\n\nIn the present work we consider only weak dot-ring couplings as in Ref.\\cite{F1}\nand we include neither the spin nor the electron-electron interaction.\nSection III contains the numerical results and their\ndiscussion. Section IV concludes the paper.\n\n\\section{The model}\n\nWe consider a mesoscopic ring connected to electron reservoirs by two\n semi-infinite leads. A noninteracting quantum dot is\nembedded in the upper arm of the ring. The system is schematically shown\nin Fig.\\ 1, indicating as well the\nlabeling of the leads and the contact points. We use here a tight-binding representation\nso the dot is modeled by a discrete lattice. The Aharonov-Bohm interferometer (I)\n is formed by coupling two sub-systems: a truncated ring (R) and the dot (D).\n Its Hamiltonian is then conveniently written in matrix form:\n\\begin{eqnarray}\nH^I=\n\\begin{pmatrix}\n      H^D & H^{DR}\\cr\n      H^{RD} & H^R\\cr\n\\end{pmatrix},\n\\end{eqnarray}\nwhere $H^{DR}$ and $H^{RD}$ are the ring-dot and dot-ring coupling terms:\n\\begin{equation}\nH^{DR}+H^{RD}\n=\\tau\\sum_{m=a,b}(e^{-i\\varphi _m }|m\\rangle\\langle 0m|+h.c).\n\\end{equation}\nThe notation $0m$ stands for the site of the ring which is the closest\none to the dot.\n$\\tau$ gives the ring-dot coupling and\n  $\\varphi _m$ is the Peierls phase associated with the pair of sites\n$|0m\\rangle$, $|m\\rangle$. Similar Peierls phases enter in both $H^D$ and $H^R$.\nWe do not give explicitly the tight-binding Hamiltonian $H^D$ of the dot.\nWe just stress that it contains an on-site constant $-eV_g$\nsimulating the plunger voltage used in experiments. We take $e=h=1$.\n\nThe conductance (i.e transmittance) of one-dot AB interferometer was obtained via the\n Landauer-B\\\"{u}ttiker formalism in Ref. \\cite{MTAT}\nDenoting by $\\tau_L$ the lead-ring coupling and by $E_F$ the Fermi energy on leads\none gets the following\nformula for $\\alpha\\neq\\beta$ at $T=0K$\n(see Ref. \\cite{MTAT} for further details):\n\\begin{eqnarray}\\label{gab}\ng_{\\alpha\\beta}(E_F)=4\\tau_L^4\\sin^2k \\left |\n{\\tilde G}^R_{\\alpha\\beta}+\n\\tau^2e^{i\\theta_{mn}}\n{\\tilde G}^R_{\\alpha m}{\\tilde G}^D_{mn}{\\tilde G}^R_{n\\beta}\\right |^2,\n\\end{eqnarray}\nwhere $\\theta_{mn}=\\varphi_m-\\varphi_{n}$, summations over $m,n$ are understood\nand we introduced two effective Green functions describing {\\it individually} the dot\nand the truncated ring:\n\\begin{eqnarray}\\label{GReff}\n{\\tilde G}^R_{\\alpha m}(z)&:=& \\langle \\alpha|(H^R-\\Sigma^L(z)-z)^{-1}\n|0m\\rangle\\\\\\label{H_eff}\n{\\tilde G}^D_{mn}(z)&:=&\\langle m|(H^D-\\Sigma^D(z)-z)^{-1}|n\\rangle.\n\\end{eqnarray}\nHere $\\Sigma^L(z)$ is the leads' self-energy and\n$\\Sigma^D(z):=H^{DR}(H^R-\\Sigma^L-z)^{-1}H^{RD}$ is the self-energy due to the coupling to the ring. Formula (\\ref{gab}) allows us to numerically compute $g_{\\alpha\\beta}$.\nThe first term represents the free transport via the lower arm and the second term\nembodies {\\it all} the trajectories that involve at least one resonant tunneling.\nThus Eq. (\\ref{gab}) is more complicated than what one gets in a two-slit experiment.\nThe mesoscopic Fano effect differs in this respect to the original Fano effect\nwhich involves only two interfering contributions.\n\nSome of the features of the single dot interferometer\ncan be understood better if we temporarily specialize Eq. (\\ref{gab})\nto a triangular interferometer which is exactly solvable. More precisely,\nthe quantum dot is a single site (thus $0a=\\alpha, 0b=\\beta, a=b$) and the reference arm\nis reduced to the lead ends $\\alpha$ and $\\beta$.\nThen the conductance can be presented in the generalized Fano-like form:\n\\begin{equation}\\label{fano}\ng_{\\alpha\\beta}(E_F)=\\frac{4\\tau_L^4\\sin^2k}{|a^2-1|^2}\\cdot\\frac{|\\varepsilon +q|^2}{1+\\varepsilon ^2},\n\\end{equation}\nwith the following notations:\n\\begin{eqnarray}\n\\varepsilon &=&\\frac{E-V_g-E_F+{\\rm Re}A}{{\\rm Im}A},\nq=-\\frac{\\tau^2e^{-2\\pi i\\phi}+{\\rm Re}A}{{\\rm Im}A}\\\\\nA &=&\\frac{2\\tau^2(\\cos(2\\pi\\phi)-2)}{a^2-1},\\quad\na=-\\tau_L^2e^{-ik}-E_F.\n\\end{eqnarray}\nNotice that here the Fano parameter $q$ is complex and not real as in \\cite{Fano},\na fact that was suggested in the fitting conductance formula proposed in Ref.\\cite{F2}\nIt is easily seen that $q$ is a periodic function of flux.\n\nA complex $q$ was derived for a multichannel double barrier in \\cite{Wu}\nand also by Nakanishi \\cite{Nak} under a double-slit condition.\n Alternative calculations for exactly solvable models were given in \\cite{HKS,Ueda,KH}.\nIn particular, the Fano parameter obtained in Ref. \\cite{Ueda} lies on an ellipse in\nthe complex plane with the center at origin, while according to\nKobayashi {\\it et al.} \\cite{F2} the center of the ellipse is shifted from the origin.\nThis shift is captured within our model. Indeed, by straightforward calculations\none gets explicit formulae for the real and imaginary parts of the Fano parameter \\cite{ReIm}.\nClearly $q$ lies on an ellipse with the center shifted on the real axis.\nThe shift is due to the second term from ${\\rm Re}(q)$ and disappears\nonly for $E_F=0$.\nWe believe the difference from previous calculations comes from the fact\nthat in our approach the leads' self-energy $\\Sigma^L$\ndepends explicitly on energy, while in Refs.\\cite{HKS,Ueda,KH} this quantity is energy independent.\nThe real part which is responsible for the\nasymmetry of the line shape behaves roughly as $\\cos2\\pi \\phi$\n and ${\\rm Im }(q)$ as $\\sin2\\pi \\phi$ which\nagrees qualitatively with Fig.\\ 6b from Ref.\\cite{F2}.\nThe condition for a symmetric peak is achieved twice in a flux period since\n${\\rm Re}(q)$ vanishes twice.\n\n\n\\section{Results and discussion}\n\nFollowing the setup\nfrom Ref. \\cite{F1} we consider a quantum dot having 7$\\times$8 sites while the 1D ring\ncontains 140 sites. The uniform perpendicular magnetic field pierces both the ring and the QD.\nThroughout the paper the magnetic flux piercing the ring will be denoted by $\\phi$\nand expressed in quantum flux units $\\Phi_0$. Figure 2 shows several Fano line shapes of the transmittance\n$g_{\\alpha\\beta}$ as a function of the gate potential on the dot $V_g$, for fixed magnetic flux.\nThe lead-ring coupling $\\tau_L=1$.\n\nNow we look for the Aharonov-Bohm oscillations of the transmittance in order to identify\nspecific properties due to the 2D character of the dot.\nTo this end we fix first an initial magnetic flux and choose two gate\npotentials whose associated transmittance values are located on different sides of a Fano\npeak. Then we keep the gate potentials unchanged and vary instead the flux through the ring.\nThe transmittances assigned to the two gate potentials (Fig.\\ 3a) show AB\noscillations, proving that the transport through the system is coherent.\nWe notice that in the flux interval $[5,10]$ the two oscillations are not in-phase,\nwhile for $\\phi\\in [10,15]$ they are in-phase. Fig.\\ 3b helps us to explain this fact.\nIt shows that as $\\phi$ increases the Fano line moves to the right (as we shall argue below,\nthis shift is due only to the magnetic field dependence of the resonant level of the 2D dot).\nConsequently, the transmittance value corresponding to $V_g=-2.680$ passes gradually from one side\nof the peak to the other. The irregularity seen in Fig.\\ 3a for $\\phi\\in [10,12]$ is due\nto the passing through the Fano peak. When the magnetic flux is chosen such that the transmittance\nvalue for $V_g=-2.680$ is still on the right side of the peak, its Aharonov-Bohm oscillation\nis in anti-phase with the one for $V_g=-2.694$. This happens because the phase of the QD transmittance\njumps by $\\pi$ on resonance. If $\\phi$ is further increased the resonant level of the dot moves\nin magnetic field and the corresponding Fano line is pushed to the right so that the transmittance\nvalue for $V_g=-2.680$ goes to the left side of the peak.\nThen clearly the two AB oscillations are in-phase. Thus the phase of the AB oscillation can be changed not only by varying the gate potential but also by varying the magnetic flux through\nthe ring. This effect is important only if the dot levels depend in a sensible way on the magnetic flux.\n\nAnother effect coming from the spectral properties of the 2D quantum dot is the magnetic drift of the\nFano line, as already shown in Fig.\\ 3b. In order to see this more clearly\nwe have monitored the position of the Fano peak (in $V_g$ units) as function of\nthe magnetic flux in Fig.\\ 4a (the solid line). Its trajectory is rather complex:\n the global shift towards higher gate voltages is modulated by a $\\Phi_0$-periodic\noscillation. The shift comes from the magnetic field dependence\nof the dot levels. Indeed, the dotted line shows the trajectory of the eigenvalue of\nthe isolated dot which is associated to the Fano resonance located in the range $[-2.8,-2.6]$\n(see Fig.\\ 2). Clearly, its positive slope is responsible for the magnetic drift of the Fano\npeak position. Note also that the peak positions correspond to gate voltages\n$V_g\\sim E_i-E_F$. This is roughly the resonant condition for the tunneling through\nthe dot. The dotted line represents the\ntrajectory of the symmetric peak that appears when the lower arm of the ring is disconnected\n(i.e $\\tau=0$). In this case there is no interference, the Peierls phases disappear from\nEq. (\\ref{gab}) and the only flux dependence comes from the denominator of the effective\nGreen function. Because of this the symmetric peak position has a simple drift. The above\ndiscussion shows also that the additional modulation of the Fano peak position\nis due entirely to the other flux dependences in Eq. (\\ref{gab}).\n\nA parameter which provides criteria for the experimental observation of the two magnetic\nfield effects reported above is the ratio $R$ between the area of the ring $A_r$ and\nthe area of the dot $A_d$. For the system composed of a $7\\times8$ sites dot and an\n140 sites ring $R=20$ while in the experiments of Kobayashi {\\it et al.} $R\\sim 56$.\nFig.\\ 4b shows the behavior of the Fano peak position for $R\\sim 44$ and $R\\sim 70$.\nWe kept the dot dimension fixed and increased the length of the ring; for $R\\sim 44$\nthe ring has 200 sites and for $R\\sim 70$ the ring has 250 sites.\nThe data from Fig.\\ 4a were added in Fig.\\ 4b in order to make the comparison easier.\nIn the case of a 250 sites ring the Fermi level was set to $E_F=0.02$ while in\nthe other two cases $E_F=0.05$ (this causes the large shift of the peak position for\nthe 250 sites ring). There are two things to be noticed: i) as the ring gets bigger\nthe oscillation of the peak position becomes simpler - the two local maxima located on\neach side of the absolute maxima for $R\\sim 20$ are pushed towards integer flux values\nfor $R\\sim 44$ and vanish completely at $R\\sim 70$. ii)\nThe magnetic drift of the peak position is weaker for $R\\sim 44$\n(one can notice it only at rather large fluxes) and disappears for $R\\sim 70$.\nThis is easily understood since for bigger ring a flux quanta means smaller magnetic\nfields inside the dot, thus a slower deviation of its eigenvalues. Having in mind\nthe ratio $R\\sim 56$ from Ref. \\cite{F1} it is clear that\nthe oscillations like the ones depicted in Fig.\\ 4a and the magnetic drift of\nthe Fano lines were not observed due to the large ratio $R$.\n\nIn the following we discuss the magnetic field dependence of the Fano interference.\nWe look first at the Fano dip position in magnetic field.\nThe solid curve in Fig.\\, 5 gives the dip trajectory. As already\nexplained for the simple triangular interferometer the Fano peak has two domains\nof symmetry between consecutive integer flux values.\nIn this case the Fano dip cannot be defined and the eye-guiding dashed line marks the\nsudden change of the dip position from one side of the resonance to the other.\n\nThe next step of our analysis is to study the phases of the two complex terms\ncontributing in Eq. (\\ref{gab}). We shall denote by $\\varphi_{bg}$ the phase\n  of the background contribution ${\\tilde G}^R_{\\alpha\\beta}$ while\n$\\varphi_{res}$ stands for the phase of the second term. We stress that\n$\\varphi_{res}$ cannot be simply identified with the intrinsic phase of\nthe quantum dot since it is the phase accumulated along multiple bouncing\ntrajectories in the inteferometer.\nFig.\\, 6a shows the evolution of a Fano line shape from Fig. 2 for other two values of\nthe magnetic flux. The asymmetric tail changes periodically with the magnetic flux as\nreported in Ref. \\cite{F1}. More precisely, the Fano dip located on the right side of\na transmittance peak for $\\phi=5.0$ moves to the other side as the flux is increased by a\nhalf flux quanta and comes back on the RHS when $\\phi=6.0$. The filled circles mark the\ntransmittance values at $V_g=-2.9616$. This value was chosen simply because, at $\\phi=5.0$,\nit corresponds to a Fano peak. Fig.\\ 6b depicts the phases $\\varphi_{bg}$, $\\varphi_{res}$ and\ntheir difference $\\Delta\\phi=\\varphi_{bg}-\\varphi_{res}$\nas a function of flux while keeping $V_g=-2.9616$. The phase $\\varphi_{bg}$ is simply a\nlinear function of flux, decreasing by $\\pi$ as one more flux quanta pierces the ring.\n$\\varphi_{res}$ presents instead a weak oscillation and increases by $\\pi$, such that the\nphase difference decreases roughly by $2\\pi$. The investigation of Fig.\\ 6b allows us to\nexplain the transition from the Fano peak at $\\phi=5.0$ to a Fano valley at  $\\phi=5.5$\nand finally the recovery of the constructive interference at $\\phi=6.0$. It is easy to see\nthat the modulus of ${\\tilde G}^R_{\\alpha\\beta}$ in Eq. (\\ref{gab}) does not depend on\nthe magnetic field thus it is a constant complex vector which rotates clockwise according to\nits flux-dependent phase $\\varphi_{res}$.\nThe vector describing the second term in Eq. (\\ref{gab}) depends on $\\phi$ both in\namplitude and phase and rotates anti-clockwise (not shown).\nAt $\\phi=5.0$,  ${\\tilde G}^R_{\\alpha\\beta}$ lies almost on the\npositive real axis while the resonant term is in the 4-th quadrant, its phase being close to\n$\\pi\/2$. The real parts of the two contributions are both positive and the imaginary\nones negative,\nthus they interfere constructively justifying the Fano peak at $\\phi=5.0$. In contrast, for $\\phi=5.5$\nthe background vector lies on the negative imaginary axis while the resonant one comes close to the\npositive imaginary axis. Clearly the interference is destructive and the transmittance assigned to\n$V_g=-2.6916$, $\\phi=5.5$, is located in a Fano valley. Using similar arguments one can\nexplain the situation from $\\phi=6.0$. We have checked that the results mentioned above\nhold as well for all other Fano resonances from Fig.\\ 2.\n\n\n\\section{Conclusions}\n\nWe have studied the mesoscopic Fano effect in an Aharonov-Bohm ring with a two-dimensional\nquantum dot. We have considered the effect of the magnetic field on the orbital motion of\nelectrons inside the dot. It is shown that this dependence leads to a global shift\nof the Fano lines as a function of the magnetic field.\nIt would be interesting to perform experiments with bigger dots in order to observe this effect.\nHowever, if the dot area is small compared to the ring area as in Ref. \\cite{F1}\nor if the magnetic field is weak this shift can be neglected.\nWe emphasized that a continuous variation of the AB oscillation phase by\n$\\pi$ along one Fano resonance can be also obtained by varying the magnetic flux and\nkeeping the gate potential fixed. The origin of this effect lies again in the\nspecific spectral properties of the two-dimensional dot.\n\nThe mechanism leading to the magnetic control of the\nFano interference reported in the experiments of Kobayashi {\\it et al.} \\cite{F1}\nwas described by a detailed analysis of the phases of two complex contributions appearing in the\nconductance formula: the reference transmittance of the free arm and {\\it all} possible\ntunneling processes through the dot.\n\n\n\n\n\\acknowledgments{\nV.\\,M. was supported by a NATO Science Fellowship.\nWe acknowledge instructive discussions with Prof. B. R. Bu{\\l}ka. }\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe black hole stability problem, i.e. the problem of proving dynamical\nstability for the Kerr family of black hole spacetimes, is one of the central\nopen problems in General Relativity. The analysis of linear test fields on\nthe exterior Kerr spacetime is an important step towards the full non-linear\nstability problem. For test fields of spin 0, i.e. solutions of\nthe wave equation $\\nabla^a \\nabla_a \\psi = 0$, estimates proving boundedness\nand decay in time are known to hold. See \n\\cite{finster:etal:2006:MR2215614,dafermos:rodnianski:2010arXiv1010.5137D,andersson:blue:kerrwave,tataru:tohaneanu:2011}\nfor references and background. \n\nThe field equations for linear test fields of spins 1 and 2 are the \nMaxwell and linearized gravity\\footnote{Note that linearized gravity is distinct from the massless spin-2 equation. On a type D background, any solution to the massless spin-2 equation is proportianal to the Weyl tensor of the spacetime. This fact is\nreferred to as the Buchdahl constraint, cf. \\cite{Buchdahl:1958xv}, see also\nequation (5.8.2) in \\cite{PR:I}.} equations, respectively. \nThese equations imply wave equations for the Newman-Penrose Maxwell and\nlinearized Weyl scalars. \nIn particular, the Newman-Penrose scalars of spin weight zero\nsatisfy (assuming a suitable gauge condition for the case of linearized\ngravity) analogs of the Regge-Wheeler equation.\nThese wave equations take the form \n$$\n(\\nabla^a \\nabla_a + c_s \\Psi_2) \\psi_s = 0\n$$\nwhere for spin $s=1$, $c_1 = 2$, $\\psi_1 = \\Psi_2^{-1\/3} \\phi_1$, while for\nspin $s=2$, $c_2 = 8$,  \nand $\\psi_2 = \\Psi_2^{-2\/3} \\dot \\Psi_2$. Here $\\dot \\Psi_2$ is the\nlinearized Weyl scalar of spin weight zero. See\n\\cite{aksteiner:andersson:2011} for details. \nAs these scalars can \nbe used\nas potentials for the Maxwell and linearized Weyl fields, \none may apply the\ntechniques developed in the previously mentioned papers to prove estimates\nalso for the Maxwell and linearized gravity equations. This approach has been\napplied in the case of the Maxwell field on the Schwarzschild background in\n\\cite{blue:maxwell}. \n \nIn contrast to the spin-0 case, the spin 1 and 2 field equations on the Kerr exterior admit non-trivial finite energy time-independent solutions.\nWe shall refer to time-independent solutions as \nnon-radiating modes. There\nis a close relation between gauge-invariant \nnon-radiating modes and conserved charge integrals. For the Maxwell field, \nthere is a two-parameter family of  \nnon-radiating, Coulomb type solutions which carry the two\nconserved electric and magnetic charges. \nIn fact, \na Maxwell field on the Kerr exterior will disperse exactly when it has\nvanishing charges. For linearized gravity, however, there\nare both \nnon-radiating modes corresponding to \ngauge-invariant \nconserved\ncharges, and ``pure gauge'' non-radiating modes. \nThus conditions ensuring\nthat a solution of\nlinearized gravity will disperse must be a combination of charge-vanishing\nand gauge conditions. \n\nFrom the discussion above, it is clear that in order to prove \nboundedness and decay for higher spin test\nfields on the Kerr exterior,\nit is a necessary step to eliminate the non-radiating modes. Due in part to\nthis additional difficulty, decay estimates for the higher spin fields have\nbeen proved only for Maxwell test fields. See \\cite{blue:maxwell} for the\nSchwarzschild case and \\cite{andersson:blue:nicolas:maxwell} for the Kerr\ncase. \nIn view of\nthe just mentioned relation between non-radiating modes and charges, \nan essential step in doing so involves setting conserved charges to zero. \nIn order to make\neffective use of such charge vanishing conditions, it is necessary to have\nsimple expressions for the charge integrals in terms of the field strengths. \nThe main result\nof this paper is to provide an expression for the conserved charge corresponding to the\nlinearized mass, in terms of linearized curvature quantities on the Kerr\nbackground. \n\nWe start by discussing the relation between charges and non-radiating modes\nfor the case of the Maxwell field. \nLet the symmetric valence-2 spinor \n$\\phi_{AB}$ be the Maxwell spinor\\footnote{The following discussion is in terms of the 2-spinor formalism,\ncf. \\cite{PR:I, PR:II}}, \ni.e. a\nsolution of the massless spin-1 (source-free Maxwell) equation \n$$\n\\nabla_{A'}{}^A \\phi_{AB} = 0\n$$\nand let $\\mathcal{F}_{ab} = \\phi_{AB} {\\epsilon}_{A'B'}$ be the corresponding complex\nself-dual two-form. The Maxwell equation takes the form $d\\mathcal{F} = 0$ and hence the charge integral \n$$\n\\int_{S} \\mathcal{F}\n$$\ndepends only on the homology class of the surface $S$. Here real and imaginary parts correspond to electric and magnetic charges,\nrespectively. The Kerr exterior, being diffeomorphic to\n$\\mathbb R^4$ with a solid cylinder removed, contains\ntopologically non-trivial 2-spheres, and hence the Maxwell equation on the\nKerr exterior admits solutions with non-vanishing charges. In view of the\nfact that the charges are conserved, it is natural that there is a\ntime-independent solution which ``carries'' the charge. In Boyer-Lindquist\ncoordinates, this takes the\nexplicit form \n\\begin{equation}\\label{eq:coloumb}\n\\phi_{AB} = \\frac{c}{(r-ia\\cos\\theta)^2} \\iota_{(A} o_{B)} \\, ,\n\\end{equation}\nwhere $c$ is a complex number, and $\\iota_A, o_A$ are principal spinors for\nKerr. \n\nIn order to prove boundedness and decay for the Maxwell field, it is\nnecessary to make use of the above mentioned facts, see\n\\cite{andersson:blue:nicolas:maxwell}. \nIn particular, one eliminates the non-radiating modes by imposing the charge\nvanishing condition \n\\begin{equation}\\label{eq:Fchargezero}\n\\int_{S} \\mathcal{F} = 0 \\, .\n\\end{equation}\nWritten in terms of the Newman-Penrose scalars $\\phi_I$, $I = 0,1,2$, the charge vanishing condition \\eqref{eq:Fchargezero} in the Carter tetrad \\cite{Znajek:1977} takes the form \\cite{andersson:blue:nicolas:maxwell} \n\\begin{equation}\\label{eq:integrability}\n\\int_{S^2(t,r)}  2 V_L^{-1\/2} \\phi_1 \n+ ia\\sin\\theta (\\phi_0 - \\phi_2 ) \\d\\mu = 0 \\, ,\n\\end{equation} \nwhere $S^2(t,r)$ is a sphere of constant $t,r$ in the Boyer-Lindquist\ncoordinates, $V_L = \\Delta\/(r^2+a^2)^2$ and $\\d\\mu=\\sin\\theta\\d\\theta\\d\\varphi$. \nThis yields a relation between the $\\ell = 0, m=0$ spherical harmonic of \n$\\phi_1$ and the $\\ell=1, m=0$ spherical harmonics with spin weights $1$, $-1$\nof $\\phi_0$, $\\phi_2$, respectively. \n\nNext, we consider the spin-2 case. Recall that the Kerr spacetime is a vacuum space of Petrov type D and hence, in addition to the Killing vector fields $\\partial_t, \\partial_\\phi$ admits a ``hidden symmetry'' manifested by the existence of the \nvalence-2 Killing spinor $\\kappa_{AB} = \\psi \\, \\iota_{(A} o_{B)}$. \nHere the scalar $\\psi$ is determined up to a constant, \nwhich we fix by setting\\footnote{This choice has the natural (non vanishing) Minkowski limit $\\psi = r$.} $\\massADM\\psi^{-3} = -\\Psi_2$ on a Kerr background.\nIn this situation, one may consider the spin-lowered version \n$$\n\\psi_{ABCD} \\kappa^{CD} \n$$ \nof the Weyl spinor, which is again a massless spin-1 field and hence the complex\nself-dual two-form\n$$\n\\MM_{ab} = \\psi_{ABCD}\\kappa^{CD} {\\epsilon}_{A'B'}\n$$ \nsatisfies the Maxwell equations $d\\MM = 0$. \nThe charge for this field defined on any topologically non-trivial 2-sphere\nin the Kerr exterior is\n\\begin{equation}\\label{eq:mass-charge}\n\\frac{1}{4\\pi i} \\int_{S} \\MM = \\massADM \\, ,\n\\end{equation}\ncf. \\cite{Jezierski:Lukasik:2006} for a tensorial version (the calculation\nhas been done much earlier in \\cite{jordan:ehlers:sachs:1961:II}, but not in\nthe context of Killing spinors and \nspin-lowering). \nHere $\\massADM$ is the ADM mass \\cite{ADM:1962:MR0143629}\nof the Kerr spacetime\\footnote{Equivalently, the mass parameter in the Boyer-Lindquist form of the Kerr line element.}. The relation between the mass and charge for \nthe spin-lowered Weyl tensor $\\MM$ is natural in view of the fact that the divergence \n$$\n\\xi^{A'A} = \\nabla^{A'}{}_B \\kappa^{AB}\n$$\nis proportional to $\\partial_t$, see the discussion in \\cite[Chapter 6]{PR:II}. \n\nNote that the charge \\eqref{eq:mass-charge} is in general complex. The\nimaginary part corresponds to the NUT charge, which is the gravitational\nanalog of a magnetic charge. \nDetails are not discussed in this paper, see\n\\cite{Ramaswamy:Sen:1981} for the construction of charge integrals in NUT\nspacetime. \n\nFor linearized gravity on the Kerr background, the non-radiating modes include\nperturbations within the Kerr family, i.e. infinitesmal changes of mass and\naxial rotation speed. We denote the parameters for these \ndeformations $\\dot \\massADM, \\dot \\angmom$. Since $\\massADM, \\angmom$ are\ngauge-invariant quantities, it is not possible to eliminate these modes by \nimposing a gauge condition. \nA canonical\nanalysis along the lines of \\cite{iyer:wald:1994PhRvD..50..846I}, see below, \nyields conserved charges corresponding to the Killing fields\n$\\partial_t, \\partial_\\phi$, which in turn correspond to the gauge invariant\ndeformations $\\dot \\massADM, \\dot \\angmom$ mentioned above.\n\nThe infinitesimal boosts, translations and\n(non-axial) rotations of the black hole yield further non-radiating\nmodes which are, however, ``pure gauge'' in the sense that they are generated\nby infinitesimal coordinate changes. \nIf one imposes suitable regularity\\footnote{The Kerr family of\nline elements may be viewed as part of the type D family of vacuum metrics\nwhich includes, among others, the NUT and C-metrics. See section \\ref{sec:curved} for further discussion. The perturbations corresponding\ne.g. to infinitesimal deformations of the NUT parameter are singular and may\nthus be exluded by suitable regularity and decay conditions. See\n\\cite{virmani:2011PhRvD..84f4034V}, \\cite{jezierski:1995GReGr..27..821J} for\nremarks.} conditions on the perturbations\nwhich exclude e.g. those which turn on the NUT charge, a 10-dimensional\nspace of non-radiating modes remains. This is spanned by\nthe 2-dimensional space of non-gauge modes which carry the $\\dot \\massADM,\n\\dot \\angmom$ charges, together with the \n``pure gauge'' non-radiating modes, and \ncorresponds in a natural way to the \nLie algebra of the Poincare group.\nIt can be seen from this discussion that \na combination of charge vanishing conditions and gauge conditions\nallows one to eliminate all non-radiating solutions of linearized gravity. \n\nThe constraint equations implied by the Maxwell and linearized gravity\nequations are underdetermined elliptic systems, and therefore admit\nsolutions of compact support, see \\cite{delay:2010arXiv1003.0535D} and\nreferences therein. In particular, one may find solutions of the constraint\nequations with arbitrarily rapid fall-off at infinity. The corresponding\nsolutions of the Maxwell equations have vanishing charges. For the case of\nlinearized gravity, the charges corresponding to $\\dot \\massADM, \\dot\n\\angmom$ vanish for solutions of the field equations with rapid fall-off\nat infinity. For such solutions, all non-radiating modes may\ntherefore be eliminated by imposing suitable gauge conditions. \n\nThe following discussion may\neasily be extended to the Einstein-Maxwell equations. \nGiven an asymptotically flat vacuum spacetime\n$(N, g_{ab})$,\na solution of the linearized Einstein equations $\\dot g_{ab}$ (satisfying suitable asymptotic conditions) and a \nKilling field $\\xi^a \\partial_a$ we have that \nthe variation of the Hamiltonian current is an exact form, which yields \nthe relation \n\\begin{equation}\\label{eq:xicharge} \n\\dot \\PP_{\\xi; \\infty} = \\int_S \\dot \\bQ[\\xi] - \\xi \\cdot \\mathbf \\Theta \\, .\n\\end{equation} \nHere, $\\PP_{\\xi; \\infty}$ is the Hamiltonian charge at infinity, \ngenerating the action of $\\xi$, $\\bQ[\\xi]$ is the Noether charge two-form for\n$\\xi$, and $\\mathbf \\Theta$ is the symplectic current three-form, defined with respect\nto the variation $\\dot g_{ab}$. We use a $\\dot{\\ }$ to denote variations\nalong $\\dot g_{ab}$, thus $\\dot \\PP_{\\xi;\\infty}$ and $\\dot \\bQ[\\xi]$ denote the\nvariation of the Hamiltonian and the Noether two-form, respectively. The\nintegral on the right hand side of \\eqref{eq:xicharge} is evaluated over an\narbitrary sphere, which generates the second homology class.\n\nFor the case of $\\xi = \\partial_t$, and considering solutions of the\nlinearized Einstein equations on the Kerr background we have, following the\ndiscussion above, \n$$\n\\dot \\massADM = \\dot \\PP_{\\partial_t; \\infty} \n$$\nWorking with the Carter tetrad, let $\\Psi_i$, $i=0,\\cdots, 4$ be the Weyl\nscalars and let $Z^I$, $I = 0,1,2$ denote the corresponding basis \nfor the space of complex, self-dual two-forms, see section\n\\ref{sec:form} for details. \nIn this paper we shall show that the natural linearization of the\nspin-lowered Weyl tensor $\\MM$ is the two-form\n$$\n \\linmass = \\psi\\dot\\Psi_1 Z^0 + \\psi\\dot\\Psi_2 Z^1 + \\psi\\dot\\Psi_3 Z^2 +\n \\tfrac{3}{2} \\psi\\Psi_2 \\dot Z^1 .\n$$\nAs will be demonstrated, see section \\ref{sec:fack} below, \n$\\linmass$ \nis closed, and hence the integral \n\\begin{equation}\\label{eq:linmasscharge} \n\\int_S \\linmass\n\\end{equation}\ndefines a conserved charge. \nA charge vanishing condition for the linearized mass, \nanalogous to the\none discussed above for the charges of the Maxwell field, may be introduced\nby requiring that this integral vanishes. The coordinate form of this \ncharge vanishing condition is \n\\begin{align} \\label{eq:integrability2}\n\\int_{S^2(t,r)} \\big( 2 V_L^{-1\/2} \\dot{\\widehat \\Psi}_2 + \\i a \\sin \\theta\n\\dot \\Psi_{diff} \\big) (r-\\i a \\cos\\theta) \\d\\mu = 0 , \n\\end{align}\nwhich should be compared to the corresponding condition for the \nMaxwell case, cf.  \\eqref{eq:integrability}. \nHere, $\\dot{\\widehat \\Psi}_2$ and $\\dot \\Psi_{diff}$ are suitable\ncombinations of the linearized curvature scalars $\\dot \\Psi_1,\n\\dot\\Psi_2,\\dot\\Psi_3$ and linearized tetrad.\n\nLet $\\dot g_{ab}$ be a solution of the linearized Einstein equation on the\nKerr background, satisfying \nsuitable asymptotic conditions, and let $\\dot \\massADM$ be the corresponding\nperturbation of the ADM mass. Letting $S = S^2(t,r)$ and evaluating  the limit\nof \\eqref{eq:linmasscharge} as $r\\to \\infty$ one finds, in view of the fact\nthat \\eqref{eq:linmasscharge} is conserved, the identity \n$$\n\\dot \\massADM = \\frac{1}{4\\pi i} \\int_S \\linmass\n$$\nfor any smooth 2-sphere $S$ in the exterior of the Kerr black hole. \nThus we have the relation \n\\begin{equation}\\label{eq:twocharges} \n\\int_S \\dot \\bQ[\\partial_t] - \\partial_t \\cdot \\mathbf \\Theta \n= \\frac{1}{4\\pi i} \\int_S \\linmass\n\\end{equation}\nfor any surface $S$ in the Kerr exterior. We remark that the left hand side\nof \\eqref{eq:twocharges} can be evaluated in terms of the metric perturbation\nusing the expressions for $\\bQ$ and $\\mathbf \\Theta$ given in \\cite[section\n  V]{iyer:wald:1994PhRvD..50..846I}. On the other hand, the right hand side\nhas been calculated in terms of linearized\ncurvature. It would be of interest to have a direct derivation of the\nresulting identity. \n\nThe canonical analysis following \\cite{iyer:wald:1994PhRvD..50..846I}\nwhich has been discussed above shows that in addition to the conserved charge\ncorresponding to $\\dot \\massADM$, equation \\eqref{eq:xicharge} with $\\xi =\n\\partial_\\phi$, the angular Killing field,  \ngives a conserved charge integral \nfor linearized angular momentum $\\dot \\angmom$. \nIf $\\partial_\\phi$ is tangent\n  to $S$, then the term $\\partial_\\phi \\cdot \\mathbf \\Theta$ does not contribute in\n  \\eqref{eq:xicharge}. \nWe remark that an expression for $\\dot \\angmom$ for linearized gravity on the\nSchwarzschild background was given in \\cite[section 3]{jezierski:1999}.  \nA charge integral for\n$\\dot \\angmom$ for linearized gravity on the Kerr background will be \nconsidered in a future paper. \n\n\\begin{remark} \n\\begin{enumerate} \n\\item There are many candidates for a quasi-local mass expression in the\nliterature including, to mention just a few, \nthose put forward by Penrose, Brown and York, and Wang and Yau.\nSee the review of Szabados \n\\cite{szabados:2004LRR.....7....4S} for background and references. Although\nas discussed above, cf. equation \\eqref{eq:mass-charge}, \nfor a spacetime of type D, there is a quasi-local mass charge, \nit must be emphasized that \nfor a general spacetime on cannot expect the existence of \na quasi-local mass\nwhich is \\emph{conserved}, i.e. independent of the 2-surface used in\nits definition. The same is true for linearized gravity\non a general background. Thus the existence of a conserved charge integral\nfor the linearized mass is a feature which is special to linearized gravity on a\nbackground with Killing symmetries. \n\n\\item If we consider linearized\ngravity without sources, on the Minkowski background, \nthe linearized mass must vanish due to\nthe fact that Minkowski space is topologically trivial. This reflects the\nfact that when viewed as a function on the space of Cauchy data, the ADM mass\nvanishes quadratically at the trivial data,\ncf. \\cite{ChB:fischer:marsden:1979igsg.conf..396C}. \nOn the other hand, by the positive mass theorem, \nfor any non-flat spacetime, asymptotic to Minkowski space in a suitable\nsense, the ADM mass defined at infinity must be positive. \n\\end{enumerate}\n\\end{remark}\n\nThis paper is organized as follows. In section \\ref{sec:form}, we introduce\nbivector formalism. Conformal Killing Yano tensors and Killing spinors are\ndiscussed in section \\ref{sec:cky}. Section \\ref{sec:conservedcharge} deals with\nconserved charges for spin-2 fields on Minkowski (\\S \\ref{sec:flat} ) and\ntype D spacetimes (\\S \\ref{sec:curved}). The main result, a charge integral\nin terms of linearized curvature, is derived in section \\ref{sec:fack}, and\nfinally, section \\ref{sec:conclusions} contains some concluding remarks. \n\n\\section{Preliminaries and notation} \\label{sec:form}\nLet $(N,g_{ab})$ be a 4 dimensional Lorentzian spacetime of\nsignature $+---$, \nadmitting a spinor structure. \nAlthough most of the results can be generalized \nto the electrovac case with cosmological constant, we restrict in this paper\nto the vacuum case. In particular, we consider test Maxwell fields and\nlinearized gravity on vacuum type D background spacetimes.\n\nLet \n$o_A, {\\iota}_A$ be a spinor dyad, normalized so that $o_A {\\iota}^A = 1$, and let \n\\begin{align*}\n l^a = o^A \\bar o^{A'}, && m^a = o^A \\bar {\\iota}^{A'}, && \\bar m^a = {\\iota}^A \\bar o^{A'}, && n^a = {\\iota}^A \\bar {\\iota}^{A'} \n\\end{align*}\nbe the corresponding null tetrad, \nsatisfying $l^a n_a = - m^a \\bar m_a = 1$, the other inner products\nbeing zero. The 2-spinor calculus provides a powerful tool for computations\nin 4-dimensional geometry. The GHP formalism deals with dyad (or\nequivalently tetrad) components of geometric objects \nand exploits the simplifications arising by\ntaking into account the action of dyad rescalings and permutations. \nThese formalisms are closely related to the less widely used \n\\emph{bivector formalism} \\cite{jordan:ehlers:sachs:1961:II,Bichteler:1964, Cahen:Debever:Defrise:1967,israel:bivector:book}\nin which the basic quantity is a basis for the 3-dimensional space of complex\nself-dual two-forms. A two-form $Z$ is called self-dual, if $*Z = \\i Z$ and anti self-dual, if $*Z = -\\i Z$. Given a spinor dyad, a\nnatural choice\\footnote{We use the convention of \\cite{fayos:ferrando:jaen:1990}, which differs from \\cite{israel:bivector:book,fackerell:1982} by a factor of 2 in the middle component and the numbering.} \nis \n\\begin{subequations}\\begin{align} \n Z^0_{ab} &= 2 \\bar m_{[a} n_{b]} = {\\iota}_{A} {\\iota}_\\sb \\ba{\\epsilon}_{{A}'\\sb'} \\\\\n Z^1_{ab} &= 2n_{[a} l_{b]} - 2\\bar m_{[a} m_{b]}  = -2 o_{({A}} {\\iota}_{\\sb)} \\ba{\\epsilon}_{{A}'\\sb'} \\\\\n Z^2_{ab} &= 2 l_{[a} m_{b]}  = o_{A} o_\\sb \\ba{\\epsilon}_{{A}'\\sb'} \\, ,\n\\end{align}\\end{subequations}\nwhere the notation $2x_{[a} y_{b]} = x_a y_b - y_a x_b$ for anti symmetrization and $2x_{(a} y_{b)} = x_a y_b + y_a x_b$ for symmetrization is used.\nWe use capital latin indices $I,J,K$ taking values in $0,1,2$ for the\nelements in the bivector triad $Z^I$. \nThe metric $g_{ab}$ induces a triad metric $G_{IJ}$ and its inverse\n$G^{IJ}$ given by \n\\begin{align*}\nG^{IJ} = Z^I \\cdot Z^J = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & -2 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} , &&\nG_{IJ} = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & -\\tfrac{1}{2} & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} \\, .\n\\end{align*}\nHere, $\\cdot$ is the induced inner product on two-form, $ Z^I \\cdot Z^J = \\frac{1}{2} Z^I{}_{ab}  Z^{Jab}$. Triad indices are raised and lowered with this metric,\n\\begin{align*}\n Z_{0} = Z^2 , && Z_{1} = -\\tfrac{1}{2} Z^1 , && Z_{2} = Z^0 .\n\\end{align*}\nMore general we have\n\\begin{prop} \\label{prop:zz}\n\\begin{subequations}\\begin{align}\n Z^J{}_a{}^c Z^K{}_{bc} &= \\frac{1}{2} G^{JK}g_{ab} + \\epsilon^{JKL} Z_{L ab}\\\\\n Z^J{}_{[a}{}^c \\bar Z^K{}_{b]c} &= 0 \\\\\n Z^{Jab} \\bar Z^K{}_{ab} &= 0\n\\end{align}\\end{subequations}\nwith $\\epsilon^{JKL}$ the totally antisymmetric symbol fixed by $\\epsilon^{012}=1$.\n\\end{prop}\n\nA real two-form $F_{ab}$, e.g. the Maxwell field strength, has spinor\nrepresentation\n\\begin{align*}\n F_{ab} = \\phi_{{A}\\sb} {\\epsilon}_{{A}'\\sb'} + \\ba \\phi_{{A}'\\sb'} {\\epsilon}_{{A}\\sb}.\n\\end{align*}\nIt is equivalent to the symmetric 2-spinor $\\phi_{{A}\\sb} = \\phi_2 o_{A}\no_\\sb -2 \\phi_1 o_{({A}} {\\iota}_{\\sb)} + \\phi_0 {\\iota}_{A} {\\iota}_\\sb$, where the six\nreal degress of freedom of $F_{ab}$ are encoded in 3 complex scalars \n\\begin{align*}\n \\phi_0 &= \\phi_{{A}\\sb} o^{A} o^\\sb  = F_{ab}  l^a m^b = F \\cdot Z_0\\\\\n \\phi_1 &= \\phi_{{A}\\sb} {\\iota}^{A} o^\\sb = \\tfrac{1}{2} F_{ab} ( l^a n^b - m^a \\bar m^b ) = F \\cdot Z_1\\\\\n \\phi_2 &= \\phi_{{A}\\sb} {\\iota}^{A} {\\iota}^\\sb = F_{ab}  \\bar m^a n^b = F \\cdot Z_2 \\, .\n\\end{align*}\nSo the real two-form has bivector representation\n\\begin{align*}\nF = \\phi_0 Z^0 + \\phi_1 Z^1 + \\phi_2 Z^2 + \\ba \\phi_0 \\ba Z^0 + \\ba \\phi_1 \\ba Z^1 + \\ba \\phi_2 \\ba Z^2,\n\\end{align*}\nor in index notation $ \\phi_I = F \\cdot Z_I $ and $ F = \\phi_I Z^I + \\ba \\phi_I \\ba Z^I $.\n\nThe Weyl tensor is a symmetric 2-tensor over bivector space and has spinor representation\n\\begin{align*}\n -C_{abcd} = \\Psi_{{A}\\sb\\sc{D}} \\ba{\\epsilon}_{{A}'\\sb'} \\ba{\\epsilon}_{\\sc'{D}'} + \\ba\\Psi_{{A}'\\sb'\\sc'{D}'} {\\epsilon}_{{A}\\sb} {\\epsilon}_{\\sc{D}} \\, ,\n\\end{align*}\nwhere $\\Psi_{{A}\\sb\\sc{D}}$ is a completely symmetric 4-spinor. The 10 degrees of freedom of the Weyl tensor are given by 5 complex scalars\\footnote{Due to its symmetries, the Weyl tensor is a symmetric two-tensor over the space of two-forms. The induced inner product is $C \\cdot (Z_I, Z_J) = \\frac{1}{4} C_{abcd} Z_I^{ab} Z_J^{cd}$.}\n\\begin{alignat*}{3}\n \\Psi_0 &= \\Psi_{{A}\\sb\\sc{D}} \\, o^{A} o^\\sb o^\\sc o^{D} &&= -C_{abcd} l^a m^b l^c m^d &&= -C \\cdot (Z_0,Z_0)\\\\\n \\Psi_1 &= \\Psi_{{A}\\sb\\sc{D}} \\, o^{A} o^\\sb o^\\sc {\\iota}^{D} &&= -C_{abcd} l^a n^b l^c m^d &&= -C \\cdot (Z_0,Z_1)\\\\ \n \\Psi_2 &= \\Psi_{{A}\\sb\\sc{D}} \\, o^{A} o^\\sb {\\iota}^\\sc {\\iota}^{D} &&= -C_{abcd} l^a m^b \\bar m^c n^d &&= -C \\cdot (Z_0,Z_2)= -C \\cdot (Z_1,Z_1)\\\\ \n \\Psi_3 &= \\Psi_{{A}\\sb\\sc{D}} \\, o^{A} {\\iota}^\\sb {\\iota}^\\sc {\\iota}^{D} &&= -C_{abcd} l^a n^b \\bar m^c n^d &&= -C \\cdot (Z_2,Z_1)\\\\\n \\Psi_4 &= \\Psi_{{A}\\sb\\sc{D}} \\, {\\iota}^{A} {\\iota}^\\sb {\\iota}^\\sc {\\iota}^{D} &&= -C_{abcd} n^a \\bar m^b n^c \\bar m^d &&= -C \\cdot (Z_2,Z_2) \\, .\n\\end{alignat*}\nSimilarly we could have used the Weyl 2-bivector \n\\begin{align*}\nC_{IJ} = - \\frac{1}{4} C_{abcd} Z_I^{ab} Z_J^{cd} =  \n\\begin{pmatrix} \n\\Psi_0 & \\Psi_1 & \\Psi_2 \\\\  \n\\Psi_1 & \\Psi_2 & \\Psi_3 \\\\\n\\Psi_2 & \\Psi_3 & \\Psi_4\n\\end{pmatrix} \n\\end{align*}\nwhich relates to the real Weyl tensor via\n\\begin{align} \\label{eq:curv}\n-C_{abcd} = C_{IJ} Z^I_{ab} \\otimes Z^J_{cd} + \\ba C_{IJ} \\ba Z^I_{ab} \\otimes \\ba Z^J_{cd} \\, .\n\\end{align}\n\nBecause of different conventions and normalisations in the literature \\cite{jordan:ehlers:sachs:1961:II,Bichteler:1964, Cahen:Debever:Defrise:1967,israel:bivector:book}, we rederive here the equations of structure in bivector formalism. Based on Cartan's equations of structure for tetrad one-forms \n\\footnote{Connection and curvature are defined by $\\omega^a{}_{b\\mu}= e^a{}_\\nu \\nabla_\\mu e_b{}^\\nu$ and $\\Omega^a{}_{b\\mu\\nu} = 2 e^a{}_\\sigma \\nabla_{[\\mu}\\nabla_{\\nu]} e_b{}^\\sigma$, respectively.}\n\\begin{align} \\label{eq:cartan1}\n \\d e^a = - \\omega^a{}_b \\wedge e^b &&& \n \\Omega^a{}_b = \\d \\omega^a{}_b + \\omega^a{}_c \\wedge \\omega^c{}_b \\, ,\n\\end{align}\nBianchi identities\n\\begin{align} \\label{eq:bianchi1}\n \\Omega^a{}_b \\wedge e^b = 0  &&&  \n \\d \\Omega^a{}_b = \\Omega^a{}_c \\wedge \\omega^c{}_b - \\omega^a{}_c \\wedge \\Omega^c{}_b \\, ,\n\\end{align}\nand definitions of connection one-forms $\\sigma_J$ and curvature two-forms $\\Sigma_J$  in bivector formalism,\n\\begin{align} \\label{eq:conncurv}\n \\omega_{ab} \\, e^a \\wedge e^b = -2 \\sigma_J Z^J - 2 \\bar \\sigma_J \\bar Z^J &&& \n \\Omega_{ab} e^a \\wedge e^b = - 2 \\Sigma_J Z^J - 2 \\bar\\Sigma_J \\bar Z^J, \n\\end{align} \nwe find\n\\begin{prop} \\label{prop:biveq}\nThe bivector equations of structure are\n\\begin{align} \\label{eq:cartan2}\n \\d Z^J = -2 \\epsilon^{J K L} \\sigma_K \\wedge Z_L &&& \n\\Sigma_J = \\d \\sigma_J + \\frac{1}{2} \\epsilon_{JKL} \\sigma^K \\wedge \\sigma^L\n\\end{align}\nwhile the Bianchi identities read\n\\begin{align} \\label{eq:bianchi2}\n \\Sigma_{[J} \\wedge Z_{K]} = 0 &&& \n d\\Sigma_J = - \\epsilon_{JKL} \\Sigma^K \\wedge \\sigma^L \\, .\n\\end{align} \nHere $\\wedge$ is the usual wedge product of one-forms $\\sigma^J$ and two-forms $Z^J, \\Sigma^J$.\n\\end{prop}\n\\begin{proof}\nExpanding the bivectors $Z^J = \\frac{1}{2} Z^J _{ab} e^a \\wedge e^b$, we find\n\\begin{align*}\n \\d Z^J \n&= \\frac{1}{2} Z^J _{ab}  \\left( \\d e^a \\wedge e^b - e^a \\wedge \\d e^b \\right) \n= Z^J_{a b} \\d e^a \\wedge e^b \\\\\n&= - Z^J_{a b} \\, \\omega^a{}_c e^c \\wedge e^b \\\\\n&= Z^J_{a b} \\left( \\sigma_K Z^{Ka}{}_c + \\bar \\sigma_K \\bar Z^{Ka}{}_c \\right) \\wedge e^c \\wedge e^b \\\\\n&= \\epsilon^{JKL} Z_{Lbc} \\sigma_K \\wedge e^c \\wedge e^b \\\\\n&= -2 \\epsilon^{JKL} \\sigma_K \\wedge Z_L \\, \n\\end{align*}\nwhere proposition \\ref{prop:zz} has been used in the third step.\nFor the second equation of structure, we plug \\eqref{eq:conncurv} into \\eqref{eq:cartan1},\n\\begin{align*}\n -\\Sigma_J Z^J_{ab} - \\bar\\Sigma_J \\bar Z^J_{ab} \n&= -\\d\\sigma_J Z^J_{ab} - \\d \\bar\\sigma_J \\bar Z^J_{ab} + (\\sigma_J Z^J_{ac} + \\bar\\sigma_J \\bar Z^J_{ac}) \\wedge (\\sigma_K Z^{Kc}{}_{b} + \\bar\\sigma_K \\bar Z^{Kc}{}_{b}) \\, .\n\\end{align*}\nSince $Z^J \\cdot \\bar Z^K = 0$ and proposition \\ref{prop:zz}, the selfdual part reads\n\\begin{align*}\n \\Sigma_J Z^J_{ab} = \\d \\sigma_J Z^J_{ab} + \\epsilon^{KLJ} Z_{J ab} \\sigma_K \\wedge \\sigma_L \\, .\n\\end{align*}\nChanging index positions by using $\\det G_{JK} = \\tfrac{1}{2}$ gives the 2nd equation of structure. For the first Bianchi identity, look at\n\\begin{align*}\n 0 &= \\d^2 Z^J \\\\\n &= -2 \\epsilon^{JKL} \\left( \\d\\sigma_K \\wedge Z_L - \\sigma_K \\wedge \\d Z_L \\right) \\\\\n &= -2 \\epsilon^{JKL} \\left( \\Sigma_K \\wedge Z_L - \\frac{1}{2} \\epsilon_{KNM} \\sigma^N \\wedge \\sigma^M \\wedge Z_L + \\sigma_K \\wedge \\epsilon_{LNM} \\sigma^N \\wedge Z^M \\right) \\\\\n &= -2 \\epsilon^{JKL} \\Sigma_K \\wedge Z_L \\underbrace{+ \\sigma^L \\wedge \\sigma^J \\wedge Z_L - \\sigma^J \\wedge \\sigma^L \\wedge Z_L -2 \\sigma_L \\wedge \\sigma^J \\wedge Z^L}_{=0} + 2 \\underbrace{\\sigma_K \\wedge  \\sigma^K}_{=0} \\wedge Z^J\n\\end{align*}\nwhere the identity $\\epsilon^{IJK} \\epsilon_{INM} = \\delta^J_N \\delta^K_M - \\delta^J_M \\delta^K_N$ has been used. Finally, the second Bianchi identity is\n\\begin{align*}\n \\d \\Sigma_J \n &= -\\epsilon_{JKL} \\d \\sigma^K \\wedge \\sigma^L \\\\\n &= -\\epsilon_{JKL} (\\Sigma^K - \\epsilon^{KMN} \\sigma_M \\wedge \\sigma_N) \\wedge \\sigma^L \\\\\n &= -\\epsilon_{JKL} \\Sigma^K \\wedge \\sigma^L + \\underbrace{\\sigma_L \\wedge \\sigma_J \\wedge \\sigma^L}_{=0} -  \\sigma_J \\wedge \\underbrace{\\sigma_L \\wedge \\sigma^L}_{=0} \\, .\n\\end{align*}\n\\end{proof}\n\\begin{remark}\n Instead of using Cartan equations for the tetrad one could have used the bivector connection form\n\\begin{align}\n\\omega_{IJa} := {\\epsilon}_{IJK} \\sigma^K_a =  Z_{[J}^{bc} \\nabla_a Z_{I]bc} \\, .\n\\end{align}\n\\end{remark}\n\nFor later use it is convenient to write the components of the equations of structure explicitely. The connection one-forms for example can be expressed in terms of NP spin\ncoefficients,\n\\begin{subequations}\\begin{alignat}{2}\n\\sigma_{0a} &= m^b \\nabla_a l_b &&= \\tau l_a + \\kappa n_a - \\rho m_a - \\sigma \\bar m_a \\\\\n\\sigma_{1a} &= \\frac{1}{2} \\left( n^b \\nabla_a l_b - \\bar m^b \\nabla_a m_b \\right)  &&=-\\epsilon' l_a + \\epsilon n_a + \\beta' m_a - \\beta \\bar m_a \\\\\n\\sigma_{2a} &= -\\bar m^b \\nabla_a n_b &&= -\\kappa' l_a - \\tau' n_a + \\sigma' m_a + \\rho' \\bar m_a \\, .\n\\end{alignat}\\end{subequations}\nThe middle component $\\sigma_{1a}$ collects all unweighted coefficients and so can be used to define the GHP covariant derivative $\\Theta_a \\eta = (\\nabla_a - p \\sigma_{1a} - q \\ba \\sigma_{1a}) \\eta$. To avoid clutter in the notation, we write $\\Gamma:=\\sigma_0$ and $\\sigma_2 = -\\Gamma'$, where $'$ is the GHP prime operation\\cite{GHP}. Derivatives of the spinor dyad can now be written in the compact form $\\Theta_a o^A = - \\Gamma_a {\\iota}^A$ and $\\Theta_a {\\iota}^A = - \\Gamma'_a o^A$, and the components of the first equations of structure, which we present here for convenience with the usual exterior derivative and with weighted exterior derivative $\\d^\\Theta = \\d - p \\sigma_1 \\wedge - q \\ba \\sigma_1 \\wedge$, read\n\\begin{subequations}\\begin{align} \n\\d^\\Theta Z^0 &= \\Gamma' \\wedge Z^1  & \\Leftrightarrow & & \\d Z^0 &= -2 \\sigma_1 \\wedge  Z^0 + \\Gamma' \\wedge Z^1\\\\\n\\d^\\Theta Z^1 &= 2 \\Gamma \\wedge Z^0 + 2 \\Gamma' \\wedge Z^2 & \\Leftrightarrow & & \\d Z^1 &= 2 \\Gamma \\wedge Z^0 + 2 \\Gamma' \\wedge Z^2\\\\\n\\d^\\Theta Z^2 &= \\Gamma \\wedge Z^1 & \\Leftrightarrow & & \\d Z^2 &= 2 \\sigma_1 \\wedge Z^2 + \\Gamma \\wedge Z^1 .\n\\end{align}\\end{subequations}\nNote that the middle component can be simplified to $\\d Z^1 = - h \\wedge Z^1 $ with the one-form $ h =  2 (\\rho' l + \\rho n -\\tau' m - \\tau \\bar m)$. This fact and a relation between type D curvature $\\Psi_2$ and $h$ will be crucial in the derivation of the conservation law in section \\ref{sec:fack}.\n\nIn vacuum, we have for the curvature two-forms $\\Sigma_J= C_{JK} Z^K$ and the components of the second equations of structure read\n\\begin{subequations}\\begin{align}\n \\Sigma_0 &= C_{0J}Z^{J} = \\d^\\Theta \\Gamma = \\d \\Gamma - 2 \\sigma_1 \\wedge \\Gamma \\\\\n \\Sigma_1 &= C_{1J}Z^{J} = \\d \\sigma_1 - \\Gamma \\wedge \\Gamma' \\\\\n \\Sigma_2 &= C_{2J}Z^{J} = -\\d^\\Theta \\Gamma' = -\\d \\Gamma' - 2 \\sigma_1 \\wedge \\Gamma' \\, .\n\\end{align}\\end{subequations}\nFinally the Bianchi identities are\n\\begin{subequations}\\begin{align}\n\\d^\\Theta \\Sigma_0 &= -2\\Gamma \\wedge \\Sigma_1 \n& \\Leftrightarrow & &\\d \\Sigma_0 &= 2 \\sigma_1 \\wedge \\Sigma_0 - 2 \\Gamma \\wedge \\Sigma_1 \\\\\n\\d^\\Theta \\Sigma_1 &= - \\Gamma' \\wedge \\Sigma_0 - \\Gamma \\wedge \\Sigma_2  \n& \\Leftrightarrow & &\\d \\Sigma_1 &= - \\Gamma' \\wedge \\Sigma_0 - \\Gamma \\wedge \\Sigma_2 \\label{eq:bianchi}\\\\\n\\d^\\Theta \\Sigma_2 &= - 2 \\Gamma' \\wedge \\Sigma_1 \n& \\Leftrightarrow & & \\d \\Sigma_2 &= -2 \\sigma_1 \\wedge \\Sigma_2 - 2 \\Gamma' \\wedge \\Sigma_1 . \n\\end{align}\\end{subequations}\n\n\\section{Conformal Killing Yano tensors and Killing spinors} \\label{sec:cky}\n\nConformal Killing Yano tensors of rank 2 are two-forms $Y_{ab}$ solving the conformal Killing Yano equation,\n \\begin{align} \\label{eq:cyt}\n Y_{a(b;c)} = g_{bc}\\xi_a -g_{a(b}\\xi_{c)}, \\text{ where } \\xi_a = \\tfrac{1}{3} Y_a{}^b{}_{;b} .\n\\end{align}\nIt is well known, that the divergence $\\xi^a$ is a Killing vector and in\ncase it vanishes, $Y_{ab}$ is called Killing Yano tensor. The symmetrised\nproduct $X_{c(a}Y_{b)}{}^c =:K_{ab}$ of Killing Yano tensors\n$X_{ab}, Y_{ab}$ is a Killing tensor, $\\nabla_{(a} K_{bc)} = 0$,\nwhich can be used to construct a constant of motion or a symmetry operator for e.g. the scalar\nwave equation, known as Carter's constant and Carter operator, respectively. By inserting $Y_{ab} = \\kappa_{AB} {\\epsilon}_{A'B'} + \\bar\n\\kappa_{A'B'} {\\epsilon}_{AB}$ into \\eqref{eq:cyt} one can show that $\\kappa_{AB}$\nand $\\bar \\kappa_{A'B'}$ satisfy the Killing spinor equation \n\\begin{align} \\label{eq:ks}\n \\nabla_{A'(A} \\kappa_{BC)} = 0\n\\end{align}\nand its complex conjugated version. For the spinor components $\\kappa_{AB} = \\kappa_2\no_A o_B -2 \\kappa_1 o_{(A} {\\iota}_{B)} + \\kappa_0 {\\iota}_A {\\iota}_B  $ (or\nequivalently the self dual bivector components of $Y_{ab}$, we find the\nfollowing set of eight scalar equations \n\\begin{align}\n \\begin{aligned} \\label{eq:kscomp1}\n \\tho \\kappa_0 = -2 \\kappa \\kappa_1 , &&& \\edt \\kappa_0 = -2 \\sigma \\kappa_1  ,&&& \n {\\tho}' \\kappa_2 = -2 \\kappa' \\kappa_1 , &&& {\\edt}' \\kappa_2 = -2 \\sigma' \\kappa_1 \n\\end{aligned}\\\\\n\\begin{aligned} \\label{eq:kscomp2}\n({\\edt}' + 2\\tau')\\kappa_0 +2 (\\tho + \\rho)\\kappa_1 = -2 \\kappa \\kappa_2 , &&&\n({\\tho}' + 2\\rho')\\kappa_0 +2 (\\edt + \\tau) \\kappa_1 = -2 \\sigma \\kappa_0 \\\\\n({\\edt} + 2\\tau)\\kappa_2 +2 ({\\tho}' + \\rho')\\kappa_1 = -2 \\kappa' \\kappa_0 , &&&\n({\\tho} + 2\\rho)\\kappa_2 +2 ({\\edt}' + \\tau') \\kappa_1 = -2 \\sigma' \\kappa_2 \\,,\n\\end{aligned} \n\\end{align}\nby projecting \\eqref{eq:ks} into a spinor dyad. Thus, we have three different sets of equations, \\eqref{eq:cyt}, \\eqref{eq:ks}, \n(\\ref{eq:kscomp1},\\ref{eq:kscomp2}), which are equivalent and we will use the most appropriate for the problem at hand.\n\nAs spin-s fields are heavily restricted on curved backgrounds (Buchdahl constraint, see equation (5.8.2) in \\cite{PR:I}), so are Killing spinors. Consider a Killing spinor $\\kappa_{A_1 ... A_n} = \\kappa_{(A_1 ... A_n)}$ which satisfies the  Killing spinor equation of valence $n$\n\\begin{align}\n \\nabla_{B'(B}\\kappa_{A_1 ... A_n)} = 0 \\, .\n\\end{align}\nContracting a second derivative $\\nabla^{B'}{}_C$ and symmetrising gives\n\\begin{align*}\n 0 &= \\nabla^{B'}{}_{(C} \\nabla_{|B'|B} \\kappa_{A_1 ... A_n)} \\\\\n &= -\\Box_{(BC} \\kappa_{A_1 ... A_n)} \\\\\n &= \\Psi_{(BCA_1}{}^D \\kappa_{D A_2 ... A_n)} + \\dots + \\Psi_{(BCA_n}{}^D \\kappa_{A_1 ... A_{n-1}D)} \\\\\n &= n\\Psi_{(BCA_1}{}^D \\kappa_{D A_2 ... A_n)} \\, .\n\\end{align*}\nFor Killing spinors of valence 1 (satisfying the twistor equation) this yields $ 0 = \\Psi_{ABCD} \\kappa^D$ as can be found in \\cite{PR:II}, eq.(6.1.6). For 2-spinors we find\n\\begin{align} \\label{eq:IntCond}\n 0 = \\Psi_{(ABC}{}^D \\kappa_{DE)} \\, .\n\\end{align}\nFor non trivial $\\kappa$, this restricts the spacetime to be of Petrov type $D,N$ or $O$.\nFor a given spacetime of type D in a principal frame (only $\\Psi_2 \\neq 0$) \\eqref{eq:IntCond} becomes\n\\begin{align*}\n 0 &= \\Psi_2 \\, o_{(A} o_B {\\iota}_C {\\iota}_D \\left( \\kappa_0 {\\iota}^D {\\iota}_{E)} + \\kappa_1 o^D {\\iota}_{E)} + \\kappa_1 {\\iota}^D o_{E)} + \\kappa_2 o^D o_{E)} \\right) \\\\\n &= \\Psi_2 \\left( C_1 \\kappa_0 o_{(A} {\\iota}_B {\\iota}_C {\\iota}_{E)} + C_2 \\kappa_2 {\\iota}_{(A} o_B o_C o_{E)} \\right)\n\\end{align*}\nwith constants $C_1,C_2$ and it follows $\\kappa_0 \\equiv 0 \\equiv\n\\kappa_2$. The remaining component satisfies the simplified equations\n\\eqref{eq:typeDKS}, which have only one non trivial complex solution, cf. \\cite{Glass:1996} where explicit integration of the conformal Killing Yano equation was done.\n\n\\section{Conserved Charges} \\label{sec:conservedcharge} \n\n\\subsection{Conserved charges for Minkowski spacetime} \\label{sec:flat}\n\nThe Killing spinor equation or conformal Killing Yano equation on Minkowski space has been widely discussed in the literature \\cite{PR:II},\\cite{Jezierski:Lukasik:2006}, \\cite{herdegen:1991} and the explicit solution in cartesian coordinates is well known,\n\\begin{align} \\label{eq:kscart}\n \\kappa^{AB} = U^{AB} + 2 x^{A'(A} V^{B)}_{A'} + x^{A'A}x^{B'B} W_{A'B'} .\n\\end{align}\nHere  $U^{AB},W_{A'B'}$ are constant, symmetric spinors and $V_{A'}^B$ a constant complex vector which yield $2 \\cdot 6 + 8 = 20$ independent real solutions. Each solution gives a charge when contracted into a spin-2 field, e.g. the linearized Weyl tensor, and integrated over a 2-sphere. In \\cite[p.99]{PR:II}, 10 of these charges are related to a source for linearized gravity in the following sense. Given a divergence free, symmetric energy momentum tensor $T_{ab}$, one has for each Killing field $\\xi^b$ the divergence free current $J_a = T_{ab} \\xi^b$. Using linearized Einstein equations\n\\begin{align}\n \\dot G_{ab} = \\dot R_{acb}{}^c - \\frac{1}{2}g_{ab} \\dot R_{cd}{}^{cd} = -8 \\pi G \\dot T_{ab}\n\\end{align}\nand the conformal Killing Yano equation \\eqref{eq:cyt}, they showed\n\\begin{align}\n 3 \\int_{\\partial \\Sigma} \\dot R_{abcd} *\\hspace{-4pt}Y^{cd} \\d x^a \\wedge \\d x^b = 16 \\pi G \\int_\\Sigma e_{abc}{}^d \\dot T_{df} \\xi^f \\d x^a \\wedge \\d x^b \\wedge \\d x^c .\n\\end{align}\nHere $\\Sigma$ denotes a 3 dimensional hypersurface with boundary $\\partial\\Sigma$ and $e_{abcd}$ is the Levi-Civita tensor. The left hand side is the charge integral described above, while the right hand side gives the more familiar form of a conserved three-form corresponding to a linarized source and a Killing vector $\\xi^a = \\tfrac{1}{3} Y^{ab}{}_{;b}$. Note that it is the dual conformal Killing Yano tensor on the left hand side, which gives the charge associated to the isometry $\\xi^a$. In cartesian coordinates  $x^a = (t,x,y,z)$ the Poincar\\'{e} isometries read\n\\begin{align}\n \\mathcal{T}_a = \\frac{\\partial}{\\partial x^a}  && \\mathcal{L}_{ab} = x_a \\frac{\\partial}{\\partial x^b} - x_b \\frac{\\partial}{\\partial x^a}\n\\end{align}\nand the relation to the charges is listed in table \\ref{tab:isometries}. The angular momentum around the $z$-axis is found in the component $\\mathcal{L}_{xy} = \\partial_\\phi$.\n\\begin{table}\n\\caption{Poincar\\'{e} isometries and corresponding charges}\n\\begin{tabular}{llll}\n\\toprule\nlabel& isometry & charge& \\# \\\\\n\\hline\n$\\mathcal{T}_t$ & time translation & mass & 1\\\\\n$\\mathcal{T}_i$ & spatial translations & linear momenta & 3\\\\\n$\\mathcal{L}_{ij}$ & rotations &  angular momenta & 3 \\\\\n$\\mathcal{L}_{ti}$ & boosts & center of mass & 3\\\\\n\\bottomrule\n\\end{tabular} \\label{tab:isometries}\n\\end{table}\nExplicit expressions for linearized sources generating these charges can be found in \\cite[eq.27]{jezierski:1995GReGr..27..821J}.\n\nThe 10 remaining charges cannot be generated this way, since the\ncorresponding conformal Killing Yano tensors have vanishing divergence (they\nare Killing Yano tensors). One of these charges corresponds to the NUT\nparameter\\footnote{sometimes called dual mass, because of duality rotation from Schwarschild to NUT, see the appendix of \\cite{Ramaswamy:Sen:1981}}, \nand the remaining nine are three\ndual linear momenta and six ofam\\footnote{Obstructions for angular\n  momentum, see \\cite{jezierski:2002CQGra..19.4405J}.}.\nIn the expression \\eqref{eq:kscart} for a general Killing spinor, \nthey correspond to $U$ and the\nimaginary part of $V$. For a metric perturbation, which one might interpret\nas a potential for the linarized curvature, these 10 additional charges\nvanish, see \\cite[\\S 6.5]{PR:II}. \n\nTo understand the charges as projections into $l=0$ and $l=1$ mode, we rederive the complete set of solutions in spherical coordinates using spin weighted spherical harmonics. A null tetrad for Minkowski spacetime in spherical coordinates $(t,r,\\theta,\\phi)$ (symmetric Carter tetrad) is given by\n\\begin{align*}\n l^a = \\frac{1}{\\sqrt{2}} \\bigg[1,1,0,0 \\bigg] , &&\nn^a = \\frac{1}{\\sqrt{2}}\\bigg[1,-1,0,0 \\bigg] , &&\nm^a = \\frac{1}{\\sqrt{2}r}\\bigg[0,0,1,\\frac{\\i}{\\sin\\theta} \\bigg] ,\n\\end{align*}\nwith non vanishing spin coefficients\n\\begin{align*}\n \\rho = -\\frac{1}{\\sqrt{2}r} = -\\rho' , && \\beta = \\frac{\\cot\\theta}{2\\sqrt{2}r} = \\beta' .\n\\end{align*}\nA general two-form can be expanded\n\\begin{align*} \n Y =&+ \\kappa_2 \\tfrac{r}{2}(\\d r - \\d t) \\wedge (\\d\\theta + \\i  \\sin\\theta \\, \\d\\varphi )\\\\ &- \\kappa_1 ( \\d t \\wedge \\d r + \\i r^2 \\sin\\theta \\, \\d \\theta \\wedge \\d \\varphi )\\\\  &+ \\kappa_0 \\tfrac{r}{2}(\\d r + \\d t) \\wedge (\\d\\theta - \\i\\sin\\theta \\, \\d\\varphi ) + c.c.\n\\end{align*}\nand it is a conformal Killing Yano tensor, if the components $\\kappa_i$ satisfy (\\ref{eq:kscomp1},\\ref{eq:kscomp2}). The subset \\eqref{eq:kscomp1} of the Killing spinor equation becomes\n\\begin{align*}\n \\left(\\partial_t + \\partial_r \\right) \\kappa_0 = 0, && \\left(\\partial_\\theta + \\frac{\\i}{\\sin\\theta}\\partial_\\varphi - \\cot\\theta \\right) \\kappa_0 = 0, \\\\\n \\left(\\partial_t - \\partial_r \\right) \\kappa_2 = 0, && \\left(\\partial_\\theta - \\frac{\\i}{\\sin\\theta} \\partial_\\varphi - \\cot\\theta \\right)\\kappa_2 = 0,\n\\end{align*}\nso $\\kappa_0 = f_0(t-r) \\Y{1}{1}{m}$ and $\\kappa_2 = f_1(t+r) \\Y{-1}{1}{m}$ with functions $f_i$ depending on advanced and retarded coordinates only. Finally \\eqref{eq:kscomp2} can be solved for $\\kappa_1$, which is only possible for particular functions $f_i$. The result is given in table \\ref{tab:MinkKS}.\n\\begin{table} \n\\caption{Solutions to the Killing spinor equation on Minkowski spacetime in\n  spherical coordinates.} \\label{tab:KS}\n \\centering\n\\begin{tabular}{lccc||ccc} \n\\toprule\n& \\multicolumn{3}{c}{components} & & \\multicolumn{2}{c}{divergence} \\\\\nlabel & $\\kappa_0 \/ \\sqrt{2}$ & $\\kappa_1$ & $\\kappa_2 \/ \\sqrt{2}$ & combination & $\\Re$ & $\\Im$ \\\\\n\\hline\n\\rowcolor{g1}\n&&& & $\\Omega^0_{-1}$ & 0 & 0 \\\\ \\rowcolor{g1}\n$\\Omega^0_m$ & $\\Y{1}{1}{m}$ & $\\Y{0}{1}{m}$ & $\\Y{-1}{1}{m}$ & $\\Omega^0_{0}$ & 0 & 0 \\\\ \\rowcolor{g1}\n&&& & $\\Omega^0_{1}$ & 0 & 0 \\\\ \\rowcolor{g1}\n\\rowcolor{g2}\n&&& & $\\Omega^1$ & $\\mathcal{T}_t$ & 0 \\\\ \\rowcolor{g2}\n$\\Omega^1$ & 0 & r & 0 & $\\Omega^1_1 - \\Omega^1_{-1}$ & $\\mathcal{T}_x$ & 0 \\\\ \\rowcolor{g2}\n$\\Omega^1_m$ & $(t-r) \\Y{1}{1}{m}$ & $t \\Y{0}{1}{m}$ & $(t+r) \\Y{-1}{1}{m}$ & $\\i\\Omega^1_1 + \\i\\Omega^1_{-1}$ & $\\mathcal{T}_y$ & 0\\\\ \\rowcolor{g2}\n&&& & $\\Omega^1_0$ & $\\mathcal{T}_z$ & 0 \\\\ \\rowcolor{g2}\n\\rowcolor{g3}\n&&& & $\\Omega^2_1 - \\Omega^2_{-1}$ & $\\mathcal{L}_{tx}$ & $\\mathcal{L}_{yz}$ \\\\ \\rowcolor{g3}\n$\\Omega^2_m$ & $(t-r)^2 \\Y{1}{1}{m}$ & $(t^2-r^2) \\Y{0}{1}{m}$ & $(t+r)^2\\Y{-1}{1}{m}$ & $\\i\\Omega^2_1 + \\i\\Omega^2_{-1}$ &  $\\mathcal{L}_{ty}$ & $\\mathcal{L}_{xz}$ \\\\ \\rowcolor{g3}\n&&& & $\\Omega^2_0$ & $\\mathcal{L}_{tz}$ & $\\mathcal{L}_{xy}$ \\\\\n\\bottomrule\n\\end{tabular} \\label{tab:MinkKS}\n\\end{table}\n$\\Omega^1$ is one complex solution, while $\\Omega^i_m, i=0,1,2$ represent 3 complex solutions each, ($m=0,\\pm 1$). We find the following correspondence to the solutions \\eqref{eq:kscart} in cartesian coordinates\n\\begin{align*}\n \\Omega^0_m \\leftrightarrow U^{AB}, && \\Omega^1,\\Omega^1_m \\leftrightarrow V_{A'}^A, && \\Omega^2_m \\leftrightarrow W_{A'B'}.\n\\end{align*}\n\n\\subsection{Conserved charges for type D spacetimes} \\label{sec:curved}\n\nThe vacuum field equations in the algebraically special case of Petrov type D\nhave been integrated explicitly by Kinnersley \\cite{kinnersley:1969}. An\nexplicit type D line element solving the Einstein-Maxwell \nequations with cosmological\nconstant is known, from which all type D line elements of this type \ncan be derived by certain limiting procedures,\nsee \\cite[\\S 19.1.2]{exact:solutions:book}, see also \n\\cite{Debever:Kamran:McLenaghan:1984}. The family of\ntype D spacetimes contains the Kerr and Schwarzschild solutions, but also\nsolutions with more complicated topology and asymptotic behaviour, such \nas the NUT- or C-metrics, and solutions whose orbits of the isometry group are\nnull. In the following, we again restrict to the vacuum case. \n\n\nA Newman-Penrose tetrad such that the \ntwo real null vectors $l^a, n^a$ are aligned with\nthe two repeated principal null directions of a Weyl tensor of Petrov type \nD is called a principal tetrad. In this case,\n\\begin{align*}\n \\Psi_0 = \\Psi_1 = 0 = \\Psi_3 = \\Psi_4, && \\kappa = \\kappa' = 0  = \\sigma = \\sigma'\n\\end{align*}\nand $\\Psi_2 \\neq 0$. \nDue to the integrability condition \\eqref{eq:IntCond}, \nwe have $\\kappa_0 = 0 = \\kappa_2$. Hence, the\ncomponents (\\ref{eq:kscomp1},\\ref{eq:kscomp2}) of the Killing spinor equation\nsimplify  to \n\\begin{align} \\label{eq:typeDKS}\n (\\tho + \\rho)\\kappa_1 = 0 , &&\n (\\edt + \\tau) \\kappa_1 = 0, && \n ({\\tho}' + \\rho')\\kappa_1 = 0 , &&\n ({\\edt}' + \\tau') \\kappa_1 = 0 .\n\\end{align}\nComparison with the Bianchi identities\n\\begin{align} \\label{eq:typeDbianchi}\n (\\tho - 3\\rho)\\Psi_2 = 0 , &&\n (\\edt - 3\\tau) \\Psi_2 = 0, && \n ({\\tho}' - 3\\rho')\\Psi_2 = 0 , &&\n ({\\edt}' - 3\\tau') \\Psi_2 = 0, \n\\end{align}\nshows that $\\kappa_1 := \\psi \\propto \\Psi_2^{-1\/3}$ is a solution, and in fact\nup to a constant $\\kappa_{AB} = \\psi o_{(A} \\iota_{B)}$ is the only solution\nof the Killing spinor equation. \n\nThe divergence $\\xi^{AA'} =\n  \\nabla^{A'}{}_B \\kappa^{AB}$ is a Killing vector field, which is\n  proportional to a real Killing vector field for all \ntype D spacetimes except for Kinnersley class IIIB,\ncf. \\cite{collinson:1976}. If $\\xi^{AA'}$ is real, \nthe imaginary part of $\\kappa_{AB}$ \nis a Killing-Yano tensor. Spacetimes satisfying the just\nmentioned condition are called \n  \\textit{generalized Kerr-NUT} spacetimes \\cite{ferrando:saez:2007JMP....48j2504F}. \nThe square of the Killing-Yano tensor is a symmetric Killing tensor $K_{ab} =\n  Y_{ac} Y^c{}_b$ and it follows, that $\\eta^a = K^{ab} \\xi_b$ is a Killing\n  vector. On a Kerr background, $\\xi^a$ and $\\eta^a$ are linearly independent\n  and span the space of isometries, see \\cite{Hughston:Sommers:1973}. In the\n  special case of a Schwarzschild background, $\\eta^a$ vanishes, see also\n  \\cite{Collinson:Smith:1977} for details. \n\nFor Kerr spacetime in Boyer-Lindquist coordinates we find \n\\begin{align*}\n \\Psi_2 = - \\frac{M}{(r - \\i a \\cos\\theta)^3} ,  && \\psi \\propto r - \\i a \\cos\\theta\n\\end{align*}\nand we set the factor of proportionality to 1, so that the solution \n\\begin{align} \\label{eq:kssolution}\n \\kappa_0 = 0 && \\kappa_1 = \\psi && \\kappa_2 = 0\n\\end{align} \nreduces to $\\Omega^1$ as given in table \\ref{tab:KS},\nin the Minkowski limit $M,a \\to 0$.  We find\n$ \\nabla_b \\left( \\psi Z^{1ab} \\right) = 3\\left( \\partial_t \\right)^a $. \nThe Killing spinor with components given by \\eqref{eq:kssolution} is \n\\begin{equation}\\label{eq:kappa}\n\\kappa_{AB} = -2\\psi o_{(A} {\\iota}_{B)} , \n\\end{equation}\nWe have\n$ \\psi Z^1_{ab} = \\kappa_{AB} {\\epsilon}_{A'B'} $ and therefore\n$$\n\\left( \\partial_t \\right)_a = \\frac{1}{3} \\nabla^b \\left( \\psi Z^{1}_{ab} \\right) = - \\frac{2}{3} \\nabla^{B'B} ( \\kappa_{AB} {\\epsilon}_{A'B'} ) = \\frac{2}{3} \\nabla_{A'}{}^{B}  \\kappa_{AB} \\, .\n$$\nSpin lowering the Weyl spinor using \\eqref{eq:kappa} gives the Maxwell\n  field $\\psi_{ABCD} \\kappa^{CD}$, which has charges proportional to \n\\textit{mass} and \\textit{dual mass}, see also\n\\cite{jezierski:lukasik:2009}. \nLetting $\\MM(C,\\kappa)$ denote the \ncorresponding closed complex two-form we have \n\\begin{equation} \\label{eq:maxwell}\n \\MM(C,\\kappa)= \\psi \\Psi_2 Z^1 .\n\\end{equation}\nEvaluating the charge for the Kerr metric yields \n\\begin{align}\n \\frac{1}{4\\pi\\i} \\int_{S^2} \\MM(C,\\kappa) = \\frac{1}{4\\pi\\i} \\int_{S^2} - \\frac{M}{(r-\\i a \\cos\\theta)^2} (-\\i)(r^2+a^2) \\sin\\theta \\d\\theta \\wedge \\d\\varphi = M,\n\\end{align}\nwhere $M$ is the ADM mass while the dual mass is zero.\n\nThe closed two-form \\eqref{eq:maxwell} has been derived much earlier by Jordan, Ehlers and Sachs \\cite{jordan:ehlers:sachs:1961:II}. We will repeat the derivation here, since this formulation can be generalized to linearized gravity most easily. On a type D background, the curvature forms and the connection simplify to \n\\begin{align}\n \\Sigma_0 = \\Psi_2 Z^2 && \\Sigma_1 = \\Psi_2 Z^1 && \\Sigma_2 = \\Psi_2 Z^0 && \\Gamma = \\tau l - \\rho m \\, ,\n\\end{align}\nso the middle Bianchi identity \\eqref{eq:bianchi} becomes\n\\begin{align*}\n 2\\d \\Sigma_1 &= 2\\Psi_2 \\left[ (\\rho' \\bar m - \\tau' n) \\wedge l \\wedge m + (\\rho m - \\tau l) \\wedge \\bar m \\wedge n \\right] \\\\\n&=  2 \\Psi_2 (\\rho' l + \\rho n -\\tau' m - \\tau \\bar m) \\wedge Z^1 \\\\\n&= h \\wedge \\Sigma_1 \\, ,\n\\end{align*}\nwhere $ h =  2 (\\rho' l + \\rho n -\\tau' m - \\tau \\bar m)$ was used. \nAs noted in \\cite{fayos:ferrando:jaen:1990}, the Bianchi identities \\eqref{eq:typeDbianchi} can be rewritten as $2 \\d \\Psi_2 = 3 h \\Psi_2$ and one obtains \n\\begin{align*}\n\\d(\\Psi_2 Z^1) =  \\d\\Sigma_1 = \\frac{1}{2} h \\wedge \\Sigma_1 = \\frac{1}{3} \\d\\Psi_2 \\wedge Z^1 \\, .\n\\end{align*}\nWe finally end up with the \\textit{Jordan-Ehlers-Sachs conservation law}\\cite{jordan:ehlers:sachs:1961:II},\n\\begin{align} \\label{eq:jes}\n\\d \\left( \\Psi_2^{2\/3} Z^1 \\right) = 0 \\, .\n\\end{align}\nUsing $\\psi \\propto \\Psi_2^{-1\/3}$, this is the same result as \\eqref{eq:maxwell}. See also \\cite{israel:bivector:book}, where the conservation law is generalised to spacetimes of Petrov type II. The result for type D backgrounds fit into the picture of Penrose potentials\\cite{Goldberg:1990} and in the next section we will see that it generalizes to linear perturbations.\n\n\\section{Fackerell's conservation law} \\label{sec:fack}\n\nWe can of course linearize the two-form \\eqref{eq:maxwell}, which would\nprovide a charge for perturbations within the class of type D spacetimes. But\nmore generally, Fackerell \\cite{fackerell:1982} derived a closed two-form for\narbitrary linear perturbations around a type D \nbackground\\footnote{One can expect that such a structure for perturbations of\n  algebraically special solutions exists also for other signatures. A\n  classification of the Weyl tensor in Euclidean signature can be found in\n\\cite{Karlhede:1986}, see also \n  \\cite{hacyan:1979PhLA...75...23H}, a unified formulation for arbitrary signature is given\n  in \\cite{Batista:2012}.}. \nStarting from this conservation law, Fackerell and Crossmann derived field equations for\nperturbations of Kerr-Newmann spacetime. Let us give a shortened derivation\nin the vacuum case.  \n\nWhen linearizing (with parameter $\\epsilon$) the general bivector equations around a type D background in principal tetrad, we have\n\\begin{align*}\n\\Gamma = \\tau l - \\rho m + O(\\epsilon) && \\Gamma' = \\tau' n - \\rho' \\bar m + O(\\epsilon)\n\\end{align*}\nand it follows\n\\begin{align*}\n\\d^\\theta Z^0 = - \\tfrac{1}{2} h \\wedge Z^0 + O(\\epsilon) &&\n\\d^\\theta Z^2 = - \\tfrac{1}{2} h \\wedge Z^2 + O(\\epsilon) \\\\\n\\Gamma' \\wedge \\Sigma_0 = (\\tau' m - \\rho' l) \\wedge\\Sigma_1 + O(\\epsilon^2) && \\Gamma \\wedge \\Sigma_2 = (-\\tau \\bar m + \\rho n) \\wedge \\Sigma_1 + O(\\epsilon^2)\\, .\n\\end{align*}\n\\begin{proof}\nSince\n\\begin{align*}\n\\Sigma_0 &= \\Psi_0 Z^0 + \\Psi_1 Z^1 + \\Psi_2 Z^2 \\\\\n\\Sigma_1 &= \\Psi_1 Z^0 + \\Psi_2 Z^1 + \\Psi_3 Z^2 \\\\\n\\Sigma_2 &= \\Psi_2 Z^0 + \\Psi_3 Z^1 + \\Psi_4 Z^2\n\\end{align*}\nand\n\\begin{align*}\nZ^0 = \\bar m \\wedge n &&\nZ^1 = n \\wedge l - \\bar m \\wedge m &&\nZ^2 = l \\wedge m\n\\end{align*}\nwe have\n\\begin{align*}\n\\Gamma' \\wedge \\Sigma_0 \n&= (\\tau' n +\\kappa l - \\rho' \\bar m - \\sigma m) \\wedge (\\Psi_0 Z^0 + \\Psi_1 Z^1 + \\Psi_2 Z^2) \\\\\n&= \n\\Psi_0 (\\kappa l \\wedge \\bar m \\wedge n - \\sigma m \\wedge \\bar m \\wedge n) \\\\\n&\\hspace{5pt} - \\Psi_1 (\\rho' \\bar m \\wedge n \\wedge l + \\sigma m \\wedge n \\wedge l + \\tau' n \\wedge \\bar m \\wedge m +\\kappa l \\wedge \\bar m \\wedge m) \\\\\n&\\hspace{5pt} + \\Psi_2 (\\tau' n \\wedge l \\wedge m - \\rho' \\bar m \\wedge l \\wedge m) \\\\\n&= - \\Psi_1 (\\rho' \\bar m \\wedge n \\wedge l + \\tau' n \\wedge \\bar m \\wedge m) \n+ \\Psi_2 (\\tau' n \\wedge l \\wedge m - \\rho' \\bar m \\wedge l \\wedge m) + O({\\epsilon}^2) \\\\\n&= \\Psi_1(-\\rho' l + \\tau' m) \\wedge Z^0 + \\Psi_2 ( \\tau' m - \\rho' l) \\wedge Z^1 + O({\\epsilon}^2)\n\\end{align*}\nBecause $ (\\tau' m - \\rho' l) \\wedge Z^2 = 0$ this could be added which yields the result.\n\\end{proof}\n\nNow expanding the Bianchi identitiy \\eqref{eq:bianchi}, we find $ \\d \\Sigma_1 = \\tfrac{1}{2} h \\wedge \\Sigma_1 + O(\\epsilon^2) $ which can be written\n\\begin{align} \n(\\d - \\tfrac{1}{2}h \\wedge) \\Sigma_1 = O(\\epsilon^2)\n\\end{align}\nIn the background, this gives the Jordan-Ehlers-Sachs conservation law \\eqref{eq:jes}. For linearized gravity, making use of $3 h \\Psi_2 = 2\\d \\Psi_2$, we find the identity\n\\begin{equation}\\label{eq:fakint}\\begin{aligned}\n0 &= \\psi(\\d - \\tfrac{1}{2}h \\wedge) \\dot \\Sigma_1 \n- \\tfrac{1}{2}\\psi \\dot h \\wedge \\Sigma_1 \\\\\n&=\\d ( \\psi{\\dot\\Psi_1} Z^0 + \\psi{\\dot \\Psi_2}Z^1 + \\psi{\\dot \\Psi_3} Z^2 + \\psi\\Psi_2 \\dot Z^1) -\\tfrac{1}{2} \\psi\\Psi_2 \\dot h \\wedge Z^1 \\\\\n&= \\d ( \\psi{\\dot \\Psi_1} Z^0 + \\psi{\\dot \\Psi_2}Z^1 + \\psi{\\dot \\Psi_3} Z^2 + \\tfrac{3}{2} \\psi\\Psi_2 \\dot Z^1) \\, ,\n\\end{aligned}\\end{equation}\nwere the linearized version of $\\d Z^1 = -h \\wedge Z^1$ is used in the last step. Note, that also\n\\begin{align}\n0 = \\d ( \\psi{\\dot \\Psi_1} Z^0 + \\psi{\\dot \\Psi_2}Z^1 + \\psi{\\dot \\Psi_3} Z^2) -\\tfrac{3}{2} \\psi\\Psi_2 \\dot h \\wedge Z^1\n\\end{align}\nholds, which looks similar to Maxwell equations with a source. We summarize the above discussion by the following\n\\begin{thm}\nFor linearized gravity on a vacuum type D background in principal tetrad exists a closed two-form \n\\begin{align} \\label{eq:linmass}\n \\linmass = \\psi\\dot\\Psi_1 Z^0 + \\psi\\dot\\Psi_2 Z^1 + \\psi\\dot\\Psi_3 Z^2 + \\tfrac{3}{2} \\psi\\Psi_2 \\dot Z^1\n\\end{align}\nwhich can be used to calculate the ``linearized mass''. The integral\n\\begin{align} \\label{eq:1stintcond}\n \\frac{1}{4\\pi \\i} \\int_{S^2} \\linmass \n\\end{align}\nis conserved, gauge invariant and gives the linearized ADM mass.\n\\end{thm}\nThe gauge invariance follows already from its relation to the ADM mass, but the integrand itself has interesting behaviour under gauge transformations. Beside infinitesimal changes of coordinates (coordinate gauge), there are infinitesimal Lorentz transformations of the tetrad (tetrad gauge). To discuss the second one, we need some notation. Following \\cite{Crossmann:1976}, introduce 4 real functions $N_1,N_2,L_1,L_2 $ and 6 complex functions $L_3,N_3,M_i, i=1,..,4 $ to relate the linearized tetrad to the background tetrad\n\\begin{align} \\label{eq:perttetrad}\n \\begin{pmatrix}l^a \\\\ n^a \\\\ m^a \\\\ \\ba m^a \\end{pmatrix}_B =\n\\begin{pmatrix} \nL_1 & L_2 & L_3 & \\ba L_3 \\\\\nN_1 & N_2 & N_3 & \\ba N_3 \\\\\nM_1 & M_2  & M_3 & M_4\\\\\n\\ba M_1 & \\ba M_2  & \\ba M_4 & \\ba M_3\n\\end{pmatrix}_B \n\\begin{pmatrix}\n l^a \\\\ n^a \\\\ m^a \\\\ \\bar m^a\n\\end{pmatrix} .\n\\end{align}\nThese are 16 d.o.f. at a point, 10 correspond to metric perturbations and 6 are infinitesimal Lorentz transformations (tetrad gauge). The linearized tetrad one-forms have the representation\n\\begin{align} \n \\begin{pmatrix}l_a \\\\ n_a \\\\ m_a \\\\ \\ba m_a \\end{pmatrix}_B =\n\\begin{pmatrix} \n-N_2 & - L_2 & \\ba M_2 &  M_2 \\\\\n-N_1 & -L_1 & \\ba M_1 &  M_1 \\\\\n\\ba N_3 & \\ba L_3  & - \\ba M_3 & - M_4 \\\\\nN_3 & L_3  & - \\ba M_4 & - M_3\n\\end{pmatrix}_B \n\\begin{pmatrix}\n l_a \\\\ n_a \\\\ m_a \\\\ \\ba m_a\n\\end{pmatrix}.\n\\end{align}\nIt follows\n\\begin{align}\n \\dot Z^0 &= -(L_1+M_3)Z^0 + \\tfrac{1}{2}(\\ba M_1+N_3)Z^1 -\\ba M_4\\ba Z^0 - \\tfrac{1}{2} (\\ba M_1-N_3)\\ba Z^1 +N_1 \\ba Z^2 \\nonumber \\\\\n\\dot Z^1 &=  -(M_2 +\\ba L_3)Z^0 - \\tfrac{1}{2}(L_1+N_2+M_3+\\ba M_3)Z^1 -(\\ba M_1 +N_3)Z^2 \\nonumber\\\\ & \\hspace{4mm} +(L_3-\\ba M_2)\\ba Z^0 - \\tfrac{1}{2}(L_1+N_2-M_3-\\ba M_3)\\ba Z^1 +(\\ba N_3 -M_1)\\ba Z^2 \\label{eq:linZ1} \\\\\n\\dot Z^2 &= - \\tfrac{1}{2}(M_2 +\\ba L_3)Z^1 -(N_2 +\\ba M_3)Z^2 +L_2 \\ba Z^0 + \\tfrac{1}{2}(M_2-\\ba L_3)\\ba Z^1 -M_4 \\ba Z^2. \\nonumber\n\\end{align}\nLinearization of the tetrad representation of the metric yields\n\\begin{align*}\n h_{ln}  = -L_1 - N_2 &&&   \nh_{m\\bar m}  = \\ba M_3 + M_3 &&&\nh_{n\\bar m} = N_3 -\\ba M_1 &&& \nh_{lm} = \\ba L_3 - M_2 \n\\end{align*}\nand therefore $\\textrm{tr}_g h = - 2 ( L_1 + N_2 + M_3 + \\ba M_3 )$. One should also note, that the selfdual components of $\\dot Z^1$ in $\\linmass$ cancel some of the additional terms, not coming from the linearized Weyl tensor,\n\\begin{subequations}\\begin{align}\n \\dot \\Psi_1 &= - \\dot C \\cdot (Z_0,Z_1) + \\tfrac{3}{2} (\\ba L_3 + M_2) \\Psi_2  \\label{eq:psi1b}\\\\\n \\dot \\Psi_2 &=- \\dot C \\cdot (Z_1,Z_1) + (L_1 + N_2 + M_3 + \\ba M_3) \\Psi_2 \\label{eq:psi2b}\\\\\n \\dot \\Psi_3 &= - \\dot C \\cdot (Z_2,Z_1) + \\tfrac{3}{2} (N_3 + \\ba M_1) \\Psi_2 \\, . \\label{eq:psi3b} \n\\end{align}\\end{subequations}\n\nUsing these facts, we show\n\\begin{prop}\\label{prop:exact}\n  On a spacetime of Petrov type D, the two-form $\\linmass$ is tetrad gauge invariant and changes only with a term $\\chi$ which is exact, $\\chi=\\d f$, under coordinate gauge transformations.\n\\end{prop}\n\\begin{remark} In the work of Fayos et al. \\cite{fayos:ferrando:jaen:1990},\n  a gauge in which $\\d \\left( \\psi \\Psi_2\\dot Z^1 \\right) = 0$ was used. It\n  is not clear from that work \n  whether this gauge condition \nis compatible with a hyperbolic system of evolution\n  equations for linearized gravity. \n\\end{remark}\n\\begin{proof}[Proof of Proposition \\ref{prop:exact}]\nLet us first look at the coordinate gauge. Under infinitesimal coordinate transformations $x^a \\to x^a + \\xi^a$, a tensor field transforms with Lie derivative, $T \\to T - \\mathcal{L}_\\xi T$. For linearized gravity, we write this as $\\dot T \\to \\dot T + \\delta \\dot T = \\dot T - \\mathcal{L}_\\xi T$. Now look at the middle bivector component $Z^1$ and use Cartan's identity $ \\mathcal{L}_\\xi \\omega = \\d (\\xi \\invneg \\omega) + \\xi \\invneg \\d\\omega $, which holds for arbitrary forms $\\omega$. It follows for coordinate gauge transformations in $\\linmass$,\n\\begin{equation}\\label{eq:masscoordgauge}\\begin{aligned}\n \\delta \\linmass\n&= -\\psi\\xi(\\Psi_2) Z^1 - \\tfrac{3}{2}\\psi\\Psi_2 [ \\d(\\xi \\invneg Z^1) +  \\xi \\invneg \\d Z^1 ] \\\\\n&= - \\tfrac{3}{2}\\psi\\Psi_2 (\\d + h\\wedge)(\\xi \\invneg Z^1) \\\\\n&= - \\tfrac{3}{2} \\d [ \\psi\\Psi_2(\\xi \\invneg Z^1)]\n\\end{aligned}\\end{equation}\nwhere $\\xi \\invneg h = \\tfrac{2}{3} \\Psi_2^{-1} \\xi(\\Psi_2)$ and $\\xi \\invneg (h\\wedge Z^1) = (\\xi \\invneg h) Z^1 - h \\wedge (\\xi \\invneg Z^1)$ was used. The two-form \\eqref{eq:masscoordgauge} is exact and hence integrates to zero.\n\nA tetrad gauge transformation changes the tetrad \\eqref{eq:perttetrad} as follows,\n\\begin{align} \\label{eq:tetradgauge}\n \\delta\\begin{pmatrix}l^a \\\\ n^a \\\\ m^a \\\\ \\bar m^a \\end{pmatrix}_B =\n\\begin{pmatrix} \nA & 0 & \\bar b & b \\\\\n0 & -A & \\bar a & a \\\\\na & b & \\i \\vartheta & 0 \\\\\n\\bar a & \\bar b & 0 & -\\i \\vartheta\n\\end{pmatrix} \n\\begin{pmatrix}\n l^a \\\\ n^a \\\\ m^a \\\\  \\bar m^a\n\\end{pmatrix}_B\n\\end{align}\nwith $a,b$ complex and $A,\\vartheta$ real valued. It follows, that the tetrad gauge dependent terms in (\\ref{eq:psi1b},\\ref{eq:psi3b}) cancel the ones in \\eqref{eq:linZ1}. The anti selfdual part in \\eqref{eq:linZ1} is invariant, as follows from \\eqref{eq:tetradgauge}. This shows the tetrad gauge invariance of $\\linmass$ and therefore gauge invariance of \\eqref{eq:1stintcond}. \n\\end{proof}\n\nFinally, to express the charge integral in a form similar to the Maxwell case \\eqref{eq:integrability}, we need the $\\theta\\phi$ components of the bivectors,\n\\begin{align}\n Z^1_{\\theta\\phi}=-\\i (r^2+a^2)\\sin\\theta &&& Z^0_{\\theta\\phi} = -Z^2_{\\theta\\phi} = \\frac{a \\sqrt{\\Delta}}{2} \\sin^2 \\theta.\n\\end{align}\nThe charge integral becomes\n\\begin{align}\n\\frac{2\\i}{\\sqrt{\\Delta}} \\int_{S^2(t,r)} \\mathcal{\\dot M} =\n\\int_{S^2(t,r)} \\left( 2 V_L^{-1\/2} \\dot{\\widehat \\Psi}_2 + \\i a \\sin \\theta \\Psi_{diff} \\right) (r-\\i a \\cos\\theta) \\d\\mu\n\\end{align}\nwith $V_L = \\Delta\/(r^2+a^2)^2$, $\\d\\mu=\\sin\\theta\\d\\theta\\d\\varphi$ and\n\\begin{subequations}\\begin{align}\n  \\dot{\\widehat \\Psi}_2  &= \\dot\\Psi_2 -\\Psi_2(M_3 + \\bar M_3) \\\\\n \\Psi_{diff} &= \\dot\\Psi_1 - \\dot\\Psi_3 - 3\\Psi_2 \\left( \\Re(M_2 - M_1) - \\i \\Im(L_3 + N_3) \\right)  \\, .\n\\end{align}\\end{subequations}\n\n\n\\section{Conclusions} \\label{sec:conclusions} \nFor each isometry of a given background, there is a conserved charge for the linearized gravitational field. Working in terms of linearized curvature, we derived a linearized mass charge (corresponding to the time translation isometry) for Petrov type D backgrounds, by using Penrose's idea of spin-lowering with a Killing spinor. \n\n\nA second Killing spinor, corresponding to the axial isometry of Kerr spacetime does not exist, \\eqref{eq:typeDKS}. Hence spin lowering cannot be used directly to derive a linearized angular momentum charge, even tough a canonical analysis provides one in terms of the linarized metric.\n\nFor a Schwarzschild background, gauge conditions are known, which eliminate\nthe gauge dependent non-radiating modes\n\\cite{zerilli:1970,jezierski:1999}. Understanding these conditions in a\ngeometric way and generalizing them to a Kerr background needs further investigation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe field of deep learning provides a broad family of algorithms to fit an unknown target function via a deep neural network and is having an enormous success in the fields of computer vision, machine learning and artificial intelligence \\cite{mnih2015human,lecun2015deep,radford2015unsupervised,schmidhuber2015deep,goodfellow2016deep}.\nThe input of a deep learning algorithm is a training set, which is a set of inputs of the target function together with the corresponding outputs.\nThe goal of the learning algorithm is to determine the parameters of the deep neural network that best reproduces the training set.\n\nDeep learning algorithms generalize well when trained on real-world data \\cite{hardt2015train}: the deep neural networks that they generate usually reproduce the target function even for inputs that are not part of the training set and do not suffer from over-fitting even if the number of parameters of the network is larger than the number of elements of the training set \\cite{neyshabur2014search,canziani2016analysis,novak2018sensitivity,zhang2016understanding}.\nA thorough theoretical understanding of this unreasonable effectiveness is still lacking.\nThe bounds to the generalization error of learning algorithms are proven in the probably approximately correct (PAC) learning framework \\cite{vapnik2013nature}.\nMost of these bounds depend on complexity measures such as the Vapnik-Chervonenkis dimension \\cite{baum1989size,bartlett2017nearly} or the Rademacher complexity \\cite{sun2016depth,neyshabur2018towards} which are based on the worst-case analysis and are not sufficient to explain the observed effectiveness since they become void when the number of parameters is larger than the number of training samples \\cite{zhang2016understanding,kawaguchi2017generalization,neyshabur2017exploring,dziugaite2017computing,dziugaite2018data,arora2018stronger,morcos2018importance}.\nA complementary approach is provided by the PAC-Bayesian generalization bounds \\cite{mcallester1999some,catoni2007pac,lever2013tighter,dziugaite2018data,neyshabur2017pac}, which apply to nondeterministic learning algorithms.\nThese bounds depend on the Kullback-Leibler divergence \\cite{cover2012elements} between the probability distribution of the function generated by the learning algorithm given the training set and an arbitrary prior probability distribution that is not allowed to depend on the training set: the smaller the divergence, the better the generalization properties of the algorithm.\nMaking the right choice for the prior distribution is fundamental to obtain a nontrivial generalization bound.\n\nA good choice for the prior distribution is the probability distribution of the functions generated by deep neural networks with randomly initialized weights \\cite{perez2018deep}.\nUnderstanding this distribution is therefore necessary to understand the generalization properties of deep learning algorithms.\nPAC-Bayesian generalization bounds with this prior distribution led to the proposal that the unreasonable effectiveness of deep learning algorithms arises from the fact that the functions generated by a random deep neural network are biased towards simple functions \\cite{arpit2017closer,wu2017towards,perez2018deep}.\nSince real-world functions are usually simple \\cite{schmidhuber1997discovering,lin2017does}, among all the functions that are compatible with a training set made of real-world data, the simple ones are more likely to be close to the target function.\nThe conjectured bias towards simple functions has been numerically explored in \\cite{perez2018deep}, which considered binary classifications of bit strings and showed that binary classifiers with a small Lempel-Ziv complexity \\cite{lempel1976complexity} are more likely to be generated by a random deep neural network than binary classifiers with a large Lempel-Ziv complexity.\nHowever, a rigorous proof of this bias is still lacking.\n\n\\subsection{Our contribution}\nWe prove that random deep neural networks are biased towards simple functions,\nin the sense that a typical function generated is insensitive to large changes\nin the input.\nWe consider random deep neural networks with Rectified Linear Unit (ReLU) activation function and weights and biases drawn from independent Gaussian probability distributions, and we employ such networks to implement binary classifiers of bit strings.\nOur main results are the following:\n\\begin{itemize}\n\\item We prove that for $n\\gg1$, where $n$ is the length of the string, for any given input bit string the average Hamming distance of the closest bit string with a different classification is at least $\\sqrt{n\/(2\\pi\\ln n)}$ (\\autoref{thm:main}),\nwhere the Hamming distance between two bit strings is the number of different bits.\n\\item We prove that, if the bits of the initial string are randomly flipped, the average number of bit flips required to change the classification grows linearly with $n$ (\\autoref{thm:main2}).\nFrom a heuristic argument, we find that the average required number of bit flips is at least $n\/4$ (\\autoref{ssec:heuristic}), and simulations on deep neural networks with two hidden layers indicate a scaling of approximately $n\/3$.\n\\end{itemize}\nBy contrast, for a random binary classifier drawn from the uniform distribution over all the possible binary classifiers of strings of $n\\gg1$ bits, the average Hamming distance of the closest bit string with a different classification is one, and the average number of random bit flips required to change the classification is two.\nTherefore, our result identifies a fundamental qualitative difference between a typical binary classifier generated by a random deep neural network and a uniformly random binary classifier.\n\nThe result proves that the binary classifiers generated by random deep neural networks are simple and identifies the classifiers that are likely to be generated as the ones with the property that a large number of bits need to be flipped in order to change the classification.\nWhile all the classifiers with this property have a low Kolmogorov complexity\\footnote{The Kolmogorov complexity of a function is the length of the shortest program that implements the function on a Turing machine  \\cite{kolmogorov1998tables,cover2012elements,li2013introduction}.\n}, the converse is not true.\nFor example, the parity function has a small Kolmogorov complexity, but it is sufficient to flip just one bit of the input to change the classification, hence our result implies that it occurs with a probability exponentially small in $n$.  Similarly, our results explain why\n\\cite{perez2018deep} found that the look-up tables for the functions generated by random deep networks are typically highly compressible using the LZW algorithm \\cite{ziv_universal_1977}, which identifies statistical regularities, but not all functions with highly compressible look-up tables are likely to be generated.\n\nThe proofs of Theorems \\ref{thm:main} and \\ref{thm:main2} are based on the approximation of random deep neural networks as Gaussian processes, which becomes exact in the limit of infinite width \\cite{neal1996priors,williams1997computing,lee2017deep,matthews2018gaussian,garriga2018deep,poole2016exponential,schoenholz2016deep,xiao2018dynamical, jacot2018neural,novak2019bayesian,yang2019scaling,lee2019wide}.\nThe crucial property of random deep neural networks captured by this approximation is that the outputs generated by inputs whose Hamming distance grows sub-linearly with $n$ become perfectly correlated in the limit $n\\to\\infty$.\nThese strong correlations are the reason why a large number of input bits need to be flipped in order to change the classification.\nThe proof of \\autoref{thm:main2} also exploits the theory of stochastic processes, and in particular the Kolmogorov continuity theorem \\cite{stroock2007multidimensional}.\nWe stress that for activation functions other than the ReLU, the scaling with $n$ of both the Hamming distance of the closest bit string with a different classification and the number of random bit flips necessary to change the classification remain the same.\nHowever, the prefactor can change and can be exponentially small in the number of hidden layers.\n\n\nWe validate all the theoretical results with numerical experiments on deep neural networks with ReLU activation function and two hidden layers.\nThe experiments confirm the scalings $\\Theta(\\sqrt{n\/\\ln n})$ and $\\Theta(n)$ for the Hamming distance of the closest string with a different classification and for the average random flips required to change the classification, respectively.\nThe theoretical pre-factor $1\/\\sqrt{2\\pi}$ for the closest string with a different classification is confirmed within an extremely small error of $1.5\\%$.\nThe heuristic argument that pre-factor for the random flips is greater than $1\/4$ is confirmed by numerics which indicate that the pre-factor is approximately $0.33$.\nMoreover, we explore the Hamming distance to the closest bit string with a different classification on deep neural networks trained on the MNIST database \\cite{lecun_gradient-based_1998} of hand-written digits.\nThe experiments show that the scaling $\\Theta(\\sqrt{n\/\\ln n})$ survives after the training of the network and that the distance of a training or test picture from the closest classification boundary is strongly correlated with its classification accuracy, i.e., the correctly classified pictures are further from the boundary than the incorrectly classified ones.\n\n\n\n\\subsection{Further related works}\\label{sec:biblio}\nThe properties of deep neural networks with randomly initialized weights have been the subject of intensive studies \\cite{raghu2016expressive,giryes2016deep,lee2017deep,matthews2018gaussian,garriga2018deep,poole2016exponential,schoenholz2016deep,pennington2018emergence}.\nThe relation between generalization and simplicity for Boolean function was explored in \\cite{franco2006generalization}, where the authors provide numerical evidence that the generalization error is correlated with a complexity measure that they define.\nRef. \\cite{zhang2016understanding} explores the generalization properties of deep neural networks trained on partially random data, and finds that the generalization error correlates with the amount of randomness in the data.\nBased on this result, Ref. \\cite{arpit2017closer,soudry2017implicit} proposed that the stochastic gradient descent employed to train the network is more likely to find the simpler functions that match the training set rather than the more complex ones.\nHowever, further studies \\cite{wu2017towards} suggested that stochastic gradient descent is not sufficient to justify the observed  generalization.\nThe idea of a bias towards simple patterns has been applied to learning theory through the concepts of minimum description length \\cite{rissanen1978modeling}, Blumer algorithms \\cite{blumer1987occam,wolpert2018relationship} and universal induction \\cite{li2013introduction}.\nRef. \\cite{lattimore2013no} proved that the generalization error grows with the Kolmogorov complexity of the target function if the learning algorithm returns the function that has the lowest Kolmogorov complexity among all the functions compatible with the training set.\nThe relation between generalization and complexity has been further investigated in \\cite{schmidhuber1997discovering,dingle2018input}.\nThe complexity of the functions generated by a deep neural networks has also been studied from the perspective of the number of linear regions \\cite{pascanu2013number,montufar2014number,hinz2018framework} and of the curvature of the classification boundaries \\cite{poole2016exponential}.   We note that\nthe results proved here \u2014 viz., that the functions generated by random deep networks are insensitive to large changes in their inputs \u2014 implies that such functions should be simple with respect to all the measures of complexity above,\nbut the converse is not true: not all simple functions are likely to be generated by random deep networks.\n\n\n\n\n\n\n\\section{Setup and Gaussian process approximation}\\label{sec:setup}\nWe consider a feed-forward deep neural network with $L$ hidden layers, activation function $\\tau$, input in $\\mathbb{R}^{n}$ and output in $\\mathbb{R}$.\nThe most common choice for $\\tau$ is the ReLU activation function $\\tau(x)=\\max(0,x)$.\nWe stress that Theorems \\ref{thm:main} and \\ref{thm:main2} do not rely on this assumption and hold for any activation function.\nFor any $x\\in\\mathbb{R}^{n}$ and $l=2,\\ldots,L+1$, the network is recursively defined by\n\\begin{equation}\\label{eq:network}\n\\phi^{(1)}(x) = W^{(1)}x + b^{(1)}\\,,\\qquad \\phi^{(l)}(x) = W^{(l)}\\,\\tau\\left(\\phi^{(l-1)}(x)\\right) + b^{(l)}\\,,\n\\end{equation}\nwhere $\\phi^{(l)}(x),\\,b^{(l)}\\in\\mathbb{R}^{n_l}$, $W^{(l)}$ is an $n_l\\times n_{l-1}$ real matrix, $n_0=n$ and $n_{L+1}=1$.\nWe put for simplicity $\\phi=\\phi^{(L+1)}$, and we define $\\psi(x) = \\mathrm{sign}\\left(\\phi(x)\\right)$ for any $x\\in\\mathbb{R}^n$.\nThe function $\\psi$ is a binary classifier on the set of the strings of $n$ bits identified with the set $\\{-1,1\\}^{n}\\subset\\mathbb{R}^n$, where the classification of the string $x\\in\\{-1,1\\}^{n}$ is $\\psi(x)\\in\\{-1,1\\}$.\nWe choose this representation of the bit strings since any $x\\in\\{-1,1\\}^n$ has $\\|x\\|^2=n$, and the covariance of the Gaussian process approximating the deep neural network has a significantly simpler expression if all the inputs have the same norm.\nMoreover, having the inputs lying on a sphere is a common assumption in the machine learning literature \\cite{daniely2016toward}.\n\n\nWe draw each entry of each $W^{(l)}$ and of each $b^{(l)}$ from independent Gaussian distributions with zero mean and variances $\\sigma_w^2\/n_{l-1}$ and $\\sigma_b^2$, respectively.\nWe employ the Gaussian process approximation of \\cite{poole2016exponential,schoenholz2016deep}, which consists in assuming that for any $l$ and any $x,\\,y\\in\\mathbb{R}^{n}$, the joint probability distribution of $\\phi^{(l)}(x)$ and $\\phi^{(l)}(y)$ is Gaussian, and $\\phi_i^{(l)}(x)$ is independent from $\\phi_j^{(l)}(y)$ for any $i\\neq j$.\nThis approximation is exact for $l=1$ and holds for any $l$ in the limit $n_1,\\,\\ldots,\\,n_L\\to\\infty$ \\cite{matthews2018gaussian}.\nIndeed, $\\phi^{(l)}_i(x)$ is the sum of $b^{(l)}_i$, which has a Gaussian distribution, with the $n_{l-1}$ terms $\\{W^{(l)}_{ij}\\,\\tau(\\phi^{(l-1)}_j(x))\\}_{j=1}^{n_{l-1}}$ which are iid from the inductive hypothesis.\nTherefore if $n_{l-1}\\gg1$, from the central limit theorem $\\phi^{(l)}_i(x)$ has a Gaussian distribution.\nWe notice that for finite width, the outputs of the intermediate layers have a sub-Weibull distribution \\cite{vladimirova2019understanding}.\nOur experiments in \\autoref{sec:exp} show agreement with the Gaussian approximation starting from $n\\gtrsim100$.\n\n\nIn the Gaussian process approximation, for any $x,\\,y$ with $\\|x\\|^2=\\|y\\|^2=n$, the joint probability distribution of $\\phi(x)$ and $\\phi(y)$ is Gaussian with zero mean and covariance that depends on $x$, $y$ and $n$ only through $x\\cdot y\/n$:\n\\begin{equation}\\label{eq:covF}\n\\mathbb{E}\\left(\\phi(x)\\right)=0\\,,\\qquad \\mathbb{E}\\left(\\phi(x)\\,\\phi(y)\\right) = Q\\,F\\left(\\tfrac{x\\cdot y}{n}\\right)\\,,\\qquad \\|x\\|^2=\\|y\\|^2=n\\,.\n\\end{equation}\nAnalogously, $\\phi(x)$ is a Gaussian process with zero average and covariance given by the kernel $K(x,y) = Q\\,F\\left(\\tfrac{x\\cdot y}{n}\\right)$.\nHere $Q>0$ is a suitable constant and $F:[-1,1]\\to\\mathbb{R}$ is a suitable function that depend on $\\tau$, $L$, $\\sigma_w$ and $\\sigma_b$, but not on $n$, $x$ nor $y$.\nWe have introduced the constant $Q$ because it will be useful to have $F$ satisfy $F(1)=1$.\nWe provide the expression of $Q$ and $F$ in terms of $\\tau$, $L$, $\\sigma_w$ and $\\sigma_b$ in \\autoref{app:setup}, where we also prove that for the ReLU activation function $t\\le F(t) \\le 1$.\n\nThe correlations between outputs of the network generated by close inputs are captured by the behavior of $F(t)$ for $t\\to1$.\nIf $F(t)$ stays close to $1$ as $t$ departs from $1$, then the outputs generated by close inputs are almost perfectly correlated and have the same classification with probability close to one.\nOn the contrary, if $F(t)$ drops quickly, the correlations decay and there is a nonzero probability that close inputs have different classifications.\nIn \\autoref{app:setup} we prove that for the ReLU activation function we have $0<F'(1)\\le1$ and for $t\\to 1$,\n\\begin{equation}\\label{eq:FTaylor}\nF(t) = 1 - F'(1)\\left(1-t\\right) + O\\left((1-t)^\\frac{3}{2}\\right)\\,,\n\\end{equation}\nimplying strong short-distance correlations.\n\n\\section{Theoretical results}\\label{sec:main}\n\\subsection{Closest bit string with a different classification}\nOur first main result is the following \\autoref{thm:main}, which states that for $n\\gg1$, for any given input bit string of a random deep neural network as in \\autoref{sec:setup} the average Hamming distance of the closest input bit string with a different classification is $\\sqrt{n\/(2\\pi F'(1)\\ln n)}$.\nThe proof is in \\autoref{app:T1}.\n\\begin{thm}[closest string with a different classification]\\label{thm:main}\nFor any $n\\in\\mathbb{N}$, let $\\phi:\\{-1,1\\}^n\\to\\mathbb{R}$ be the output of a random deep neural network as in \\autoref{sec:setup}.\nLet $a>0$ and let $h_n = \\lfloor a\\sqrt{n\/\\ln n}\\rfloor$, where $\\lfloor t\\rfloor$ denotes the integer part of $t\\ge0$.\nLet us fix $x\\in\\{-1,1\\}^n$ and $z>0$, and let $N_n(a,z)$ be the average number of input bit strings $y\\in\\{-1,1\\}^n$ with Hamming distance $h_n$ from $x$ and with a different classification from $x$, conditioned on $\\phi(x) = \\sqrt{Q}\\,z$:\n\\begin{equation}\nN_n(a,z) = \\mathbb{E}\\left(\\#\\left\\{y\\in\\{-1,1\\}^n:h(x,y)=h_n\\,,\\phi(y)<0\\right\\}\\left|\\phi(x) = \\sqrt{Q}\\,z\\right.\\right)\\,.\n\\end{equation}\nHere $h(x,y)$ is the Hamming distance between $x$ and $y$ and we recall that $Q=\\mathbb{E}(\\phi(x)^2)$.\nThen, for $n\\to\\infty$\n\\begin{equation}\n\\ln N_n(a,z) = \\frac{a}{2}\\sqrt{n\\ln n}\\left(1 - \\frac{z^2}{4F'(1)a^2} + \\frac{\\ln\\frac{\\ln n}{a^2}}{\\ln n} + O\\left(\\tfrac{1}{\\sqrt[4]{n\\ln n}}\\right)\\right)\\,.\n\\end{equation}\nIn particular,\n\\begin{equation}\n\\lim_{n\\to\\infty}N_n(a,z) = 0 \\quad\\textnormal{for}\\quad a<\\frac{z}{2\\sqrt{F'(1)}}\\,,\\qquad \\lim_{n\\to\\infty}N_n(a,z) = \\infty \\quad\\textnormal{for}\\quad a\\ge\\frac{z}{2\\sqrt{F'(1)}}\\,.\n\\end{equation}\n\\end{thm}\n\\autoref{thm:main} tells us that, if $n\\gg1$, for any input bit string $x\\in\\{-1,1\\}^n$, with very high probability all the input bit strings $y\\in\\{-1,1\\}^n$ with Hamming distance from $x$ lower than\n\\begin{equation}\\label{eqn:phi_vs_h}\nh^*_n(x) = \\frac{\\left|\\phi(x)\\right|}{2\\sqrt{Q\\,F'(1)}}\\sqrt{\\frac{n}{\\ln n}}\n\\end{equation}\nhave the same classification as $x$, i.e., $\\phi(y)$ has the same sign as $\\phi(x)$.\nMoreover, the number of input bit strings $y$ with Hamming distance from $x$ higher than $h_n^*(x)$ and with a different classification than $x$ is exponentially large in $n$.\nTherefore, with very high probability the Hamming distance from $x$ of the closest bit string with a different classification is approximately $h_n^*(x)$.\nSince $\\mathbb{E}(|\\phi(x)|) = \\sqrt{2Q\/\\pi}$, the average Hamming distance of the closest string with a different classification is\n\\begin{equation}\\label{eq:main}\n\\mathbb{E}\\left(h_n^*(x)\\right) = \\sqrt{\\frac{n}{2\\pi F'(1)\\ln n}} \\ge \\sqrt{\\frac{n}{2\\pi\\ln n}}\\,,\n\\end{equation}\nwhere the last inequality holds for the ReLU activation function and follows since in this case $F'(1)\\le1$.\n\\begin{rem}\nWhile \\autoref{thm:main} holds for any activation function, the property $F'(1)\\le1$ may not hold for activation functions different from the ReLU.\nFor example, in the case of $\\tanh$ there are values of $\\sigma_w$ and $\\sigma_b$ such that $F'(1)$ grows exponentially with $L$ \\cite{poole2016exponential}.\nIn this case, the Hamming distance of the closest string with a different classification still scales as $\\sqrt{n\/\\ln n}$, but the prefactor can be exponentially small in $L$.\nTherefore with the $\\tanh$ activation function, for finite values of $L$ and $n$, $F'(1)$ may become comparable with\n$\\sqrt{n\/\\ln n}$ and significantly affect the Hamming distance.\n\\end{rem}\n\n\n\\subsection{Random bit flips}\\label{ssec:flips}\nLet us now consider the problem of the average number of bits that are needed to flip in order to change the classification of a given bit string.\nWe consider a random sequence of input bit strings $\\{x^{(0)},\\,\\ldots,\\,x^{(n)}\\}\\subset\\{-1,1\\}^n$, where at the $i$-th step $x^{(i)}$ is generated flipping a random bit of $x^{(i-1)}$ that has not been already flipped in the previous steps.\nAny sequence as above is geodesic, i.e., $h(x^{(i)},\\,x^{(j)}) = |i-j|$ for any $i,\\,j=0,\\,\\ldots,\\,n$.\nThe following \\autoref{thm:main2} states that the average Hamming distance from $x^{(0)}$ of the closest string of the sequence with a different classification is proportional to $n$.\nThe proof is in \\autoref{app:T2}.\n\\begin{thm}[random bit flips]\\label{thm:main2}\nFor any $n\\in\\mathbb{N}$, let $\\phi:\\{-1,1\\}^n\\to\\mathbb{R}$ be the output of a random deep neural network as in \\autoref{sec:setup}, and let $\\{x^{(0)},\\,\\ldots,\\,x^{(n)}\\}\\subset\\{-1,1\\}^n$ be a geodesic sequence of bit strings.\nLet $h_n$ be the expected value of the minimum number of steps required to reach a bit string with a different classification from $x^{(0)}$:\n\\begin{equation}\nh_n = \\mathbb{E}\\left(\\min\\left\\{\\min\\left\\{1\\le i \\le n:\\phi(x^{(0)})\\phi(x^{(i)})<0\\right\\},\\,n\\right\\}\\right).\n\\end{equation}\nThen, there exists a constant $t_0>0$ which depends only on $F$ such that $h_n \\ge n\\,t_0$ for any $n\\in\\mathbb{N}$.\n\\end{thm}\n\\begin{rem}\nSince the entry of the kernel \\eqref{eq:covF} associated to two inputs lying on the sphere is a function of their squared Euclidean distance, which coincides with the Hamming distance in the case of bit strings, Theorems \\ref{thm:main} and \\ref{thm:main2} may be generalized to continuous inputs on the sphere by replacing the Hamming distance with the squared Euclidean distance.\n\\end{rem}\n\n\\subsection{Heuristic argument}\\label{ssec:heuristic}\nFor a better understanding of Theorems \\ref{thm:main} and \\ref{thm:main2}, we provide a simple heuristic argument to their validity.\nThe crucial observation is that, if one bit of the input is flipped, the change in $\\phi$ is $\\Theta(1\/\\sqrt{n})$.\nIndeed, let $x,\\,y\\in\\{-1,1\\}^n$ with $h(x,y)=1$.\nFrom \\eqref{eq:covF}, $\\phi(y)-\\phi(x)$ is a Gaussian random variable with zero average and variance\n\\begin{equation}\\label{eq:step}\n\\mathbb{E}\\left((\\phi(y)-\\phi(x))^2\\right) = 2Q\\left(1-F\\left(1-\\tfrac{2}{n}\\right)\\right)\\simeq 4QF'(1)\/n\\,.\n\\end{equation}\n\nFor any $i$, at the $i$-th step of the sequence of bit strings of \\autoref{ssec:flips}, $\\phi$ changes by the Gaussian random variable $\\phi(x^{(i)}) - \\phi(x^{(i+1)})$, which from \\eqref{eq:step} has zero mean and variance $4Q\\,F'(1)\/n$.\nAssuming that the changes are independent, after $h$ steps $\\phi$ changes by a Gaussian random variable with zero mean and variance $4h\\,Q\\,F'(1)\/n$.\nRecalling that $\\mathbb{E}(\\phi(x^{(0)})^2)=Q$ and that $F'(1)\\le 1$ for the ReLU activation function, approximately $h\\approx n\/(4F'(1)) \\ge n\/4$\nsteps are needed in order to flip the sign of $\\phi$ and hence the classification.\n\nLet us now consider the problem of the closest bit string with a different classification from a given bit string $x$.\nFor any bit string $y$ at Hamming distance one from $x$, $\\phi(y)-\\phi(x)$ is a Gaussian random variable with zero mean and variance $4Q\\,F'(1)\/n$.\nWe assume that these random variables are independent, and recall that the minimum among $n$ iid normal Gaussian random variables scales as $\\sqrt{2\\ln n}$ \\cite{bovier2005extreme}.\nThere are $n$ bit strings $y$ at Hamming distance one from $x$, therefore the minimum over these $y$ of $\\phi(y)-\\phi(x)$ is approximately $-\\sqrt{8Q\\,F'(1)\\ln n\/n}$.\nThis is the maximum amount by which we can decrease $\\phi$ flipping one bit of the input.\nIterating the procedure, the maximum amount by which we can decrease $\\phi$ flipping $h$ bits is $h\\sqrt{8Q\\,F'(1)\\ln n\/n}$.\nSince $\\mathbb{E}(\\phi(x^{(0)})^2)=Q$, the minimum number of bit flips required to flip the sign of $\\phi$ is approximately $h\\approx \\sqrt{n\/(8F'(1)\\ln n)} \\ge \\sqrt{n\/(8\\ln n)}$, where the last inequality holds for the ReLU activation function.\nThe pre-factor $1\/\\sqrt{8}\\simeq 0.354$ obtained with the heuristic proof above is very close to the exact pre-factor $1\/\\sqrt{2\\pi}\\simeq 0.399$ obtained with the formal proof in \\eqref{eq:main}.\n\n\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{0.55\\linewidth}\n\\centering\n\\includegraphics[height=0.6\\linewidth,width=\\linewidth]{greedy_v2.pdf}\n\\caption{}\n\\label{fig:greedy}\n\\end{subfigure}%\n\\begin{subfigure}[b]{0.45\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{PhiVsH_v2_cropped.pdf}\n\\caption{}\n\\label{fig:phi_vs_x}\n\\end{subfigure}\n\\caption{(a) Average Hamming distance to the nearest differently classified input string versus the number of input neurons for the neural network. The Hamming distance to the nearest differently classified string scales as $\\sqrt{n\/(2\\pi\\ln n).}$ with respect to the number of input neurons. Left: the results of the simulations clearly show the importance of the $\\ln n$ term in the scaling. Right: the empirically calculated value $0.405$ for the pre-factor $a$ is close to the theoretically predicted value of $1\/\\sqrt{2 \\pi}$. Each data point is the average of 1000 different calculations of the Hamming distance for randomly sampled bit strings. Each calculation was performed on a randomly generated neural network. Further technical details for the design of the neural networks are given in \\autoref{exp:structure}.\\\\\n(b) The linear relationship between $|\\phi(x)|$ and $h^*_n(x)$ is consistent across neural networks of different sizes. To calculate the average distance at values of $|\\phi(x)|$ within an interval, data was averaged across equally spaced bins of 0.25 for values of $|\\phi(x)|$. Averages for each bin are plotted at the midpoint of the bin. Points are only shown if there are at least 10 samples within the bin.}\n\\end{figure}\n\n\\section{Experiments}\\label{sec:exp}\n\\subsection{Closest bit string with a different classification}\\label{exp:thm:main}\nTo confirm experimentally the findings of \\autoref{thm:main}, Hamming distances to the closest bit string with a different classification were calculated for randomly generated neural networks with parameters sampled from normal distributions (see \\autoref{exp:structure}). This distance was calculated using a greedy search algorithm (\\autoref{fig:greedy}).\nIn this algorithm, the search for a differently classified bit string progressed in steps, where in each step, the most significant bit was flipped. This bit corresponded to the one that produced the largest change towards zero in the value of the output neuron when flipped. To ensure that this algorithm accurately calculated Hamming distances, we compared the results of the greedy search algorithm to those from an exact search which exhaustively searched all bit strings at specified Hamming distances for smaller networks where this exact search method was computationally feasible. Comparisons between the two algorithms in \\autoref{table} show that outcomes from the greedy search algorithm were consistent with those from the exact search algorithm.\nThe results from the greedy search method confirm the $\\sqrt{n\/\\ln n}$ scaling of the average Hamming distance starting from $n\\gtrsim100$.\nThe value of the pre-factor $1\/\\sqrt{2 \\pi}$ is also confirmed with the high precision of $1.5\\%$.\n\\autoref{fig:phi_vs_x} empirically validates the linear relationship between the value of the output neuron $|\\phi(x)|$ and the Hamming distance to bit strings with different classification $h^*_n(x)$ expressed by \\eqref{eqn:phi_vs_h}. This linear relationship was consistent with all neural networks empirically tested in our analysis. Intuitively, $|\\phi(x)|$ is an indication of the confidence in classification. The linear relationship shown here implies that as the value of $|\\phi(x)|$ grows, the confidence of the classification of an input strengthens, increasing the distance from that input to boundaries of different classifications.\n\n\n\\subsection{Random bit flips}\\label{exp:flips}\n\\autoref{fig:bit_flip} confirms the findings of \\autoref{thm:main2}, namely that the expected number of random bit flips required to reach a bit string with a different classification scales linearly with the number of input neurons.\nThe pre-factor found by simulation is $0.33$, slightly above the lower bound of $0.25$ estimated from the heuristic argument.\nOur results show that, though the Hamming distance to the nearest classification boundary scales on average at a rate of $\\sqrt{n\/\\ln n}$, the distance to a random boundary scales linearly and more rapidly.\n\n\n\\begin{figure}[ht]\n  \\begin{minipage}[c]{0.5\\textwidth}\n    \\includegraphics[width=\\textwidth]{pseudorandom_v2.pdf}\n  \\end{minipage}\\hfill\n  \\begin{minipage}[c]{0.5\\textwidth}\n    \\caption{The average number of random bit flips required to reach a bit string with different classification scales linearly with the number of input neurons. Each point is averaged across a sample of 1000 neural networks, where the Hamming distances to differently classified bit strings for each network are tested at a single random input bit string.}\n\\label{fig:bit_flip}\n  \\end{minipage}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Analysis of MNIST data}\\label{exp:MNIST}\nOur theoretical results hold for random, untrained deep neural networks. It is an interesting question whether trained deep neural networks exhibit similar properties for the Hamming distances to classification boundaries. Clearly some trained networks will not: a network that has been trained to return as output the final bit of the input string has Hamming distance one to the nearest classification boundary. For networks that are trained to classify noisy data, however, we expect the trained networks to exhibit relatively large Hamming distances to the nearest classification boundary. Moreover, if a `typical' network can perform the noisy classification task, then we expect training to guide the weights to a nearby typical network that does the job, for the simple reason that networks that exhibit $\\Theta(\\sqrt{n\/\\ln n})$ distance to the nearest boundary and an average distance of $\\Theta(n)$ to a boundary under random bit flips have much higher prior probabilities than atypical networks.\n\nTo determine if our results hold for models trained on real-world data, we trained 2-layer fully-connected neural networks to categorize whether hand-drawn digits taken from the MNIST database \\cite{MNIST} are even or odd.\nImages of hand drawn digits were converted from their 2-dimensional format (28 by 28 pixels) into a 1-dimensional vector of 784 binary inputs. The starting 8 bit pixel values were converted to binary format by determining whether the pixel value was above or below a threshold of 25. Networks were trained to determine whether the hand-drawn digit was odd or even. All networks followed the design described in \\autoref{exp:structure}. 400 Networks were trained for 20 epochs using the Adam optimizer \\cite{kingma2014adam}; average test set accuracy of 98.8\\% was achieved.\n\nFor these trained networks, Hamming distances to the nearest bit string with a different classification were calculated using the greedy search method outlined in \\autoref{exp:thm:main}. These Hamming distances were evaluated for three types of bit strings: bit strings taken from the training set, bit strings taken from the test set, and randomly sampled bit strings where each bit has equal probability of 0 and 1.  For the randomly sampled bit strings, the average minimum Hamming distance to a differently classified bit string is very close to the expected theoretical value of $\\sqrt{n\/(2\\pi\\ln n)}$ (\\autoref{fig:MNISTgreedy}).\nBy contrast, for bit strings taken from the test and training set, the minimum Hamming distances to a classification boundary were on average much higher than that for random bits, as should be expected: training increases the distance from the data points to the boundary of their respective classification regions and makes the network more robust to errors when classifying real-world data compared with classifying random bit strings.\n\nFurthermore, even for trained networks, a linear relationship is still observed between the absolute value of the output neuron (prior to normalization by a sigmoid activation) and the average Hamming distance to the nearest differently classified bit string (\\autoref{fig:MNISTPhiVsH}). Here, the slope of the linear relationship is larger for test and training set data, consistent with the expectation that training should extend the Hamming distance to classification boundaries for patterns of data found in the training set.\n\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}{0.4\\linewidth}\n\\includegraphics[width=0.9\\linewidth]{MNIST_greedy_v3.pdf}\n\\caption{}\n\\label{fig:MNISTgreedy}\n\\end{subfigure}%\n\\begin{subfigure}{0.5\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{MNIST_PhiVsH.pdf}\n\\caption{}\n\\label{fig:MNISTPhiVsH}\n\\end{subfigure}\n\\caption{(a) Average Hamming distance to the nearest differently classified input bit string for MNIST trained models calculated using the greedy search method. The average distance calculated for random bits is close to the expected value of approximately 4.33.\nFurther technical details for the design of the neural networks are given in \\autoref{exp:structure}.\\\\\n(b) The linear relationship between $|\\phi(x)|$ and $h^*_n(x)$ is consistent for networks trained on MNIST data. To calculate the average distance at values of $|\\phi(x)|$ within an interval, data was averaged across equally spaced bins of 2.5 for values of $|\\phi(x)|$. Averages for each bin are plotted at the midpoint of the bin. Points are only shown if there are at least 25 samples within the bin.}\n\\end{figure}\n\n\nFinally, we have explored the correlation between the distance of a training or test picture from the closest classification boundary with its classification accuracy.\n\\autoref{fig:generalization} shows that the incorrectly classified pictures tend to be significantly closer to the classification boundary than the correctly classified ones: the average distances are $1.42$ and $10.61$, respectively, for the training set, and $2.30$ and $10.47$, respectively, for the test set.\nTherefore, our results show that the distance to the closest classification boundary is empirically correlated with the classification accuracy and with the generalization properties of the deep neural network.\n\n\\begin{figure}[ht]\n  \\begin{minipage}[c]{0.6\\textwidth}\n    \\includegraphics[width=\\textwidth]{distance_histogram2.pdf}\n  \\end{minipage}\\hfill\n  \\begin{minipage}[c]{0.4\\textwidth}\n    \\caption{\n       Histogram counting instances of correctly and incorrectly classified MNIST pictures shows that trained neural networks are far more likely to misclassify points closer to a classification boundary for both the training and test sets. Results are aggregated across 20 different trained neural networks trained to classify whether digits are even or odd. Networks are trained for 10 epochs using the Adam optimizer.\n    } \\label{fig:generalization}\n  \\end{minipage}\n\\end{figure}\n\n\n\n\\subsection{Experimental apparatus and structure of neural networks}\\label{exp:structure}\nWeights for all neural networks are initialized according to a normal distribution with zero mean and variance equal to $2\/n_{in}$, where $n_{in}$ is the number of input units in the weight tensor. No bias term is included in the neural networks.\nAll networks consist of two fully connected hidden layers, each with $n$ neurons (equal to number of input neurons) and activation function set to the commonly used Rectified Linear Unit (ReLU). All networks contain a single output neuron with no activation function.\nIn the notation of \\autoref{sec:setup}, this choice corresponds to $\\sigma_w^2=2$, $\\sigma_b^2=0$, $n_0=n_1=n_2=n$ and $n_3=1$, and implies $F'(1)=1$.\nSimulations were run using the python package Keras with a backend of TensorFlow \\cite{tensorflow2015-whitepaper}.\n\n\n\n\n\\section{Conclusions}\\label{sec:concl}\nWe have proved that the binary classifiers of strings of $n\\gg1$ bits generated by wide random deep neural networks with ReLU activation function are simple.\nThe simplicity is captured by the following two properties.\nFirst, for any given input bit string the average Hamming distance of the closest input bit string with a different classification is at least $\\sqrt{n\/(2\\pi\\ln n)}$.\nSecond, if the bits of the original string are randomly flipped, the average number of bit flips needed to change the classification is at least $n\/4$.\nFor activation functions other than the ReLU both scalings remain the same, but the prefactor can change and can be exponentially small in the number of hidden layers.\n\n\nThe striking consequence of our result is that the binary classifiers of strings of $n\\gg1$ bits generated by a random deep neural network lie with very high probability in a subset which is an exponentially small fraction of all the possible binary classifiers.\nIndeed, for a uniformly random binary classifier, the average Hamming distance of the closest input bit string with a different classification is one, and the average number of bit flips required to change the classification is two.\nOur result constitutes a fundamental step forward in the characterization of the probability distribution of the functions generated by random deep neural networks, which is employed as prior distribution in the PAC-Bayesian generalization bounds.\nTherefore, our result can contribute to the understanding of the generalization properties of deep learning algorithms.\n\nOur analysis of the MNIST data suggests that, for certain types of problems, the property that many bits need to be flipped in order to change the classification survives after training the network.\nBoth our theoretical results and our experiments are completely consistent to the empirical findings in the context of adversarial perturbations \\cite{szegedy2013intriguing,goodfellow2014explaining,peck2017lower,madry2017towards,tsipras2018robustness,nakkiran2019adversarial}, where the existence of inputs that are close to a correctly classified input but have the wrong classification is explored.\nAs expected, our results show that as the size of the input grows, the average number of bits needed to be flipped to change the classification increases in absolute terms but decreases as a percentage of the total number of bits.\nAn extension of our theoretical results to trained deep neural networks would provide a fundamental robustness result of deep neural networks with respect to adversarial perturbations, and will be the subject of future work.\n\nMoreover, our experiments on MNIST show that the distance of a picture to the closest classification boundary is correlated with its classification accuracy and thus with the generalization properties of deep neural networks, and confirm that exploring the properties of this distance is a promising route towards proving the unreasonably good generalization properties of deep neural networks.\n\n\n\nFinally, the simplicity bias proven in this paper might shed new light on the unexpected empirical property of deep learning algorithms that the optimization over the network parameters does not suffer from bad local minima, despite the huge number of parameters and the non-convexity of the function to be optimized \\cite{choromanska2015loss,choromanska2015open,kawaguchi2016deep,baity2018comparing,mehta2018loss}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}