Retarded Greens Functions In Perturbed Spacetimes For Cosmology andGravitational Physics

Yi-Zen Chu1,2 and Glenn D. Starkman1Center for Particle Cosmology, Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA,

2Physics Department, Arizona State University, Tempe, AZ 85287, USACERCA, Physics Department, Case Western Reserve University, Cleveland, OH 44106-7079, USA

Electromagnetic and gravitational radiation do not propagate solely on the null cone in a genericcurved spacetime. They develop tails, traveling at all speeds equal to and less than unity. Ifsizeable, this off-the-null-cone effect could mean objects at cosmological distances, such as super-novae, appear dimmer than they really are. Their light curves may be distorted relative to theirflat spacetime counterparts. These in turn could affect how we infer the properties and evolutionof the universe or the objects it contains. Within the gravitational context, the tail effect inducesa self-force that causes a compact object orbiting a massive black hole to deviate from an other-wise geodesic path. This needs to be taken into account when modeling the gravitational wavesexpected from such sources. Motivated by these considerations, we develop perturbation theory forsolving the massless scalar, photon and graviton retarded Greens functions in perturbed spacetimesg = g + h , assuming these Greens functions are known in the background spacetime g .In particular, we elaborate on the theory in perturbed Minkowski spacetime in significant detail;and apply our techniques to compute the retarded Greens functions in the weak field limit of theKerr spacetime to first order in the black holes mass M and angular momentum S. Our methodsbuild on and generalizes work appearing in the literature on this topic to date, and lays the foun-dation for a thorough, first principles based, investigation of how light propagates over cosmologicaldistances, within a spatially flat inhomogeneous Friedmann-Lematre-Robertson-Walker (FLRW)universe. This perturbative scheme applied to the graviton Greens function, when pushed to higherorders, may provide approximate analytic (or semi-analytic) results for the self-force problem in theweak field limits of the Schwarzschild and Kerr black hole geometries.

I. INTRODUCTION AND MOTIVATION

This paper is primarily concerned with how to solvefor the retarded Greens functions of the minimallycoupled massless scalar , photon A, and graviton in spacetimes described by the perturbed metricg = g+h , if the solutions are known in the back-ground metric g . One important instance is that ofMinkowski spacetime, where these Greens functions1

are known explicitly. Here, we will carry out the anal-ysis in detail for the 4 dimensional case, and obtainO[h]-accurate solutions to the Greens functions in per-turbed Minkowski spacetime up to quadrature. Ourmethods are akin to the Born approximation employedin quantum theory, where one first obtains an integralequation for the Greens functions, and the O[hN ]-accurate answer is gotten after N iterations, followedby dropping a remainder term. We are not the first todevelop perturbation theory for solving Greens func-tions about weakly curved spacetimes. DeWitt andDeWitt [1], Kovacs and Thorne [2], and more recently,

1 Since we will be dealing exclusively with retarded Greensfunctions, we will drop the word retarded from henceforth.Despite this restriction, our methods actually apply for ad-vanced Greens functions too.

Pfenning and Poisson [3] have all tackled this problemusing various techniques which we will briefly compareagainst in the conclusions. As far as we are aware,however, our approach is distinct from theirs and havenot appeared before in the gravitational physics andcosmology literature.

Greens functions play crucial roles in understand-ing the dynamics of both classical and quantum fieldtheories. The Greens function depends on the coor-dinates of two spacetime locations we will denote asx (t, ~x) and x (t, ~x),2 and respectively identifyas the observer and source positions. At the classicallevel, which will be the focus of this paper, it can beviewed as the field measured at the spacetime point xproduced by a spacetime-point source with unit chargeat x. To understand this, consider some spacetimeregion V between two constant time hypersurfaces tand t, with t > t. In this paper we assume thatspacetime is an infinite (or, in the cosmological con-

2 The spacetime coordinates in this paper will take the formx, x, x, etc. Instead of displaying the dependence on thesecoordinates explicitly, we will put primes on the indices oftensorial quantities to indicate which of the variables are tobe associated with them. For example G = G [x, x

], denotes the covariant derivative with respect to x, g isthe determinant of the metric at x and g at x, etc.

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2text, semi-infinite) manifold. Let there be some fieldproducing source J present in the volume V , and non-trivial initial conditions for the fields at t, for example,[x0 = t] and 0[x0 = t]. Denote the scalar, pho-ton, and graviton Greens functions as Gx,x , G andG respectively. Then the scalar field evaluatedat some point x lying on the t surface can be writtenas3

x =

V

ddx|g| 12Gx,xJx (1)

+

dd1~x|g| 12

(Gx,x0x 0Gx,xx

)x0=t

while the photons vector potential evaluated at x canbe written as

A =

V

ddx|g| 12GJ (2)

+

dd1~x|g| 12

(G0A 0GA

)x0=t

.

In the same vein, the graviton field at x reads4

=

V

ddx|g| 12GJ (3)

+

dd1~x|g| 12

(GP

0

0GP

)x0=t

,

where P

(1/2)( + gg).

From equations (1), (2) and (3), we see that thephysical solution of a linear field theory can be ex-pressed as the sum of two integrals of the Greens func-tion (and its gradient), one weighted by the sourcespresent in the system at hand and the other weightedby the initial conditions of the fields themselves. Inparticular, the d-dimensional volume integrals (withrespect to x) of the Greens functions, weighted by thefield-producing Js, reaffirms the interpretation thatthe Greens function is the field of a spacetime-pointunit charge because these volume integrals corre-sponds to calculating the field at x by superposing the

3 These are known as the Kirchhoff representations. We referthe reader to the review by Poisson [4] for their derivation. Inthis paper, whenever a formula holds in arbitrary spacetimedimensions greater or equal to 4, we will use d to denote thedimensions of spacetime. Summation convention is in force:Greek letters run from 0 to d1 while small English alphabetsrun from 1 to d 1.

4 We will not be concerned with the nonlinear self-interactionof the gravitons in this paper. However, these nonlinear termsmay be considered to be part of J , since gravity gravitates.

xt

t

J[x]

x

V

FIG. 1: The spacetime region V is the volume containedwithin the two constant time surfaces at times t and t,where t > t. The observer is located at x (t, ~x). Thedark oval is the region defined by the intersection betweenthe past light cone of x and its interior with that of theconstant time surface at t. On it, we allow some non-trivialfield configuration to be present, and through the Kirchhoffrepresentations in equations (1), (2) and (3), the Greensfunctions evolves it forward in time. We emphasize thatthe causal structure of the Greens function, as exhibitedby equations (4), (5) and (6) means, in a generic curvedspacetime, the observer receives fields not only from herpast light cone (edge of the dark oval), but also its interior(dark oval itself). In addition, there is some (scalar, photonor graviton)-producing source J which sweeps out a worldtube, and our observer receives radiation from the portionof this world tube that lies on and within the interior of herpast light cone. The picture here is to be contrasted againstthe Minkowski one, where observers only detect fields fromtheir past null cone.

field produced by all the charges Js present in thesystem.

Moreover, that the Greens function yields a causal-ity respecting solution can be seen from the following.In a generic curved spacetime, if x,x (usually knownas Sygnes world function) denotes half the square ofthe geodesic distance between x and x, a general anal-ysis in 4 dimensions tells us that the Greens functionin a generic curved spacetime consists of two terms.5

One of them is proportional to [tt][x,x ], and de-scribes propagation of the fields on the null cone. The

5 See Poissons review [4] for the Hadamard construction of theGreens functions in (4), (5) and (6) below. We note in pass-ing that, in higher than 4 dimensions, the general form ofthe Greens function will be more complicated, containing notonly and terms, but derivatives of -functions too.

3other is proportional to [t t][x,x ], and describespropagation on the interior of the future light cone ofx.

Gx,x =[t t]

4pi(Ux,x[x,x ] + Vx,x[x,x ]) ,

(4)

G =[t t]

4pi(U[x,x ] + V[x,x ]) , (5)

G =[t t]

4pi

(U[x,x ]

+ V[x,x ]). (6)

In Fig. (1), we illustrate the Kirchhoff representationsin (1), (2) and (3).Tails in curved spacetime The presence of the

two terms in equations (4), (5) and (6), the propertythat for some fixed x, the Greens functions of mass-less fields are non-zero for all x both on and insidethe future null cone for x, teaches us an importantdifference between the propagation of electromagneticand gravitational wave signals in a curved versus flat4 dimensional spacetime. In the latter, signals travelstrictly on the null cone, and the radiation received atsome location x is related to the source at retardedtime t t = |~x ~x|. In the former, signals travel atall speeds equal and less than unity.6 (We are settingc = 1.) This off-the-light-cone piece of massless radi-ation is known in the literature as the tail (or, some-times, wake); and is often touted as a violation of Huy-gens principle in curved spacetime. As elucidated byDeWitt and Brehme [5], this implies the electrodynam-ics of even a single electrically charged particle dependson its entire past history: it exerts a force upon itself(a self-force), in addition to the one already presentin flat Minkowski spacetime, because the electromag-netic fields it produces travels away from it but thenscatters off the geometry of spacetime and returns tointeract with it at some later time.Gravitational Dynamics This tail-induced self-

force finds an analog in the gravitational dynamics ofcompact objects orbiting massive black holes, becausethe gravitational waves they generate scatter off thenon-trivial background geometry and return to nudgetheir trajectories away from a geodesic one. The grav-itational radiation signals of such systems are believedto be within reach of future gravitational wave detec-tors, and there is currently intense theoretical workdone to understand their dynamics. Perturbation the-ory in the weak field limit of Schwarzschild and Kerr

6 This is barring special properties, such as the conformal sym-metry enjoyed by the Maxwell action, which says that light isblind to conformal factors of the metric: a2g and g areequivalent in its eyes. We will shortly elaborate on this point.

may thus provide us with approximate but concreteresults from which we can gain physical insight from(and possibly serve as a check against numerical calcu-lations). For instance, that the tail effect is the resultof massless fields scattering off the background geome-try will be manifest within the perturbative frameworkwe are about to undertake; this point has already beennoted by DeWitt and DeWitt [1] and Pfenning andPoisson [3].Cosmology Turning our attention now to cos-

mology, the past decades have provided us with ob-servational evidence that we live in a universe that is,at the roughest level, described by the spatially flatFLRW metric. In conformal coordinates, it is

g = a2 , diag[1,1,1,1], (7)

where a tells us the relative size of the universe at var-ious times along its evolution. Most of our inferenceof the properties of the universe come from examininglight emanating from objects at cosmological or astro-physical distances, and furthermore our interpretationof electromagnetic signals are based on the assump-tion that they travel on null geodesics. This statementis precisely true when the metric is (7) because theMaxwell action that governs the dynamics of photonsin vacuum, is insensitive to the conformal factor a2.Specifically, in 4 dimensional spacetime, SMaxwell[]and SMaxwell[a

2] are exactly the same object; the con-formal factor a2 drops out.

S(d=4)Maxwell[a

2] = S(d=4)Maxwell[] (8)

= 14

d4x

FF

This means electromagnetic radiation in 4 dimen-sional spatially flat FLRW universes behaves no dif-ferently from how it does in 4 dimensional Minkowskispacetimes. In particular, it travels only along nullgeodesics. However, cosmological and astrophysicalobservations have become so sensitive that it is nolonger sufficient to model our universe as the exactlysmooth and homogeneous spacetime in (7). Rather,one needs to account for the metric perturbations,

g = a2 ( + h) . (9)

Because the a2 drops out of the Maxwell action, we rec-ognize that a first principles theoretical investigationof the propagation of light over cosmological distancesis equivalent to the same investigation in perturbedMinkowski spacetime. Moreover, since the geometry isnow curved (albeit weakly so), light traveling over cos-mological length scales should therefore develop tails.As already mentioned in the abstract, if a significantportion of light emitted from a supernova at cosmolog-ical distances leaks off the light cone, then the observer

4on Earth may mistakenly infer that it is dimmer thanit actually is, as some of the light has not yet arrived.This leakage may also modify the light curves of theseobjects at cosmological distances. To our knowledge,the size of the electromagnetic tail effect in cosmologyhas not been examined before. Our development ofperturbation theory for the photon Greens function(and confirmation of DeWitt and DeWitts first orderresults [1]) in perturbed Minkowski, is therefore thefirst step to a thorough, first principles based, under-standing of the properties of light in the cosmologicalcontext. This may in turn affect how we interpret cos-mological and astronomical observations.7

JWKB Now, the JWKB approximation whereone assumes that the wavelength of the massless fieldsare extremely small relative to the characteristic lengthscales of the spacetime geometry (which, in term, usu-ally amounts to neglecting all geometric terms relativeto the in the wave equation), is often used to justifythat null cone propagation is the dominant channelof travel for massless fields in generic curved space-times. (See for example Misner, Thorne and Wheeler[6].) Here, we caution that, even in cases where theJWKB approximation yields exact results, it does notimply that light travels solely on the light cone. Such acounterexample is that of odd dimensional Minkowskispacetimes, where the momentum vector k satisfiesthe exact dispersion relation kk = 0, but theGreens functions of massless fields develop power lawtails: for odd d, V [x, x] ((tt)2(~x~x)2)(d2)/2.(See Soodak and Tiersten [7] for a pedagogical discus-sion on tails of Greens functions in Minkowski space-times.) This tells us that, even for 4 dimensional flatspacetime, the rigorous way to prove that light travelson the null cone is by computing the photon Greensfunction, since it is the Greens function (via the Kirch-hoff representations in (1), (2) and (3)) that deter-mines how physical signals propagate away from theirsources.

In the next section, we will review the general theoryof Greens functions and some geometrical constructsrelated to them. Perturbation theory for Greens func-tions will then be delineated in the subsequent two sec-tions; following that, we will apply the technology tocalculate the Greens functions in the Kerr black holespacetime, up to first order in its mass and angular mo-mentum. We will conclude with thoughts on possiblefuture investigations.

7 We emphasize here that, we are not, as yet, claiming that thetail effect is significant in the cosmological context. Rather,this paper is laying down the groundwork the computationof the photon Greens function in order to investigate thisissue from first principles.

II. GENERAL THEORY

This section will summarize the key technical fea-tures of Greens functions we will need to understandin the development of perturbation theory in the fol-lowing two sections. We refer the reader to Poissonsreview [4] for an in-depth discussion. We first exam-ine the world function x,x , van Vleck determinantx,x and the parallel propagator g , which are ge-ometrical objects needed for the formal constructionof the Greens functions themselves. We will recordthe equations obeyed by the Greens functions, andthen describe the coefficients of [x,x ] and [x,x ]in (4), (5) and (6). Finally we will compute the x,x ,x,x and g in Minkowski and perturbed Minkowskispacetimes.World Function The world function x,x de-

fined in the introduction is half the square of thegeodesic distance between x and x. Assuming thereis a unique geodesic whose worldline has coordinates{[]| [0, 1]; [0] = x, [1] = x}, it has theintegral representation

x,x =1

2

10

g []d (10)

with d/d.van Vleck Determinant Closely related to x,x

is the van Vleck determinant x,x

x,x = det[x,x ]

|gg|1/2 . (11)

Parallel Propagator The parallel propagatorg is formed by contracting two sets of orthonormalbasis tangent vector fields {A|A, = 0, 1, 2, 3, . . . , d1}, one based at x and the other at x. (The A-index israised and lowered with AB and the -index is raisedand lowered with the metric.)

g [x, x] AB A [x] B [x], (12)

with the boundary conditions that the metric be re-covered at coincidence x = x,

g [x, x] = g [x], g [x, x] = g [x]. (13)

The defining property of these vector fields {A} andhence the parallel propagator itself, is that for a fixedpair of x and x, the {A} are parallel transportedalong the geodesic joining x to x. That is, A =0 and consequently

g = 0. (14)Greens Function Equations Next we record

the equations defining the Greens function. For the

5massless scalar,

xGx,x = xGx,x =d[x x]|gg|1/4 (15)

with x g and x g . For theLorenz gauge photon (A = 0),8

xG R G = xG R G

= gd[x x]|gg|1/4 . (16)

DeWitt and Brehme [5] points out that the divergence(with respect to x) of the Lorenz gauge photon Greensfunction is the negative gradient (with respect to x)of the massless scalar Greens function

G = Gx,x . (17)We will later note that our perturbative result satis-fies (17). Proceeding to the de Donder gauge graviton( = /2, with g),(

1

2

({

} gg

)(R+ 2) + 2R

+R{} gR gR

)G

= ;d[x x]|gg|1/4 (18)

where we have included a non-zero cosmological con-stant . The ; is built out of the parallel prop-agator g [x, x

],

; 12

(gg + gg) . (19)

Greens functions are bitensors. Coordinate transfor-mations at x can be carried out independently fromx (and vice versa). Derivatives with respect to x areindependent of that with respect to x, so for instance,G = G G .Hadamard form We now have sufficient vocabu-

lary to describe the coefficients of [x,x ] and [x,x ]in the Greens functions in equations (4), (5) and (6).Assuming x and x lie in a region of spacetime where

8 Our Christoffel symbol is = (1/2)g({g} g);

Riemann tensor is R = +

( ); the

Ricci tensor and scalar R = R , R = gR. Sym-

metrization is denoted, for example, by T{} = T + T.Antisymmetrization is denoted, for example, by T[] =T T. Whenever we are performing an expansion inseries of h , the metric perturbation, indices of tensors areto be lowered and raised with the background metric g .

there is a unique geodesic joining them, in 4 dimen-sional spacetimes, the null cone pieces are built out ofthe van Vleck determinant and the parallel propaga-tors

Ux,x =

x,x (20)

U =

x,xg (21)

U =

x,xP (22)

where

P 12

(gg + gg gg) . (23)

The tail portions of the Greens functions satisfythe homogeneous equations, for example, xVx,x =xVx,x = 0; Poisson [4] explains the appropriate non-trivial boundary conditions the tail function V s mustsatisfy. Moreover, the derivation of (20), (21) and (22)shows that the geometric tensors in the wave equationfor photons and gravitons only contribute to the tailportion of the field propagation; while it is the differ-ential operator, namely , that contributes to boththe behavior of the null propagation and that of thetail piece. We will also witness this in the perturbativeframework we are about to pursue.

It is appropriate at this point to highlight that thesegeometrical constructs, from which the light cone pieceof the Greens functions are built, have physical mean-ing for the cosmologist. For example, the world func-tion obeys the following equation involving the vanVleck determinant

xx,x +x,x ln x,x = d. (24)Because x,x is proportional to the tangent vec-tor at x (it points in the direction of greatest rate

of change in geodesic distance), xx,x de-scribes the rate of change of the cross sectional area ofthe congruence of geodesics (the expansion) throughthe neighborhood of x, which via (24) is related to thegradient of x,x along the geodesics. This expansionscalar is related to the evolution of the angular di-ameter distance, which then in turn is related to theluminosity distance relation. (See, for example. Visser[8] and Flanagan et al. [9].) Along similar lines, ini-tially parallel null rays from an extended source be-come deflected due to gravitational effects (weak lens-ing). Since the parallel propagator describes the paral-lel transport of an orthonormal reference frame alongthese trajectories, namely

g [x, x]

A[x] = A[x] (see (12)), (25)

they ought to contain physical content regarding polar-ization, rotation and shear of null bundles of photons.To sum, the light cone part of the massless scalar and

6photon Greens function should provide an alternatemeans, from the standard ones in use by cosmologiststoday, of getting at the physics of null light travelingthrough the universe. This warrants more study.

Before moving on to develop our perturbation the-ory, let us take a few moments to calculate the worldfunction, van Vleck determinant and parallel propaga-tor up to first order in h in perturbed Minkowskispacetime. This will allow us to construct the nullcone piece of the scalar, photon and graviton Greensfunction, and in turn, serve as a consistency check onour first Born approximation results below.9 In fact,this was how Kovacs and Thorne [2] constructed thenull cone piece of their Greens functions, by calculat-ing separately the van Vleck determinant and Syngesworld function. But we shall argue that this is not nec-essary. The Born series scheme we have devised givesus a single coherent framework where all three geo-metric objects appearing in the null cone piece of theGreens function are byproducts of the computation.Specifically, the van Vleck determinant and the worldfunction can be read off the massless scalar Greensfunction Gx,x , and the parallel propagator can be readoff the Lorenz gauge photon Greens function G ., and g in Minkowski The geodesic equa-

tion in Minkowski spacetime is

d2

d2= 0, (26)

with boundary conditions [0] = x and [1] = x.The solution is

[] = x + (x x). (27)Inserting this into (10) yields the world function

x,x =1

2

, (28)

where we have defined

(t t, ~x ~x), (29)which is not to be confused with the van Vleck deter-minant (we will always place the spacetime coordinatesas subscripts for the latter). Since

x,x = , x,x = x,x = (30)

by equation (11), the van Vleck determinant is unity.In Cartesian coordinates, the parallel propagator is

9 Some of the results here can be found in Kovacs and Thorne[2] and Pfenning and Poisson [3], but we include them so thatthe discussion is self-contained.

numerically equal, component-by-component, to theMinkowski metric .

x,x = 1, g = . (31)

We shall soon be making heavy use of the mass-less scalar Gx,x , photon G , and graviton GGreens functions in 4 dimensional Minkowski space-time, so let us record their explicit expressions here

Gx,x =[t t][x,x ]

4pi

=[t t][t t |~x ~x|]

4pi|~x ~x| (32)G = Gx,x (33)

G = PGx,x (34)

with

P =1

2

({}

). (35)

The photon here obeys the Lorenz gauge A =0 while the graviton the de Donder gauge h =

h/2. For computational purposes, we recordthat the P has the following symmetries

P = P = P = P. (36)

, and g in perturbed MinkowskiTo tackle these geometric entities in perturbedMinkowski, we start by noting that the integral in(10) defines a variational principle for geodesics. Forfixed end points x and x, and an affine parame-ter, the paths which extremizes the integral in (10)are the geodesics. Let be the geodesic in perturbedMinkowski spacetime joining x to x. If we were tosolve it perturbatively, we can try = + , where can be viewed as a small displacement, and plugthis ansatz into the integral in (10). But since the in-tegral defines a variational principle, that means thefirst order variation of the integrand, due to the O[]deviation of the geodesic from the Minkowski one, iszero. To first order in h , the world function can thusbe obtained from (10) by simply setting = .

x,x x,x + I(0) , (37)where

I(0) 1

2

10

h []d. (38)

(The reason for the name I(0) will be clear later.) Nowput (37) into (11), and employ (30). Then use thefollowing relation, that for matrices A and B such thatB is a small perturbation relative to A,

det[A+B] = det[A](1 + Tr[A1B] + . . .

). (39)

7(Tr denotes trace, and A1 is the inverse of A.) Wethen deduce the square root of the van Vleck determi-nant is

x,x (

1 14h 1

4h + I(0) (40)

+ (

)I(0)

1

2 I(0)

).

Here, h h and I(0) I(0) .In [8], Visser developed perturbation theory for solv-

ing the van Vleck determinant. In particular, heshowed that the O[h] accurate x,x is given by (hisequation 61)

x,x 1 +

2

10

d(1 )(R|1) [], (41)

where (R|1) [] is the linearized Ricci tensor evalu-ated on the unperturbed geodesic .

Let us show the equivalence of (40) and (41). First,we write down the explicit form of the linearized Riccitensor in Cartesian coordinates. One is lead to theexpression

x,x 1 +

2

10

d(1 )(h

12h

) 1

2 I(0) , (42)

where we have employed (1) = and =. In the first line, the

= d/d. This can be

integrated-by-parts, and the resulting (1 2) is , and can be pulled out of the integral,

2

10

d(1 )h = ( )I(0) .(43)

What remains is to demonstrate that

4

10

d(1 )h = 14h 1

4h + I(0).

This relation can be reached by recognizingh = d

2h/2, followed by integrating-by-

parts the d2/d2.Equation (14) says the parallel propagator is paral-

lel propagated along . If we write g = + hand keep only the O[h] terms in the Christoffel symbolin h = h h , (14) is then approxi-mately equivalent to

d

dh [[], x

] (44)

=1

2

(h [] + h [] h []

)[]

where the derivatives are with respect to ; for ex-ample, /. Since has to begin at O[h],that means to the first order, we can replace with = . Recognizing

dh []

d= h [] (45)

and recalling the boundary conditions (13) then allowus to integrate (44) to deduce

g + 12

(h + h) +

2

10

[h] []d.

(46)

As can be checked explicitly,

h [] = ( + )h . (47)

We may thus re-write (46) in terms of I(0) in (38),

g + 12

(h + h) + ([ + [)I(0)].

(48)

To be clear, h and h are the metric perturbationsat x and x respectively; while +h is the parallelpropagator in perturbed Minkowski spacetime.

III. PERTURBATION THEORY

We now describe the Born series method to solve theGreens functions in a formal power series in h , themetric perturbation.Scalar The quadratic action of the minimally cou-

pled massless scalar field evaluated in the perturbedmetric g = g + h reads

S[g] 12

ddx|g| 12 (49)

while the same action evaluated in the backgroundmetric g , with denoting the covariant derivativewith respect to it, is

S[g] 12

ddx|g| 12

. (50)

In S[g], if we replace one field with Gx,x , the Greensfunction in g , and the other with Gx,x , the Greensfunction in g , upon integration-by-parts, and using(15), we see that

2S[g; Gx,x , Gx,x ] (51)

=

ddx|g| 12Gx,xGx,x = Gx,x .

8Similarly, by replacing one of the fields in S[g] withGx,x and the other with Gx,x , one obtains

2S[g;Gx,x , Gx,x ] (52)

=

ddx|g| 12

Gx,xGx,x = Gx,x .

The surface terms incurred during integration-by-parts in (51) and (52) are zero because the sur-

face integrands at hand, namely Gx,xGx,x andGx,x

Gx,x , due the causal structure of the

Greens functions, are non-zero only in the spacetimeregion defined by the intersection of the interiors ofthe past light cone of x with that of the future nullcone of x. As Fig. (2) informs us, this intersection isalways a finite region of spacetime. As long as we aredealing with a spacetime manifold that is infinite (orsemi-infinite) in extent, this finite region of intersec-tion lies deep inside the region enclosed by the surfaceat infinity, and hence does not contribute to the sur-face integral itself. Subtracting the equations (51) and(52) then hands us an integral equation for Gx,x :

Gx,x Gx,x (53)=

ddx|g| 12 gGx,xGx,x

ddx|g| 12 gGx,xGx,x .

First Born Approximation Perturbation theory maynow be carried out by iterating (53) as many times asone wishes (followed by dropping the remainder inte-gral terms containing Gx,x), and expanding

g = g h + . . . (54)

|g| 12 = |g| 12(

1 +1

2h+ . . .

); h gh (55)

to as high an order in h as desired. (We are now rais-ing and lowering all indices with the background met-ric g .) To obtain the first Born approximation, theO[h]-accurate result for Gx,x , one replaces the Gx,xoccurring within the integrals in (53) with Gx,x and

only need to expand the |g|1/2 and g to first or-der. The result is

Gx,x Gx,x +

ddx|g| 12 (56)

{Gx,x

(1

2hg

h)Gx,x

}with h g h . In perturbed Minkowski space-time, we set g = , employ Cartesian coordinates,and then use the spacetime translation symmetry re-flected by the Greens function Gx,x for any d, namely

Gx,x = Gx,x , (57)

xt

t x

V

xt

t x

V

FIG. 2: Top Panel : The intersection of the interiors ofthe future null cone of x and that of the past null cone ofx always defines a finite (as opposed to infinite) region ofspacetime. Moreover, if (and only if) x lies on or within theinterior of the backward light cone of x (or, equivalently, ifand only if x lies on or within the interior of the forwardlight cone of x), then there is a non-trivial intersection (in-dicated by the dark dashed oval) between the forward lightcone of x and backward light cone of x, which in 3-spacewe shall show is a prolate ellipsoid, when the background isMinkowski. Bottom Panel : If x lies outside the backwardlight cone of x (or, equivalently, if x lies outside the forwardlight cone of x), then there is no intersection between theforward light cone of x and backward light cone of x.

to pull the two derivatives out of the integral

Gx,x Gx,x (58)

+

ddxGx,x

(1

2h h

)Gx,x

with h h . This matches equation 2.27 ofDeWitt and DeWitt [1], if we note that their Greens

9function is negative of ours.Photon Next, we turn to the photon. The

Maxwell action in terms of electric and magnetic fieldsF is

SMaxwell = 14

ddx|g| 12 ggFF .

(59)

We have already noted in the introduction, that thisaction SMaxwell enjoys a conformal symmetry in 4 di-mensions, namely, it evaluates to the same object inboth the metric g and the metric a

2g ; the con-formal factor a2 drops out. Whenever there is such aconformal factor, for instance, as in the context of aspatially flat inhomogeneous FLRW universe describedby the metric in (9) we will choose the Lorenz gaugewith respect to g and not a

2g :

A 1|g| 12 (|g| 12 gA

)= 0 (60)

so that the dynamics of A will also be blind to a2.

The quadratic action for the photons vector potentialA evaluated in the metric g = g + h is

SA[g] = 12

ddx|g| 12

(AA (61)

+RAA

).

Via steps analogous to the ones taken to obtain theintegral equation for the scalar Greens function, re-placing one field with G and the other with G, we canwrite down the corresponding integral equation for thephoton Greens function G in the perturbed space-time g = g + h :

G G (62)

=

ddx|g| 12

(gg

GG

+RGG

)

ddx|g| 12(g g

GG

+ RGG

).

Here and below, the barred geometric tensors such asR are built out of g ; whereas the un-barred onesare built out of g .

First Born Approximation Like the scalar case,one may now pursue perturbation theory of the pho-ton Greens function by iterating the integral equation(62) however many times (followed by dropping the re-mainder integral terms containing G) and performthe expansion in (54) and (55), and of the Christoffelsymbols

[g] [g] (63)=

1

2(g h + . . . ) ({h} h)

to whatever order in h one wishes. To O[h], wemerely need to replace the G occurring under theintegral sign in (62) with G and develop the neces-sary expansion to linear order in h . The additionalcomplication in the photon case here, and the gravi-ton case below, is that one has to deal with integralsof the schematic form

G(|1)G, arising from the

covariant differentiation of the Greens functions. The(|1) is the first order in h variation of the Christoffelsymbol,

(|1) =1

2g

({h} h) . (64)For such terms, we will choose to integrate by parts,moving all the (single) derivatives acting on the hsin the (|1) onto the un-perturbed Greens functionsG. (As already argued, there are no surface terms.)The ensuing manipulations require the use of equations(16) and (17). About a generic perturbed spacetimeg = g + h , we then gather that

G G + 12Gh

+

1

2h G

+

ddx|g| 12

(G

(1

2hg

g h g

)G

+1

2Gh

G 12Gh

Gx,x

+1

2Gx,xhG 1

2Gh

G

10

12Gx,xhG + 1

2Gh

G

12GhG + 1

2GhGx,x

+ G(hR

+ R

h

)G

+ G

((R|1) + 1

2hR

)G

). (65)

In (65), we are again raising and lowering all in-dices with the background metric g . Here andbelow, (R|n), (R|n) and (R|n) are the por-tion of the respective geometric tensors (built out ofg = g + h) containing precisely n powers of theperturbation h .

When the background is Minkowski g = allthe barred geometric tensors are identically zero. Likein the scalar case, we employ Cartesian coordinatesand the spacetime translation symmetry property ofGx,x in (57) to massage (65) into

G Gx,x + 12Gx,x (h + h) (66)

+

ddx

{Gx,x

(1

2h

h)Gx,x

+1

2( )

([ + [

)Gx,xh

]Gx,x + Gx,x (R|1) Gx,x

}.

This matches equation 2.23 of DeWitt and DeWitt[1], up to a sign error, if we take into account boththeir R and Greens function are negative of ours.(Their sign error10 is the following: the two terms onthe line right before the last line (involving the Riccitensor), should both carry a negative sign each, sincethey must have come from integrating by parts theterm []h,, [].) As a consistency check ofthis result, one may perform a direct computation toshow that the G in (66) satisfies (17) to first orderin h .Graviton Gravitation as encoded in the Einstein-

Hilbert action

SEH 116piGN

ddx|g| 12 (R 2) (67)

is a nonlinear theory. (GN is Newtons constant and is the cosmological constant.) One can insert the

metric g +

32piGN into the Einstein-Hilbert ac-tion (67) and find a resulting infinite series in .The quadratic piece, which will determine for us theGreens function of the graviton, is

S [g] =1

2

ddx|g| 12

( 1

2

2R 2 R + 2 R+

(

122)

(R 2)), (68)

where we have chosen the de Donder gauge =12, with g . (The geometric tensors in(68), such as R, are built out of g .) From (68)and following the preceding analysis for the scalar andphoton, we may write down the integral equation in-volving the graviton Greens functions

G G

=

ddx|g| 12

( Gg

(gg

12gg

)G

11

+ G

((gg

12gg

)

(R 2) 2R

Rg Rg +Rg +Rg)G

)

ddx|g| 12( G g

(g g

12g g

)G

+G

((g g

12g g

)(R 2) 2R

R g R g + R g + R g)G

). (69)

First Born Approximation Because of the numberof terms and the plethora of indices in (69), the per-turbation theory about a generic background g andarbitrary dimensions d is best left for a computer al-gebra system to handle. We shall be content with thecase of 4 dimensional perturbed Minkowski spacetime,and also set the cosmological constant to zero for now.To O[h], we replace in (69) all the G occurringunder the integral sign with PGx,x (see (34)) andexpand all quantities about Minkowski spacetime upto first order in perturbations. Let us employ Carte-sian coordinates, raise and lower indices with , andintegrate-by-parts the derivatives acting on h occur-ring within the Christoffel symbols,

ddxGx,xhGx,x (70)

= ( + )

ddxGx,xhGx,x ,

where we have invoked (57). It helps to exploit the

symmetries of the Riemann tensor indices (R =R = R = R), those of P recorded in(36), and to recognize that, in d = 4 dimensions,

P P =1

2{

}. (71)

For reasons to be apparent in the next section, we shallre-express all the

as

=

1

2( + )(

+ ) 1

2

12

12

( + )2 122 1

22

followed by using the Minkowski version of (15),namely

2Gx,x = 2Gx,x = d[x x]. (72)

We then arrive at

G Gx,x(P +

1

4((h + h) + (h + h) + (h + h) + (h + h))

12h 1

2h

)+

d4x

{PGx,x

(1

2h h

)Gx,x

+1

4

(

)(([ + [)Gx,xh] Gx,x + ([ + [)Gx,xh] Gx,x

+ ([ + [)Gx,xh] Gx,x + ([ + [)Gx,xh] Gx,x

)+ Gx,x

(P(R|1) + (R|1) + (R|1)

12{(R|1)} 1

2{(R|1)} + (R|1){}

)Gx,x

}. (73)

12

One scattering approximation Let us examine(58), (66) and (73). The terms that do not involveany integrals can be viewed as the propagation of nullsignals, modulated by the metric perturbations mul-tiplying the Gx,x . The terms involving integrals, go

schematically as xx

d4xGx,xh[x]Gx,x . Dueto the causal structure of the Gs, this can be inter-preted as a scattering process. The Gx,x tells us ourmassless field begins at the source x and travels alonga null ray to x; the h[x] says it then scatters off themetric perturbations (and its derivatives) at x; andthe Gx,x informs us that it then propagates along anull path from x to reach the observer at x. The full(scattered) signal consists of integrating over all thex from which the signal can scatter off. This is theperturbative picture for the origin of tails of masslessfields in weakly curved spacetime.11 From this heuris-tic point of view, we can already anticipate that highorder perturbation theory will involve more than onescattering events contributing to the tail effect. Thisscattering picture may also help us estimate its sizewithout detailed calculations, and deserves some con-templation.

IV. [] AND [] DECOMPOSITION IN 4DIMENSIONAL PERTURBED MINKOWSKI

In this section we will restrict ourselves to 4 dimen-sions and analyze further the first order results forthe scalar (58), photon (66) and graviton (73) Greensfunctions we have obtained in perturbed Minkowskispacetime, and show that to O[h], concrete results forthe Greens functions can be gotten once a single ma-trix of integrals (involving h) can be performed. Wewill also decompose these scalar, photon and gravitonGreens functions into their null cone and tail pieces.As a consistency check of our Born approximation, weshow that their null cone pieces matches the Hadamardform described by equations (20), (21) and (22); thisgeneralizes the analysis carried out in Pfenning andPoisson [3] to the case of arbitrary perturbations h .

In the scalar (58), photon (66) and graviton (73)Greens functions results, we have to deal with deriva-tives (with respect to x or x) acting on the following

11 We are being slightly inaccurate here, in that some of thexx

d4xGx,xh[x]Gx,x terms also contribute to null

propagation, as we will see in the next section. But we wantto introduce this scattering picture here, because it is easierto see it from (58), (66) and (73), written in terms of theMinkowski Greens function Gs, than from (84), (86), and(87) below, which are expressed in terms of [x,x ] and the

I-integrals in (78).

matrix integral

1

4piI

d4xGx,xhGx,x (74)

with the Gx,x from (32). Because of (70), even thegeometric tensor terms can be expressed as sum ofderivatives with respect to x or x acting on (74). Forexample,

d4xGx,x(R|1)Gx,x (75)

= ++

d4xGx,x(h

h)Gx,x

where h is the trace of h and

+ + . (76)In appendix (A) we show that I involves the inte-gral of h (but in Euclidean 3-space) over the surfacegenerated by rotating the ellipse with foci at ~x and ~xand semi-major axis (t t)/2, about the line joining~x and ~x. (This is the dashed oval in Fig. (2).)

I [x, x] [t t][x,x ]I [x, x] (77)with

I [x, x] (78)

=1

2

S2

d

4pih

[t+ t

2+|~|2

cos ,~x+ ~x

2+ ~x

].

The infinitesimal solid angle is d = d cos d, and theCartesian components of ~x are given by

~x (

x,x

2sin cos,

x,x

2sin sin,

0

2cos

).

(79)

To separate the light cone versus tail pieces of theGreens functions, we now carry out the necessaryderivatives on (77) as they occur in (58), (66) and (73).There is no need to differentiate the [t t], becausethat would give us [t t] and its derivatives. Sincethis would be multiplied by either [x,x ] or possi-

bly [x,x ], [x,x ], etc., while x,x ~2/2 < 0,

these , , . . . terms can never be non-zero when t =t. Schematically, therefore, the derivatives now read[t t]([]I) (where the two derivatives are bothwith respect to either x or x or one each), which inturn would yield two types of terms. One is the tail

term, proportional to []I and the other the nullcone ones, proportional to either []I, []I,or []I. Following that, we would impose theconstraint x,x = 0 on the coefficients of the [] and

13

[] terms. This requires that we develop a power se-ries in x,x of I . Since there is at most one deriva-tive acting on I, however, we only need to do so up tolinear order. (Higher order terms would automaticallyvanish once we put = 0.) In appendix (A) we find

I = I(0) + x,x I(1) + . . . (80)

=(

1 x,x2

) 1

2

10

h []d+ . . . .

The I(0) = 12 10h []d has already been quoted

previously in (38).Scalar By pulling out one factor of (4pi)1 from

one of the Gs (see (32)), our result for the masslessscalar Greens function in (58) can be written as

Gx,x [t t]

4pi

([x,x ]

+

({1

2 I I

}[x,x ]

)), (81)

with I I. Carrying out the derivatives using(30) would give us, amongst other terms, the following terms:

[x,x ](x,x I I

). (82)

The first term is [x,x ] I if we employ the identityz[z] = [z]. The second term can be considered theO[h] term of [+ I ] = []+[] I+. . . .

Moreover, invoking (45) and the chain rule also in-forms us that one of the terms multiplying [x,x ] is

12

( )I(0) = 1

4h 1

4h + I(0) . (83)

Altogether, the Born approximation, O[h]-accurateanswer, for the massless scalar Greens function maynow be decomposed into its null cone and tail piecesas

Gx,x [t t]

4pi

{[x,x +

I(0)](

1 14h 1

4h + I(0) +

(

)I(0)

1

2

I(0)

)+

[x,x +

I(0)](1

2 I I

)}, h h ; h h . (84)

As already advertised earlier, comparison with (37)and (40) tells us the null cone portion of our mass-less scalar Greens function is indeed consistent withthe Hadamard form in (4) and (20).Photon and Graviton For the photon G (66)

and graviton G (73) Greens functions, we firstobserve that they contain respectively and Pmultiplied by (84), the massless scalar Gx,x . (Specif-ically, first term on the first line, and the second lineof (66) for the photon; and first term on the first line,and third line of (73) for the graviton.) The light coneportions of these terms therefore contain the first or-der van Vleck determinant. For the rest of the inte-gral terms, we first make the observation that +acting on a function whose argument is the differencex x, is identically zero. The immediate corollaryis that all the geometric terms, via (70), do not con-tribute to the null cone piece of the photon and gravi-ton Greens function because the derivatives acting onthe [] leads to zero. The remaining terms containing

derivatives take the form

1

2

(

) ([ + [

) ([x,x ]I]

)= [x,x ]

2

10

[h] []d (85)

+ [x,x ]1

2

(

) ([ + [

) I] ,where we have utilized (30) and the chain rule. Re-

calling (46) tells us the [] terms on the right sideof (85), when added to the non-integral O[h] ones al-ready multiplying Gx,x i.e., the first line of (66) andfirst two lines of (73) would give us the necessaryfirst order parallel propagators to once again ensureconsistency with the Hadamard form in (5), (21), (6)and (22). That is, we may now use the expressions forx,x (37),

x,x (40) and g (46) and decompose

the photon and graviton Greens function into theirnull cone and tail pieces. To first order in h ,

14

G [t t]

4pi

{g

x,x [x,x ] (86)

+ [x,x ]

(

(1

2 I I

)+

1

2( )

([ + [

) I] + (R|1))},G [t t

]4pi

{P

x,x [x,x ]

+ [x,x ]

(P

(1

2 I I

)+

1

4

(

)(([ + [)I] + ([ + [)I]

+ ([ + [)I] + ([ + [)I] )

+ P (R|1) + (R|1) + (R|1) 1

2{ (R|1)}

1

2{ (R|1)}

+ (R|1){})}

. (87)

The geometric terms (R|1) , (R|1) , and (R|1)in (86) and (87) can be obtained by taking the corre-sponding linearized tensors in terms of the perturba-

tion h , and replacing all the h with I and allderivatives with + .

(R|1) 1

2

(+[+ I] +[+ I]

)(88)

(R|1) 1

2

(+{+ I }

++ I ++ I)

(89)

(R|1) ++(I I

), (90)

with + + . Even though these terms involv-ing geometric curvature are best evaluated by differen-

tiating I , it is necessary to record here their analogsto (78). For instance, if (R|1) is the linearized Riccitensor, we have

1

4pi[t t][x,x ](R|1)

=

d4xGx,x(R|1)Gx,x , (91)

where

(R|1) [x, x] (92)

=1

2

S2

d

4pi(R|1)

[t+ t

2+|~|2

cos ,~x+ ~x

2+ ~x

].

At the first Born approximation, therefore, we see thata concrete expression from the perturbative solution ofthe scalar (84), photon (86), and graviton (87) can be

obtained once the matrix integral I in (78) is evalu-ated. We also note that, suppose I in (78) has beenevaluated; then at least when h is time-independent(space-independent), there is no need to perform the

line integral I(0) in (80); rather, I(0) is gotten by re-placing t t |~x ~x| (|~x ~x| t t). In suchcases, the Born series method advocated here allowsone to read off, as a byproduct of a single coherentcalculation, the world function and van Vleck determi-nant from, respectively, the argument and coefficientof the -function in the massless scalar Greens func-tion; while the parallel propagator can be read off thecoefficient of the -function in the Lorenz gauge photonGreens function.Gauge dependence The skeptic may wonder if

the gauge dependence of the vector potential couldrender the tail piece of the photon Greens functionin (86) un-physical. To that end, we note that, forfixed x, the only pure gradient tail term in (86) is(1/2)()I . Hence, the rest of the tail termsdo not have zero curl the corresponding electromag-netic fields are non-zero. This provides strong theoreti-cal evidence that the wake effect is present for photonspropagating in perturbed Minkowski, and by confor-mal symmetry, in our universe too.Geometry and tails Let us notice that it was all

the differentiation that took place in our work on theperturbative solution of the Greens functions, which

15

can be traced to the operator, that gave us boththe terms in the arguments and coefficients of the -functions in the scalar, photon, and graviton Greensfunction. In turn, we have identified them as variousterms in the world function, the van Vleck determi-nant and the parallel propagator (in their perturba-tive guises). This re-affirms our assertion earlier thatit is the differential operator that is solely responsi-ble for the behavior of massless radiation on the lightcone. On the other hand, because of (70), at the levelof the Born approximation, we see that the geometrictensors contribute only to the tail piece of the Greensfunction.

V. SCHWARZSCHILD AND KERRGEOMETRIES

As a concrete application of our formalism, in thissection we will calculate the null cone and tail pieces ofthe Greens functions in the weak field limit of the Kerrgeometry, to first order in the black holes mass M andangular momentum S. Setting S to zero would thengive us the first order in mass result for the weak fieldSchwarzschild geometry. These results, when pushedto higher orders in M and S, would provide us withconcrete expressions for the Greens functions to inves-tigate the tail induced self force and more generally,the gravitational n-body problem, in the weak fieldlimit background of astrophysical black holes. Strictlyspeaking, because S M2, a consistent answer for theGreens functions would require at least a second orderin M calculation, but since this constitutes a signifi-cant computational effort, we shall leave it for a futurepursuit.Schwarzschild We begin with a discussion of

the Schwarzschild case. If we choose to writethe Schwarzschild black hole metric in (Cartesian)isotropic coordinates (t, ~x), so that there are no offdiagonal terms, we may express

g = + h (93)

where

h00 (

1 M2r1 + M2r

)2 1 (94)

= 4M2r

+ 8

(M

2r

)2 12

(M

2r

)3+ . . . ,

hij ij((

1 +M

2r

)4 1)

(95)

= ij

(4M

2r+ 6

(M

2r

)2+ 4

(M

2r

)3+ . . .

),

h0i = 0. (96)

Here r ijxixj , M is the mass of the black hole,and we have set Newtons constant to unity, GN = 1.The power series expansion of h00 and hij can be sub-

stituted into the I-integral in (78). At order (M/2r)2and beyond, the solution would of course receive con-tributions from more iterations and high order h termsfrom the integral equations (53), (62), and (69), and

would likely involve two or more overlapping I-typeintegrals. Here we will focus on the first Born approx-imation.

Within the one scattering approximation, the maintechnical hurdle to overcome is therefore the class ofintegrals

I(n) 14pi

+11

2pi0

d(cos )d

|~x[, , ] ~z[s, +, +]|n(97)

where n is a positive integer, ~x has Cartesian compo-nents defined in (79) (so that, in particular, = 0 =t t), and

~z ~x+ ~x

2. (98)

Because we choose our coordinate system such that

~x ~x = |~|e3, where e3 is the unit vector in the 3-direction, we have the following equalities (see (A4)),

~z[s, +, +] =1

2

(s2 |~|2 sin + cos+,

s2 |~|2 sin + sin+,

s cos +

)=|~|2e3 ~x, (99)

from which we can deduce that

s = r + r, cos + =r r|~x ~x| . (100)

(The other solution (s, cos +) = (|rr|,(r+r)/|~x~x|) is inadmissible because (r+r)/|~x~x| 1.) Here,r |~x| and r |~x|, and the azimuth angles of ~x and~x are both equal to + + pi.

For the moment, it helps to think of ~x and ~z as in-dependent vectors which we have chosen to write theirCartesian components in terms of ellipsoidal coordi-nates (, , ) and (s, +, +); we will also take R in(A5) to be simply a constant, not necessarily equal to

|~|.The n = 1 case has been evaluated by both DeWitt

and DeWitt [1] and Pfenning and Poisson [3] by per-forming a prolate ellipsoidal harmonics expansion ofthe inverse Euclidean distance |~x ~z|1. An alter-nate means of getting the same result is, as already

noted by DeWitt and DeWitt, to recognize that I(1) is

16

the Columb (electric) potential of a charged perfectlyconducting ellipsoid defined by ~x.12 By definition,the conducting surface is an equipotential one. This

implies that the answer to I(1) has to depend on thes-coordinate of ~z only, for that would automatically bea constant on the ellipsoidal surface. For ~z lying awayfrom the ellipsoidal surface, our integral must satisfy

Poissons equation gijzizj I(1)[s] = 0 (with the in-verse metric gij of (A5)), which in turn is equivalentto the ordinary differential equation

0 = (1 2)d2I(1)[]d2

2dI(1)[]d

, s/R. (101)

The general solution is a linear combination ofa constant and the Legendre function Q0[s/R] =(1/2) ln[((s/R) + 1)/((s/R) 1)]. But the asymptoticboundary condition implied by the integral represen-tation in (97) is

lims I(1) lims

1

4pi|~z|S2

d 2s. (102)

(When s R, (A4) says s/2 |~z|; s/2 essentiallybecomes the spherical radial coordinate.) The asymp-totic limit

limsQ0[s/R]

R

s(103)

then tells us the solution for ~z located outside the el-lipsoid is

1

4pi

+11

2pi0

d(cos )d

|~x[, , ] ~z[s > , +, +]|= I(1)[s > ] =

1

Rln

[s+R

sR]. (104)

A conducting surface forms a Faraday cage, so for ~zlying inside the ellipsoid, the potential is position in-dependent and the same as that on the surface,

1

4pi

+11

2pi0

d(cos )d

|~x[, , ] ~z[s , +, +]|= I(1)[s ] = 1

Rln

[+R

R]. (105)

12 The Columb potential at ~z can be obtained by the Greensfunction type integral

d2x

g2[~x

]/(4pi|~x ~z|), whereg2 is the determinant of the induced metric on the ellipsoidalsurface and the surface charge density is the normal deriva-tive of the electric potential, = N ii, evaluated on the saidsurface. Because N ii is a unit normal, one would find thatthe combination d2x

g2N

ii is equal to the infinitesimalsolid angle d in 3 spatial dimensions, up to overall constantfactors.

Reinstating the relationships ~z = (~x + ~x)/2, R =|~x ~x|, = t t and s = r + r, we gather

I(1) = |~x ~x|1 (106)

(

[r + r (t t)] ln[r + r + |~x ~x|r + r |~x ~x|

]+ [t t (r + r)] ln

[t t + |~x ~x|t t |~x ~x|

]).

For later use, let us record the following symmetrized

spatial derivative on I(1), keeping in mind that (i+i)acting on any function that depends on the spatialcoordinates solely through the difference ~x~x is zero:

(i + i )I(1) = |~x ~x|1[r + r (t t)] (107)

(i + i) ln[r + r + |~x ~x|r + r |~x ~x|

](Note that the two -function terms arising from dif-

ferentiating the [r+r(tt)] and [tt(r+r)]in I(1) cancel each other, upon setting t t = r + rin their respective coefficients.)

When n > 1, this conducting ellipsoid interpretationfor the n = 1 case does not continue to hold; but onemay attempt to derive a partial differential equation interms of the variables (s, +, +), such that some dif-ferential operator D acting on the kernel |~x ~z|n iszero. (Note that if ~x and ~z lived in n+2 spatial dimen-sions, D would be the (n + 2)-dimensional Laplacian,but implementing this scheme would involve introduc-ing an additional n 1 fictitious angles and Cartesiancomponents for ~x and ~z.) The general solutions ofthis partial differential equation may either help leadto a physical interpretation just as one was found forthe n = 1 case or a harmonics expansion analogousto the one used by DeWitt and DeWitt, so that theresulting series can be integrated term-by-term. Be-

cause of the cylindrical symmetry of the integral I(n),the final result should not depend on +. We shallleave these pursuits for future work, and merely sum

up the O[M/|~|] results here. Recalling the relation-ship between h and I from (78):

I = M I(1) +O[(M/|~x ~x|)2

], (108)

with I(1) given by (106).Kerr Let us now turn our attention to a Kerr

black hole with mass M and angular momentum S,with its spin axis aligned along the 3-direction.13

13 This 3-direction is not to be confused with the 3-direction ofthe prolate ellipsoidal coordinate system invoked during theevaluation of I in (78).

17

Starting from the Kerr metric written in Boyer-Lindquist coordinates (see equation 33.2 of [6]), we firstperform the following transformation on the r coordi-nate

r r(

1 +M

2r

)2. (109)

(This coordinate transformation would yield, whenS = 0, the Schwarzschild metric in isotropic coordi-nates.) Denoting the unit vector in the 3-direction ase3 and further define

~S Se3, (110)to first order in both S and M , we may then write theKerr metric as

g = + h (111)

where

h [t, ~x] 2(M +

0{

i}

(~S

~x

)i

)1

r.

(112)

In a Cartesian basis,(~S

~x

)i

= S

( x2

,

x1, 0

)i

. (113)

The off diagonal nature of 0{i} implies that the first

order in mass I00 and Iij for the Kerr black hole areidentical to that of the Schwarzschild case. As for I0i,by referring to (74), integrating by parts the spatialgradient acting on r1, and using (57) to pull the re-sulting two derivatives out of the integral, we observethat it can be gotten by acting

Ji {~S

(

~x+

~x

)}i

(114)

on the n = 1 integral in (97). That is,

I0i = JiI(1) + . . . (115)Altogether, to first order in mass M and angular mo-mentum S, the Kerr spacetime hands us

I = (M +

0{

i}Ji

)I(1), (116)

with I(1) given by (106).We will now construct the null cone portion of the

Greens functions by computing the world function,van Vleck determinant, and the parallel propagator.From (37), (40) and (48), we recall that these objects

may be gotten once I(0) is known. I(0) is related to

I , as can be inferred from (78), by replacing the|~x ~x| in the time argument of h with t t;and replacing the t t in the spatial arguments ofh with |~x ~x|. Since the h at hand does nothave any time dependence, this means I(0) is given byreplacing every t t with |~x ~x| in (116). Because|~x~x| r+r, this means the [t t (r+r)] termin (106) may be dropped and the [r + r (t t)]set to unity.14

I(0) = 1

|~x ~x|(M +

0{

i}Ji

) ln

[r + r + |~x ~x|r + r |~x ~x|

](117)

World Function The world function is x,x x,x +

I(0) . Some calculus reveals

x,x x,x (

1

r+

1

r

)2(t t)~S (~x ~x)rr (1 + x x) (118)

M|~x ~x|((t t)2 + (~x ~x)2) ln [r + r + |~x ~x|

r + r |~x ~x|],

with ~S (~x ~x) = S(x1x2 x1x2), x ~x/r, x ~x/r and x x ij xixj being the Euclidean dotproduct.van Vleck Determinant Because the Kerr space-

time is a vacuum solution to Einsteins equationsR = 0, the Ricci tensor to first order in mass and an-gular momentum must vanish, at least away from thespatial origin ~x 6= ~0. Vissers result (41) then informsus that the van Vleck determinant must remain unityto this order,

x,x 1. (119)

We may also confirm this by computing the van Vleckdeterminant from the world function in (118) using(11), or by a direct differentiation (see (40))(

+ (

) 1

2

)I(0)

=1

4(h[x] + h[x]) = M

(1

r+

1

r

). (120)

14 The following remark is in order. Because |rr| |~x~x| r + r, the only way r + r = t t = |~x ~x| can be satisfiedsimultaneously is when a null signal is sent from ~x to ~x withthe spatial origin (i.e. the spatial location of the black hole)lying on the straight line joining them (as viewed in Euclidean3space), so that ~x ~x = rr. But we do not expect anysignal to be able to pass through the black hole; hence, allterms implying such a configuration may be discarded.

18

Parallel Propagator According to (46), the sym-metric portion of the parallel propagator can be readoff the metric perturbations, namely

1

2(g + g) (121)

= M(

1

r+

1

r

)

+ 0{}i

(1

r3

(~S ~x

)i+

1

r3(~S ~x

)i)At this point, it is convenient to define

Vj xj + x

j

rr (1 + x x) . (122)

(One may need to recognize (r+r)|~x~x|2 = 2rr(1+x x).) By a direct calculation, one may show that Vjis divergence-less.

iVi = iVi = 0 (123)and it also satisfies

~S (

~x ~x

)Vj = 0. (124)

The antisymmetric portion of the parallel propagatoris given by

1

2(g g) = [+ I(0)]. (125)

In terms of Vi, its non-zero components are1

2(g0j gj0) =

(M(t t) + iJi

)Vj (126)and

1

2(gjk gkj) =

(M[k (t t)J[k

)Vj]. (127)Tails in Kerr By recalling (107), I0i = JiI(1)

reads

I0i = |~x ~x|1[r + r (t t)]

Ji ln[r + r + |~x ~x|r + r |~x ~x|

]+O

[(S/|~x ~x|2)2] (128)

To first order in angular momentum S, therefore, theKerr spacetime does add non-trivial terms to the nullcone portion of the Greens functions of massless fieldsin a Schwarzschild spacetime. However, the tail partof these Greens functions only receives additional con-tributions from the Kerr spacetime within the regionr + r t t > |~x ~x| near the null cone; no contri-butions due to angular momentum arise deeper inside

the null cone, t t > r+ r. This latter observation isconsistent with Poissons findings in [12]. In fact, weshall find that the tail of the scalar and photon Greensfunctions are only altered by angular momentum pre-cisely at t t = r+ r, corresponding to the reflectionof null rays off the black hole. Only the tail of thegraviton Greens function, which is sensitive not onlyto the Ricci curvature but to Riemann as well, experi-ence angular momentum effects throughout the region|~x ~x| < t t r + r.

Let us now proceed to compute the various piecesof the tail portion of the Greens functions. Equations(84), (86), and (87) tell us there are only three distinctbuilding blocks. Employing (116), these are as follows.The first is

I(S) 12 I I

=1

2(t t)

(4M[t t (r + r)]

(t t)2 |~x ~x|2)

(129)

+2~S (x x)rr (1 + x x)

[t t (r + r)],

The second is

I(A) 1

2[+ I], (130)

where . In terms of Vj in (122), thenon-zero components of I(A) are then

I(A)0j = M[r + r (t t)]Vj , (131)where (124) was used to set the angular momentumterms to zero, and

I(A)jk =M

2[k

([r + r (t t)]Vj]

)(132)

+ J[k([r + r (t t)]Vj]

).

The third and final building blocks are the geomet-ric curvature terms. The non-zero components of theRiemann terms are

(R|1)0i0j =M

4{i+

([r + r (t t)]Vj}

), (133)

(R|1)0ijk = 1

2[j+Jk] ([r + r (t t)]Vi) ,(134)

(R|1)ijkl = M

2[k+

(l]i[r + r

(t t)]Vj)

(i j) (135)where the (i j) means one has to take the precedingterm and swap the indices i and j. Performing theappropriate contractions and utilizing (123) yields theRicci tensor and scalar terms

(R|1) =(M +

0{

i}SJi

) [t t (r + r)]rr

(136)

19

(R|1) = 2M [t t (r + r)]rr

(137)

We note that these -functions (the [t t (r+ r)]and its derivative) arise from null rays scattering offthe point mass (i.e., the black hole) at the spatial ori-gin. For instance, one may also arrive at (136) by

recalling from (91) and (92) that the (R|1) is an in-tegral involving the Ricci tensor over a prolate ellipsoidcentered at (~x+ ~x)/2 and whose foci are at ~x and ~x.Since the linearized Ricci tensor for the metric in (111)and (112) is

(R|1) [~x] = 4pi(M +

0{

i}

(~S

~x

)i

)(3)[~x]

(the 4pi(3)[~x] comes from ijijr1) we have

(R|1) = 4pi(M +

0{

i}Ji

) 1

2

d

4pi(3)

[~x+ ~x

2+ ~x

],

with the ~x in (79). This integral leads us to (136),if we re-express (3)[~x ~z] = [s ][cos cos +][+ + pi ](

det gij/ sin

)1, using (A6).Greens Functions We may now put together the

minimally coupled massless scalar Greens function ina weak field Kerr spacetime, with the geometry de-scribed in (111) and (112), to first order its mass Mand angular momentum S.

Gx,x [t t]

4pi

{ [x,x ] + [x,x ] I(S)

}(138)

The Lorenz gauge photon counterpart is

G [t t]

4pi

{g [x,x ] (139)

+ [x,x ]( I(S) + I(A) + (R|1)

)},

while the de Donder gauge gravitons is

G [t t]

4pi

{P [x,x ] + [x,x ]

(PI(S) + 1

2

(I(A) + I(A) + I(A) + I(A)

)+ P (R|1) + (R|1) + (R|1)

1

2{ (R|1)}

1

2{ (R|1)} + (R|1){}

)}.

We remind the reader that the van Vleck determinantis unity; whereas the world function x,x can be foundin (118), the parallel propagator g components in

(121), (126) and (127), I(S) in (129); the componentsof I(A) in (131) and (132); and the components of thegeometric terms such as (R|1) in (133) through(137). As a consistency check of the building blocks

I(S), I(A) and (R|1) , by employing the identities,

[z] = [z], [z] = [z], [z] = [z], (140)

we have verified that the tail part of our massless scalarand photon Greens functions satisfy (17), or equiva-lently,

+ I(S) + (I(A) + (R|1)

)= 0. (141)

Causal structure Notice the I(S) in (129) is zeroclose to the light cone, and only non-zero for late times:t t r + r. This in turns indicates the tail part ofthe massless scalar Greens function is non-zero onlyafter the time elapsed t t equals or exceeds the time

needed for a null ray to travel from the source at x,reflect off the black hole, and reach the observer at x.Furthermore it is sensitive to first order spin effectsonly exactly at t t = r + r. On the other hand,the photon Greens functions contain, in addition to

I(S), the I(A) in (131) and (132) and (R|1) in (136);the graviton Greens function contain all three build-

ing blocks, I(S), I(A) and the curvature terms in (133)through (137). The I(A) and curvature terms are non-zero only at early times |rr| |~x~x| < tt r+r;mathematically this is because all these terms contain

the derivative j+ I(1) [r+r(t t)]Vj . However,it may be worthwhile to search for a more physical ex-planation, for it could lead us to a deeper understand-ing of the tail effect. In any case, this means both thephoton and graviton Greens functions carry non-zerotails throughout the entire interior of the future nullcone of x, though their behaviors are altered abruptlywhen the time elapsed tt changes from tt < r+rto t t > r + r. This is because, in the former, theycontain effects described by I(A) and geometric cur-vature; while in the latter region they are, like their

20

r'r

r'

t-t'

FIG. 3: Time elapsed (t t) vs. radial distance (r) view ofthe causal structure of the Greens functions in the weaklycurved limit of Kerr spacetime. The spacetime point sourceis located at radial coordinate distance r from the blackhole. The dark grey area represents the early time tail|r r| |~x ~x| t t r + r; and the light greyregion is the late time tail t t r + r. The dashed lineis tt = r+r; the two solid black lines are tt = |rr|.As already noted by DeWitt and DeWitt [1] and Pfenningand Poisson [3], the Greens functions undergo an abruptchange in behavior when time elapsed transitions from tt < r+ r to t t > r+ r. The region t t r+ r onlyreceives contribution from I(S) in (129), which containsangular momentum S terms only when t t is exactlyequal to r+r; while the dark grey region |rr| t t r+r only receives contribution from I(A) in (130) and thegeometric curvature terms in (133) through (137). The tailpart of the massless scalar Greens functions is only non-zero for t t r + r, because it is entirely governed byI(S). Because the photon and graviton Greens functionsdepend on I(A) and the geometric terms, they are non-zerowithin |rr| tt r+r. However, their behaviors arealtered once t t r+r, since they too become governedsolely by I(S).

scalar cousin, governed solely by I(S). (We illustratethis abrupt change in behavior of the Greens functionsin Figure (3).) Finally, we observe that spin effects arepresent on the null cone and, in the tail, exactly att t = r+ r, for the scalar and photon Greens func-tions. Only the graviton is sensitive to the full Rie-mann curvature of spacetime, which unlike the Riccitensor and scalar, is non-zero everywhere. This is whythe tail of the graviton Greens function contain spineffects within the whole region of |~x~x| < tt r+r.

VI. SUMMARY AND CONCLUDINGTHOUGHTS

In this paper, we have developed a general Born se-ries expansion for solving the minimally coupled mass-less scalar, photon, and graviton Greens function inperturbed spacetimes described by the metric g =g + h . The key starting points are the integralequations for the scalar (53), photon (62) and gravi-ton (69) cases, which were gotten from the quadraticportions of the actions of the respective field theories.From these, one performs a power series in the per-turbation h and iterate these equations (followed bydropping the remainder terms) however many timesnecessary to achieve the desired accuracy. We deriveda first order integral representation for the scalar (56)and photon (65) Greens functions in generic back-grounds, and for scalar (58), photon (66) and gravi-ton (73) in a Minkowski background. Furthermore, in(84), (86), and (87), we decomposed these perturbedMinkowski results into their light cone and tail pieces,showing their consistency with the Hadamard form.We reiterate that, at first order in metric perturba-tions, the solution of the scalar, photon and gravitonGreens functions is reduced to the evaluation of thesingle matrix integral in (78); the remaining work ismere differentiation. Even though we have applied ourperturbation theory only to massless scalars, photonsand gravitons, because all we have exploited are thequadratic actions of the field theories involved, ourmethods should in fact apply to any field theory whosequadratic action is hermitian.

As a concrete application of our formalism, we havecalculated the Greens functions of the massless scalar(138), photon (139), and graviton (140) in the weakfield limit of the Kerr black hole geometry, to first or-der in its mass M and angular momentum S. A subsetof these weak field results for the Schwarzschild casehave previously been obtained by DeWitt and DeWitt[1], and Pfenning and Poisson [3]. Our Kerr calcula-tion shows that, to first order in angular momentumS, there will be rotation-induced corrections to theseSchwarzschild Greens functions, only on and near thenull cone, namely |~x ~x| t t r + r (wherer |~x| and r |~x|). Beyond that, t t > r + r,the behavior of the Greens functions changes abruptlyand is governed solely by the mass of the black hole.

Of the previous approaches we have studied De-Witt and DeWitt [1], Kovacs and Thorne [2] and Pfen-ning and Poisson [3] DeWitt and DeWitts seems tobe the most general. They utilized Julian Schwingersperspective that the Greens function is an operatorin a fictitious Hilbert space, for example, Gx,x =

x|G|x, from which they found its variation. How-ever, on the level of classical field theory, the mainconcern of this paper, our methods do not require any

21

additional structure than the quadratic action of thefield theory at hand. Hence, we hope it is accessibleto a wider audience.15 Our null cone versus tail de-composition was modeled after the work of Pfenningand Poisson [3] (except we generalized it to arbitrarymetric perturbations), who in turn state that theirwork was based on calculations by Kovacs and Thorne[2]. In Pfenning and Poissons work, they wrote downa perturbative version of the differential equations in(15), (16) and (18) for a weakly curved spacetime withonly scalar perturbations , and derived integral rep-resentations of the solutions using the flat spacetimeGreens function Gx,x ; their methods can very likelybe generalized to arbitrary perturbations. However,repeated (and un-necessary) use was made of the equa-tions obeyed by the gravitational potential . We feelthis obscures the fact that the solution of the Greensfunction of some field theory depends on the geometrybut not on the underlying dynamics of the geometryitself.

Cosmology We close with some thoughts on ap-plying our work to cosmological physics. We have al-ready shown that the classical theory of light in a spa-tially flat inhomogeneous FLRW universe is equivalentto that in a perturbed Minkowski spacetime. Considera source of photons that turns on for a finite durationof time, say a gamma ray burst at redshift z = 6. Wedisplay in Fig. (4) that not only would these photonssweep out a null cone of finite thickness proportional tothe duration of the burst, but they will also fill its in-terior. If t is the present time, the dark oval representsthe light that has leaked off the light cone. From ourcalculation in (86), the tail part of the Greens function

and hence the vector potential A(tail) begins at O[h].

Because the components of the stress energy tensor ofthe electromagnetic fields in an orthonormal frame T(which is what an observer can measure) is quadraticin the derivatives of the potential, T a4(A)2,this means deep in the interior of the null cone Titself must be quadratic in the metric perturbationsh .

16 In cosmology, because the metric perturbationsh are believed to be sourced by quantum fluctua-tions of fields in the very early universe, the h at a

15 At the same time, we should mention that Schwingers [10]initial value formulation of quantum field theory (nowadaysknown as the Schwinger-Keldysh formalism), has in fact beenemployed to tackle the post-Newtonian program in generalrelativity, itself a weak field, perturbative problem about flatspacetime. See, for instance, Galley and Tiglio [11]. There isvery likely a position-space diagrammatic calculation one cando to reproduce (58), (66) and (73).

16 A consistent O[h2] calculation of the stress energy tensorthat is valid everywhere, both near the light cone and deepwithin it, would therefore require the knowledge of the photonGreens function to O[h2].

tTail

X

FIG. 4: At X, let there be a burst of photons from asource of finite duration. If there were no tails, these pho-tons would sweep out a light cone of non-zero thicknessproportional to the duration of the event itself. Becauseof the metric perturbations h , we have shown via ourphoton Greens function calculation that light develops atail in a spatially flat inhomogeneous FLRW universe. The

dark oval represents the tail of the photon field A(tail) at the

present time t. Deep within the light cone, we argue thatthe size of the tail effect in our universe is primarily gov-erned by the power spectrum of the metric perturbations,which is currently being probed by large scale structureobservations.

particular point in spacetime is a random variable andto obtain concrete results one would have to discussthe statistical average of the product of h with it-self, i.e. h [x]h [x], the power spectrum. Thescalar sector of this power spectrum is being probedby the observations currently underway of large scalestructure in the universe, and one would have to foldthese data into a theoretical investigation of how largethe tail effect is in our universe.

One way to proceed is perhaps, following Poisson[12], to start with some generic localized wave packetto mimic the light from a finite duration event such asour gamma ray burst at z = 6. We may then invokethe Kirchhoff representation in (2) (with no current,

J

= 0) to evolve this wave packet forward in time. Atsome later time, one can compute the ratio of energiesin the tail piece to that still remaining on the null cone

T(tail)

00

det[ij + hij ]d3xT(light cone)

00

det[ij + hij ]d3xt

. (142)

This will indicate if there is a significant correctionfactor that needs to be applied to observations of ob-jects at cosmological distances, when inferring theirtrue brightness. Because of the integrals encountered

22

in (2) and the complicated terms in (86), however, thisis a difficult calculation. We hope to report on this lineof investigation in a future publication.

VII. ACKNOWLEDGEMENTS

YZC was supported by funds from the Universityof Pennsylvania; and by the US Department of En-ergy (DOE) both at Arizona State University duringthe 2010-2011 academic year and, before that, at CaseWestern Reserve University (CWRU), where this workwas started. GDS was supported by a grant from theDOE to the particle-astrophysics group at CWRU.

YZC would like to acknowledge discussions with nu-merous people on the tail effect in cosmology, per-turbation theory for Greens functions, and relatedissues. A non-exhaustive list includes: Niayesh Af-shordi, Yi-Fu Cai, Shih-Hung (Holden) Chen, ScottDodelson, Sourish Dutta, Chad Galley, Ted Jacobson,Justin Khoury, Harsh Mathur, Vincent Moncrief, EricPoisson, Zain Saleem, and Tanmay Vachaspati.

Appendix A: The Matrix I

The primary objective in this section is the analysisof I in (74), including its behavior near the null conex,x = 0. A similar discourse may be found in Pfen-ning and Poisson [3], but ours is more general becausewe performed it for arbitrary metric perturbations.

Let us first display the integral in its most explicitform, using the second equality of (32):

I = 14pi

d4xh [t, ~x]

[t t |~x ~x|][t t |~x ~x|]

|~x ~x||~x ~x| . (A1)

We may integrate over t immediately, so that t =t |~x ~x| = t + |~x ~x|. This in turns yieldsthe constraint that, viewed in Euclidean 3-space, theobserver at ~x and the emitter at ~x form the foci of aprolate ellipsoid, with semi-major axis 0/2, definedby

t t = |~x ~x|+ |~x ~x|. (A2)This implies, to get a non-zero I , x needs to lie inthe future light cone of x,

t t |~x ~x|. (A3)For by Cauchys inequality, |~x~x| |~x~x|+|~x~x|,which means, outside the light cone t t < |~x ~x| |~x ~x|+ |~x ~x| and no solution can be found. Fig.(2) illustrates the situation at hand: we see that the

product Gx,xhGx,x , due to the causal structureof the Greens function, is non-zero if and only if thex lie both on the future null cone of x and on the pastnull cone of x. This can be satisfied if and only if x lieson or within the past light cone of x or equivalently, ifand only if x lies on or within the future light cone ofx.

If we now assume that (A3) holds, then it is the sur-face of the ellipsoid in (A2) that we need to integrateover, weighted by h [t

, ~x]. To see this, let us em-ploy ellipsoidal coordinates centered at (1/2)(~x + ~x),i.e. put ~x (1/2)(~x + ~x) + ~x, with

~x[s, , ] =(

(s/2)2 (R/2)2 sin cos,(s/2)2 (R/2)2 sin sin,

(s/2) cos

). (A4)

These coordinates fix the foci to be at ~x and ~x butallow the size of the ellipse to vary with s. (The 1-

and 2-components of ~x tell us

(s/2)2 (R/2)2 actas the radial coordinate in the 12-plane, and hence weshall require s R. This means all volume integralsinvolve s would have limits

R

ds.) The Euclideanspatial metric in 3 dimensions goes from gij = ij forCartesian coordinates to

gij = diag

[(s/2)2 (R/2)2 cos2

s2 R2 ,

(s/2)2 (R/2)2 cos2 ,((s/2)

2 (R/2)2)

sin2

],

(s, , ), s R, (A5)

where R |~| = |~x ~x|. The Euclidean volumemeasure is

det[gij ] =1

2

((s/2)

2 (R/2)2 cos2 )

sin . (A6)

Using the expressions for the components of ~x in (A4),we may obtain

|~x ~x| = s2 R

2cos[], |~x ~x| = s

2+R

2cos[].

This means the argument in the remaining -functionof the I-integrand is t t s, and the det gij |~x ~x|1|~x~x|1 = (sin )/2. The integral over s can beperformed immediately, and because the lower limit isR, it gives us [ttR] = [tt][x,x ] multiplyingh with s = t t. We are left with the angularintegration,

I [t t][x,x ]I

23

= [t t][x,x ] (A7)

12

S2

d

4pih

[t+ t

2+R

2cos[],

~x+ ~x

2+ ~x

],

where now ~x = ~x[t t, , ].Small x,x expansion For small x,x , we may

develop I as a series expansion in powers of x,x .Right on the null cone tt = |~x~x|, and if we lie ~x~xalong the positive 3-axis, the spacetime arguments ofh take on the Cartesian components

(t+ t

2+

0

2cos , 0, 0,

x3 + x3

2+R

2cos

)(A8)

which is equivalent to [cos ] = (1/2)(x + x) +(1/2)(x x) cos . This is a straight line joining xto x. Let us expand about this straight line by ex-pressing the time component (1/2)(t+ t +R cos ) as

t+ t

2+

(t t

2

)2 x,x

2cos (A9)

and the 3-component (1/2)(x3 + x3 + 0 cos ) as

x3 + x3

2+

(R

2

)2+x,x

2cos . (A10)

Perform a change of variables in (A7) cos 21 andTaylor expand h in powers of x,x in the time and3-components and in powers of

x,x in the remaining

orthogonal directions. One would find it is necessaryto expand the orthogonal directions up to second orderto achieve a non-zero result. With = x + (x x),

the expansion of (A7) is

I = 12

10

dh[]

(A11)

+x,x

4

10

d

(2 1R

3h 2 1t t 0h

)+x,x

4

10

d(1 ) (21 + 22)h + . . .where 21 +

22 is the Laplacian involving only the

directions orthogonal to ~x ~x. The arguments of thehs on the second and third lines have been sup-pressed; they are the same as that of the first line.The single derivative terms can be converted into dou-ble derivatives by using (45) and integrating-by-parts.Up to a remainder that is of O[2], one can show

x,x

4

10

d

(2 1R

Zh 2 1t t th

) x,x

4

10

d(1 ) (2Z 2t)h .(A12)

By the chain rule,

(1 ) 2h []

=

2h []

xx, (A13)

and hence we gather, as x,x 0,

I (

1 x,x2

)(1

2

10

h []d

).

(A14)

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I Introduction and MotivationII General TheoryIII Perturbation TheoryIV [] and [] decomposition in 4 dimensional perturbed MinkowskiV Schwarzschild and Kerr GeometriesVI Summary and concluding thoughtsVII AcknowledgementsA The Matrix I References