-
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
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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