Blanchet L., Spallicci A., Whiting B. (Eds.) Mass and Motion in General Relativity
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Radiation Reaction and Energy–Momentum Conservation 381
G
H
D
1
.2/
d
d 2
X
kD0
u
k
d 1k
C v ln C w
!
; (54)
where u
0
D
1=2
and we denoted v D u
d 1
; w D u
d
. Applying with respect to
x
, we obtain the system of recurrent differential equations for u
i
.x; x
0
/.Integrat-
ing them along the geodesic connecting the points x; x
0
, one can uniquely express
u
1
.x; x
0
/ through u
0
.x; x
0
/.Furthermore,u
2
is expressed through u
1
,etc.
To relate the coefficient functions u
D
i
.x; x
0
/ in different dimensions we first ob-
serve that u
0
D
1=2
for any D. It is worth noting, however, that in the expansion
in terms of
u
D
0
D
1=2
D 1 C 1=12 R
˛ˇ
˛
ˇ
C; (55)
where the tensor indices ˛; ˇ run through all values corresponding to D. With this
in mind, we can write u
D
0
0
D u
D
0
for any D; D
0
. The next equation in the recurrence
gives for D; D
0
> 5,
u
D
0
1
D u
D
1
d 2
d
0
2
: (56)
Similarly for D > 7, D
0
D D C 2 one obtains:
u
DC2
2
D u
D
2
d 3
d 1
: (57)
Continuing this process further one finds
1
d 1
@G
D
H
@
D .2/G
dir DC2
H
; (58)
where the “direct” part of the Hadamard function is
G
H dir
D
1
.2/
d
d 2
X
kD0
u
k
d 1k
: (59)
4.2 Divergences
For the retarded Green’s function one finds:
G
ret
D
1
2.2/
d 1
.x
0
; ˙.x//
d 2
X
mD0
.1/
m
u
d 2m
ı
.m/
./
mŠ
v./
!
: (60)
The first term constitutes the direct part of the retarded function with the support on
the light cone:
G
dir
D
1
2.2/
d 1
Œ˙
d 2
X
mD0
.1/
m
u
d 2m
ı
.m/
./
mŠ
: (61)
382 D. Gal’tsov
Similarly to the recurrent relations of the previous section, we obtain for the
retarded Green’s functions:
@G
D
ret
@
D2 .d 1/ G
DC2
dir
: (62)
Using the regularization ı./ D lim
"!C0
ı. E /; where E D "
2
=2, we obtain
for the direct part of the reaction force
f
dir
DC2
D
1
D 1
@f
dir
D
.s; E /
@E
: (63)
The limit E !C0 has to be taken after the differentiation. The direct force is
due to the light cone part of the retarded Green’s function. This is not the full local
contribution to the Lorentz–Dirac force. An additional contribution comes from the
differentiation of the theta-function in the tail term v ./. In the scalar case this
contribution vanishes, but in the electromagnetic case an extra local term arises:
f
loc
D e
2
Œv
˛
Pz
Œv
˛
Pz
Pz
Pz
˛
; (64)
where the coincidence limit Œv
˛
depends on the dimension. The remaining contri-
bution from the tail term will have the form of an integral along the past half of the
particle world line.
The direct part of the Lorentz–Dirac force contains divergences. To separate the
divergent terms one can use the decomposition of the retarded potential suggested
in the case of four dimensions by Detweiler and Whiting [3, 4]. In higher even di-
mensions we can follow essentially the same procedure. We define the “singular”
part G
S
of the retarded Green’s function as the sum of the symmetric part (self) and
the tail function v as follows
G
S
.x; x
0
/ D G
self
.x; x
0
/ C
v.x; x
0
/
4.2/
d 1
D G
self dir
.x; x
0
/ C
v.x; x
0
/./
4.2/
d 1
: (65)
Here the direct part of the self function means its part without the tail v-term. The
remaining part of the Green’s function G
R
.x; x
0
/ D G
ret
.x; x
0
/ G
S
.x; x
0
/ satisfies
a free wave equation and is regular. Taking into account v D 0, it is clear that G
S
satisfies the same inhomogeneous equation as G
self
.Thev-term in the second line
of Eq. 65 is localized outside the light cone. Therefore the corresponding field (for
instance, scalar), at an arbitrary point x will be given by
S
.x/ D
self dir
C
m
0
q
4.2/
d 1
adv
Z
ret
v.x; z.//d; (66)
where the retarded and advanced proper time values
ret
.x/;
adv
.x/ are the inter-
section points of the past and future light cones centered at x with the world line.
Radiation Reaction and Energy–Momentum Conservation 383
Inserting (66) into the Lorentz–Dirac force defined on the world line x D z./,
we observe that the integral contribution from the tail term vanishes and so the di-
vergent part is entirely given by
self dir
: Using for delta-functions the point-splitting
regularization we find that all divergent terms arise as negative powers of ". To prove
the existence of the counter-terms we consider the interaction term in the action sub-
stituting the field as the S-part of the retarded solution to the wave equation. In the
scalar case we will have:
S
S
D
1
2
Z
G
S
.x; x
0
/.x/.x
0
/
p
g.z/g.z
0
/dxdx
0
; (67)
where a factor one-half is introduced to avoid double counting when self-interaction
is considered. Substituting the currents we get the integral over the world line. Since
the Green’s function is localized on the light cone we expand the integrand in terms
of the difference t D
0
around the point z./:
S
S
Z
d
Z
X
k;l
B
kl
./ı
.k/
.t
2
"
2
/t
l
dt: (68)
Here the coefficients B
kl
.z/ depend on the curvature, while the delta-functions are
flat: ı
.k/
.t
2
"
2
/. By virtue of parity, the integrals with odd l vanish, so only the odd
inverse powers of " will be present in the expansion. Moreover, once we know the
divergent terms in some dimension D, we can obtain by differentiation all divergent
terms in D C 2, except for 1=" term. The linearly divergent term corresponds to
l D 2k. The integral is equal to
1
Z
0
ı
.k/
.t
2
"
2
/t
2k
dt D
.1/
k
.2k 1/ŠŠ
2
kC1
1
"
:
In four dimensions this term is unique. Applying our recurrence chain we obtain the
inverse cubic divergence in six dimensions and calculate again the linearly divergent
term.Thus,inD D 2d dimensions we will get d 1 divergent terms from which
d 2 can be obtained by the differentiation of the previous-dimensional divergence,
and the linearly divergent will be new. This linearly divergent self-action term in the
action will have generically the form
S
.1/
S
1
"
Z
d 2
X
kD0
.1/
k
.2k 1/ŠŠ
2
kC1
B
k;2k
./d:
Here the coefficient functions are obtained taking the coincidence limits of the two-
point tensors involved in the expansion of the Hadamard solution. They actually
depend on the derivatives of the world-line embedding function z./ as well as the
curvature terms taken on the world line:
B
k;2k
./ D B
k;2k
.Pz; Rz;:::;R.z.//; R
.z.//; : : :/:
384 D. Gal’tsov
The vector case is technically the same, now one has to expand in powers of t in
the integrand of
S
S
D
1
2
Z
G
S
˛
.x; x
0
/j
.x/j
˛
.x
0
/
p
g.z/g.z
0
/dxdx
0
:
From this analysis it follows that in any dimension the highest divergent term
can be absorbed by the renormalization of the mass as in the generating four-
dimensional case. To absorb the remaining d 2 divergences one has to add to the
initial action the sum of d 2 counter-terms depending on higher derivatives of
the particle velocity. Typically the counter-terms depend on the Riemann tensor of
the background.
4.3 Four Dimensions
In four dimensions the scalar retarded Green’s function contains a single direct term
localized on the light cone and a tail term:
G
ret
D
1
4
Œ˙.x/;x
0
h
1=2
ı./ v./
i
: (69)
The retarded solution for the scalar field reads
ret
.x/ D m
0
q
ret
.x/
Z
1
h
1=2
ı./ C v./
i
d
0
: (70)
Differentiating this expression we obtain
D @
on the world line:
.z.// D m
0
q
Z
1
Œ
1=2
ı
0
./
1=2
I
ı./ vı./
C v
d
0
; (71)
where integration is performed along the past history of the particle. All the two-
point functions are taken on the world line at the points x D z./ (observation
point) and z
0
D z.
0
/ (emission point).
To compute local contributions from the terms proportional to a delta-function
or its derivative, it is enough to expand the integrand in terms of the difference
s D D
0
around the point z./. The Taylor (covariant) expansion of the
fundamental biscalar .z./; z.
0
// is given by [2]:
.z./; z.
0
// D
1
X
kD0
1
kŠ
D
k
.; /.
0
/
k
; (72)
Radiation Reaction and Energy–Momentum Conservation 385
wherewedenoteasQ.;
0
/ the quantity Q.z./; z.
0
// for any Q taken on the
world line, and D is a covariant derivative along the world line (also denoted by
a dot):
D P D
˛
Pz
˛
;D
2
R D
˛ˇ
Pz
˛
Pz
ˇ
C
˛
Rz
˛
; etc:
Such an expansion exists since the difference s D
0
is a two-point scalar itself:
this is the integral from the scalar function
R
.Pz
2
/
1=2
ds; along the world line from
z./ to z.
0
/. Taking the limits and using Pz
2
./ D1, we find:
.s/ D
s
2
2
Rz
2
./
s
4
24
C O.s
5
/: (73)
To obtain an expansion of the derivative of over z
./ one can expand
.; s/
in powers of s. This quantity transforms as a vector at z./ and a scalar at z.
0
/,
.s/ D s
Pz
Rz
s
2
C«z
s
2
6
C O.s
4
/; (74)
where the index corresponds to the point z./:
D @.z; z
0
/=@z
. Recall that
the initial Greek indices correspond to z.
0
/. The expansion of ı./ will read:
ı./ D ı.s
2
=2/ C s
4
Rz
2
./
24
ı
0
.s
2
=2/ C; (75)
where the derivative of the delta-function is taken with respect to the full argu-
ment. Since the most singular term is
1=2
ı
0
./
, the maximal order giving the
nonzero result after the integration is s
3
. (Note, that in order to use our dimen-
sional recurrent relations to obtain a reaction force in higher dimensions we should
perform an expansion up to higher orders in s.) Thus, with the required accuracy,
ı./ D 2ı.s
2
/; and all the integrals for the delta-derivatives are the same as in the
flat space-time. This allows us to use the same regularization of the delta-functions
with double roots ı.s
2
/ by the point-splitting. Expanding the biscalar
1=2
and its
gradient at z we have:
1=2
D 1 C
s
2
12
R
Pz
Pz
;@
1=2
D
s
6
R
Pz
: (76)
Combining all the contributions we obtain finally for the field strength on the world
line:
ˇ
ˇ
ˇ
z./
D m
0
q
0
@
1
2"
Rz
1
3
«z
1
6
R
Pz
1
6
R
ı
Pz
Pz
ı
Pz
C
1
12
RPz
C
Z
1
v
d
0
1
A
:
(77)
After the renormalization of mass one recovers the familiar equation.
386 D. Gal’tsov
Similarly, in the electromagnetic case one finds the retarded Maxwell tensor on
the world line x D z./:
F
ret
ˇ
ˇ
ˇ
z./
D e
Z
1
Œu
˛I
ı./ Cu
˛
ı
0
./ Cv
˛I
Cv
˛
ı./ f $ gPz
˛
d
0
:
(78)
Performing expansions on the world line, one easily recover the DeWitt–Brehme–
Hobbs result
f
em
D e
2
2
4
Rz
2"
C ˘
2
3
«z
C
1
3
R
˛
Pz
˛
CPz
./
Z
1
.v
˛I
v
I
˛
/Pz
˛
.
0
/d
0
3
5
;
(79)
where ˘
D g
Pz
Pz
. For geodesic motion the entire reaction force is given
by the tail term.
4.4 Self and Radiative Forces in Curved Space-Time
Apart from the tail term, another new feature of the instantaneous momentum
balance between the radiating charge and radiation is the presence of the finite
contribution in the self part of the reaction force originating from the half-sum of
the retarded and advanced potentials. As we have seen in the previous section, in the
case of Minkowski space the self part in four and other even dimensions is a pure
divergence which has to be absorbed by renormalization of the mass and (in higher
dimensions) the bare coupling constants in the higher derivative counter-terms.
In curved space, as was first shown in [5], the tail term in the equation for radiating
charge moving along the geodesic with nonrelativistic velocity (in weak gravita-
tional field) contains apart from the dissipative term also the conservative force.
This conservative force was later found for a static charge in the Schwarzschild
metric [26]. Here we would like to explore the significance of this result in view
of the above analysis of self/radiative decomposition. In the paper [6] the splitting
of the retarded field into the self and radiative parts was not used. Considering the
geodesic motion we have the equation:
mRz
˛
D e
2
Pz
ˇ
Z
1
v
˛
retˇ
Pz
.s
0
/ds
0
: (80)
Splitting the tail function with respect to T-parity
v
˛
retˇ
Pz
D v
˛
selfˇ
Pz
C v
˛
radˇ
Pz
; (81)
Radiation Reaction and Energy–Momentum Conservation 387
and repeating the calculations along the lines of [6] one finds in the weak-field
nonrelativistic case the corresponding (spatial) parts of the reaction force in terms
of the flat space theory:
f
self
D f
div
C f
DDSW
; f
rad
D f
Schott
; (82)
where the divergent term is the same as in the flat space. The two finite terms
f
DDSW
D
GM e
2
r
4
r; f
Schott
D
2
3
e
2
P
a (83)
are the DDSW force and the Schott force. Therefore, the DDSW force is a finite part
of the T-even (self) contribution to the tail, while the T-odd (rad) part reproduces the
Schott term which is precisely the same as in the flat space. Similar result holds for
the scalar radiation.
5 Gravitational Radiation
Gravitational radiation in the framework of linearized gravity on the curved back-
ground may seem similar to scalar or electromagnetic radiation, but the similarity
is incomplete. The difference is that radiation of nongravitational nature emitted by
bodies moving along geodesics in the fixed background is not influenced by the non-
linear nature of the Einstein equations. For gravitational radiation this is not so. The
scalar or vector linear field equations in curved space imply vanishing of the covari-
ant divergenceof the field stress tensor of matter field with respect to the background
metric. In the gravitational case one has the Bianchi identity that generally does not
imply the covariant conservation of the matter perturbation stress tensor alone un-
less the background is vacuum. In most of the literature on gravitational radiation
reaction the background is assumed to be vacuum. This seems to be satisfactory for
the Schwarzschild or Kerr metrics. But physically we have to deal not with an eter-
nal black hole, but with the collapsing body that is not globally vacuous. Meanwhile
the conservation laws that may hold in the asymptotically flat case has to be consid-
ered globally. It turns out that we must take into account the contribution from the
(perturbed) source of the background field as well. This is directly implied by the
Bianchi identities.
5.1 Bianchi Identity
Let the backgroundbe generated by the stress tensor
B
T
. We are interested by grav-
itational radiation emitted by a point particle of mass m moving along the geodesic
of the background. Perturbations caused by the particle are assumed to be small so
the particle stress tensor,
388 D. Gal’tsov
m
T
D m
Z
Pz
./Pz
./
ı.x z.//
p
g
d; (84)
is the first-order quantity with respect to
B
T
. By construction, the tensor (84)is
divergence free provided the particle follows the geodesic in the space-time. Since
it is the first-order quantity, it must be divergence free with respect to the background
covariant derivative up to the terms of the second order. Expanding the full metric
Og
D g
C h
,where
2
D 8G, and the Einstein tensor
O
G
D
B
G
C
1
G
; (85)
we could naively expect the full Einstein equations to be of the form
O
G
D
2
B
T
C
1
T
: (86)
Since
B
G
D
2
B
T
; we then should have
1
G
D
2
m
T
: (87)
But the left-hand side of this equation is not divergence free with respect to the
background covariant derivative. Expanding the full covariant derivative as
O
r
D
B
r
C
1
r
; (88)
and taking into account the Bianchi identity for the background
B
r
B
G
D 0,we
obtain in the first order
B
r
1
G
D
1
r
B
G
: (89)
Therefore
B
r
m
T
D
1
r
B
T
; (90)
where the right-hand side is the first-order quantity. Thus Eq. 87 is contradictory.
Physically the reason is that we have to take into account the perturbation of the
background ıT
caused by the particle, so that the correct equation should be
1
G
D
2
m
T
C ıT
: (91)
But this is not an equation for h
, since in order to find ıT
one has to consider
the matter field equations for the background metric. The problem thus becomes
essentially nonlocal. This nonlocality does not reduce to that of the tail term in the
DeWitt–Brehme equation.
Radiation Reaction and Energy–Momentum Conservation 389
5.2 Vacuum Background
If
B
T
D 0 the above obstacle is removed and one can proceed further with the
linearized equations for h
. The derivation of the reaction force initiated in [21]
and completed in [20] was based on the DeWitt–Brehme type calculation involving
the integration of the field momentum over a small tube surrounding the particle
world line. As was noted later [25], this derivation had some drawbacks (for more
recent discussion and further references see [15]). One problem consisted in com-
puting the contributions of “caps” at the ends of the chosen tube segment which
were not rigorously calculated. Another problem was the singular integral over the
internal boundary of the tube which was simply discarded. In addition, the usual
mass renormalization is not directly applicable in the gravitational case: due to the
equivalence principle the mass does not enter into the geodesic equations.
In [14] the local derivation of the gravitational reaction force was given which is
free from the above problems and, in addition, is much simpler technically. It deals
with the quantities defined only on the world line and does not involve the am-
biguous volume integrals over the world-tube at all. The elimination of divergences
amounts to the redefinition of the affine parameter on the world-line.
We start with reparametrization invariant form of the particle action introducing
the einbein e./ on the world line acting as a Lagrange multiplier:
SŒz
;e D
1
2
Z
e./ g
Pz
Pz
C
m
2
e./
d: (92)
Variation with respect to z
./ and e./ gives the equations
D
d
.ePz
/ D 0; e D
m
p
g
Pz
Pz
; (93)
and we obtain the geodesic equation in a manifestly reparametrization invariant
form
D
d
Pz
p
g
Pz
Pz
!
D 0: (94)
Assuming now that the particle motion with no account for radiation reaction is
geodesic on the background metric, the perturbed equation in the leading order in
will read
Rz
D
2
g
Pz
Pz
Pz
2
.h
I
2h
I
/Pz
Pz
; (95)
where contractions are with the background metric.
The particle energy–momentum tensor in our formulation will read
m
T
D
Z
e./Pz
./Pz
./
ı.x z.//
p
g
d; (96)
390 D. Gal’tsov
and we choose the non-perturbed ein-bein e
0
D const as a bare parameter. After the
calculation similar to that of the preceding section we obtain
Rz
D
2
e
0
8
<
:
7
2"
Rz
C
1
4
˘
Z
1
h
4v
˛ˇI
2
g
v
˛ˇI
C v
˛ˇ I
g
v
˛ˇI
i
Pz
0˛
Pz
0ˇ
Pz
Pz
d
0
9
=
;
: (97)
Renormalization of the einbein is
1
e
0
7
2
2"
Rz
D
1
e
Rz
: (98)
Finally, we choose the renormalized affine parameter so that Pz
2
D1,whichis
equivalent to setting e D m and obtain the MiSaTaQuWa equation. As was shown
in [14] this equation remains valid in a class of non-vacuum metrics, in particular,
for Einstein spaces.
5.3 Gravitational Radiation for Non-Geodesic Motion
If the particle world line is non-geodesic, the radiation reaction force contains a
putative antidamping term which is a local part of the rad contribution to the self-
force:
f
rad
D
11
2
3
.g
Pz
Pz
/«z
C tail: (99)
The reason is simply that the source of gravitational radiation is incomplete and the
stress tensor is not divergence free as required. Indeed, if the force is nongravita-
tional, one has to take into account the contribution of stresses of the field causing
the body to accelerate. For instance, to describe gravitational radiation of an electron
in the atom, one has to add the contribution from the Maxwell field stresses (spatial
components, nonrelativistic motion):
ij
D
2
GT
ij
;T
ij
D
m
T
ij
C
st
T
ij
; (100)
where
m
T
ij
D
X
aD1;2
m
a
Pz
i
a
Pz
j
a
ı
3
.X
a
/;
st
T
ij
D
e
1
e
2
4
X
i
1
X
j
2
.X
2
1
X
2
2
/
3=2
C .i $ j/; (101)