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Computational Methods for the Self-Force in Black Hole Spacetimes 361
results – a particularly important prospect given the intricate nature of the numerics
involved and the fact that SF calculations explore a new territory in black hole
physics, yet uncharted by neither PN theory nor Numerical Relativity.
Indeed, the field has by now matured enough that such cross-validation exercises
are becoming possible: A first such comparison between results from different cal-
culations (using different gauges and different numerical methods) was presented
very recently in Ref. [109]. We are now able to use SF codes to explore, for the
first time, the conservative post-geodesic dynamics of strong-field orbits around a
Schwarzschild black hole [25,41,42,109]. We can compute physical gauge-invariant
post-geodesic effects and test them directly against results from PN theory in the
weak-field regime [41, 42, 109]. Indeed, we can now start to use strong-field SF
results in order to calibrate PN methods and assess their performance [25]. The
exciting prospects for synergistic interaction between SF and PN theories are be-
ginning to materialize, with much scope for important developments in the coming
years (see Detweiler’s contribution in this volume).
At present, the state of the art in SF calculations is a code that can calculate
the gravitational SF along any bound geodesic of the Schwarzschild geometry (cur-
rently at great computational cost, which future developments may help reduce).
This code, as many others mentioned in our review, is an implementation of the
mode-sum regularization method, which has proven a useful framework for calcu-
lations in Schwarzschild.
The Kerr problem, on the other hand, has hardly been tackled so far, and it rep-
resents the next significant challenge. Although the standard mode-sum method is
in principle applicable to the gravitational SF in Kerr spacetime, the details of its
numerical implementation in this case are yet to be worked out. It is possible that
higher-dimensiontechniques (like the m-mode scheme discussed in Section 5) could
provide an attractive alternative to standard mode-sum in the Kerr case.
In the short term, activity is likely to focus on the following tasks: (i) continue to
improve the computational efficiency of SF calculations using advanced numerical
techniques (mesh refinement, spectral methods, etc.); (ii) tackle the Kerr problem;
(iii) use SF codes to study post-geodesic physical effects (such as the finite-mass
correction to the orbital precession rate), and in particular assess the relative im-
portance of conservative SF effects in the EMRI problem; (iv) explore what can be
learned from a comparison between SF and PN results.
Within the context of the LISA EMRI problem, the computation of the SF on
momentarily geodesic particles is only one essential ingredient. There is still much
more to understand before a faithful model of an astrophysical inspiral can be an-
nounced. Most crucially, a reliable method must be devised for calculating the
long-term evolution of the inspiral orbit. Work to address this problem has begun
recently [57, 65,99,100] but much further study is needed.
Acknowledgements We acknowledge support from PPARC/STFC through grant number
PP/D001110/1.
362 L. Barack
Appendix: The Regularization Parameter B
˛
in Kerr
We state here the explicit value of the regularization parameter B
˛
for the gravita-
tional SF at a point (t
0
;r
0
;
0
;'
0
) along a generic (geodesic) orbit in Kerr. In what
follows u
˛
represents the four-velocity at that point, and g
˛ˇ
and
˛ˇ
are the Kerr
metric and connection coefficients there. The derivation of all regularization param-
eters in Kerr was reported in Ref. [22], with the explicit value of B
˛
(reproduced
here) given in the online arXiv version of that paper [20].
Using the method sketched in Section 2.3, one obtains
B
˛
D
2
.2/
1
P
˛
abcd
I
abcd
; (65)
where hereafter roman indices (a; b; c; :::) run over the two Boyer–Lindquist angular
coordinates ;'. The coefficients P
˛
abcd
are given by
P
˛
abcd
D .3P
˛
a
P
be
P
˛
e
P
ab
/C
e
cd
C
1
2
3P
˛
d
P
abc
.2P
˛
ab
C P
˛
ab
/P
cd
; (66)
where
P
˛ˇ
g
˛ˇ
C u
˛
u
ˇ
;P
˛ˇ
u
u
˛ˇ
C g
˛ˇ;
=2
; (67)
and
C
''
D
1
2
sin
0
cos
0
;C
'
'
D C
'
'
D
1
2
cot
0
; (68)
with all other coefficients C
e
cd
vanishing. The quantities I
abcd
are
I
abcd
D .sin
0
/
N
Z
2
0
G./
5=2
.sin /
N
.cos /
4N
d;
where N N.abcd/ is the number of times the index ' occurs in the combination
.a;b;c;d/, namely
N D ı
a
'
C ı
b
'
C ı
c
'
C ı
d
'
; (69)
and
G./ D P
Q' Q'
sin
2
C 2P
Q'
sin cos C P
cos
2
; (70)
where
P
Q' Q'
P
''
= sin
2
0
;P
Q'
P
'
= sin
0
: (71)
To write down I
abcd
more explicitly, denote
˛ P
=P
Q' Q'
1; ˇ 2P
Q'
=P
Q' Q'
: (72)
Then I
abcd
takes the form
Computational Methods for the Self-Force in Black Hole Spacetimes 363
I
abcd
D
.sin
0
/
N
d
h
QI
.N /
K
O
K.w/ C I
.N /
E
O
E.w/
i
; (73)
where
Q D ˛ C 2 .˛
2
C ˇ
2
/
1=2
; (74)
d D 3P
5=2
Q' Q'
.˛
2
C ˇ
2
/
2
.4˛ C 4 ˇ
2
/
3=2
.Q=2/
1=2
; (75)
O
K.w/ and
O
E.w/ are complete elliptic integrals of the first and second kinds, respec-
tively, and the argument is
w D
2.˛
2
C ˇ
2
/
1=2
˛ C 2 C .˛
2
C ˇ
2
/
1=2
: (76)
The ten coefficients I
.N /
K
;I
.N /
E
are given by
I
.0/
K
D 4
12˛
3
C ˛
2
.8 3ˇ
2
/ 4˛ˇ
2
C ˇ
2
.ˇ
2
8/
;
I
.0/
E
D16Œ8˛
3
C ˛
2
.4 7ˇ
2
/ C ˛ˇ
2
.ˇ
2
4/ ˇ
2
.ˇ
2
C 4/; (77)
I
.1/
K
D 8ˇ
9˛
2
2˛.ˇ
2
4/ C ˇ
2
;
I
.1/
E
D4ˇŒ12˛
3
˛
2
.ˇ
2
52/ C ˛.32 12ˇ
2
/ C ˇ
2
.3ˇ
2
C 4/; (78)
I
.2/
K
D4
8˛
3
˛
2
.ˇ
2
8/ 8˛ˇ
2
C ˇ
2
.3ˇ
2
8/
;
I
.2/
E
D 8Œ4˛
4
C ˛
3
.ˇ
2
C 12/ 2˛
2
.ˇ
2
4/
C3˛ˇ
2
.ˇ
2
4/ C 2ˇ
2
.3ˇ
2
4/; (79)
I
.3/
K
D 8ˇ
˛
3
7˛
2
C ˛.3ˇ
2
8/ C ˇ
2
;
I
.3/
E
D4ˇŒ8˛
4
4˛
3
C ˛
2
.15ˇ
2
44/ C 4˛.5ˇ
2
8/
Cˇ
2
.3ˇ
2
C 4/; (80)
I
.4/
K
D4Œ4˛
4
4˛
3
C ˛
2
.7ˇ
2
8/ C 12˛ˇ
2
ˇ
2
.ˇ
2
8/;
I
.4/
E
D 16Œ4˛
5
C 4˛
4
C ˛
3
.7ˇ
2
4/ C ˛
2
.11ˇ
2
4/
C.2˛ C 1/ˇ
2
.ˇ
2
C 4/: (81)
364 L. Barack
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Radiation Reaction and Energy–Momentum
Conservation
Dmitri Gal’tsov
Abstract We discuss subtle points of the momentum balance for radiating particles
in flat and curved space-times. An instantaneous balance is obscured by the presence
of the Schott term which is a finite part of the bound field momentum. To establish
the balance, one has to take into account the initial and final conditions for acceler-
ation, or to apply averaging. In curved space-time, an additional contribution arises
from the tidal deformation of the bound field. This force is shown to be the finite
remnant from the mass renormalization and it is different both from the radiation re-
coil force and the Schott force. For radiation of nongravitational nature from point
particles in curved space-time the reaction force can be computed by substituting
the retarded field directly to the equations of motion. A similar procedure is appli-
cable to gravitational radiation in a vacuum space-time, but fails in the non-vacuum
case. The existence of the gravitational quasilocal reaction force in this general case
seems implausible, though it still exists in the nonrelativistic approximation. We
also explain the putative antidamping effect for gravitational radiation under non-
geodesic motion and derive the nonrelativistic gravitational quadrupole Schott term.
Radiation reaction in curved space of dimension other than four is also discussed.
1 Introduction
One of the major tasks of gravitational wave astronomy is the precise theoretical pre-
diction and observational measurement of gravitational waveforms from an inspiral
fall of compact bodies into a supermassive black hole. This requires knowledge of
the orbits with account for gravitational radiation reaction. Radiation gives rise to
the reaction force, which can be incorporated into the equations of motion. The stan-
dard strategy to get the reaction force consists in substitution of the retarded field
produced by the body into its equation of motion. The resulting equation is believed
D. Gal’tsov (
)
Department of Physics, Moscow State University, Vorobjevy Gory, Moscow 119899, Russia
e-mail: galtsov@phys.msu.ru
L. Blanchet, A. Spallicci, and B. Whiting (eds.), Mass and Motion in General Relativity,
Fundamental Theories of Physics 162, DOI 10.1007/978-90-481-3015-3
13,
c
Springer Science+Business Media B.V. 2011
367
368 D. Gal’tsov
to give a correct description of the instantaneous effect of radiation on the motion
of the body. If the effect of reaction is small with respect to the main force, one
can treat it adiabatically. But when the reaction force is not small, the situation is
more subtle: the balance equations involve not only the kinetic energy–momentum
of the body and that of radiation, but also a variable contribution of the bound field.
Generically, an instantaneous loss of the energy–momentumby the body is not equal
to the energy–momentum carried away by radiation.
This situation looks particularly simple in flat space Maxwell–Lorentz electro-
dynamics, where due to linearity of the equations one can use the notion of the
point-like particle described by the delta-function. The Lorentz–Dirac equation ob-
tained by substituting the retarded field into the equation of motion and performing
the mass renormalization is expected to describe the motion of the particle with
account for radiation loss. However, it turns out that the momentum loss due to ra-
diation gives only a part of the reaction force. Although the initial system of the
charges and the Maxwell field obey the overall momentum conservation equations,
the Lorentz–Dirac equation, treated as the particle equation of motion, violates the
naively expected balance between the particle momentum and the momentum of
radiation. The difference is given by the so-called Schott term, which is the third-
order total derivative of the coordinate. A careful comparison of the Dirac derivation
[7, 17] and the Rohrlich analysis [8,23,24] has led Teitelboim [28](seealso[29]) to
interpret the Schott term as the finite contribution of the bound (non-radiated) mo-
mentum of the charge remaining after the mass renormalization (an explicit proof of
this claim was given in [11]). This may look strange, since we are used to thinking
about the bound field as a stable Coulomb “coat,” which is spherically symmetric
in the instantaneous rest frame, or pancake shaped for a relativistic charge. But this
simple picture is valid only for constant velocity. Once acceleration is nonzero, the
energy–momentum carried by the Coulomb coat becomes variable. The most sur-
prising fact is that this momentum is simply proportional to the acceleration, and
thus its derivative (the force) is the third derivative of the particle coordinate. Of
course, the split of the total field into radiation and the bound field has to be done
at all distances from the charge, not just in the wave zone. This split was given by
Rohrlich [24]; for more recent discussion see the book by B. Kosyakov [19]. It is
worth noting that in this case the finite part of the reaction force is entirely given by
the time-antisymmetric part of the particle field (half-difference of the retarded and
advanced potentials).
Thus, the energy–momentum balance of the system consisting of the accelerated
charge and its Maxwell field includes three, not just two, ingredients: the particle
momentum, the momentum carried by radiation, and the bound electromagnetic
momentum. The radiation momentum can be extracted both from the particle mo-
mentum and, indirectly, from the bound momentum. This explains the origin of
radiation of the uniformly accelerated charge, in which case the total reaction force
is zero and thus the kinetic particle momentum is constant. While the charge is un-
dergoing a constant acceleration, its bound electromagnetic momentum decreases
and is transferred to radiation. Physically, however, the acceleration has to start at
some moment and to finish at some moment, and during the stages of acquiring
Radiation Reaction and Energy–Momentum Conservation 369
and loosing the acceleration the bound momentum is exchanged with the kinetic
momentum. Therefore, the total energy–momentum loss of the charge will be equal
to the momentum carried away by radiation. But an instantaneous balance is ob-
scured by the presence of the Schott term. Another simple situation is periodic
motion. Since the ambiguous Schott term is a total derivative, its contribution van-
ishes if one integrates over the period, or, equivalently, averages over the period.
For radiation of linear fields of nongravitational nature or of the linearized grav-
itational field in curved space-time the situation is more complicated because there
are no local conservation laws unless Killing symmetries are present, and because
the tail term cannot be found in the closed form [5, 16] (for a review see [22]).
Another new feature in this case is that the reaction force contains a finite time-
symmetric contribution coming from the half-sum of the retarded and advanced
potentials – the force found by DeWitt–DeWitt [6] in linearized gravity and later re-
discovered in the full General Relativity by Smith and Will [26] (the DDSW force).
This force, however, has nothing to do with radiation and so the work done by this
force is not expected to contribute to the energy balance for radiation. In the sta-
tionary space-time the total energy of the particle and the field is conserved, so one
can expect that the energy balance between radiation and the particle energy loss
will hold integrally or in average. In the case of axial symmetry, similar consider-
ations apply to an associated angular momentum. An explicit proof of this balance
for scalar, electromagnetic, and linearized gravitational radiation in the Kerr space-
time was given in [9]. Apparently this work was not properly understood and was
criticized in a number of papers until 2005, when analogous calculations were per-
formed by other people (for a review and further references see [27]).
The case of linearized gravity is not entirely similar to other linear fields in the
curved background, however. In fact, linearized gravity on the non-vacuum back-
ground is not a consistent linear theory since the full Bianchi identities do not allow
for the harmonic gauge necessary to locally disentangle the linearized Einstein equa-
tions [14]. Physically, this means that the proper source of the gravitational radiation
from the point particle is not just its energy–momentum tensor, but it also includes
a contribution from the perturbed source of the background. Thus, the total source
is nonlocal, which raises doubts that there might exist a local equation describing
radiation reaction (here we mean the nonlocality stronger than that of the tail term,
which is still localized on the world line of the point particle).
2 Energy–Momentum Balance Equation
We start by considering an interacting system of N point charges and the Maxwell
field in Minkowski space-time, which is described by the coupled system of
equations
@
F
D 4
Z
N
X
aD1
Pz
ı.x z
a
.//d; m
0
a
Rz
a
D e
a
F
Pz
a
: (1)
370 D. Gal’tsov
It has 3N C1degrees of freedom, where 1 stands for the Maxwell field. The
corresponding energy–momentum conservation equation is
@
m
T
C
F
T
D 0; (2)
where
m
T
D
Z
N
X
aD1
m
0
a
Pz
Pz
ı.x z
a
.//d;
F
T
D
1
4
F
F
C
4
F
˛ˇ
F
˛ˇ
:
(3)
We have introduced the bare masses m
0
a
in anticipation of mass renormalization.
Since the Maxwell equation is linear, one can decompose the total field in the
vicinity of any given charge e
a
into the sum of the field generated by the other
N 1 charges, F
ext
(regular at its location) and the retarded field of e
a
;F
a ret
.
In spite of the fact that the total field acting on e
a
;F
a
D F
ext
C F
a ret
diverges at
x
D z
a
, energy–momentum conservation is ensured by the equations of motion.
The required mass renormalization does not change this statement.
The field F
a ret
describes radiation and the bound field, which both contribute to
the field energy–momentum.The overall conservation equation does not distinguish
between these two parts, so an additional analysis is needed. The retarded potential
at a given point x of space-time depends on the world-line variables taken at the
moment of proper time s
ret
.x/ defined as the solution to the equation
R
R
D 0; R
D x
z
.s
ret
/; (4)
satisfying x
0
> z
0
. The advanced solution to the same equation with z
0
>x
0
refers
to the advanced proper time s
adv
.x/. Introducing the invariant distance
D v
.s
ret
/R
; v
D
dz
ds
; (5)
which is equal to the spatial distance jRjDjx z.s
ret
/j between the points of
emission and observation in the momentarily co-moving Lorentz frame at the
time moment x
0
Dz
0
.s
ret
/, one can present the retarded potential as (we omit the
index a):
A
ret
.x/ D
ev
ˇ
ˇ
ˇ
s
ret
.x/
: (6)
Introduce the normalized null vector c
D R
=, such that vc D 1, and the unit
space-like vector u
D c
v
; u
2
D1. The following differentiation rules then
hold:
c
D @
s
ret
.x/; @
D v
C c
;@
c
D
1
ı
v
c
c
v
c
c
;
(7)