Blanchet L., Spallicci A., Whiting B. (Eds.) Mass and Motion in General Relativity
Подождите немного. Документ загружается.
The State of Current Self-Force Research 411
between a theoretical template accurate to O.=M/
1
, and a template accurate
to O.=M/
0
. V=.r!/ is related to the rate of change of the envelope of the chirp
wave. Notice that d N
cyc
=d.ln f/ !
2
=Œ.2/ P! is gauge independent. The differ-
ence in these quantities between the O.=M/
0
expressions and their O.=M/
1
counterparts can be expanded using the expressions above. We find that
d N
cyc
d.ln f/
D
2
3
r
M
r
"
3
2
r
3
.1/
MV
.1/
C
1
3
r
4
0
.1/
MV
.1/
V
.2/
V
2
.1/
#
; (38)
and
V
r!
.
V
r!
2
D
r
M
r
"
1
2
r
3
.1/
MV
.1/
V
.2/
V
2
.1/
#
: (39)
While the LHS of Eq. 38 is gauge invariant, the RHS is written as a sum of three
terms, each of which is gauge dependent: it is only their sum which is gauge in-
dependent, in the same sense as the discussion following Eq. 33. Notice that the
last two quantities are at O.=M/
0
. Notice also that Eqs. 38 and 39 do not de-
pend on
.2/
. All the second-order effects are included in V
.2/
,andtheyallarise
from f
.2/
t
and f
.2/
r
. Recall that these components of the self-force are the dissi-
pative ones. Therefore, in the circular orbit Schwarzschild case, the post-adiabatic
effects originate in the first-order self-force in addition to the dissipative piece of the
second-order self-force. We have no reason to expect that this division persists also
in more general cases. The dependence of Eqs. 36 and 37 on V
.2/
implies that the
second-order self-forces are necessary for the calculation of the O.=M/
0
effects
in the waveforms. Schematically, denoting the 4-acceleration by a, the dissipative
part of a
.n/
with a subscript “D” and the conservative part with a subscript “C ”,
where n is the order of the self-force, we find that for circular Schwarzschild orbits
the last terms that determine the self-force are given by
a D a
.0/
„ƒ‚…
D0
C Œa
.1/
D
„ƒ‚…
Waveform to
O.=M/
1
C a
.1/
C
M
C Œa
.2/
D
„ ƒ‚ …
Waveform to O.=M/
0
C a
.2/
C
M
2
C
„
ƒ‚ …
Waveform to O.=M/
: (40)
Here, we braced together the terms that contribute to the waveforms (specifically,
the phase of the waveforms) at various orders in the mass ratio, specifically at or-
der O.=M/
0
which is important for parameter estimation. The first term on the
RHS in Eq. 40 is the vanishing (in the absence of nongravitational forces) geodesic
acceleration, that is included here for completeness.
A more general approach for the post-adiabatic EMRI inspiral was recently pre-
sented by Hinderer and Flanagan [30]. Hinderer and Flanagan wrote the EMRI
equations of motion in action-angle variables and obtained the post-adiabatic cor-
rections to the orbital motion. The method presented in [30] is more general than
the argument we presented above, that was made simple by making the geodesic
orbit circular. For that case, [30] found results in agreement with those of [14].
Specifically, for circular and for equatorial orbits, the leading order corrections
412 L.M. Burko
are suppressed by one power of the mass ratio, and give rise to phase errors of
O.=M/
0
over a complete inspiral through the relativistic regime. These post-
1-adiabatic corrections are generated by the fluctuating, dissipative piece of the
first-order self-force, by the conservative piece of the first-order self-force, and by
the orbit-averaged, dissipative piece of the second-order self-force.
5.2 Conservative Self-Force Effects
We have argued that the O.=M/
0
self-force effects (at least for the simple case
of circular Schwarzschild orbits) include contributions from the dissipative second-
order self-force and the conservative first-order self-force. What are these effects on
the EMRI gravitational waveform?
The traditional approach to the self-force was to consider the adiabatic wave-
forms as a satisfactory approximation for the waveform, and to ignore conservative
effects, mostly for the argument that they do not accumulate secularly. Conservative
effects are in practice ignored when one is using the so-called radiation reaction
without radiation-reaction forces. In that approach one uses the fluxes of fields
associated with otherwise conserved quantities, and integrates them over a large
sphere, taken in practice to be at infinity. As the fields associated with conserva-
tive effects drop off quicker at infinity than those associated with dissipative effects,
one in practice discards all conservative effects when one integrate over a distant
surface. The complementary viewpoint is that of the self-force. The adiabatic self-
force is equivalent in practice to the force obtained from the “half-retarded minus
half-advanced” potential. As both the advanced and the retarded potentials include
the same divergent piece on the world line, their difference is finite, and is equal to
the radiation-reaction potential. However, this difference does not capture the con-
servative piece of the self-force.
The “half-retarded minus half-advanced” approach, while equivalent to the adi-
abatic self-force [27,43], suffers from a fundamental difficulty. Physical interaction
is normally believed to be retarded, for reasons of causality. The explicit inclusion
of the advanced potential in the physical self-force makes the self-force expres-
sion acausal, and in some sense also anti-causal. While the “half-retarded minus
half-advanced” prescription undoubtedly leads to the correct adiabatic results (ne-
glecting for the moment the discarded conservative contributions), the perception of
physical reality it leads to is quite different from the accepted one, and required the
reality of advanced interactions.
This question of the conservative contributions to the waveforms was considered
first in [13]. The model used in [13] was a simple one, intended to only raise this
question. Specifically, [13] argued that in addition to the dissipative forces, there
is also a conservative force, that in the circular orbit Schwarzschild case is radial.
This radial force depends only on the distance from the central black hole, and
therefore affects the orbit like any other component to the total radial force that
is different from the inverse-square law. Specifically, it causes precession of the
The State of Current Self-Force Research 413
periastron. In the absence of specific information on the functional dependence of
the radial component of the self-force, [13] instead considered the effect of various
post-Newtonian orders of the radial self-force. It was shown that the effect would
be a retrograde contribution to the periastron precession at O.=M/, in addition
the usual general relativistic periastron precession, which is a geodesic effect at
O.=M/
0
.
The effect of such an excess periaston precession on the waveform is therefore
dephasing of the latter: as the orbit is dephased by the periastron precession, the
gravitational waveform suffers similar dephasing. As this dephasing is at O.=M/
0
it may have an importance implications on EMRI gravitational wave detection and
precise parameter estimation.
The simple argument of [13] was revisited in much greater depth in [41]. In [41]a
simple toy problem of an electric charge moving slowly in the weak background of a
central object, subjected to its electromagnetic self force. In this toy model the self-
force is known analytically, and the dissipative and conservative contributions can
be clearly separated. Pound, Poisson, and Nickel then made use of the conceptual
clarity of their model, and eloquently showed how the conservative self-force effects
contribute to the dephasing of the orbits, by changing the positional orbital elements
while keeping the constant of the motion (the three principal orbital elements in the
language of [41]) unchanged, precisely what one would expect of the effects of the
conservative self-force effects on the orbit to be. The gravitational self-force case
was considered in [40]. It was shown in [40] that the conservative effects cause
significant long-term changes in the waveforms, by changing the phasing and the
time dependence of the orbit and consequently also those of the waveform.
Another important conservative effect was studied recently in [9]. It was found
that the innermost stable circular orbit (ISCO) in Schwarzschild is altered by
the conservative self-force, so that r
isco
D3:27 in the Lorenz gauge, and
˝
isco
=˝
isco
D 0:487 =M (gauge independent). These results provide an accu-
rate strong-field benchmark against which various approximation methods can be
tested.
Acknowledgements The author was supported in part by NSF grant No. PHY–0757344 and
NASA/SSC grant No. NNX07AL52A.
References
1. T. Apostolatos, D. Kennefick, A. Ori, E. Poisson, Phys. Rev. D 47, 5376 (1993)
2. J.G. Baker, W.D. Boggs, J. Centrella, B.J. Kelly, S.T. McWilliams, M. Coleman Miller, J.R. van
Meter, Astrophys. J. 682, L29 (2008)
3. J.G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, J. van Meter, Phys. Rev. Lett. 96, 111102
(2006)
4. L. Barack, L.M. Burko, Phys. Rev. D 62, 084040 (2000)
5. L. Barack, D.A. Golbourn, Phys. Rev. D 76, 044020 (2007)
6. L. Barack, D.A. Golbourn, N. Sago, Phys. Rev. D 76, 124036 (2007)
7. L. Barack, C.O. Lousto, Phys. Rev. D 66, 061502 (2002)
414 L.M. Burko
8. L. Barack, A. Ori, Phys. Rev. Lett. 90, 111101 (2003)
9. L. Barack, N. Sago, Phys. Rev. Lett. 102, 191101 (2009)
10. L. Barack, N. Sago, Phys. Rev. D 75, 064021 (2007)
11. J.L. Barton, D.J. Lazar, D.J. Kennefick, G. Khanna, L.M. Burko, Phys. Rev. D 78, 064042
(2008)
12. L.M. Burko, Phys. Rev. Lett. 84, 4529 (2000)
13. L.M. Burko, Int. J. Mod Phys. A 16, 1471 (2001)
14. L.M. Burko, Phys. Rev. D 67, 084001 (2003)
15. L.M. Burko, G. Khanna, Europhys. Lett. 78, 60005 (2007)
16. M. Campanelli, C.O. Lousto, Y. Zlochower, Phys. Rev. D 73, 061501 (2006)
17. M. Campanelli, C.O. Lousto, Y. Zlochower, D. Merritt, Astrophys. J. 659, L5 (2007)
18. C. Cutler, L.S. Finn, E. Poisson, G.J. Sussman, Phys. Rev. D 47, 1511 (1993)
19. M. Davis, R. Ruffini, W.H. Press, R.H. Price, Phys. Rev. Lett. 27, 1466 (1971)
20. S.L. Detweiler, Astrophys. J. 225, 687 (1978)
21. S. Detweiler, Class. Q. Grav. 22, S681 (2005)
22. S. Detweiler, Phys. Rev. D 77, 124026 (2008)
23. S. Detweiler, E. Messaritaki, B.F. Whiting, Phys. Rev. D 67, 104016 (2003)
24. S. Detweiler, B.F. Whiting, Phys. Rev. D 67, 024025 (2003)
25. S. Drasco, Phys. Rev. D 79, 124016 (2009)
26. L.S. Finn, K.S. Thorne, Phys. Rev. D 62, 124021 (2000)
27. D.V. Gal’tsov, J. Math. Phys. A. Math. Gen. 15, 3737 (1982)
28. K. Glampedakis, S.A. Hughes, D. Kennefick, Phys. Rev. D 66, 064005 (2002)
29. K. Glampedakis, D. Kennefick, Phys. Rev. D 66, 044002 (2002)
30. T. Hinderer,
´
E.
´
E. Flanagan, Phys. Rev. D 78, 064028 (2008)
31. C. Hopman, T. Alexander, Astrophys. J. 629, 362 (2005)
32. S.A. Hughes, Phys. Rev. D 61, 084004 (2000); Erratum, ibidem 63, 049902(E) (2001)
33. S.A. Hughes, Phys. Rev. D 64, 064004 (2001)
34. G. Khanna, Phys. Rev. D 69, 024016 (2004)
35. K. Lackeos, G. Khanna, L. Barack, L.M. Burko, work in progress
36. K. Lackeos, G. Khanna, L.M. Burko, work in progress
37. R. Lopez-Aleman, G. Khanna, J. Pullin, Class. Q. Grav. 20, 3259 (2003)
38. E. Poisson, Phys. Rev. D 47, 1497 (1993); ibidem 48, 1860 (1993); ibidem
52, 5719 (1995);
Erratum and Addendum, ibidem 55, 7980 (1997)
39. E. Poisson, M. Sasaki, Phys. Rev. D 51, 5753 (1995)
40. A. Pound, E. Poisson, Phys. Rev. D 77, 044013 (2008)
41. A. Pound, E. Poisson, B.G. Nickel, Phys. Rev. D 72, 124001 (2005)
42. F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005)
43. T.C. Quinn, R.M. Wald, Phys. Rev. D 60, 064009 (1999)
44. E. Rosenthal, Class. Q. Grav. 22, S859 (2005); Phys. Rev. D 74, 084018 (2006)
45. N. Sago, L. Barack, S. Detweiler, Phys. Rev. D 78, 124024 (2008)
46. M. Shibata, M. Sasaki, H. Tagoshi, T. Tanaka, Phys. Rev. D 51, 1646 (1995)
47. P.A. Sundararajan, G. Khanna, S.A. Hughes, Phys. Rev. D 76, 104005 (2007)
48. P.A. Sundararajan, G. Khanna, S.A. Hughes, S. Drasco, Phys. Rev. D 78, 024022 (2008)
49. S. Suzuki, K. Maeda, Phys. Rev. D 58, 023005 (1998)
50. T. Tanaka, Y. Mino, M. Sasaki, M. Shibata, Phys. Rev. D 54, 3762 (1996)
51. T. Tanaka, M. Shibata, M. Sasaki, T. Nakamura, Prog. Theor. Phys. 90, 65 (1993)
52. S.A. Teukolsky, Astrophys. J. 185, 635 (1973)
High-Accuracy Comparison Between
the Post-Newtonian and Self-Force Dynamics
of Black-Hole Binaries
Luc Blanchet, Steven Detweiler, Alexandre Le Tiec, and Bernard F. Whiting
Abstract The relativistic motion of a compact binary system moving in circular
orbit is investigated using the post-Newtonian (PN) approximation and the pertur-
bative self-force (SF) formalism. A particular gauge-invariant observable quantity
is computed as a function of the binary’s orbital frequency. The conservative ef-
fect induced by the gravitational SF is obtained numerically with high precision,
and compared to the PN prediction developed to high order. The PN calculation
involves the computation of the 3PN regularized metric at the location of the par-
ticle. Its divergent self-field is regularized by means of dimensional regularization.
The poles / .d 3/
1
that occur within dimensional regularization at the 3PN
order disappear from the final gauge-invariant result. The leading 4PN and next-to-
leading 5PN conservative logarithmic contributions originating from gravitational
wave tails are also obtained. Making use of these exact PN results, some previously
unknown PN coefficients are measured up to the very high 7PN order by fitting to
the numerical SF data. Using just the 2PN and new logarithmic terms, the value
of the 3PN coefficient is also confirmed numerically with very high precision. The
consistency of this cross-cultural comparison provides a crucial test of the very dif-
ferent regularization methods used in both SF and PN formalisms, and illustrates the
complementarity of these approximation schemes when modeling compact binary
systems.
L. Blanchet and A. Le Tiec (
)
G R"CO, Institut d’Astrophysique de Paris, C.N.R.S. & Universit´e Pierre et Marie Curie,
98
bis
Boulevard Arago, 75014 Paris, France
e-mail: blanchet@iap.fr; letiec@iap.fr
S. Detweiler and B.F. Whiting
Institute for Fundamental Theory, Department of Physics, University of Florida,
Gainesville, FL 32611-8440, USA
e-mail: det@phys.ufl.edu; bernard@phys.ufl.edu
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
15,
c
Springer Science+Business Media B.V. 2011
415
416 L. Blanchet et al.
1 Introduction and Motivation
For the gravitational wave observatories LIGO/Virgo/GEO on Earth and LISA
in space, routine identification of inspiralling compact binaries (binary systems
composed of neutron stars and/or black holes) will require high-accuracy predic-
tions from general relativity theory [24, 49]. Providing such predictions represents
a formidable task that can be addressed using approximation schemes in gen-
eral relativity. The two main approximation schemes available are: (i) the post-
Newtonian expansion, well suited to describe the inspiralling phase of compact
binaries in the slow motion and weak-field regime independently of the mass ra-
tio; and (ii) the self-force approach, based on perturbation theory, which gives
an accurate description of extreme mass ratio binaries even in the strong-field
regime.
The post-Newtonian (PN) templates for compact binary inspiral have been de-
veloped to 3.5PN order in the phase [10, 14, 17] and 3PN order in the amplitude
[15, 18] (see Blanchet’s contribution in this volume).
1
These are suitable for the
inspiral of two neutron stars in the frequency bandwidth of LIGO and Virgo detec-
tors. For detection of black-hole binaries (with higher masses), the construction of
template banks either requires the matching of the PN waveform with full numerical
simulations for the merger phase and the ringdown of the final black-hole [1, 20],
or using the effective-one-body formalism [21] (see also Damour’s and Nagar’s
contribution in this volume).
In a completely different parameter regime, gravitational self-force (SF) analysis
[30,33,36,40,43] (see also Poisson’s contribution in this volume) is expected to pro-
vide templates for extreme mass ratio inspirals (EMRIs) anticipated to be present in
the LISA frequency bandwidth. SF analysis is a natural extension of first-order per-
turbation theory, and the latter has a long history of comparisons with PN analysis
[25, 37–39, 41, 45–47]. SF analysis itself, however, is just now mature enough to
present some limited comparisons with PN analysis, although it is not yet ready for
template generation.
In recent works [11, 12] (hereinafter referred to as Papers I and II, respectively)
we performed a high-accuracy comparison between the PN and SF analyses in their
common domain of validity, that of the slow motion weak-field regime of an ex-
treme mass ratio binary (see illustration of various methods in Fig. 1). The problem
was tackled previously by Detweiler [28], who computed numerically within the
SF approach a certain gauge invariant quantity (called the redshift observable), and
compared it with the 2PN prediction extracted from existing PN results [16]. We
then extended this comparison in Papers I and II up to higher PN orders. This
required an improvement in the numerical resolution of the SF calculation in or-
der to distinguish more accurately the various contributions of very high PN order
terms. However, our primary difficulty has been that the relevant PN results for the
1
As usual the nPN order refers to terms equivalent to .v=c/
2n
beyond Newtonian theory, where v
is a typical internal velocity of the material system and c is the speed of light.
High-Accuracy Comparison Between the PN and SF Dynamics of Black-Hole Binaries 417
Numerical
Relativity
Perturbation
Theory
Post-Newtonian
Theory
log
10
(m
2
/m
1
)
log
10
(r
12
/m)
01 23
0
1
2
3
Post-Newtonian
Theory
&
Perturbation
Theory
4
4
Fig. 1 Different analytical approximation schemes and numerical techniques are used to study
black-hole binaries, depending on the mass ratio m
1
=m
2
and the orbital velocity v
2
Gm=r
12
,
where m D m
1
C m
2
. The post-Newtonian theory and black hole perturbation theory can be
compared in the slow motion regime (v c equivalent to r
12
Gm=c
2
for circular orbits) of an
extreme mass ratio (m
1
m
2
)binary
metric were previously not available beyond the 2PN order, and had to be care-
fully derived. We have finally demonstrated an excellent agreement between the SF
contribution to the analytical PN result (derived through 3PN order, with inclusion
of specific logarithmic terms at 4PN and 5PN orders) and the exact numerical SF
result.
In this article we present a summary of the Papers I and II. The plan is as fol-
lows: After having introduced in Section 2 the coordinate-invariant relation used to
perform the comparison, we describe in Section 3 how the divergent self-field of
point particles is regularized in both SF and PN formalisms. Section 4 provides a
brief overview of how the SF computation proceeds. Sections 5 and 6 present the
PN computations of the 3PN regularized metric, as well as of the 4PN and 5PN
logarithmic contributions. The PN results for the gauge-invariant relation are dis-
cussedinSection7. Finally, Section 8 is devoted to the comparison of these PN
results with the SF numerical data, and the measurement of unknown high-order PN
coefficients.
418 L. Blanchet et al.
2 The Gauge-Invariant Redshift Observable
We consider a system of two (non-spinning) compact objects with masses m
1
and
m
2
, and moving on slowly inspiralling quasi-circular orbits. In the PN analysis, let
m
1
and m
2
be arbitrary; in the SF analysis, further assume that m
1
m
2
. We can
then call m
1
the “particle,” and m
2
the “black hole.”
SF analysis shows that the dissipative parts of the SF for a circular orbit are
the t and ' components. These result in a loss of energy and angular momentum
from the small mass at the same precise rate as energy and angular momentum
are radiated away [28]. In addition, earlier perturbative calculations of energy and
angular momentum fluxes [25,37–39,41,45–47] for this situation show them to be
equivalent to the results of the PN analysis in their common domain of validity.
Hence, by invoking an argument of energy and angular momentum balance, we
know that the PN results also agree with the dissipative parts of the SF, and further
comparison can reveal nothing new.
For our PN–SF comparison, we shall thus neglect the dissipative, radiation-
reaction force responsible for the inspiral, and restrict ourselves to the conservative
part of the dynamics. In PN theory this means neglecting the dissipative radiation-
reaction force at 2.5PN and 3.5PN orders, and considering only the conservative
dynamics at the even-parity 1PN, 2PN, and 3PN orders. This clean separation be-
tween conservative even-parity and dissipative odd-parity PN terms is correct up
to 3.5PN order. However, such split breaks at 4PN order, since at that approxi-
mation arises a contribution of the radiation-reaction force, which originates from
gravitational wave tails propagating to infinity [7] (this will be further discussed
in Section 6). In SF theory, there is also a clean split between the dissipative and
conservative parts of the SF (see e.g. [4]). This split is particularly transparent for a
quasi-circular orbit, where the r component is the only nonvanishing component of
the conservative SF.
Henceforth, the orbits of both masses are assumed to be and to remain circular,
because we are ignoring the dissipative radiation-reaction effects. For our compari-
son we require two physical quantities which are precisely defined in the context of
each of our approximation schemes. The orbital frequency ˝ of the circular orbit as
measured by a distant observer is one such quantity. The second quantity is defined
as follows.
With circular orbits and no dissipation, the geometry has a helical Killing vector
field k
˛
. A Killing vector is only defined up to an overall constant factor. In our case
k
˛
extends out to a large distance where the geometry is essentially flat. There,
k
˛
@
˛
D @
t
C ˝@
'
; (1)
in any natural coordinate system which respects the helical symmetry [44]. We let
this equality define the overall constant factor, thereby specifying the Killing vector
field uniquely.
An observer moving with the particle m
1
, while orbiting the black hole m
2
,
would detect no change in the local geometry. Thus, the four-velocity u
˛
1
of the
High-Accuracy Comparison Between the PN and SF Dynamics of Black-Hole Binaries 419
particle is tangent to the Killing vector k
˛
evaluated at the location of the particle,
which we denote by k
˛
1
. A second physical quantity is then defined as the constant
of proportionality, call it u
T
1
, between these two vectors, namely
u
˛
1
D u
T
1
k
˛
1
: (2)
The four-velocity of the particle is normalized so that .g
˛ˇ
/
1
u
˛
1
u
ˇ
1
D1; .g
˛ˇ
/
1
is
the regularized metric at the particle’s location, whereas the metric itself is formally
singular at the particle m
1
in both PN and SF approaches.
If we happen to choose a coordinate system such that (1) is satisfied everywhere,
then in particular k
t
1
D 1, and thus u
T
1
u
t
1
,thet component of the four-velocity
of m
1
. The Killing vector on the particle is then k
˛
1
D u
˛
1
=u
t
1
, and simply reduces
to the particle’s ordinary PN coordinate velocity v
˛
1
=c. In such a coordinate system,
the description of the invariant quantity we are thus considering is
u
T
1
u
t
1
D
.g
˛ˇ
/
1
v
˛
1
v
ˇ
1
c
2
1=2
: (3)
It is important to note that this quantity is precisely defined in both PN and SF
frameworks, and it does not depend upon the choice of coordinates or upon the
choice of perturbative gauge. The quantity u
T
1
represents the redshift of light rays
emitted from the particle and received on the helical symmetry axis perpendicular
to the orbital plane [28]; we shall refer to it as the redshift observable.
3 Regularization Issues in the SF and PN Formalisms
The redshift observable (3) depends upon using a valid method of regularization.
The regularized metric .g
˛ˇ
/
1
is defined with very different prescriptions in the SF
and PN approaches. Both analyses require subtle treatment of singular fields at the
location of the masses. Subtracting away the infinite part of a field while carefully
preserving the part which is desired is always a delicate task.
In the SF prescription, the regularized metric reads
g
SF
˛ˇ
.x/ DNg
˛ˇ
.x/ C h
R
˛ˇ
.x/; (4)
where Ng
˛ˇ
denotes the background Schwarzschild metric of the black hole, and
where the “Regular” metric perturbation h
R
˛ˇ
is smooth in a neighborhood of the
particle, and given by the difference
h
R
˛ˇ
D h
ret
˛ˇ
h
S
˛ˇ
(5)
between the retarded metric perturbation h
ret
˛ˇ
and the purely locally deter-
mined “Singular” field h
S
˛ˇ
. Following the Detweiler–Whiting prescription [30],
420 L. Blanchet et al.
a Hadamard expansion of Green’s functions in curved spacetime provides an
expansion for h
S
˛ˇ
. In a neighborhood of the particle with a special, locally inertial
coordinate system, h
S
˛ˇ
appears as the m
1
=r part of the particle’s Schwarzschild
metric along with its tidal distortion caused by the background geometry of the
large black hole. Details of the expansion are given in Section 6.1 of [27]. The spe-
cial locally inertial coordinates for a circular geodesic in the Schwarzschild metric
are given as functions of the Schwarzschild coordinates in Appendix B of [29]. (See
also Detweiler’s and Barack’s contributions in this volume.) Since the metric (4)is
regular at the particle’s position y
˛
1
,wesimplyhave
g
SF
˛ˇ
1
D g
SF
˛ˇ
.y
1
/: (6)
In the perturbative SF analysis we are only working through first order in the mass
ratio q m
1
=m
2
, and at that level of approximation h
R
˛ˇ
D O.q/.Thenu
T
1
can be
computed accurately to the same perturbative order and compares well with the PN
result to 2PN order [28]. The regularized 2PN metric is known [16], and therefore
the comparison is straightforward.
In the PN prescription on the other hand, one first computes the metric g
PN
˛ˇ
.x;t/
using an iterative PN procedure at any field point outside the particle, in a coordinate
system x
˛
Dfct;x
i
g. That metric is generated by the two particles, and includes
both regular and singular contributions around each particle. Such iterative PN cal-
culation is a very long and intricate procedure up to say 3PN, at which order it
will be partly based on existing computations of the 3PN equations of motion using
Hadamard [13] and dimensional [9] regularizations. Then we compute the regular-
ized metric at the location of the particle by taking the limit when x ! y
1
.t/,where
y
1
.t/ is the particle’s trajectory. In 3 spatial dimensions, that limit is singular. In
order to treat the infinite part of the field, we extend the computation in d spatial di-
mensions, following the prescription of dimensional regularization [19, 48], which
is based on an analytic continuation (AC) in the dimension d viewed as a complex
number. We do not use Hadamard’s regularization that found its limit at 3PN order.
Considering the analytic continuation in a neighborhood of " d 3 ! 0,we
define
g
PN
˛ˇ
1
D AC
"!0
h
lim
x!y
1
g
PN
˛ˇ
.x;t/
i
: (7)
The limit " ! 0 does not exist in general due to the presence of poles / "
1
occurring at the 3PN order; we thus do not take the strict limit " ! 0 but compute
the singular Laurent expansion when " ! 0. At 3PN order the result takes the
schematic form
g
PN
˛ˇ
1
D
1
"
g
.1/
˛ˇ
.y
1
;t/C g
.0/
˛ˇ
.y
1
;t/C O."/; (8)
where g
.1/
˛ˇ
.y
1
;t/ denotes the pole part, which is purely of 3PN order, and
g
.0/
˛ˇ
.y
1
;t/is the finite part. At higher PN orders we expect the presence of multipole