PHYSICAL REVIEW D
VOLUME 55, NUMBER 10
15 MAY 1997
Second post-Newtonian gravitational radiation reaction for two-body systems:
Nonspinning bodies
A. Gopakumar and Bala R. Iyer
Raman Research Institute, Bangalore 560 080, India
Sai Iyer
Physical Research Laboratory, Ahmedabad 380 009, India
~Received 20 January 1997!
Starting from the recently obtained post-post-Newtonian ~2PN! accurate forms of the energy and angular
momentum fluxes from inspiraling compact binaries, we deduce the gravitational radiation reaction to 2PN
order beyond the quadrupole approximation—4.5PN terms in the equation of motion—using the refined balance method proposed by Iyer and Will. We explore critically the features of their construction and illustrate
them by contrast with other possible variants. The equations of motion are valid for general binary orbits and
for a class of coordinate gauges. The limiting cases of circular orbits and radial infall are also discussed.
@S0556-2821~97!04410-X#
PACS number~s!: 04.25.Nx, 04.30.Db, 97.60.Jd, 97.60.Lf
I. INTRODUCTION
Inspiraling compact binaries are the most promising
sources of gravitational radiation in the near future for
ground-based laser interferometric detectors such as the Laser Interferometric Gravitational Wave Observatory ~LIGO!
@1#, VIRGO @2#, GEO600 @3#, and TAMA @4#. The method
of matched filtering will be employed to search for the inspiral waveforms and extract the information they carry
@5,6#. For this method to be successful, one needs to use
templates that are extremely accurate in their description of
the evolution of the orbital phase, which, in turn, requires a
detailed understanding of how radiation damping ~reaction!
influences orbital evolution @7–10#.
The idea of a damping force associated with an interaction that propagates with a finite velocity was first discussed
in the context of electromagnetism by Lorentz @11#. He obtained it by a direct calculation of the total force acting on a
small extended particle due to its self-field. The answer was
incorrect by a numerical factor and the correct result was first
obtained by Planck @12# using a ‘‘heuristic’’ argument based
on energy balance which prompted Lorentz @13# to reexamine his self-field calculations and confirm Planck’s result,
F i5
2 e2 i
v̈ ,
3 c3
where v i is the velocity of the particle. The relativistic generalization of the radiation reaction by Abraham @14# based
on arguments of energy and linear momentum balance preceded by a few years the direct relativistic self-field calculation by Schott @15# and illustrates the utility of this heuristic,
albeit less rigorous, approach @16#.
The argument based on energy balance proceeds thus: A
nonaccelerated particle does not radiate and satisfies Newton’s ~conservative! equation of motion. If it is accelerated, it
radiates, loses energy and this implies damping terms in the
equation of motion. Equating the work done by the reactive
force on the particle in a unit time interval to the negative of
0556-2821/97/55~10!/6030~24!/$10.00
55
the energy radiated by the accelerated particle in that interval
~Larmor’s formula! the reactive acceleration is determined
and one is led to the Abraham-Lorentz equation of motion
for the charged particle. The direct method of obtaining radiation damping, on the other hand, is based on the evaluation of the self-force. Starting with the momentum conservation law for the electromagnetic fields one rewrites this as
Newton’s equation of motion by decomposing the electromagnetic fields into an ‘‘external field’’ and a ‘‘self-field.’’
Expanding the self-field in terms of potentials, solving for
them in terms of retarded fields and finally making a retardation expansion, one obtains the required equation of motion when one goes to the point particle limit @17#.
As in the electromagnetic case, the approach to gravitational radiation damping has been based on the balance
methods, the reaction potential or a full iteration of Einstein’s equation. The first computation in general relativity
was by Einstein @18# who derived the loss in energy of a
spinning rod by a far-zone energy flux computation. The
same was derived by Eddington @19# by a direct near-zone
radiation damping approach. He also pointed out that the
physical mechanism causing damping was the effect discussed by Laplace @20#, that if gravity was not propagated
instantaneously, reactive forces could result. A useful development was the introduction of the radiation reaction potential by Burke @21# and Thorne @22# using the method of
matched asymptotic expansions. In this approach, one derives the equation of motion by constructing an outgoing
wave solution of Einstein’s equation in some convenient
gauge and then matching it to the near-zone solution. Restricting attention only to lowest order Newtonian terms and
terms sensitive to the outgoing ~ingoing! boundary conditions and neglecting all other terms, one obtains the required
result. The first complete direct calculation in the manner of
Lorentz of the gravitational radiation reaction force was by
Chandrasekhar and Esposito @23#. Chandrasekhar and collaborators @24,25# developed a systematic post-Newtonian
expansion for extended perfect fluid systems and put together correctly the necessary elements like the Landau6030
© 1997 The American Physical Society
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
Lifshiftz pseudotensor, the retarded potentials and the nearzone expansion. These works established the balance
equations to Newtonian order, albeit for weakly selfgravitating fluid systems. The revival of interest in these issues following the discovery of the binary pulsar and the
applicability of these very equations to binary systems of
compact objects follows from the works of Damour @16,26#
and Damour and Deruelle @27#.
In the context of the binary pulsar timing, the accuracy
reached by the Newtonian balance equations is amply adequate. The case of inspiraling binaries as sources for the
interferometric gravitational wave detectors is very different.
The extremely high phasing accuracy requirement makes
mandatory the control of reactive terms way beyond the
Newtonian. This has prompted on the one hand, work on
generation aspects to compute the far-zone flux of energy
and angular momentum carried by gravitational waves and
on the other, work on the radiation reaction aspects to compute the effect on the orbital motion of the emission of gravitational radiation. As in the electromagnetic case, the computation of the reactive acceleration assuming balance
equations is simpler than the computation of the damping
terms by a direct near-field iteration. The computation of the
energy and angular momentum fluxes at the lowest Newtonian order ~quadrupole equation! requires the equation of motion at only Newtonian order. Assuming the balance equations one can infer the lowest order ~2.5PN! radiation
damping whose direct computation, as mentioned before, requires a 2.5PN iteration of the near-zone equations. Similarly, the computation of the 1PN corrections to the lowest
order quadrupole luminosity requires the 1PN accurate equations of motion, but is potentially equivalent to the 3.5PN
terms in the equation of motion. This motivated Iyer and
Will ~IW! @28,29# to propose a refinement of the textbook
@30# treatment of the energy balance method used to discuss
radiation damping. This generalization uses both energy and
angular momentum balance to deduce the radiation reaction
force for a binary system made of nonspinning structureless
particles moving on general orbits. Starting from the 1PN
conserved dynamics of the two-body system, and the radiated energy and angular momentum in the gravitational
waves, and taking into account the arbitrariness of the ‘‘balance’’ up to total time derivatives, they determined the
2.5PN and 3.5PN terms in the equations of motion of the
binary system. The part not fixed by the balance equations
was identified with the freedom still residing in the choice of
the coordinate system at that order. Thus, starting from the
far-zone flux formulas, one deduces a formula that is suitable
for evolving general orbits of compact binaries of arbitrary
mass ratio and that includes 1PN corrections to the dominant
Newtonian radiation reaction terms. Blanchet @31,32#, on the
other hand, obtained the post-Newtonian corrections to the
radiation reaction force from first principles using a combination of post-Minkowskian, multipolar, and post-Newtonian
schemes together with techniques of analytic continuation
and asymptotic matching. By looking at ‘‘antisymmetric’’
waves—a solution of the d’Alembertian equation composed
of retarded wave minus advanced wave, regular all over the
source—and matching, one obtains a radiation reaction tensor potential that generalizes the Burke-Thorne reaction potential @33#, in terms of explicit integrals over matter fields in
6031
the source. The validity of the balance equations up to 1.5PN
is also proved. By specializing this potential to two-body
systems, Iyer and Will @29# checked that this solution indeed
corresponds to a unique and consistent choice of coordinate
system. This provides a delicate and nontrivial check on the
validity of the 1PN reaction potentials and the overall consistency of the direct methods based on iteration of the nearfield equations and indirect methods based on energy and
angular momentum balance.
As emphasized earlier, much better approximations are
needed to reach the precision of future gravitational-wave
astronomy @7#. In the limit where one mass is much smaller
than the other, numerical and analytical computations based
on black hole perturbation theory have been performed to the
5.5PN order @34–39#, a recent result being the analytical
expression to 5.5PN order for the energy flux from a test
particle moving in a circular orbit around a Schwarzschild
black hole @39#. Ryan @40,41# has investigated the effect of
gravitational radiation reaction, first on circular, and later
even for nonequatorial orbits around a spinning black hole.
Recently Mino, Sasaki, and Tanaka @42# have derived the
leading order correction to the equation of motion of a particle which presumably describes the effect of gravitational
radiation reaction by two methods: an extension of the
Dewitt-Brehme formalism and the method of asymptotic
matching.
On the other hand, for bodies of comparable masses, recently two independent teams @43–47# have derived the 2PN
accurate gravitational waveform and the associated energy
and angular momentum fluxes for inspiraling compact binaries through 2PN order by two independent methods: the
Blanchet-Damour-Iyer ~BDI! approach based on a mixed
multipolar post-Minkowskian and post-Newtonian framework together with asymptotic matching and analytic continuation @48# and the recently improved Epstein-Wagoner
~EW! @49# formalism by Will and Wiseman @46# which provides a method to carefully handle the divergences of the
older EW treatment. In view of the above discussion it is
natural to investigate the possibility of extending the treatment of Iyer and Will to 2PN accuracy beyond the Newtonian ~2.5PN! radiation reaction and this is what we propose to
take up in this paper. The knowledge of the reactive acceleration beyond the lowest order could also have practical
uses. For instance, Lincoln and Will @50# have studied the
late-time orbital evolution of compact binaries with arbitrary
mass ratios. They described the orbit using the osculating
orbital elements of celestial mechanics and used the
Damour-Deruelle two-body equations of motion including
Newtonian radiation reaction terms @27,16# to evolve these
orbital elements. The extension of this work to include 1PN
radiation reaction is still not available. Recently, a 2PN accurate description for the motion of spinning compact binaries of arbitrary mass ratio was obtained in a generalized
quasi-Keplerian parametrization initially suggested by
Damour and Schäfer @51–54#. These orbital elements have
also not been evolved to 2PN radiation reaction order. Our
present computation is a step in that direction. These attempts to study the evolution of binary orbits would be
complementary to those using the test particle limit @40,41#.
To summarize: Starting from 2PN accurate energy and
angular momentum fluxes for compact binaries of arbitrary
6032
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
mass ratio moving in quasielliptical orbits @47,46#, we obtain
the 4.5PN reactive terms in the equations of motion by an
extension of the IW method. Schematically, the equations of
motion for spinless bodies of arbitrary mass ratio are
a[
d 2x
mx
'2 3 @ 11O ~ e ! 1O ~ e 2 ! 1O ~ e 2.5! 1O ~ e 3 !
dt 2
r
1O ~ e 3.5! 1O ~ e 4 ! 1O ~ e 4.5! 1••• # ,
~1.1!
where x and r5 u xu denote the separation vector and distance
between the bodies, and m5m 1 1m 2 denotes the total mass.
The quantity e is a small expansion parameter that satisfies
e ;( v /c) 2 ;Gm/(rc 2 ), where v and r are the orbital velocity and separation of the binary system. The symbols O( e )
and O( e 2 ) represent post-Newtonian ~PN!, post-postNewtonian ~2PN! corrections and so on. Gravitational radiation reaction first appears at O( e 2.5) beyond Newtonian
gravitation, or at 2.5PN order. We call this the ‘‘Newtonian’’
radiation reaction. ‘‘Post-Newtonian’’ radiation reaction
terms, at O( e 3.5), were obtained by Iyer and Will @28,29# and
Blanchet @31,32#. Here we obtain the 2PN radiation reaction,
at O( e 4.5). The 4.5PN reactive terms are determined in terms
of 12 arbitrary parameters, which along the lines of @28,29#,
are associated with the possible residual ‘‘gauge’’ choice at
the 4.5PN order. These results valid for general orbits are
specialized to the two complementary cases of circular orbits
and radial infall. The expressions for ṙ and v̇ for the quasicircular orbits and ż for radial infall to 4.5PN order are in
agreement with @43,55# as required. We next examine critically the origin of the ‘‘redundant’’ equations in the formalism and examine our understanding of this redundancy by
exploring variant schemes which differ from the original IW
scheme in their choice of the functional forms for the arbitrary terms in energy and angular momentum.
The paper is organized as follows. In Sec. II, we describe
the IW method to obtain the 2PN reactive terms. Section III
examines the question of redundant equations and explores
‘‘variants’’ of the original IW scheme that differ in their
choice of the ambiguities in energy and angular momentum.
Section IV discusses the question of the undetermined parameters and arbitrariness in the choice of the gauge, in particular at 4.5PN order. Section V is devoted to the particular
cases of quasicircular orbits and head-on infall. Section VI
contains some concluding remarks. In the Appendix, for
mathematical completeness, we prove that the far-zone flux
formulas and the balance equations admit more general solutions if one relaxes the requirement that the reactive acceleration be a power series in the individual masses of the
binary or, equivalently, that it be nonlinear in the total mass.
II. IW METHOD FOR REACTIVE TERMS IN THE
EQUATIONS OF MOTION
A. The procedure
We consider only two-body systems containing objects
that are sufficiently small that finite-size effects, such as
spin-orbit, spin-spin, or tidal interactions can be ignored. The
dynamics of such systems is well studied and the two-body
equations of motion conveniently cast into a relative onebody equation of motion is given by
1!
2!
2.5!
3!
3.5!
4!
4!
4.5!
a5aN 1a~PN
1a~2PN
1a~RR
1a~3PN
1a~1RR
1a~4PN
1a~tail
1a~2RR
1O ~ e 5 ! ,
HF
SD
G F
2!
52
a~2PN
HF
~2.1!
where the subscripts denote the nature of the term, postNewtonian ~PN!, post-post-Newtonian ~2PN!, Newtonian radiation reaction ~RR!, post-Newtonian radiation reaction
~1RR!, 2PN radiation reaction ~2RR!, tail radiation reaction,
and so on; and the superscripts denote the order in e . For our
purpose we need to know explicitly the acceleration terms
through 2PN order and they are given by @27,56,50#
(G5c51)
aN 52
1!
52
a~PN
55
m
n,
r2
~2.2a!
G
J
m
m
3 2
2
2 n 22 ~ 21 h ! 1 ~ 113 h !v 2 h ṙ 22 ~ 22 h ! ṙv ,
r
r
2
m
3
m
n ~ 12129h !
r2
4
r
2
1 h ~ 324 h !v 4 1
~2.2b!
15
3
1
m
h ~ 123 h ! ṙ 4 2 h ~ 324 h !v 2 ṙ 2 2 h ~ 1324 h ! v 2
8
2
2
r
m
m
1
2 ~ 2125h 12 h 2 ! ṙ 2 2 ṙv h ~ 1514 h !v 2 2 ~ 4141h 18 h 2 ! 23 h ~ 312 h ! ṙ 2
r
2
r
where m [m 1 m 2 /m is the reduced mass, with h 5 m /m, and
n5x/r. The n.5PN reactive accelerations are determined by
following the ‘‘What else can it be ?’’ procedure employed
in IW which we summarize here. One writes down a general
form for the Newtonian ( e 2.5), 1PN ( e 3.5), and 2PN ( e 4.5)
radiation-reaction terms in the equations of motion for two
GJ
,
~2.2c!
bodies, ignoring tidal and spin effects. For the relative acceleration a[a1 2a2 , one assumes the provisional form
8
a52 h ~ m/r 2 !~ m/r !@ 2 ~ A 2.51A 3.51A 4.5! ṙn
5
1 ~ B 2.51B 3.51B 4.5! v# .
~2.3!
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
The form of Eq. ~2.3! is dictated by the fact that it must be a
correction to the Newtonian acceleration ~i.e., be proportional to m/r 2 ), must vanish in the test body limit when
gravitational radiation vanishes ~i.e., be proportional to h ),
must be dissipative, or odd in velocities ~i.e., contain the
factors ṙ, n, and v linearly! and finally, must be related to the
emission of gravitational radiation or be nonlinear in Newton’s constant G ~i.e., contain another factor m/r). The last
condition may be more precisely stated by requiring that the
reactive acceleration be a power series in the individual
masses m 1 and m 2 @57#. For spinless, structureless bodies,
the acceleration must lie in the orbital plane ~i.e., depend
only on the vectors n and v). The prefactor 8/5 is chosen for
convenience. To make the leading term of O( e 2.5) beyond
Newtonian order, A 2.5 and B 2.5 must be of O( e ). For this
structureless two-body system the only variables in the problem of this order are v 2 , m/r, and ṙ 2 . Thus A 2.5 and B 2.5 can
each be a linear combination of these three terms; to those
terms we assign six ‘‘Newtonian radiation reaction’’ parameters. Proceeding similarly, A 3.5 and B 3.5 must be of O( e 2 ),
hence must each be a linear combination of the six terms
v 4 , v 2 m/r, v 2 ṙ 2 , ṙ 2 m/r, ṙ 4 , and (m/r) 2 . To these we assign
12 ‘‘1PN RR’’ parameters. And finally, A 4.5 and B 4.5 must
be of O( e 3 ), each a linear combination of the 10 terms v 6 ,
v 4 ṙ 2 , v 4 m/r, v 2 ṙ 4 , v 2 (m/r) 2 , v 2 ṙ 2 (m/r), ṙ 6 , ṙ 4 (m/r),
ṙ 2 (m/r) 2 , and (m/r) 3 to which we assign 20 ‘‘2PN RR’’
parameters. The 6 Newtonian RR and 12 post-Newtonian
RR parameters were first determined in IW @28,29#. This
E N5 m
E PN5 m
E 2PN5 m
H
H
6033
solution has been checked and reproduced in the preliminary
part of this investigation and constitutes an input to supplement the conservative acceleration terms in Eq. ~2.3! for the
present study. Our aim is to evaluate these 20 parameters
appearing in A 4.5 and B 4.5 that will determine the 2PN radiation reaction. It is worth pointing out that in the calculation
we are setting up, the terms in the equations of motion of
O( e 3 ) and O( e 4 ) beyond Newtonian order do not play any
role. The former is nondissipative but not yet computed; the
latter on the other hand includes dissipative parts due to the
‘‘tail’’ effects @58–61# which have been separately balanced
by the tail luminosity in the works of Blanchet and Damour
@58,32#. However all the radiation-reaction results will remain as ‘‘partial results’’ in the saga of equations of motion
until a complete treatment of Chandrasekhar @23# and
Damour @16# is available through 3PN order and later
through 4PN order.
Through 2PN order, the equations of motion can be derived from a generalized Lagrangian that depends not only
on positions and velocities but also on accelerations. To this
order, that is in the absence of radiation reaction, the Lagrangian leads to a conserved energy and angular momentum
given by @27,56,62#
E5E N 1E PN1E 2PN ,
~2.4a!
J5JN 1JPN1J2PN ,
~2.4b!
where
S
D
1 2 m
,
v 2
2
r
~2.5a!
S DJ
2
3
m 1 m
1
1 m
~ 123 h !v 4 1 ~ 31 h !v 2 1 h ṙ 2 1
8
2
r 2 r
2 r
~2.5b!
,
SD
1
1
3
1
5
m
m
m
m
~ 127 h 113h 2 !v 6 1 ~ 21223h 227h 2 ! v 4 1 h ~ 1215h ! v 2 ṙ 2 2 h ~ 123 h ! ṙ 4 2 ~ 2115h !
16
8
r
4
r
8
r
4
r
SD
m
1
1 ~ 14255h 14 h 2 !
8
r
2
SD J
m
1
v 2 1 ~ 4169h 112h 2 !
8
r
2
~2.5c!
ṙ 2 ,
JN 5LN ,
JPN5LN
J2PN5LN
H
H
3
~2.5d!
J
1 2
m
,
v ~ 123 h ! 1 ~ 31 h !
2
r
~2.5e!
SD
1
m
m
m
1
1
~ 7210h 29 h 2 ! v 2 2 h ~ 215 h ! ṙ 2 1 ~ 14241h 14 h 2 !
2
r
2
r
4
r
2
J
3
1 ~ 127 h 113h 2 !v 4 ,
8
~2.5f!
and where LN [ m x3v.
Through 2PN order, the orbital energy and angular momentum per unit reduced mass, Ẽ[E/ m 5 21 v 2
2m/r1O( e 2 )1O( e 3 ), J̃5x3v@ 11O( e )1O( e 2 ) # , are constant, and correspond to asymptotically measured quantities.
However, the radiation reaction terms lead to nonvanishing expressions for dẼ/dt and dJ̃/dt containing the 20 undetermined
parameters. Following IW, starting from the 2PN-conserved expressions for Ẽ and J̃ we calculate dẼ/dt and dJ̃/dt using the
2PN two-body equations of motion @27,56,50# supplemented by the radiation-reaction terms of Eq. ~2.3!. In
6034
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
55
the balance approach, this time variation of the ‘‘conserved’’ quantities is equated to the negative of the flux of energy and
angular momentum carried by the gravitational waves to the far zone. Thus in addition to the EOM and conserved quantities
we need the 2PN accurate expressions for the far-zone fluxes of energy and angular momentum for a system of two particles
moving on general quasielliptic orbits. The waveform, energy, and angular momentum flux have been computed by Gopakumar and Iyer @47# using the BDI @48,45# formalism, and independently the waveform and energy flux by Will and Wiseman
@46# using their new improved version of the EW @49# formalism. We quote below the final results for the fluxes per unit
reduced mass:
S D
S D
dE
dt
dJ
dt
5ĖN 1Ė1PN1Ė1.5PN1Ė2PN ,
~2.6a!
far zone
5L̃N @ J̇N 1J̇1PN1J̇1.5PN1J̇2PN# ,
~2.6b!
far zone
where
S
D
8 m2 m
11
ĖN 5 h 3
4 v 2 2 ṙ 2 ,
5 r r
3
~2.7a!
F
m
8 m2 m 1
1
40
Ė1PN5 h 3
~ 7852852h !v 4 2 ~ 148721392h !v 2 ṙ 2 2 ~ 172 h !v 2
5 r r 84
42
21
r
1
S DG
4
m
m
2
1
~ 6872620h ! ṙ 4 1 ~ 367215h ! ṙ 2 1 ~ 124 h !
28
21
r 21
r
2
~2.7b!
,
F
8 m2 m 1
1
Ė2PN5 h 3
~ 169225497h 14430h 2 !v 6 2 ~ 1719210 278h 16292h 2 !v 4 ṙ 2
5 r r 126
42
2
1
m
m
1
1
~ 444625237h 11393h 2 !v 4 1 ~ 2018215 207h 17572h 2 !v 2 ṙ 4 1 ~ 498728513h 12165h 2 !v 2 ṙ 2
63
r 42
21
r
1
m
1
~ 281 473181 828h 14368h 2 !v 2
2268
r
SD
114 290h 2 ! ṙ 4
2
2
1
1
~ 2501220 234h 18404h 2 ! ṙ 6 2
~ 33 510260 971h
126
189
SD
1
m
m
2
~ 106 31919798h 15376h 2 ! ṙ 2
r 756
r
S
2
2
S DG
m
2
~ 25321026h 156h 2 !
189
r
3
,
D
8 m m
m
J̇N 5 h 2
2 v 2 23ṙ 2 12 ,
5 r r
r
F
~2.7c!
~2.7d!
1
8 m m 1
1
1
m
J̇1PN5 h 2
~ 3072548h !v 4 2 ~ 742277h !v 2 ṙ 2 2 ~ 58195h !v 2 1 ~ 952360h ! ṙ 4
5 r r 84
14
21
r 28
1
S DG
1
m
m
1
~ 3721197h ! ṙ 2 2 ~ 74522 h !
42
r 42
r
2
~2.7e!
,
F
8 m m 1
1
1
J̇2PN5 h 2
~ 2665212 355h 112 894h 2 !v 6 2
~ 2246212 653h 115 637h 2 !v 4 ṙ 2 1
~ 1652491h
5 r r 504
168
504
14022h 2 !v 4
1
1
m
m
1
1
~ 3575216 805h 115 680h 2 !v 2 ṙ 4 1
~ 21 853221 603h 12551h 2 !v 2 ṙ 2 2
~ 10 651
r 168
504
r 252
SD
SD
m
r
2
210 179h 13428h 2 !v 2
m
r
2
225 102h 14587h 2 ! ṙ 2
2
1
m
5
1
~ 392163h 197h 2 ! ṙ 6 2
~ 22 312241 398h 19695h 2 ! ṙ 4 1
~ 8436
18
504
r 252
1
m
1
~ 170 362170 461h 11386h 2 !
2268
r
S DG
3
.
~2.7f!
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
In the above expressions L̃N 5LN / m and the tail terms are
not listed. It is important to emphasize that the ‘‘tail’’ contribution to the reaction force is such that the balance equation for energy is verified for the tail luminosity @58,32#. This
corresponds to the ‘‘tail’’ acceleration at 4PN. With this part
independently accounted for, in our analysis we focus on the
‘‘instantaneous’’ terms without loss of generality. It is worth
recalling that the ‘‘balance’’ one sets up in the above treatment is always modulo total time derivatives of the variables
involved. This is crucial to realize and in IW this was systematically accounted for by noting that at orders of approximation beyond those at which they are strictly conserved
~and thus well defined!, Ẽ and J̃ are ambiguous up to such
terms. Consequently, we have the freedom to add to Ẽ and
J̃ arbitrary terms of order e 2.5, e 3.5, and e 4.5 beyond the Newtonian expressions without affecting their conservation at
2PN order. There are 3 such terms of the appropriate general
form at O( e 2.5) in each of Ẽ and J̃, respectively, 6 each at
O( e 3.5), and 10 each at O( e 4.5), resulting in 6 additional
Newtonian RR parameters, 12 additional 1PN RR parameters, and 20 additional 2PN RR parameters, respectively. As
discussed in detail in the following section, these numbers
are very much tied up with the ‘‘functional form’’ we assume for the ambiguous terms and in this section we follow
IW in close detail. Equating time derivatives of the resulting
generalized energy and angular momentum expressions Ẽ *
and J̃* ~rather than only the conserved expressions! to the
negative of the far-zone flux formulas and comparing them
term by term one seeks to determine the extent to which one
can deduce the 4.5PN reactive acceleration terms by the refined balance approach.
6035
m
1
A 2.553 ~ 11 a 3 !v 2 1 ~ 2316 b 2 29 a 3 ! 25 a 3 ṙ 2 ,
3
r
~2.8a!
m
B 2.55 ~ 21 b 2 !v 2 1 ~ 22 b 2 ! 23 ~ 11 b 2 ! ṙ 2 , ~2.8b!
r
SD
A 3.55 f 1 v 4 1 f 2 v 2
m
m
m 2
1 f 3 v 2 ṙ 2 1 f 4 ṙ 2 1 f 5 ṙ 4 1 f 6
,
r
r
r
~2.8c!
B 3.55g 1 v 4 1g 2 v 2
m
m
m 2
1g 3 v 2 ṙ 2 1g 4 ṙ 2 1g 5 ṙ 4 1g 6
,
r
r
r
~2.8d!
SD
where
f 15
f 2 52
1
3
~ 1171132h ! 2 a 3 ~ 123 h ! 13 j 2 23 r 5 ,
28
2
~2.9a!
1
3
~ 2972310h ! 23 b 2 ~ 124 h ! 2 a 3 ~ 7113h !
42
2
22 j 1 23 j 2 13 j 5 13 r 5 ,
f 35
5
5
~ 19272h ! 1 a 3 ~ 123 h ! 25 j 2 15 j 4 15 r 5 ,
28
2
~2.9c!
f 4 52
1
1
~ 6872368h ! 26 b 2 h 1 a 3 ~ 54117h !
28
2
22 j 2 25 j 4 26 j 5 ,
B. The 2PN RR computation and results
The above procedure is implemented order by order. All
the computations were done with MAPLE @63# and independently checked by MATHEMATICA @64#. At the leading order,
when the flux is given by the quadrupole equation, one deduces the ‘‘Newtonian RR’’ or 2.5PN term in the acceleration. In this case, in addition to the six unknowns in the
reactive acceleration, one has three unknowns each for the
possible 2.5PN ambiguities in the Ẽ * and J̃*. As demonstrated in IW, the balance equations yield 12 constraints on
these 12 Newtonian RR parameters. Of the 12 constraints,
only 10 are linearly independent, and thus finally one obtains
10 linear inhomogeneous equations for 12 Newtonian radiation reaction variables. Solving these equations one obtains
explicit forms for A 2.5 , B 2.5 and Ẽ 2.5 , J̃ 2.5 in terms of two
2.5PN arbitrary parameters. To get the 3.5PN reactive terms,
one adopts the above solution and extends the calculation to
O( e 3.5) after introducing Ẽ 3.5 and J̃ 3.5 with 12 additional
1PN RR parameters. At 3.5PN there are 20 constraints on
the 24 post-Newtonian radiation reaction parameters; of the
20 only 18 are linearly independent; the solution to this system yields explicit forms for A 3.5 , B 3.5 and Ẽ 3.5 , J̃ 3.5 in
terms of six 3.5PN arbitrary parameters. Since we need these
results for the present computation, we reproduce them from
IW @65#:
f 5 527 j 4 ,
f 6 52
3
g 1 523 ~ 123 h ! 2 b 2 ~ 123 h ! 2 j 1 ,
2
g 2 52
g 45
~2.9d!
~2.9e!
1
~ 15331498h ! 2 b 2 ~ 1419 h ! 13 a 3 ~ 714 h !
21
22 j 3 23 j 5 ,
g 35
~2.9b!
~2.9f!
~2.9g!
1
1
~ 1391768h ! 2 b 2 ~ 5117h ! 1 j 1 2 j 3 ,
84
2
~2.9h!
1
3
~ 3692624h ! 1 ~ 3 b 2 12 a 3 !~ 123 h ! 13 j 1 23 r 5 ,
28
2
~2.9i!
1
1
~ 2952335h ! 1 b 2 ~ 38211h ! 23 a 3 ~ 123 h ! 12 j 1
42
2
14 j 3 13 r 5 ,
g 55
5
~ 19272h ! 25 a 3 ~ 123 h ! 15 r 5 ,
28
~2.9j!
~2.9k!
6036
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
g 6 52
1
~ 634266h ! 1 b 2 ~ 713 h ! 1 j 3 .
21
~2.9l!
A 4.55h 1 v 6 1h 2 v 4 ṙ 2 1h 3 v 4
The quantities a 3 , b 2 , j 1 , j 2 , j 3 , j 4 , j 5 , and r 5 are parameters that represent the unconstrained degrees of freedom that
correspond to gauge transformations. In addition to the reactive terms listed above, one of the coefficients that determine
the 2.5PN ambiguity in Ẽ and J̃ and three of the coefficients
that determine the corresponding 3.5PN ambiguity are nonvanishing. We list these also since they are needed for setting
up the 4.5PN computation:
a 1 52 ~ 21 b 2 ! ,
j 6 52
1h 6 v 2 ṙ 2
1
~ 3072548h ! ,
84
~2.10c!
r 65 j 32
1
~ 2712214h ! .
42
~2.10d!
We now adopt the 2.5PN and 3.5PN solutions given by
Eqs. ~2.8!, ~2.9!, and ~2.10!. Following the IW strategy, we
assume the 4.5PN terms in the equations of motion to be of
the form
B 4.55k 1 v 6 1k 2 v 4 ṙ 2 1k 3 v 4
1k 6 v 2 ṙ 2
SD
2
J̃*[J̃N 1J̃PN1J̃2PN1J̃2.51J̃3.51J̃4.5
S D
SD
m
r
2
1 c 6 v 2 ṙ 2
SD
8 m
2 h
5
r
2
m
r
1k 10
SDF
2
ṙ j 1 v 4 1 j 2 v 2 ṙ 2 1 j 3 v 2
F
SD
3
.
2
ṙ c 1 v 6 1 c 2 v 4 ṙ 2 1 c 3 v 4
m
r
S D S DG
2
1 c 10
m
r
m
r
3
~2.12a!
,
S DG
S D S DG
m
m
m 1
m
m 1
m
8
8
5J̃N 1J̃PN1J̃2PN1 h L̃N ṙ b 2
2 h L̃N ṙ
~ 3072548h 184j 1 !v 2 1 r 5 ṙ 2 2 ~ 2712214h 242j 3 !
5
r
r
5
r 84
r
r 42
r
m
m
m
8
2 h L̃N ṙ x 1 v 6 1 x 2 v 4 ṙ 2 1 x 3 v 4 1 x 4 v 2 ṙ 4 1 x 5 v 2
5
r
r
r
,
2
SD SD
m
m
m
1 c 7 ṙ 6 1 c 8 ṙ 4 1 c 9 ṙ 2
r
r
r
F
3
We also assume for the ambiguity in Ẽ 4.5 and J̃ 4.5 the restrictions and functional forms adopted in IW and also require
that J̃ remain a pseudovector. The ‘‘generalized energy’’ and
‘‘angular momentum’’ through 4.5PN are thus given as sums
of the conserved parts, Eqs. ~2.5!, the ‘‘most general’’ 2.5PN
and 3.5PN contributions, i.e., with coefficients determined
by the Newtonian RR and 1PN RR calculations, and arbitrary 4.5PN terms. We use Ẽ * and J̃* to distinguish these
quantities from the conserved energy and angular momentum. We get ~per unit reduced mass!
S DG S D F
2
m
r
1h 10
m
m
1k 4 v 2 ṙ 4 1k 5 v 2
r
r
m
m
m
1k 7 ṙ 6 1k 8 ṙ 4 1k 9 ṙ 2
r
r
r
8 m
ṙ @~ 21 b 2 !v 2 2 a 3 ṙ 2 # 2 h
5
r
m 4
m
2 ~ 124 h !
r 21
r
1 c 4 v 2 ṙ 4 1 c 5 v 2
2
~2.11b!
Ẽ * [Ẽ N 1Ẽ PN1Ẽ 2PN1Ẽ 2.51Ẽ 3.51Ẽ 4.5
8 m
5Ẽ N 1Ẽ PN1Ẽ 2PN1 h
5
r
2
SD SD
m
m
m
1h 7 ṙ 6 1h 8 ṙ 4 1h 9 ṙ 2
r
r
r
~2.10b!
r 35 j 11
1 j 4 ṙ 4 1 j 5 ṙ 2
SD
m
m
1h 4 v 2 ṙ 4 1h 5 v 2
r
r
~2.11a!
~2.10a!
4
~ 124 h ! ,
21
55
2
1 x 6 v 2 ṙ 2
m
m
m
1 x 7 ṙ 6 1 x 8 ṙ 4 1 x 9 ṙ 2
r
r
r
2
1 x 10
m
r
2
3
,
~2.12b!
We now compute the 4.5PN terms in dẼ * /dt and dJ̃*/dt using the identities
1 dv2
[v•a,
2 dt
~2.13a!
d ~ x3v!
[x3a,
dt
~2.13b!
v 2 1r•a2ṙ 2
,
r
~2.13c!
r̈[
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
6037
where a is given by Eqs. ~2.1!, ~2.2!, ~2.3!, ~2.8!, ~2.9!, and ~2.11!. To compute E8 * and JP * to O( e 4.5), one needs to evaluate
(E8 N ,JP N ), (E8 1PN , JP 1PN), and (E8 2PN , JP 2PN) by using a to O( e 4.5), O( e 3.5) and O( e 2.5), respectively. On the other hand, for time
derivatives of the ‘‘ambiguity parts,’’ (Ẽ 4.5 , J̃4.5), (Ẽ 3.5 , J̃3.5), and (Ẽ 2.5 , J̃2.5), the relevant accelerations are the ‘‘conservative’’ accelerations to order Newtonian, post-Newtonian, and second post-Newtonian, respectively. Schematically, we get
S
8 m2 m
m
dẼ *
52 h 3
~ 12v 2 211ṙ 2 ! 1
dt
15 r
r
r
H F
1
m
~ 7852852h !v 4 12 ~ 2148711392h !v 2 ṙ 2 1160~ 2171 h ! v 2
28
r
S D GJ
D H F
SS
8
m m
m
m
dJ̃*
52 h L̃N 2
2 v 2 12 23ṙ 2 1
dt
5
r r
r
r
15
2
m
m
13 ~ 6872620h ! ṙ 18 ~ 367215h ! ṙ 2 116~ 124 h !
r
r
4
1
Y [4]
( R[4.5]
i
i
i51
D
~2.14a!
,
1
m
~ 3072548h !v 4 16 ~ 2741277h !v 2 ṙ 2 24 ~ 58195h ! v 2
84
r
S D GJ
15
2
m
m
13 ~ 952360h ! ṙ 4 12 ~ 3721197h ! ṙ 2 12 ~ 274512 h !
r
r
1
Y [4]
( S [4.5]
i
i
i51
D
~2.14b!
,
where
F SD SD SD SD
SD SDSD SD SD G
8 6
Y [4]
i ~ i51, . . . ,15 ! 5 v , v
3v2
2
m
m
, v 6 ṙ 2 , v 4
r
r
m 4 m
ṙ ,
r
r
4
,
m
r
, v 4 ṙ 4 , v 4
3
ṙ 2 ,
m
r
m 2 2 m
ṙ , v
r
r
2
ṙ 4 ,
3
, v 2 ṙ 6 , v 2
SD
m
r
2
ṙ 2 ,
m 6 8
ṙ ,ṙ ,
r
~2.15!
and S [4.5]
consist of combinations of the parameters h i and k i from A 4.5 and B 4.5 , c i , x i combined with functions
and R[4.5]
i
i
of h from Ẽ 4.5 and J̃4.5 , j 1 , j 2 , j 3 , j 4 , j 5 , r 5 combined with functions of h from 1PN corrections of 3.5PN terms and a 3 and
b 2 combined with functions of h from 2PN corrections of 2.5PN terms. We equate dẼ * /dt and dJ̃*/dt thus obtained to the
negative of the 2PN far-zone fluxes given by Eqs. ~2.7!. This results in 30 constraints on the 40 parameters h i , k i , c i , and
x i . Two of these constraints being redundant, of the 30 constraints only 28 are linearly independent. The system of 28 linear
inhomogeneous equations for 40 variables is therefore underdetermined to the extent of 12 arbitrary parameters, and we choose
these to be c 1 ••• c 9 , x 6 , x 8 , and x 9 . With this choice, the coefficients in Eq. ~2.11! determining the 4.5PN reactive acceleration are given by
h 1 52
h 25
h 35
1
3
3
~ 12122278h 14012h 2 ! 2 a 3 ~ 129 h 121h 2 ! 2 ~ j 2 2 r 5 !~ 123 h ! 13 c 2 23 x 6 ,
168
8
2
~2.16a!
5
5
5
~ 32921487h 11244h 2 ! 1 a 3 ~ 129 h 121h 2 ! 1 ~ j 2 2 j 4 2 r 5 !~ 123 h ! 25 c 2 15 c 4 15 x 6 25 x 8 , ~2.16b!
84
8
2
1
3
1
3
~ 7692287 429h 111 218h 2 ! 1 a 3 ~ 1297h 125h 2 ! 1 b 2 ~ 323 h 219h 2 ! 13 j 1 ~ 124 h ! 2 ~ j 2 2 r 5 !~ 7113h !
504
8
4
2
3
2 j 5 ~ 123 h ! 22 c 1 23 c 2 13 c 6 13 x 6 23 x 9 ,
2
h 4 52
h 5 52
5
7
~ 392163h 197h 2 ! 1 j 4 ~ 123 h ! 27 c 4 17 c 7 17 x 8 ,
18
2
~2.16c!
~2.16d!
1
1
~ 37 089264 005h 111 297h 2 ! 19 a 3 ~ 2113h 12 h 2 ! 1 b 2 ~ 482121h 254h 2 ! 1 j 1 ~ 1419 h !
252
4
3
13 ~ j 2 2 r 5 !~ 714 h ! 13 j 3 ~ 124 h ! 2 j 5 ~ 7113h ! 22 c 3 23 c 6 13 c 9 13 x 9 ,
2
~2.16e!
6038
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
h 6 52
1
1
3
1
~ 45 4752219 535h 143 121h 2 ! 2 a 3 ~ 142403h 177h 2 ! 2 b 2 h ~ 7213h ! 16 h j 1 1 j 2 ~ 6829 h !
504
4
2
2
5
1
2 j 4 ~ 7113h ! 13 j 5 ~ 123 h ! 2 r 5 ~ 62119h ! 24 c 2 25 c 4 26 c 6 15 c 8 12 x 6 15 x 8 16 x 9 ,
2
2
h 7 529 c 7 ,
h 85
h 95
1
1
1
~ 181 3712342 479h 142 598h 2 ! 2 a 3 ~ 1171109h 16 h 2 ! 2 b 2 ~ 281245h 120h 2 ! 12 h j 1
756
2
4
~2.16j!
3
3
k 1 5 ~ b 2 12 !~ 12 h 211h 2 ! 1 j 1 ~ 123 h ! 2 c 1 ,
8
2
~2.16k!
1
3
9
3
~ 49922656h 2146h 2 ! 2 a 3 ~ 123 h 23 h 2 ! 2 b 2 ~ 12 h 211h 2 ! 2 ~ 3 j 1 22 j 2 2 r 5 !~ 123 h ! 13 c 1 23 x 6 ,
168
2
8
2
~2.16l!
k 35
1
1
1
3
~ 8129127h 214 482h 2 ! 2 b 2 ~ 31121h 17 h 2 ! 1 j 1 ~ 5117h ! 1 j 3 ~ 123 h ! 1 c 1 2 c 3 ,
504
8
2
2
~2.16m!
5
5
5
~ 32921487h 11244h 2 ! 1 a 3 ~ 123 h 23 h 2 ! 2 ~ 2 j 2 22 j 4 1 r 5 !~ 123 h ! 15 x 6 25 x 8 ,
84
2
2
~2.16n!
11
1
1
~ 11072805h 2508h 2 ! 1 b 2 ~ 161255h 122h 2 ! 2 j 1 ~ 713 h ! 1 j 3 ~ 5117h ! 1 c 3 2 c 5 ,
252
4
2
~2.16o!
k 45
k 5 52
1
3
1
1
~ 1797154 816h 222 463h 2 ! 1 a 3 ~ 113 h 15 h 2 ! 2 b 2 ~ 422485h 1173h 2 ! 2 j 1 ~ 56249h ! 23 ~ j 2 12 j 3 2 j 5 !
504
2
4
2
3
3~ 123 h ! 1 r 5 ~ 7111h ! 14 c 1 14 c 3 13 x 6 23 x 9 ,
2
k 7 52
k 8 52
5
~ 392163h 197h 2 ! 27 j 4 ~ 123 h ! 17 x 8 ,
18
~2.16p!
~2.16q!
1
1
3
~ 39 808292 788h 124 563h 2 ! 1 a 3 ~ 142105h 159h 2 ! 2 b 2 h ~ 69113h ! 23 h j 1 2 ~ 2 j 2 15 j 4 16 j 5 !~ 123 h !
504
2
8
1
2 r 5 ~ 6213 h ! 12 x 6 15 x 8 16 x 9 ,
2
k 95
~2.16i!
1
3
1
~ 265 2651262 230h 115 072h 2 ! 2 a 3 ~ 1021177h 116h 2 ! 1 b 2 ~ 2001325h 140h 2 ! 1 j 3 ~ 1419 h !
756
4
4
13 j 5 ~ 714 h ! 22 c 5 23 c 9 ,
k 65
~2.16g!
~2.16h!
1
1 ~ 2 j 2 15 j 4 !~ 714 h ! 17 h j 3 1 j 5 ~ 60121h ! 13 h r 5 22 c 6 25 c 8 27 c 9 ,
2
k 2 52
~2.16f!
1
1
33
1
~ 5002236 589h 14496h 2 ! 2 a 3 h ~ 233263h ! 1 b 2 h ~ 123 h ! 13 h j 2 1 j 4 ~ 82123h ! 15 h r 5
252
8
4
2
22 c 4 27 c 7 28 c 8 ,
h 105
55
~2.16r!
1
1
1
~ 831927683h 111 809h 2 ! 13 a 3 ~ 3213h 2 h 2 ! 2 b 2 ~ 1941215h 124h 2 ! 2 ~ 2 j 1 13 r 5 !~ 713 h ! 2 j 3 ~ 4429 h !
252
4
2
23 j 5 ~ 123 h ! 12 c 3 15 c 5 13 x 9 ,
~2.16s!
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
k 105
1
1
~ 425 4131111 636h 26912h 2 ! 2 b 2 ~ 531103h 14 h 2 ! 2 j 3 ~ 713 h ! 1 c 5 .
2268
2
At the 4.5PN order, four parameters determining Ẽ 4.5 and
J̃4.5 are nonvanishing and are given by
1
1
2 ~ 123 h ! j 3 2 ~ 3252155h 2595h 2 ! ,
2
42
7
1
x 55 c 31 b 2h 2
~ 52424483h 13675h 2 ! ,
2
126
1
x 6 → x̄ 6 5 x 6 2 ~ 123 h ! r 5 ,
2
~2.17!
A final minor remark is with regard to the two possible ways
one may implement the requirement that the ambiguity in
J̃* be a pseudovector. One may either choose it proportional
to L̃N as in the treatment above or to the conserved angular
momentum J̃. At 2.5PN order both choices are identical. At
the 3.5PN order, the two choices lead to an identical system
of linear equations barring a translation in the values of r 3
and r 6 by an amount given by the coefficients of v 2 and
m/r in J1PN :
1
r 3 → r̄ 3 5 r 3 1 ~ 123 h ! b 2 ,
2
r 6 → r̄ 6 5 r 6 1 ~ 31 h ! b 2 .
1
1
~ 3072548h ! 1 ~ 123 h ! b 2 ,
84
2
r̄ 6 5 j 3 2
1
~ 2712214h ! 1 ~ 31 h ! b 2 .
42
1
x 9 → x̄ 9 5 x 9 2 ~ 215 h ! h b 2 2 ~ 31 h ! r 5 ,
2
1
x 10→ x̄ 105 x 102 ~ 22165h ! b 2 2 ~ 31 h ! j 3
4
1
1
~ 569122597h 21498h 2 ! .
294
~2.20!
Consequently, in terms of the above ‘‘shifted’’ variables, the
solutions for the reactive accelerations are identical. As x 6
and x 9 are among the independent parameters that determine
the reactive acceleration, in terms of x 6 and x 9 the two
choices yield equivalent but different looking solutions for
the 4.5PN reactive terms in the equations of motion.
Of the two choices, the second choice is more convenient
for calculations by hand since dJ/dt50 to O( e 2 ), but has no
special advantage when the calculation is done on a computer.
~2.18!
Since r 3 and r 6 are not among the arbitrary parameters determining the solution, the solution determining the reactive
terms and j 6 is unchanged. Only the expressions for r 3 and
r 6 are changed to
r̄ 3 5 j 1 1
1
~ 30721469h 11644h 2 ! ,
168
1
x 5 → x̄ 5 5 x 5 1 ~ 116 h 23 h 2 ! b 2 2 ~ 31 h ! j 1
2
1
~ 2665212 355h 112 894h 2 ! ,
504
7
1
x 105 c 5 2 b 2 h 1
~ 77523939h 12942h 2 ! .
2
252
~2.16t!
1
1
x 3 → x̄ 3 5 x 3 1 ~ 129 h 121h 2 ! b 2 2 ~ 123 h ! j 1
8
2
2
1
c 105
~ 36221548h 1400h 2 ! ,
189
x 35 c 11
6039
~2.19!
At 4.5PN order, however, the situation is different. Indeed,
as before, the two choices lead to an identical system of
linear equations barring a translation in the values of the five
parameters x 3 , x 5 , x 6 , x 9 , and x 10 :
III. REDUNDANT EQUATIONS
AND RELATED VARIANT SCHEMES
It was noticed in IW that both at the 2.5PN and at the
3.5PN order, the ‘‘balance procedure’’ leads to two redundant constraint equations @29#. Here, at 4.5PN order, we once
again obtain two redundant constraint equations. In this section, we examine critically the origin of these redundant
equations.
In implementing the ‘‘refined balance procedure’’ for the
general orbits, IW @29# balance the ‘‘energy flux’’ and ‘‘angular momentum flux’’ completely independently of each
other. However, for circular orbits, these fluxes are not independent but related @66# via
S D
dE
dt
5 v 2 J̇,
far zone
where J̇ is defined by the equation
6040
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
S D
dJ
dt
5LN J̇.
far zone
The general balance should reflect this limit and we find that
for Newtonian RR a linear combination of the six equations
representing energy balance and another linear combination
of the six equations representing angular momentum balance
are indeed identical and given by
55
e 1 1e 2 2450 .
Similarly at 3.5PN we have
g 1 1g 2 1g 6 2 ~ 32 h ! b 2 1
1
~ 29272252h ! 50 ,
84
~3.2!
and finally at 4.5PN order the ‘‘degenerate’’ equation is
1
1
k 1 1k 3 1k 5 1k 101 ~ 32 h !~ j 1 1 j 3 ! 1 ~ 90113h 16 h 2 ! b 2 2
~ 635 7711297 117h 281 000h 2 ! 50.
4
4536
Thus we can trace the existence of one of the redundant
equations in the IW procedure to the fact that for circular
orbits the energy and angular momentum fluxes are not independent but proportional to each other.
The mystery of the other redundant equation was not so
easy to resolve but after a careful examination of the system
of equations and ‘‘experiments’’ in modifying the system,
we could finally track it back to its source. The observation
that this redundant equation relates the coefficients of the
polynomial representing the ambiguity in J̃ led us to examine
the functional form that IW proposed as the starting ansatz
for the calculation. A comparison of the functional forms for
the ambiguity in Ẽ and J̃, Eqs. ~2.12! reveal that indeed IW
assume a more general possibility for J̃ than required. The
ambiguity in angular momentum leads to terms more general
than required by the far-zone flux formula and time derivative of the leading term using the reactive acceleration. The
absence of such terms in the far-zone flux then yields only
the trivial solution for these additional variables in J̃, and the
second redundant equation is just a homogeneous linear
combination of these trivial solutions. Thus the second redundant equation in the IW scheme is due to the fact that the
IW scheme—extended here to 4.5PN order—is not a ‘‘minimal’’ one.
To verify this ‘‘conjecture’’ we experimented with alternatives for the functional form that one assumes as the starting expression for the ambiguity in Ẽ and J̃ — the 2.5PN,
3.5PN, and 4.5PN order terms. In the first instance, we replace the IW scheme—labeled for clarity of reference by
IW21—by the ‘‘minimal’’ variant in Eq. ~2.12!—labeled by
IW22. The notation IW21 indicates, e.g., that (m/r) 2 is
pulled out in Ẽ while only (m/r) 1 is pulled out in J̃. As
explained above, the minimal choice for J̃* is obtained by
pulling out the factor (8/5) h L̃N (m/r) 2 ṙ from arbitrary terms
in J̃*, rather than the factor (8/5) h L̃N (m/r)ṙ as in the IW
scheme for J̃*. This reduces by one the order of the polynomial in v 2 , ṙ 2 , and m/r that constitutes the arbitrariness, and
consequently implies a reduction in the number of variables
that characterize the ambiguity in J̃ to one for J̃2.5 , three in
J̃3.5 and six in J̃4.5 . Thus in the IW22 scheme, at the 2.5PN
level we have six variables in the reactive acceleration, three
variables determining the energy ambiguity Ẽ 2.5 and 1 vari-
~3.1!
~3.3!
able determining the ambiguity in J̃2.5, i.e., 10 variables in
all. The balance equations lead to nine equations—six from
energy and three from angular momentum—of which eight
are linearly independent. In other words, there is only one
redundant equation. The linear system of 8 equations for 10
variables is then the same as before and leads to the IW21
solution in terms of 2 arbitrary parameters. ~The two extra
variables in IW21 are identically zero.! Similarly, at the
3.5PN level we have 12 variables in the reactive acceleration, 6 variables determining the energy ambiguity Ẽ 3.5 and 3
variables determining the ambiguity in J̃3.5 , i.e., 21 variables
in all. The balance equations lead to 16 equations — 10 from
energy and 6 from angular momentum — of which 15 are
linearly independent, leaving only one redundant equation.
The linear system of 15 equations for 21 variables is then the
same as before and leads to the IW21 solution in terms of 6
arbitrary parameters. ~The three extra variables in IW21 are
identically zero.! Finally, at the 4.5PN level, we have 20
variables in the reactive acceleration, 10 variables determining the energy ambiguity Ẽ 4.5 and 6 variables determining
the ambiguity in J̃4.5 , i.e., 36 variables in all. The balance
equations lead to 25 — 15 from energy and 10 from angular
momentum — equations of which 24 are linearly independent, again leaving only one redundant equation. The linear
system of 24 equations for 36 variables is the same as before
and leads to the solution obtained in the previous section in
terms of 12 arbitrary parameters. ~The four extra variables in
the IW21 scheme are identically zero.! The IW22 ~minimal!
scheme thus confirms the conjecture that the occurrence of
the second redundant equation is special to the IW scheme
~IW21! and is related to the choice they make for the functional form of the J̃ ambiguity by pulling out only one factor
of nonlinearity m/r rather than its square — the minimal
choice. To double check the above explanation, we performed another experiment by examining a variant that
would generate an increased number of redundant or degenerate equations. This scheme denoted by IW11 differs from
IW21 in that the ambiguity in Ẽ* is assumed to have
(8/5) h (m/r)ṙ as the common factor, i.e., by pulling out only
one order of nonlinearity m/r rather than its square as in
IW21; the polynomial representing the ambiguity in Ẽ is
consequently of one order more than in IW21. In this case, at
2.5PN order one has 61613515 variables and
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
TABLE I. Comparison of four alternative schemes: IW21, IW22
~minimal!, IW11, and IW00. N denotes the order of approximation,
NV the number of variables, NC the number of constraints coming
from balance equations, ND the number of degenerate equations,
NI the number of independent equations, and NA the number of
arbitrary parameters determining the solution. In the NV column,
a1b1c means a variables of reactive acceleration, b in energy
ambiguity, and c in angular momentum ambiguity.
N
NV
NC
ND
NI
NA
10
18
28
2
6
12
1
1
1
8
15
24
2
6
12
3
3
3
13
22
33
2
6
12
IW21: IW scheme
2.5PN
3.5PN
4.5PN
61313
121616
20110110
12
20
30
2
2
2
IW22: Minimal scheme
2.5PN
3.5PN
4.5PN
61311
121613
2011016
9
16
25
IW11 scheme
2.5PN
3.5PN
4.5PN
61613
1211016
20115110
16
25
36
IW00 scheme
2.5PN
611016
25
5
20
2
1016516 equations of which 3 are redundant. The 13
equations for 15 variables thus yield the required solution in
terms of 2 arbitrary parameters and similarly for higher orders. One may also explore the most general of choices in
which only (8/5) h is pulled outside and the ambiguity is the
highest order polynomial consistent with the order of the
approximation. We studied one such scheme ~IW00! in the
Newtonian RR case. For convenience, the various experiments are summarized in Table I.
To conclude: at 2.5PN, 3.5PN, and 4.5PN orders all variants of IW examined in this subsection with different forms
of the ambiguities in Ẽ and J̃—minimal ~IW22! or IW11—
lead to identical reactive accelerations including their gauge
arbitrariness.
At this juncture one may wonder about the issues of the
‘‘uniqueness’’ and ‘‘ambiguities’’ of the schemes discussed
earlier. In this regard, we would like to make the following
general remarks. For general orbits, in addition to the balance of energy one must take into account the balance of
angular momentum. Thus, schemes involving only energy
balance are not relevant except in special cases like ‘‘circular
orbits’’ and ‘‘radial infall’’ ~see Sec. V!. Can one have
schemes where one implements both energy and angular momentum balance but does not take into account the possible
ambiguities in Ẽ and J̃? One can show that even at the 2.5PN
level this system of equations is inconsistent. Further, is the
ambiguity necessary both in Ẽ and J̃? If one examines a
scheme with both energy and angular momentum balance
taking account of the ambiguity only in Ẽ one does obtain a
consistent solution up to 4.5PN order but with only half the
6041
number of arbitrary parameters as in the IW scheme. The
reduced ‘‘gauge’’ freedom is not adequate to treat as special
cases the Burke-Thorne gauge at the 2.5PN level or the
Blanchet choice at the 3.5PN level. And finally, in a scheme
with both energy and angular momentum balance taking account of the ambiguity only in J̃ one obtains a consistent
solution at 2.5PN order containing no arbitrary parameters at
all. No solution is possible at higher orders.
On general considerations, the reactive acceleration
should be a power series in the individual masses m 1 and
m 2 or equivalently, it should be nonlinear in the total mass
m as assumed in earlier sections. It is interesting to investigate whether the functional forms of the far-zone fluxes and
the balance procedure necessarily lead to such ‘‘physical’’
solutions alone or whether they are consistent with more
general possibilities. In the Appendix, for mathematical completeness @67# we investigate this question in detail and
prove that the flux formulas and balance equations do not
constrain the reactive acceleration to their ‘‘physical’’ forms
alone but allow for a more general form for the reactive
acceleration.
IV. ARBITRARINESS IN REACTIVE TERMS
AND GAUGE CHOICE
It is well known that the formulas for the energy and
angular momentum fluxes in the far zone are gauge invariant, i.e., independent of the changes in the coordinate system
that leave the spacetime asymptotically flat. On the other
hand, the expressions for the reactive force are ‘‘gauge dependent’’ and consequently, e.g., the Chandrasekhar form is
different from the Burke-Thorne or Damour-Deruelle forms.
In IW it was shown that the Burke-Thorne gauge corresponds to the values b 2 54 and a 3 55, while the DamourDeruelle choice corresponds to b 2 521 and a 3 50. It was
further shown that the reactive acceleration implied by
Blanchet’s first principles determination of the 1PN radiation
reaction indeed corresponds to a particular choice of the arbitrary parameters in the IW solution. One of the satisfactory
aspects of IW was the demonstration that the part of the
reactive acceleration not determined by the balance requirement was precisely related to the possible ambiguity in the
choice of the gauge at that order. ~The flux is equal to the
time variation of the conserved quantities only up to total
time derivatives; this ambiguity may be absorbed in a
‘‘change’’ in the relative separation vector as discussed below.!
Following IW, we seek to establish the correspondence
between the arbitrary parameters contained in the radiation
reaction terms and the residual gauge freedom in the construction. The residual gauge freedom arises from the fact
that the far-zone fluxes, Eqs. ~2.6! and ~2.7!, are independent
of changes in the coordinate system that leave the spacetime
asymptotically flat. These coordinate changes will induce a
change in x which is the difference between the centers of
mass of the two bodies x1 (t) and x2 (t) at coordinate time
t. Following IW, we choose the transformation to be of the
form x→x8 5x1 d x, where d x can depend only on the two
vectors x and v,
d x5 ~ f 2.51 f 3.51 f 4.5! ṙx1 ~ g 2.51g 3.51g 4.5! rv. ~4.1!
6042
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
In order that d x/x be O( e 2.5), O( e 3.5) and O( e 4.5), f 2.5 and
g 2.5 must be O( e 2 ), f 3.5 and g 3.5 must be O( e 3 ) and f 4.5 and
g 4.5 must be O( e 4 ). As for the other variables, the f ’s and
g’s will also be polynomials in the variables m/r, v 2 , and
ṙ 2 . As pointed out in @29#, we do not independently take into
account changes in the coordinate time t since the
v-dependent term in d x includes this contribution via
x(t1 d t);x(t)1vd t.
In @29# it was proved that to cancel the dependence on the
two 2.5PN arbitrary parameters and the six 3.5PN arbitrary
parameters, d x should be chosen such that
f 2.55
SD
m
8
h
15
r
g 2.55
S DF
S DF
8 m
f 3.55 h
5
r
2
8 m
g 3.55 h
5
r
2
SD
m
8
h
15
r
2
P 2252
F
S DF
SD
S DF
SD
8 m
f 4.55 h
5
r
1 P 45
~4.2a!
8 m
g 4.55 h
5
r
~ 2 a 3 23 b 2 ! ,
~4.2b!
1Q 45
2
P 21v 2 1 P 22
SD
SD
Q 21v 2 1Q 22
G
G
m
1 P 23ṙ 2 ,
r
~4.2c!
m
1Q 23ṙ 2 ,
r
~4.2d!
F
2
~4.3f!
P 41v 4 1 P 42v 2
G
SD
m
m
1 P 43v 2 ṙ 2 1 P 44
r
r
m 2
ṙ 1 P 46ṙ 4 ,
r
2
~4.4a!
SD
2
m
m
1Q 43v 2 ṙ 2 1Q 44
r
r
Q 41v 4 1Q 42v 2
G
m 2
ṙ 1Q 46ṙ 4 .
r
2
~4.4b!
The change in the 2PN equations of motion Eqs. ~2.2! produced by this change of variable Eq. ~4.1! can be determined
using the known form of d x up to 3.5PN order Eqs. ~4.2! and
~4.3!, the provisional form chosen above for the 4.5PN terms
Eq. ~4.4! and the transformations given below:
G
~4.3a!
v→v8 5v1 d v5
G
F
1
3
3
1
j 2 1 j 4 2 j 5 2 r 5 2 b 2 h 1 a 3 ~ 4111h ! ,
6
2
2
2
~4.3b!
1
P 235 j 4 ,
5
~4.3c!
F
G
2
8
1
Q 215 j 1 1 j 2 1 j 4 1 ~ 3 b 2 22 a 3 !~ 123 h ! ,
3
15
2
~4.3d!
1
3
63
Q 2252 6 j 1 15 j 2 23 j 3 15 j 4 2 j 5 1 r 5 2 b 2 h
6
2
2
G
1
2 a 3 ~ 4255h ! ,
2
P 4152
G
x→x8 5x1 d x,
1
2
1
j 1 j 2 r 5 2 a 3 ~ 123 h ! ,
3 2 5 4
2
F
F
1 2
j 1 r 5 2 a 3 ~ 123 h ! .
3 5 4
We provisionally choose the 4.5PN part of d x to be of the
form
a3 ,
where P ab ’s and Q ab ’s are given by
P 215
Q 235
55
~4.3e!
dx d d x
1
,
dt
dt
r→r 8 5r 11
G
n• d x
,
r
x8
x d x pn
p 5 p 1 p 2 p ~ n• d x ! ,
r8
r
r
r
F
v 2 → v 8 2 5 v 2 1 2v•
ṙ→ṙ 8 5
F
G
ddx
,
dt
G
1
ddx
2 ~ n• d x! ṙ .
rṙ1 d x•v1x•
r
dt
~4.5!
The gauge change generates reactive terms and the requirement that this change should cancel the dependence of the
radiation-reaction terms on arbitrary parameters dictates that
1
1
1
2
8
1
2
a 3 ~ 129 h 121h 2 ! 2 ~ 5 j 2 12 j 4 25 r 5 !~ 123 h ! 1 c 2 1 c 4 1
c 72 x 62 x 8,
24
30
3
15
105
3
15
~4.6a!
1
3
1
1
1
1
1
1
4
P 4252 a 3 ~ 31 h 2 ! 1 b 2 h 2 j 1 h 1 j 2 ~ 3223h ! 1 j 4 ~ 19277h ! 2 j 5 ~ 123 h ! 2 r 5 ~ 3222h ! 2 c 2 2 c 4
6
8
4
12
60
8
12
3
15
1
1
1
1
4
1
1 c 62 c 71 c 81 x 61 x 82 x 9 ,
4
5
12
3
15
4
P 4352
1
1
4
1
j ~ 123 h ! 1 c 4 1 c 7 2 x 8 ,
10 4
5
35
5
~4.6b!
~4.6c!
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
P 445
1
1
1
1
1
a 3 ~ 13112h 116h 2 ! 1 b 2 ~ 1112h 22 h 2 ! 1 ~ j 1 22 j 3 ! h 1 ~ j 2 1 j 4 !~ 9131h ! 2 j 5 ~ 7113h !
30
5
10
30
20
2
1
2
2
1
2
1
1
2
2
1
r ~ 9128h ! 1 c 2 1 c 4 2 c 6 1 c 7 2 c 8 1 c 9 2 x 6 2 x 8 1 x 9 ,
30 5
15
15
10
15
10
5
15
15
10
P 455
Q 4252
1
1
1
1
1
1
2
1
1
a h 2 2 b 2 h ~ 123 h ! 1 j 2 h 2 j 4 ~ 117 h ! 2 r 5 h 2 c 4 2 c 7 1 c 8 1 x 8 ,
12 3
4
6
15
3
15
15
6
15
~4.6d!
~4.6e!
1
P 465 c 7 ,
7
~4.6f!
1
1
2
8
16
Q 415 ~ 2 a 3 23 b 2 !~ 12 h 211h 2 ! 2 ~ 15j 1 110j 2 18 j 4 !~ 123 h ! 1 c 1 1 c 2 1 c 4 1 c 7 ,
8
10
3
15
35
~4.6g!
1
1
1
1
1
a 3 ~ 1082331h 1197h 2 ! 1 b 2 ~ 482121h 163h 2 ! 1 j 1 ~ 9228h ! 1 j 2 ~ 492142h ! 2 ~ 6 j 3 13 j 5 22 r 5 !
24
8
2
12
8
1
5
1
47
1
22
1
1
1
j ~ 2312653h ! 22 c 1 2 c 2 1 c 3 2 c 4 1 c 6 2 c 7 1 c 8 2 x 6 2 x 8 ,
60 4
3
2
30
4
15
6
6
10
~4.6h!
1
1
2
16
1
2
Q 435 a 3 ~ 123 h 23 h 2 ! 2 ~ 2 j 2 12 j 4 1 r 5 !~ 123 h ! 1 c 4 1
c 71 x 61 x 8 ,
6
6
15
105
3
15
~4.6i!
3 ~ 123 h ! 1
Q 445
6043
1
1
1
1
1
a 3 ~ 32173h 1254h 2 ! 2 b 2 ~ 511157h 1258h 2 ! 2 j 1 ~ 102307h ! 2 j 2 ~ 192279h ! 2 j 3 ~ 15159h !
30
30
30
30
30
2
1
1
1
4
6
1
6
1
7
6
7
j 4 ~ 192279h ! 2 j 5 ~ 9191h ! 1 r 5 ~ 9128h ! 1 c 1 1 c 2 2 c 3 1 c 4 1 c 5 2 c 6 1 c 7 2 c 8
30
60
30
3
5
3
5
3
30
5
30
1
2
2
2
1
c 1 x 1 x 2 x ,
15 9 15 6 15 8 10 9
Q 4552
1
~4.6j!
1
33
3
1
1
1
3
1
1
a ~ 24229h 291h 2 ! 2 b 2 h 2 1 j 1 h 1 j 2 ~ 21 h ! 1 j 4 ~ 6213h ! 2 j 5 ~ 123 h ! 2 r 5 h 2 c 4 2 c 7
24 3
8
4
12
15
4
4
10
5
1
1
7
1
c 82 x 62 x 81 x 9 ,
12
6
30
4
~4.6k!
1
2
1
Q 4652 j 4 ~ 123 h ! 1 c 7 1 x 8 .
5
35
5
The above computation shows that as at the 3.5PN order the
~12-parameter! arbitrariness in the 4.5PN radiation reaction
formulas reflects the residual freedom that is available to one
in the choice of a 4.5PN accurate ‘‘gauge.’’ Every particular
4.5PN accurate radiation reaction formula should correspond
to a particular choice of these 12 parameters.
V. PARTICULAR CASES: QUASICIRCULAR ORBITS
AND HEAD-ON INFALL
In this section we specialize our solutions valid for general orbits to the particular case of quasicircular orbits and
radial infall and verify that they indeed reproduce the simpler
reactive solutions one would obtain if one formulated the
problem ab initio appropriate to these two special cases. We
~4.6l!
first consider the quasicircular limit that is of immediate relevance to sources for the ground based interferometric gravitational wave detectors. In this particular case, the reactive
acceleration may be deduced using only the energy balance.
Using the reactive acceleration we compute the 4.5PN contribution to ṙ and v̇ . We also discuss the complementary
case of the radial infall of two compact objects of arbitrary
mass ratio and determine the 4.5PN contribution to the radial
infall velocity for the two special cases: radial infall from
infinity and radial infall with finite initial separation.
A. Quasicircular inspiral
Using our general reactive solution we can compute the
physically relevant quantities ṙ and v̇ for quasicircular in-
6044
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
spiral, where r and v are the orbital separation and the orbital angular frequency in harmonic coordinates, respectively. As would be expected, these results are independent
of the arbitrary parameters that are present in the reactive
solution. We obtain the radiation reaction contribution to a
up to 4.5PN for quasicircular inspiral by setting
ṙ501O( e 2.5) and using
v 25
S
F
m
m
41h
1h2
12 ~ 32 h ! 1 61
r
r
4
DS D G
m
r
2
~5.1!
in Eqs. ~2.3!, ~2.8!, ~2.11!, and ~2.16!. We get
aRR 52
1
F S
D
3431 5 m
32h m 3 v
2
12
5r 4
336 4 r
S
7
794 369 26 095
1
h2 h2
18 144
2016
4
DS D G
m
r
~5.2!
F
G
m
1
5 v 2 11 ~ 32 h !v 2 1 ~ 48289h 14 h 2 !v 4 . ~5.3!
r
4
Differentiating Eq. ~5.3! with respect to t and noting that the
a that appears is the total acceleration ~conservative 1 reactive! we get, after some rearrangement
S DF S
1
S
3
D
DS D G
1751 7 h m
1
12
336
4 r
303 455 40 981h h
1
1
18 144
2016
2
2
m
r
2
~5.4!
.
Using Eq. ~5.4! and the expression for angular velocity
( v [ v /r)
v 25
S
F
m
41h
m
1h2
12 ~ 32 h ! 1 61
r3
r
4
DS D G
H
ż 2
3 ~ 123 h ! ż 4 ~ 312 h ! g ż 2 g 2
2g1
1
1
2
8
2
2
1
5 ~ 127 h 113h 2 ! ż 6 3 ~ 728 h 216h 2 ! g ż 4
1
16
8
1
~ 917 h 18 h 2 ! g 2 ż 2 ~ 2115h ! g 3
2
,
4
4
J
where g 5m/z. Unlike the quasicircular inspiral, for head-on
infall we can distinguish between two different cases. Following @55# we denote them by (A) and (B), respectively,
and list the expressions relevant for our computations. In
case (A), the radial infall proceeds from rest at infinite initial
separation, E(z)5E(`)50, and inverting Eq. ~5.7! we get
ż52
H F S D S
2m
h
81h
15 h 2
125 g 12 1 g 2 132
z
2
4
F
S
D
D
D GJ
1/2
.
~5.8!
In case (B), the radial infall proceeds from rest at finite
initial separation z 0 , which implies
H
E ~ z ! 5E ~ z 0 ! 52 m g 0 2
S
g 20 g 30
15h
1
11
2
2
2
, ~5.5!
H
F S D S
D S
D GJ
h
ż52 2 ~ g 2 g 0 ! 125 g 12
v̇ 96
743 11
1 h
h ~ m v ! 5/3 12 ~ m v ! 2/3
25
v
5
336 4
~5.7!
DJ
. ~5.9!
We obtain as in case (A), an expression for ż given by
2
m
r
we may express v̇ as
S
Recently Simone, Poisson, and Will @55# have obtained to
2PN accuracy the gravitational wave energy flux produced
during head-on infall and starting from these formulas one
can deduce ab initio the reactive acceleration in this limit
adapting IW to the radial infall case. As required, these results match exactly with expressions obtained by applying
radial infall limits to the general orbit solutions and we summarize the relevant formulas in this limit in what follows.
Equations representing the head-on infall can be obtained
from the general orbit expressions by imposing the restrictions, x5zn̂, v5żn̂, r5z, and v 5ṙ5ż. For radial infall the
conserved energy Eq. ~2.5! to 2PN order then becomes
2
It is worth noting that for quasicircular inspiral the energy
flux determines the reactive acceleration without any gauge
ambiguity. All the arbitrary terms in energy are proportional
to ṙ and hence play no role in this instance. Inverting Eq.
~5.1!, we get
64 m
ṙ52 h
5
r
B. Head-on infall
E~ z !5m
.
55
S
S
1 g 2 132
G
59
34 103 13 661
1
1
h 1 h 2 ~ m v ! 4/3 . ~5.6!
18 144
2016
18
The results Eqs. ~5.4! and ~5.6! are in agreement with @43#
as expected and required, suggesting that the reactive terms
obtained here could be used to evolve orbits in the more
general case also @68# .
1 g 20 12
2
1 g 0 12
9h
2
D
81h
173h
15 h 2 2 gg 0 52
113h 2
4
4
5h
18 h 2
4
D
1/2
,
~5.10!
where g 0 5m/z 0 . We first compute the 4.5PN contribution to
z̈ for case (B), the radial infall from finite initial separation.
We use the radial infall restriction along with Eq. ~5.10! in
Eqs. ~2.3!, ~2.8!, ~2.11!, and ~2.16! to obtain 4.5PN terms in
z̈ as
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
z̈5
H
1
8hg3
~ 2 g 22 g 0 ! 1/2 ~ 241121z 1 ! g 1 ~ 824 z 1 ! g 0 1
5m
3
S
FS
1 2
1
FS
6045
D
1
1
~ 551028849h ! 1 ~ 4022643h ! z 1 226z 2 26 z 3 gg 0 1 ~ 362126h 2 ~ 18263h ! z 1 18 z 2 ! g 20
28
4
2
G
1
1
1
~ 30 549 820254 233 376h 115 776 427h 2 ! 1 ~ 27 156249 816h 115 057h 2 ! z 1 2 ~ 7662527h ! z 2
18 144
32
2
D S
1
1
2 ~ 5462417h ! z 3 122z 4 144z 5 111z 6 g 3 1
~ 6 314 916220 766 190h 18 663 249h 2 !
4
3024
2
D
1
1
~ 18 054213 231h ! 2 ~ 4382331h ! z 1 118z 2 19 z 3 g 2
84
4
D
1
1
~ 17 052256 198h 123 811h 2 ! z 1 1 ~ 6802759h ! z 2 1 ~ 5462855h ! z 3 234z 4 2104z 5 28 z 6 g 2 g 0
16
4
S
1 2
1
1
1
~ 1 521 30827 938 232h 15 800 187h 2 ! 1 ~ 12 372264 104h 146 641h 2 ! z 1 2 ~ 68221315h ! z 2
2016
32
2
D S
D GJ
1
1
2 ~ 542189h ! z 3 112z 4 176z 5 gg 20 1 ~ 34822016h 13339h 2 !~ 22 z 1 ! 1 ~ 442162h ! z 2 216z 5 g 30
2
8
.
~5.11!
To obtain the 2PN reactive terms for case (A), the radial infall from infinity, we use in Eqs. ~2.3!, ~2.8!, ~2.11!, and ~2.16! the
radial infall restriction and Eq. ~5.8!. The expression thus obtained is the same as obtained by putting g 0 50 in Eq. ~5.11!. The
z ’s in Eq. ~5.11! are given by
z 15 a 32 b 2 ,
z 25 j 11 j 21 j 4 ,
z 35 j 31 j 5 ,
z 45 c 31 c 61 c 8 ,
z 55 c 11 c 21 c 41 c 7 ,
z 65 c 51 c 9 .
~5.12!
We have also computed the 2PN reactive terms for cases (A) and (B) ab initio using the IW method adapted to radial infall.
In this case, only energy balance is needed as J50 for head-on infall. The result thus obtained is in agreement with Eq. ~5.11!.
Equation ~5.11! may be integrated straightforwardly to obtain the 4.5PN contribution to ż 2 in case (B) and it yields
H
16~ 2 g 22 g 0 ! 3/2h 1
4
8 2
ż 5
gg 0 2
g 1
~ 41221z 1 ! g 2 2
5
21
105
315 0
2
3 g 31
1
S
S
D S
D G FS
1
1
~ 26841682h ! 1
~ 144021680h ! z 1 g 30 1
945
4725
D
D
1
1 091 065 564 931h 5 258 809h 2
2
1
2
~ 27 156
7128
2079
66 528
352
D S
1
1
21 548 237
~ 7662527h ! z 2 1 ~ 5462417h ! z 3 22 z 4 24 z 5 2 z 6 g 4 1 2
22
44
224 532
1
1
1
26 019 487h 2 750 389h 2
2
1
~ 26 3162139 638h 167 231h 2 ! z 1 2 ~ 441626241h ! z 2 2
~ 54622023h ! z 3
49 896
11 088
528
99
132
D
12 z 4 18 z 5 g 0 g 3 1
1
1
1
~ 218 054113 231h ! 1 ~ 4382331h ! z 1 22 z 2 2 z 3
756
36
1
1
1
1
~ 192627597h ! 1
~ 66022534h ! z 1 12 z 2 g 0 g 2
~ 23421341h ! 1
~ 2402280h ! z 1 g 20 g
252
168
315
525
249 816h 115 057h 2 ! z 1 1
1
FS
S
1
3 823 453 1 681 430h 4 399 627h 2
2
1
2
~ 30 8282146 592h 1244 127h 2 ! z 1
149 688
14 553
22 176
2464
D
S
1
567 739 608 992h 228 227h 2
1
1
1
2
2
~ 53 2622202 741h ! z 2 1 ~ 24228h ! z 3 24 z 5 g 20 g 2 1
~ 50411080h
5082
77
187 110
72 765
27 720
385
21622h 2 ! z 1 2
D S
1
567 739 1 217 984h 228 227h 2
1
1
1
2
2
~ 105621232h 2 ! z 2 1
~ 962112h ! z 3 g 30 g 1
~ 1008
2541
385
280 665
218 295
41 580
1155
12160h 23244h 2 ! z 1 2
D GJ
1
1
~ 633627392h ! z 2 1
~ 10 560212 320h ! z 3 g 40
22 869
63 525
.
~5.13!
6046
55
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
We obtain the 4.5PN contribution to ż 2 for case (A) by putting g 0 50 in Eq. ~5.13!. Unlike in the case of quasicircular
inspiral the expressions in the head-on or radial infall cases
are dependent on the choice of arbitrary variables or the
choice of ‘‘gauge.’’
VI. CONCLUDING REMARKS
Starting from the 2PN accurate energy and angular momentum fluxes for structureless nonspinning compact binaries of arbitrary mass ratio moving on quasielliptical orbits
we deduce the 4.5PN reactive terms in the equation of motion by an application of the IW method. The 4.5PN reactive
terms are determined in terms of twelve arbitrary parameters
which are associated with the possible residual choice of
‘‘gauge’’ at this order. These general results could prove
useful to studies of the evolution of the orbits. The limiting
and complementary cases of circular orbits and head-on infall have also been examined.
We have systematically and critically explored different
facets of the IW choice like the functional form of the reactive acceleration and provided a better understanding of the
origin of redundant equations by studying variants obtained
by modifying the functional forms of the ambiguities in Ẽ *
and J̃*. The main conclusions we arrive at by this analysis
are the following.
In terms of the number of arbitrary parameters and the
corresponding gauge transformations, the IW scheme exhibits remarkable stability for a variety of choices for the form
of the ambiguity in energy and angular momentum. The different choices merely produce different numbers of degenerate equations. This indicates the essential validity and soundness of the scheme. These solutions are general enough to
treat as special cases any particular solutions obtained from
first principles in the future.
Relaxing the requirement of nonlinearity in m or more
precisely the power series behavior in m 1 and m 2 permits
mathematically more general solutions for the reactive accelerations involving more arbitrary parameters. Solutions more
general than the ones discussed in the Appendix, e.g., a solution involving six parameters at the Newtonian level, cannot be gauged away either by gauge transformations of the
form discussed by IW or by more general gauge transformations that differ in their powers of nonlinearity (m/r dependence!. However, none of these solutions are of ‘‘physical’’
interest to describe the gravitational radiation reaction of
two-body systems.
ACKNOWLEDGMENTS
We are particularly grateful to Thibault Damour for discussions clarifying the nonlinear structure of the reactive acceleration ansatz. We thank Luc Blanchet, Clifford Will, and
Rajaram Nityananda for their valuable comments. One of us
~S.I.! would like to thank the Raman Research Institute for
hospitality during the initial stages of this collaboration.
APPENDIX: THE GENERAL SOLUTION TO THE
BALANCE METHOD
1. The 2.5PN reactive solution
It should be noted that all the discussion in Sec. III follows only after one has assumed a functional form for the
reactive acceleration — in particular, the intuitive requirement that it be nonlinear, i.e., contain an overall factor of
m/r. It is pertinent to ask whether more general possibilities
obtain, consistent with the far-zone fluxes, if one relaxes this
requirement. We have explored this question in detail at the
2.5PN level and we summarize the results in what follows.
In this instance the reactive acceleration is assumed to be
SD
8
m
a52 h 2 @ 2 ~ A2.5! ṙn1 ~ B2.5! v# ,
5 r
A2.55a 81 v 4 1a 82 v 2
1a 58
2
SD
m 2
ṙ 1a 68 ṙ 4 ,
r
B2.55b 81 v 4 1b 82 v 2
1b 58
SD
m
m
1a 83 v 2 ṙ 2 1a 84
r
r
SD
m
m
1b 83 v 2 ṙ 2 1b 84
r
r
2
SD
m 2
ṙ 1b 68 ṙ 4 ,
r
~A1!
i.e., it is determined by 12 reactive coefficients instead of the
earlier 6. Recall that the nomenclature IW22, IW21, and
IW11 refers to the functional forms chosen for the ambiguity
in energy and angular momentum and we introduce similar
notation EJ22, EJ21, and EJ11, respectively, in this appendix, where the acceleration has a more general form as given
by Eq. ~A1!. With this form of the reactive acceleration,
however, one gets, e.g., in the EJ21 scheme at 2.5PN,
SDS
8 m
Ẽ * [Ẽ N 1Ẽ 2.55Ẽ N 2 h
5
r
2
ṙ a 1 v 2 1 a 2
D
m
1 a 3 ṙ 2 ,
r
~A2a!
S
D
m
m
8
J̃*[L̃N 1J̃2.55L̃N 1 h L̃N ṙ b 1 v 2 1 b 2 1 b 3 ṙ 2 .
5
r
r
~A2b!
The derivatives of Ẽ * and J̃* with the new form of the
reactive acceleration are given by
TABLE II. Comparison of the four alternative schemes: EJ21,
EJ22, EJ11, and EJ00 at 2.5PN level. The notation is as in Table I.
In the NC column, a1b indicates that a constraints arise from
energy balance and b from angular momentum balance.
Scheme
EJ22
EJ21
EJ11
EJ00
NV
NC
ND
NI
NA
121311
121313
121613
1211016
1016
1016
1016
15110
2
1
1
3
14
15
15
22
2
3
6
6
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
F
8 m
dẼ *
m
52 h 2 ~ b 18 !v 6 1 ~ b 28 1 a 1 ! v 4 1 ~ 2a 18 1b 38 ! ṙ 2 v 4
dt
5 r
r
SD
SD
SD
1 ~ b 84 2 a 1 1 a 2 !
m
r
v 2 1 ~ 2a 83 1b 86 ! ṙ 4 v 2 1 ~ 2a 82
SD
SD
3
m 2 2
m
ṙ v 2 a 2
r
r
1b 85 23 a 1 13 a 3 !
14 a 2 13 a 3 !
2
m
r
2
2 ~ a 84 12 a 1
G
m 4
ṙ 2a 86 ṙ 6 ,
r
ṙ 2 2 ~ a 85 15 a 3 !
~A3a!
S DF
dJ̃*
8
m
52 h L̃N 2
dt
5
r
~ b 81 2 b 1 !v 4 1 ~ b 82 1 b 1 2 b 2 !
1 ~ b 38 12 b 1 23 b 3 ! ṙ 2 v 2 1 ~ b 48 1 b 2 !
13 b 2 13 b 3 !
SD
SD
G
m
r
SD
m 2
v
r
1 ~ b 58 12 b 1
m 2
ṙ 1 ~ b 86 14 b 3 ! ṙ 4 .
r
a 82 53 ~ 11 a 3 2 b 3 ! ,
a 48 523/323 a 3 12 b 2 ,
~A3b!
b 18 50,
b 28 521 b 2 ,
a 83 524 b 3 ,
a 58 525 a 3 ,
b 38 53 b 3 ,
b 58 523 ~ 11 b 2 1 b 3 ! ,
general form of reactive acceleration ~see Table II! can be
gauged away? We find that at 2.5PN order, though this is
possible with the three parameters of the EJ21 scheme, it is
not true for the six arbitrary parameters in the EJ11 and EJ00
schemes. For this reason the EJ11 and EJ00 schemes are not
satisfactory and we discuss them no further. We present here
for the EJ21 scheme details of the gauge calculation at 2.5PN
order. We choose d x to be
d x5
SD
8h m
~ f 82.5ṙx1g 82.5rv! ,
5 r
a 68 50 ,
~A4a!
b 48 522 b 2 ,
b 68 524 b 3 .
~A4b!
This construction can be generalized to 3.5PN and 4.5PN
orders in which cases the number of arbitrary parameters are
8 and 15, respectively. The EJ11 and EJ00 schemes, on the
other hand, lead to a solution with six arbitrary parameters at
the 2.5PN level. However, not all these solutions are similar
in regard to the possibility of gauging away all the arbitrary
parameters they contain.
~A5!
where f 82.5 and g 82.5 are given by
8 5 P 01
8
f 2.5
2
Using Eqs. ~A2! and ~A3! one can understand the counts of
the various variables summarized in Table II.
One can explain the new counts for the arbitrary parameters by comparing, e.g., the EJ21 scheme with a general
form for the reactive acceleration as in this section with the
IW21 scheme with the restricted form for reactive acceleration as in Sec. III. One has six extra variables and 4 extra
equations. However one gains an extra equation because one
of the degeneracies is lifted. The resulting five equations for
six variables lead to an extra arbitrary parameter resulting in
a three-parameter solution in this instance. All the other entries in Table II can be similarly understood by comparison
of Tables I and II.
The reactive solution resulting from the EJ22 scheme in
this instance is exactly the same as the IW21 reactive solution discussed earlier. From the EJ21 scheme one obtains a
solution with three arbitrary parameters given by
a 81 53 b 3 ,
6047
g 82.55Q 801
SD
SD
m
8 v 2 1 P 03
8 ṙ 2 ,
1 P 02
r
m
1Q 802v 2 1Q 803ṙ 2 .
r
~A6!
For the reactive acceleration given by Eqs. ~A1! and ~A4! we
obtain
1
P 8015 ~ a 3 2 b 3 ! ,
3
~A7a!
1
8 5 b3 ,
P 02
2
~A7b!
P 80350,
~A7c!
1
Q 8015 ~ 2 a 3 23 b 2 1 b 3 ! ,
3
~A7d!
Q 80250,
~A7e!
1
Q 80352 b 3 .
2
~A7f!
The EJ21 scheme leads to a more general solution to the
balance equations, and as in IW all the arbitrary parameters
that appear in its solution can be associated with a residual
choice of gauge. It has been explored in detail up to 4.5PN
and the results are summarized below. We list the new general reactive solutions and the corresponding gauge transformations for the arbitrary parameters they contain. For brevity, the solutions are presented in the form: ‘‘New solution’’
5 ‘‘old solution’’ 1 ‘‘difference.’’
3. The 3.5PN and 4.5PN reactive solutions
The reactive acceleration is assumed to have the following general form:
2. The 2.5PN gauge arbitrariness
8 m
a52 h 2 @ 2 ~ A2.51A3.51A4.5! ṙn1 ~ B2.51B3.51B2.5! v# ,
5 r
~A8!
We have also investigated the question whether all the
extra arbitrary parameters appearing in schemes with the
with A2.5 and B2.5 given in Eqs. ~A1! and ~A4! and
A3.5 ,B3.5 ,A4.5 , and B4.5 given by
6048
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
A3.55 f 81 v 6 1 f 82 v 4
SD
SD
m
m
m
1 f 83 v 4 ṙ 2 1 f 84 v 2 ṙ 2 1 f 85 v 2 ṙ 4 1 f 86 v 2
r
r
r
2
m
m
m
1g 38 v 4 ṙ 2 1g 48 v 2 ṙ 2 1g 58 v 2 ṙ 4 1g 68 v 2
B3.55g 18 v 1g 28 v
r
r
r
6
4
A4.55h 18 v 8 1h 28 v 6 ṙ 2 1h 38 v 6
1h 812ṙ 2
SD
m
r
3
1h 813ṙ 4
SD
m
r
B4.55k 81 v 8 1k 82 v 6 ṙ 2 1k 83 v 6
8 ṙ 2
1k 12
SD
m
r
3
8 ṙ 4
1k 13
SD
m
m
1h 48 v 4 ṙ 4 1h 58 v 4
r
r
2
1h 814ṙ 6
m
r
SD
2
8 ṙ 6
1k 14
1h 68 v 4 ṙ 2
1 f 87
SD
SD
m 4
m
ṙ 1 f 88
r
r
2
m
m
1g 78 ṙ 4 1g 88
r
r
SD
SD
2
ṙ 2 1 f 89
m
r
m
ṙ 1g 98
r
2
SD
m
m
m
1h 78 v 2 ṙ 6 1h 88 v 2 ṙ 4 1h 98 v 2 ṙ 2
r
r
r
3
1 f 810ṙ 6 ,
~A9a!
8 ṙ 6 ,
1g 10
~A9b!
3
2
8 v2
1h 10
SD SD
3
m
r
8
1h 11
m
1h 815ṙ 8 ,
r
1k 86 v 4 ṙ 2
SD
m
m
m
1k 87 v 2 ṙ 6 1k 88 v 2 ṙ 4 1k 89 v 2 ṙ 2
r
r
r
2
1k 810v 2
SD SD
m
r
3
m
8 ṙ 8 .
1k 15
r
FS D
FS
m
r
15
2
~ 12v 211ṙ ! 1
2
2
D
Y [4]
( R8[3.5]
i
i
i51
10
G
G
F
6 4
Y [3]
i ~ i51, . . . ,10 ! 5 v , v
1k 811
m
r
4
~A10a!
,
8
m m
m
dJ̃*
52 h L̃N 2
2 v 2 12 23ṙ 2 1
S8[3.5]
Y [3]
,
i
i
dt
5
r r
r
i51
is given by Eqs. ~2.15!,
where Y [4]
i
4
~A9d!
With this form of the acceleration we have, at 3.5PN,
8 m
dẼ *
52 h 2
dt
15 r
m
r
~A9c!
2
m
m
1k 84 v 4 ṙ 4 1k 85 v 4
r
r
SD
2
2
55
SD
m 4 2 2 m
, v ṙ , v
r
r
2
,v2
(
SDSD
m 2 2 4 m
ṙ , v ṙ ,
r
r
3
,
m
r
2
~A10b!
m
ṙ 2 , ṙ 4 ,ṙ 6
r
G
~A11!
and R8[3.5]
, S8[3.5]
consist of corresponding linear combinai
i
tions of the parameters involved. Repeating the procedure
explained in the text, the 3.5PN reactive solution obtained is
g 81 50 ,
~A12k!
g 28 5g 1 ,
~A12l!
3
f 18 52 ~ 123 h ! b 3 23 r 2 ,
2
~A12a!
3
g 83 52 ~ 123 h ! b 2 23 r 2 ,
2
~A12m!
1
f 28 5 f 1 2 ~ 21139h ! b 3 13 r 2 ,
2
~A12b!
1
g 48 5g 3 2 ~ 21133h ! b 3 13 r 2 ,
2
~A12n!
f 83 52 ~ 123 h ! b 3 14 r 2 25 r 4 ,
~A12c!
g 58 52 ~ 123 h ! b 3 14 r 2 25 r 4 ,
~A12o!
1
f 48 5 f 3 1 ~ 56115h ! b 3 12 r 2 15 r 4 ,
2
~A12d!
g 86 5g 2 ,
~A12p!
f 58 56 r 4 ,
~A12e!
1
g 87 5g 5 1 ~ 561 h ! b 3 12 r 2 15 r 4 ,
2
~A12q!
f 86 5 f 2 1 ~ 21112h ! b 3 ,
~A12f!
g 88 5g 4 1 ~ 2119 h ! b 3 ,
~A12r!
f 87 5 f 5 24 h b 3 ,
~A12g!
g 89 5g 6 ,
~A12s!
f 88 5 f 4 23 h b 3 ,
~A12h!
g 81056 r 4 ,
~A12t!
f 98 5 f 6 ,
~A12i!
8 50 ,
f 10
~A12j!
where f i , g i are given by Eqs. ~2.9!. The solution corresponding to Eqs. ~2.10! remains identical.
Similarly at 4.5PN we have
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
8 m
dẼ *
52 h 2
dt
15 r
SS D
m
r
2
S DH F
m
~ 12v 211ṙ ! 1
r
2
2
2
1
m
~ 7852852h !v 4 12 ~ 2148711392h !v 2 ṙ 2 1160~ 2171 h ! v 2
28
r
S D GJ
D H F
SS
15
2
m
m
13 ~ 6872620h ! ṙ 18 ~ 367215h ! ṙ 2 116~ 124 h !
r
r
4
8
m m
m
m
dJ̃*
52 h L̃N 2
2 v 2 12 23ṙ 2 1
dt
5
r r
r
r
6049
1
Y [4]
( R8 [4.5]
i
i
i51
D
~A13a!
,
1
m
~ 3072548h !v 4 16 ~ 2741277h !v 2 ṙ 2 24 ~ 58195h ! v 2
84
r
S D GJ
m
m
13 ~ 952360h ! ṙ 4 12 ~ 3721197h ! ṙ 2 12 ~ 274512 h !
r
r
15
2
1
Y [4]
( S8 [4.5]
i
i
i51
D
~A13b!
,
where
F
SD
SD SD
10 8
Y [5]
i ~ i51, . . . ,21 ! 5 v , v
v
2
m 8 2 6 m
, v ṙ , v
r
r
m
r
3
ṙ 2 , v 2
m
r
2
,v6
2
ṙ 4 , v 2
SD SD
SDSD SD SD
3
m 2 6 4 4 m
ṙ , v ṙ , v
r
r
m 6 2 8 m
ṙ , v ṙ ,
r
r
and Y [4]
is given by Eq. ~2.15!.
i
,S8[4.5]
consist of linear combinations of the
Here R8[4.5]
i
i
parameters involved. The 4.5PN reactive solution reads as
1
3
h 81 52 ~ 3227h 163h 2 ! b 3 1 ~ 123 h ! r 2 23 x 2 ,
8
2
~A15a!
m
r
,v4
5
,
m
r
4
ṙ 2 ,
2
ṙ 2 , v 4
m
r
3
ṙ 4 ,
m
r
~A15b!
1
1
h 38 5h 1 1 ~ 32207h 175h 2 ! b 3 1 ~ 21139h ! r 2 13 x 2 ,
8
2
~A15c!
h 84 523 ~ 123 h ! r 4 16 x 4 27 x 7 ,
~A15d!
1
1 ~ 35165h ! r 4 14 x 2 15 x 4 ,
2
h 87 58 x 7 ,
~A15f!
12 x 4 17 x 7 ,
~A15h!
G
~A14!
~A15i!
1
8 5h 5 2 ~ 3061489h 148h 2 ! b 3 ,
h 10
4
~A15j!
h 8115h 10 ,
~A15k!
1
h 8125h 9 2 ~ 12187h 224h 2 ! b 3 ,
4
~A15l!
1
h 8135h 8 2 ~ 8149h 134h 2 ! b 3 12 h r 2 15 h r 4 ,
2
~A15m!
h 8145h 7 16 h @~ 123 h ! b 3 1 r 4 # ,
~A15n!
8 50 ,
h 15
~A15o!
k 18 50 ,
~A15p!
1
3
k 82 52 ~ 3227h 163h 2 ! b 3 1 ~ 123 h ! r 2 23 x 2 ,
8
2
~A15q!
k 83 5k 1 ,
~A15g!
1
1
h 88 5h 4 1 ~ 23531195h ! h b 3 1 h r 2 2 ~ 84125h ! r 4
8
2
m
ṙ 6 , ṙ 8 ,ṙ 10
r
,
2 ~ 35120h ! r 4 ,
h 85 5h 3 1 ~ 18196h 118h 2 ! b 3 2 ~ 21112h ! r 2 ,
~A15e!
1
1
h 86 5h 2 2 ~ 242397h 195h 2 ! b 3 2 ~ 70211h ! r 2
4
2
2
4
1
h 89 5h 6 2 ~ 2601119h 130h 2 ! b 3 2 ~ 1415 h ! r 2
4
1
5
h 28 5 ~ 129 h 121h 2 ! b 3 22 ~ 123 h ! r 2 1 ~ 123 h ! r 4
2
2
14 x 2 25 x 4 ,
SD
m 4 4 6 2 m
ṙ , v ṙ , v
r
r
~A15r!
1
5
k 48 5 ~ 129 h 121h 2 ! b 3 22 ~ 123 h ! r 2 1 ~ 123 h ! r 4
2
2
14 x 2 25 x 4 ,
~A15s!
6050
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
k 85 5k 3 ,
~A15t!
1
k 813 5k 8 2 ~ 1441179 h 140 h 2 ! b 3 2 ~ 1416 h ! r 2
2
3
1
k 86 5k 2 1 ~ 1281h 113h 2 ! b 3 1 ~ 21133h ! r 2 13 x 2 ,
8
2
~A15u!
k 87 523 ~ 123 h ! r 4 16 x 4 27 x 7 ,
2 ~ 35115 h ! r 4 ,
23hr212x417x7 ,
1
1
k 88 5k 4 2 ~ 242421h 2 h 2 ! b 3 2 ~ 70225h ! r 2
4
2
~A15cc!
k 81558 x 7 ,
~A15w!
~A15dd!
where h i , k i are given by Eqs. ~2.16! of the text and Eqs.
~2.17! remain the same.
1
k 89 5k 6 1 ~ 841525h 154h 2 ! b 3 2 ~ 2119 h ! r 2 ,
4
~A15x!
4. The 3.5PN and the 4.5PN gauge arbitrariness
8 5k 5 ,
k 10
~A15y!
Finally it can be shown that all the arbitrary parameters in
the reactive solution may be absorbed in a choice of
‘‘gauge’’ of the form
8 5k 10 ,
k 11
~A15z!
d x5 h
1
k 8125k 9 2 ~ 1591288h 112h 2 ! b 3 ,
2
~A15aa!
F
f 83.55 P 821v 4 1 P 822v 2
F
g 83.55 Q 821v 4 1Q 822v 2
F
8 5 Q 41
8 v 6 1Q 42
8
g 4.5
SD
SD
8
5
m
~ f 8 1 f 8 1 f 8 ! ṙx1 ~ g 82.51g 83.51g 84.5! rv, ~A16!
r 2.5 3.5 4.5
8 and g 2.5
8 are given by Eqs. ~A6! and ~A7!, while
where f 2.5
8 , f 4.5
8 , g 3.5
8 , and g 4.5
8 have the form
f 3.5
SD
SD
m
m
m
1 P 823v 2 ṙ 2 1 P 824 ṙ 2 1 P 825
r
r
r
m
m
m
1Q 823v 2 ṙ 2 1Q 824 ṙ 2 1Q 825
r
r
r
m 4
m
v 1 P 843v 4 ṙ 2 1 P 844v 2
r
r
2
m 4
m
8 v 4 ṙ 2 1Q 44
8 v2
v 1Q 43
r
r
2
f 84.55 P 841v 6 1 P 842
F
~A15bb!
1
1
8 5k 7 2 ~ 3171105h!hb32 ~8413h!r4
k 14
8
2
~A15v!
1
1 ~ 35155h ! r 4 14 x 2 15 x 4 ,
2
55
m
r
2
2
m
r
1Q 826ṙ 4 ,
1 P 845v 2 ṙ 2 1 P 846v 2 ṙ 4 1 P 847 ṙ 4 1 P 848
8 v2
1Q 45
G
G
SD
SD
1 P 826ṙ 4 ,
m
r
m 2
m
m
8 v 2 ṙ 4 1Q 47
8 ṙ 4 1Q 48
8
ṙ 1Q 46
r
r
r
2
ṙ 1 P 849
2
2
SD
SD
ṙ 2 1Q 849
m
r
m
r
G
3
1 P 8410ṙ 6 ,
3
G
8 ṙ 6 .
1Q 410
~A17!
At 3.5PN we have
1
1
P 82152 ~ 123 h ! b 3 2 ~ 2 r 2 1 r 4 ! ,
4
4
~A18a!
1
1
8 5 P 211 ~ 3210h ! b 3 1 ~ 20r 2 117r 4 ! ,
P 22
3
30
~A18b!
1
8 52 r 4 ,
P 23
4
~A18c!
1
~ 5 h b 31 r 4 !,
10
~A18d!
1
1
~ 2125h ! b 3 2 ~ r 2 1 r 4 ! ,
12
3
~A18e!
P 8245 P 231
8 5 P 221
P 25
55
SECOND POST-NEWTONIAN GRAVITATIONAL . . .
6051
P 82650 ,
~A18f!
8 50 ,
Q 21
~A18g!
1
1
8 5Q 212 ~ 123 h ! b 3 2 ~ 10r 2 17 r 4 ! ,
Q 22
2
30
~A18h!
1
1
Q 8235 ~ 123 h ! b 3 1 ~ 2 r 2 1 r 4 ! ,
4
4
~A18i!
1
1
8 5Q 232 ~ 328 h ! b 3 2 ~ 10r 2 113r 4 ! ,
Q 24
6
30
~A18j!
Q 8255Q 222
1
1
~ 2125h ! b 3 1 ~ r 2 1 r 4 ! ,
12
3
1
8 5 r4 .
Q 26
4
~A18k!
~A18l!
Similarly at 4.5PN we have
P 84152
P 8425 P 412
1
1
1
1
~ 129 h 121h 2 ! b 3 1 ~ 123 h ! r 2 1 ~ 123 h ! r 4 2 ~ 12x 2 16 x 4 16 x 7 ! ,
16
4
8
24
~A19a!
1
1
1
1
~ 115524817h 1367h 2 ! b 3 2 ~ 1302318h ! r 2 2 ~ 532117h ! r 4 1
~ 105x 2 184x 4 168x 7 ! ,
840
60
30
105
~A19b!
1
1
P 8435 ~ 123 h ! r 4 2 ~ 6 x 4 14 x 7 ! ,
8
24
P 8445 P 421
~A19c!
1
1
1
1
~ 42021917h 1967h 2 ! b 3 1 ~ 552228h ! r 2 1
~ 2202834h ! r 4 2 ~ 15x 2 114x 4 113x 7 ! ,
120
30
120
15
~A19d!
P 8455 P 432
1
4
1
1
h ~ 47148h ! b 3 2 h r 2 2 ~ 8217h ! r 4 1
~ 21x 4 132x 7 ! ,
140
5
20
105
1
8 52 x 7 ,
P 46
6
~A19g!
1
1
1
~ 30014935h 21360h 2 ! b 3 1 h ~ 12r 2 2 r 4 ! 2 ~ x 4 12 x 7 ! ,
600
20
15
~A19h!
1
1
2
~ 921121h 1309h 2 ! b 3 1 ~ 1152h !~ r 2 1 r 4 ! 1 ~ x 2 1 x 4 1 x 7 ! ,
60
15
5
~A19i!
P 8485 P 451
8 5Q 411
Q 42
~A19f!
27
1
1
h ~ 123 h ! b 3 2 h r 4 1 x 7 ,
56
4
21
P 8475 P 462
8 5 P 442
P 49
~A19e!
P 841050 ,
~A19j!
Q 84150 ,
~A19k!
1
1
7
1
19
x ,
~ 1052659h 2347h 2 ! b 3 1 ~ 123 h ! r 2 1 ~ 123 h ! r 4 2 ~ 10x 2 17 x 4 ! 2
840
2
20
30
105 7
~A19l!
A. GOPAKUMAR, BALA R. IYER, AND SAI IYER
6052
Q 8435
8 5Q 422
Q 44
1
1
1
~ 129 h 121h 2 ! b 3 2 ~ 123 h !~ 2 r 2 1 r 4 ! 1 ~ 12x 2 16 x 4 14 x 7 ! ,
16
8
24
1
1
1
1
~ 42021604h 11434h 2 ! b 3 2 ~ 802301h ! r 2 2 ~ 402131h ! r 4 1 ~ 10x 2 19 x 4 18 x 7 ! ,
240
60
30
15
Q 8455Q 431
55
~A19m!
~A19n!
1
1
1
1
~ 1752639h 1146h 2 ! b 3 1 ~ 50299h ! r 2 1 ~ 942183h ! r 4 2 ~ 14x 2 114x 4 112x 7 ! , ~A19o!
140
30
60
21
1
1
Q 84652 ~ 123 h ! r 4 1 ~ 6 x 4 14 x 7 ! ,
8
24
1
3
1
1
26
h ~ 1212363h ! b 3 1 h r 2 1 ~ 528 h ! r 4 2 x 4 2
x ,
280
10
20
10
105 7
~A19q!
1
1
1
1
~ 135264h 211h ! b 3 2 ~ 302119h ! r 2 2 ~ 15279h ! r 4 1 ~ 5 x 2 16 x 4 17 x 7 ! ,
60
60
30
15
~A19r!
1
1
2
~ 921121h 1309h 2 ! b 3 2 ~ 1152h !~ r 2 1 r 4 ! 2 ~ x 2 1 x 4 1 x 7 ! ,
60
15
5
~A19s!
Q 8475Q 461
8 5Q 452
Q 48
~A19p!
8 5Q 441
Q 49
1
8 5 x7 .
Q 410
6
~A19t!
In the above, the P ab and Q ab are given by Eqs. ~4.3! and ~4.6! of the text.
To conclude, the far-zone flux formulas and the balance equations by themselves do not constrain the reactive acceleration
to be a power series in m 1 and m 2 , or equivalently nonlinear in the total mass m, as assumed in the paper, following IW. They
are also consistent with the more general form of the reactive acceleration discussed in this appendix.
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@65# For consistency of notation we have renamed the variables in
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variable and the corresponding notation used here, denoted as
‘‘old’’ → ‘‘new.’’ A 5/2 →A 2.5 ,B 5/2 →B 2.5 ,A 7/2 →A 3.5 ,
B 7/2 →B 3.5 , c i → f i , d i →g i , d i → j i , « i → r i , where
g → a 1 , d → a 2 , b → a 3,
and
i51, . . . ,6,
k → b 1 , a → b 2 , e → b 3.
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ansatz for acceleration is proportional to m 1 m 2 or G 2 while the
combination h (m/r 2 ) chosen in the Appendix is proportional
to m 1 m 2 (m 1 1m 2 ) 21 . Unlike the former choice, the latter is
not a power series in m 1 and m 2 and hence while the former is
the ‘‘physically relevant solution’’ the latter is of ‘‘mathematical interest.’’
@68# It is also possible to obtain ṙ and v̇ using 2PN representation
of orbits @51–54#. Here one writes down ‘‘r’’ in harmonic
coordinates in terms of conserved energy E. To compute ṙ one
then requires to calculate Ė using, acceleration to O( e 4.5).