Leading-order spin-orbit and spin(1)-spin(2) radiation-reaction Hamiltonians

0.58 MB·English

by  Han Wang

Leading-order spin-orbit and spin(1)-spin(2) radiation-reaction Hamiltonians
Han Wang (王涵),1 Jan Steinhoff,1,2,∗ Jing Zeng (曾靓),3,† and Gerhard Sch¨afer1,‡
1Theoretisch–Physikalisches Institut, Friedrich–Schiller–Universita¨t, Max–Wien–Platz 1, 07743 Jena, Germany, EU
2Centro Multidisciplinar de Astrof´ısica — CENTRA, Departamento de F´ısica,
Instituto Superior T´ecnico — IST, Universidade T´ecnica de Lisboa,
Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal, EU
3Key Laboratory for Research in Galaxies and Cosmology,
Shanghai Astronomical Observatory, Nandan Road 80, Shanghai, 200030, China
(Dated: January 31, 2012)
In the present paper, the leading-order post-Newtonian spin-orbit and spin(1)-spin(2) radiation-
reaction Hamiltonians are calculated. We utilize the canonical formalism of Arnowitt, Deser, and
Misner (ADM), which has shown to be valuable for this kind of calculation. The results are valid
for arbitrary many objects. The energy loss is then computed and compared to well-known results
for the energy flux as a check.
PACSnumbers: 04.25.Nx,04.20.Fy,04.25.-g,97.80.-d
Keywords: post-Newtonian approximation; ADM canonical formalism; Gravitational radiation reaction;
Binariesandmultiplestars;Spinningbodies
2
1 I. INTRODUCTION been proved to be astrophysically realistic [7]. There-
0 fore, the derivation of the Hamiltonians with spin effects
2 Gravitational radiation reaction is a problem of great is necessary for the detection of gravitational waves.

n interest in the detection of gravitational waves. For sec- In this paper, we utilize the canonical formalism of
a ond and third generations of gravitational wave detec- Arnowitt, Deser, and Misner (ADM), which has not
J
tors, a leading candidate source is the radiation-reaction only shown to be valuable for calculating the conser-
0
induced inspiral and merger of two compact objects like vative dynamics within the post-Newtonian and post-
3
black holes or neutron stars. Moreover, the effects of Minkowskian approximations (see, e.g., [8–10]) but also

spins are important for the emission of gravitational for the dissipative part of the dynamics [2] (with mis-
]
c wavesfromsuchsystems. Thusinordertodevelophighly prints corrected in [11]). Notice that the ADM formal-
q
accurate theoretical templates for gravitational wave de- ismwasextendedfrompoint-massestoobjectswithspins
-
r tectors, one must study the gravitational radiation reac- only recently [12] (see also [1, 13, 14]). This extension
g tion from compact binary systems with spin effects. is valid to linear order in the single spins of the ob-
[
Inthepresentpaper,theleading-orderpost-Newtonian jects,whichnotonlyincludesspin-orbitbutalsospin(1)-

2 (PN) spin-orbit and spin(1)-spin(2) radiation-reaction, spin(2)interactions. Theremarkablestructureoftheex-
v i.e., dissipative, Hamiltonians are calculated. This is the tendedADMformalismoftheinclusionofthematterinto
2 continuation of previous work in [1], where the formal- the canonical field momentum [see Eq. (2.6)] is passing
8
ismwasprepared,andalsoextendsthecalculationofthe an excellent test in the present paper. For Hamiltonians
1
3.5PN point-mass Hamiltonian in [2] to that of spinning of higher orders in spins see [13, 15–18].
1
. objects. The contributions of the spin-dependent Hamil- Energy and angular momentum flux relevant for the
9
tonians derived in the present paper to the equations of PN order in question has been well known (see [19], for
0
motion are 2.5PN orders weaker than the corresponding thenext-to-leading-ordercalculationsee[20,21]). Based
1
1 leading-order conservative ones. Recently, the contribu- on these results, secular equations of motion for the or-
: tions to the motion of spinning objects have just come bital elements corresponding to the leading-order spin-
v
within reach of experimental verifications [3, 4]. A fur- orbit and spin(1)-spin(2) radiation-reaction equations of
i
X therincreaseinprecisionofexperimentaltestsofgeneral motion were obtained in [22–24]. The general equations
r relativity will become available by creation and subse- of motion at this order were calculated in [25–27] within
a quentimprovementofgravitationalwaveastronomy[5,6] theharmonicgauge. TheHamiltonianscalculatedinthe
in the future. For compact binary systems detectable by present paper provide a compact expression which con-
gravitational wave detectors, the Hamiltonians derived tainsthesegeneralequationsofmotion(butwithinadif-
in the present paper become relevant in the late inspi- ferent gauge). And most importantly, the results in the
ral phase if one or more of the binary’s constituents is present paper are valid for arbitrary many object sys-
rapidly rotating. And rapidly rotating black holes have tems. The derived Hamiltonians are then applied to the
calculation of the energy loss of a binary system, which
is then compared with the well-known energy flux as a
∗ jan.steinhoff@ist.utl.pt check.
† [email protected] The conservative leading-order (PN) spin interactions
‡ [email protected] for self-gravitating objects were derived some time ago
2
[28–31], see also [32, 33]. For the leading-order spin(1)- A 3-vector xi is also denoted by x. Square brackets de-
spin(1) radiation-reaction level calculations see, e.g., noteindexantisymmetrizationandroundbracketsindex
[23, 33, 34]. However, only recently the conservative symmetrization, i.e., a(µbν) = 1(aµbν +aνbµ).
2
next-to-leading-orderspineffectscouldbetreated,start-
ing with the spin-orbit equations of motion in harmonic
gauge [35] (with some extensions and misprints cor- II. THE ADM FORMALISM
rected in [36]). A corresponding conservative Hamil-
tonian in the ADM gauge was obtained in [37]. The In this section, we provide a short overview of the
complete next-to-leading-order spin(1)-spin(2) conserva- ADMcanonicalformalismaftergaugefixing[53],seealso
tive Hamiltonian was first given in [38]. Other deriva- [54, 55]. The Hamiltonian is given by the ADM energy
tions of the conservative next-to-leading-order spin-orbit expressed in terms of certain canonical variables, which
andspin(1)-spin(2)dynamicscanbefoundin[39–43]and alsorequiresa(atleastapproximate)solutionofthefield
a generalization to arbitrary many objects succeeded in constraints.
[44]. Notice that the results given in the present paper The constraints of the gravitational field read
are already valid for arbitrary many objects. Also the
(cid:20) (cid:21)
conservative next-to-leading-order spin(1)-spin(1) inter- 1√ γR+ 1(cid:0)γ πij(cid:1)2−γ γ πikπjl =Hmatter,
actionofblackholeand/orneutronstarbinarieswasde- 16πG γ 2 ij ij kl
rivedrecently[16–18,45,46]. Thelatterrequiresamod- (2.1)
eling of the spin-induced quadrupole deformation, see
1
[33, 47]. Very recently, the conservative spin-dependent −8πGγijπjk;k =Himatter, (2.2)
part of the post-Newtonian Hamiltonian was extended
even to next-to-next-to-leading order for both the spin- with the definitions
orbit [48] and the spin(1)-spin(2) [49] cases. A poten- √
πij =− γ(γikγjl−γijγkl)K , (2.3)
tialforthespin(1)-spin(2)casewassimultaneouslycalcu- kl
√
lated within an effective field theory approach [50]. No- Hmatter = γT nµnν, (2.4)
µν
tice that the conservative next-to-next-to-leading-order √
Hmatter =− γT nν. (2.5)
spin(1)-spin(2)Hamiltonianandthespin-orbitradiation- i iν
reaction Hamiltonian derived in the present paper are They arise as certain projections of the Einstein field
both of the order 4PN for maximally rotating objects. equations with respect to a timelike unit 4-vector n
µ
However,notallspin-dependentHamiltoniansupto4PN withcomponentsn =(−N,0,0,0)ornµ =(1,−Ni)/N.
µ
for maximally rotating objects are known yet. We will Here, γ is the induced three-dimensional metric of the
ij
in most cases use the phrase ”formal n-th PN order” to hypersurfaces orthogonal to n , γ its determinant, R
µ
representourcountingofPNordersinthepresentpaper. the three-dimensional Ricci scalar, K = −(1γ −
ij 2 ij,0
ThisgivesPNordersdifferentfromthemaximallyrotat- N )/N the extrinsic curvature, N the lapse function,
ingcase,whichwealsooccasionallyrefertointhepresent N(ii;tjh)e shift vector, √γT the stress-energy tensor den-
µν
paper (for a more detailed discussion see, e.g., Appendix
sity of the matter system, and semicolon denotes the
A of [1]). But one should be aware that the spins are
three-dimensional covariant derivative. Partial coordi-
in fact further (independent) expansion variables. Spin
nate derivatives ∂ are also indicated by commas.
i
effectswerealsoconsideredwithinthepost-Minkowskian For nonspinning objects, 1 πij is the canonical mo-
approximation [51, 52]. 16πG
mentum conjugate to γ before gauge fixing. For spin-
ij
The paper is organized as follows. First the ADM for- ning objects, the canonical field momentum has to be
malism is reviewed in Sec. II. Then formal expressions adapted, see [1, 12]. We write
for the radiation-reaction Hamiltonians in question are
derived in Sec. III. Integrals appearing in these formal πij =πij +πij , (2.6)
can matter
expressions are performed in Sec. IV. In Sec. V, the de-
rived Hamiltonians are applied to the calculation of the whereπij containsspin-corrections. Throughoutthis
matter
energyloss,whichisthencomparedwiththeenergyflux. paper we use the ADM transverse-traceless (TT) gauge,
Finally, conclusions are given in Sec. VI. which is defined by:
Our units are such that c = 1, but for the Newtonian
∂ (γ − 1δ γ )=0, (2.7)
gravitationalconstantGnoconventionwillbeused. This j ij 3 ij kk
allows an easy transition to the different conventions for πciian =0. (2.8)
G used in [2] and [1]. For the signature of spacetime, we
Here,δ istheKroneckerdelta. Andonehasthedecom-
choose +2. Latin indices from the beginning of the al- ij
positions:
phabet, such as a, b, label the individual objects. Greek
indicesrunover0,1,2,3. Latinindicesfromthemiddleof (cid:18) φ(cid:19)4
the alphabet run over 1,2,3. Round brackets around an γ = 1+ δ +hTT, (2.9)
ij 8 ij ij
index denote a local basis, while round brackets around
a number denote the formal order in c−1, as in [1, 14]. πij =πijTT+π˜ij . (2.10)
can can can
3
Noticethati,j,andk,etc.,runover1,2,3,andupperor III. RADIATION-REACTION HAMILTONIANS
lower an index is from now on done with the flat metric, UP TO FORMAL 3.5PN LEVEL
thus changes nothing in the equations. We will ignore
the difference of upper and lower indexes later and any In this section, we generalize the derivation of the
two identical indexes can contract with no need to be radiation-reaction Hamiltonians up to the formal 3.5PN
one upper and one lower. hTT and πijTT are transverse- level performed in [2] so it becomes applicable to the
ij can
traceless, e.g., hTT =hTT =0, andπ˜ij isrelatedtothe spinning case.
ii ij,j can
vector potentials Vi and π˜i by:
can can
2 A. Interaction Hamiltonian and wave equation
π˜ij =Vi +Vj − δ Vk , (2.11)
can can,j can,i 3 ij can,k
1 1 We split the ADM Hamiltonian H into matter,
=π˜i +π˜j − δ π˜k − ∆−1π˜k . ADM
can,j can,i 2 ij can,k 2 can,ijk field, and interaction parts, i.e.,
(2.12)
H =Hmatter+Hfield+Hint, (3.1)
ADM
It holds that:
where the matter part Hmatter is independent of the
(cid:18) (cid:19) (truly dynamical) canonical field variables hTT and
1 ij
Vcian = δij − 4∂i∂j∆−1 π˜cjan, (2.13) πcijaTnT,thefieldpartHfieldisindependentofthecanonical
matter variables and reads explicitly:
π˜i =∆−1πij =∆−1π˜ij , (2.14)
can can,j can,j 1 (cid:90) (cid:20)1 (cid:21)
πijTT =δTTijπkl , (2.15) Hfield = d3x (hTT)2+(πijTT)2 , (3.2)
can kl can 16πG 4 ij,k can
with the inverse Laplacian ∆−1 and and the interaction part Hint depends on both canonical
matter and field variables. The interaction Hamiltonian
δiTjTkl = 12[(δil−∆−1∂i∂l)(δjk−∆−1∂j∂k) uptoandincludingtheformal3.5PNlevelreads([1],see
+(δ −∆−1∂ ∂ )(δ −∆−1∂ ∂ ) (2.16) also [2])
ik i k jl j l
−(δ −∆−1∂ ∂ )(δ −∆−1∂ ∂ )]. 1 (cid:90) (cid:20)(cid:16) (cid:17)
kl k l ij i j Hint = d3x B +Bˆ hTT
16πG (4)ij (6)ij ij
The canonical field variables after gauge fixing are hTijT −2πGHmatter(cid:0)hTT(cid:1)2− 1φ (cid:0)hTT(cid:1)2
and πcijaTnT. (2) ij 4 (2) ij,k
InordertoobtaintheADMHamiltonian,thefourfield (cid:21)
constraints must be solved for the four variables φ and +2(V(i3)φ(2),j −π(i5j)matter)πcijaTnT , (3.3)
π˜i in terms of hTT, πijTT and canonical matter vari-
can ij can
ables, which enter through the stress-energy tensor via where
dtheerssoouurrcceetteerrmmssinHsmpaitnt,erseaen[d1,H12im,a1tt3e]ra(nfdoratlshoe[1li4n]e)a.rTohre- δ(cid:16)(cid:82) d3xH(m8a)tter(cid:17) 1
canonical matter variables are the canonical position zˆai, B(4)ij =16πG δhTT − 8φ(2),iφ(2),j, (3.4)
momentum P , and spin-tensor S of the a-th ob- ij
ai a(i)(j)
ject. An analytic solution for φ and π˜cian, however, can and V(i3) is a field quantity which will be discussed in
in general only be given in some approximation scheme.
Sec. IV. Bˆ is given by a similar expression [see
The ADM Hamiltonian is then given by: (6)ij
(5.14) in [1]]. For comparison with [2], notice that
H =− 1 (cid:90) d3x∆φ[zˆi,P ,S ,hTT,πijTT], 2δkTlTij(V(k3)φ(2),l) = −δkTlTij(φ(2)π˜(k3l)) [π˜(k3l) is another
ADM 16πG a ai a(i) ij can field quantity which will be discussed later]. Further,
(2.17) in [2] the quantity A(4)ij =2B(4)ij is used in this paper.
where S = 1(cid:15) S and (cid:15) is the completely The equations of motion for the canonical field vari-
a(i) 2 ijk a(j)(k) ijk
antisymmetric Levi–Civita symbol. H is the ADM ables follow from the ADM Hamiltonian by virtue of the
ADM
energyexpressedintermsofthecanonicalvariablesmen- Poisson brackets (2.18) as:
tioned above. The Poisson brackets read
1 δH
h˙TT =δTTij ADM , (3.5)
{hTT(x),πklTT(x(cid:48))}=16πGδTTklδ(x−x(cid:48)), (2.18) 16πG ij kl δπckalTnT
ij can ij
1 δH
{zˆi,P }=δ , (2.19) π˙ijTT =−δTTij ADM . (3.6)
a aj ij 16πG can kl δhTT
{S ,S }=(cid:15) S , (2.20) kl
a(i) a(j) ijk a(k)
Here the dot over a variable denotes the partial time
all others are zero. derivative ∂ ≡ ∂ . For quantities not depending on the
t ∂t
4
hypersurface coordinate x, this is to be understood as where the TT-projector was pulled in front of the re-
the ordinary time derivative. In terms of the interaction tarded solution and the integral operator L is defined
n
Hamiltonian Hint, the field equations read by
161πG(cid:3)hTijT =δkTlTij(cid:20)2δδHhTinTt − ∂∂tδδπHkliTntT(cid:21) , (3.7) (Lnf)(x,t)=−4π1n!(cid:90) d3x(cid:48)|x−x(cid:48)|n−1f(x(cid:48),t). (3.15)
kl can
161πGπcijaTnT = 12(cid:20)161πGh˙TijT−δkTlTijδδπHkliTntT(cid:21) , (3.8) N∆o−t1i.ce that L2n = ∆−1−n for n ∈ N, in particular L0 =
can
Using the PN-expanded source of the wave equation
with (cid:3) = ∆−∂2. To arrive at these expressions, the from Eq. (3.9) one may arrange the near-zone expansion
t
explicit form of Hfield is used as in Eq. (3.2). Notice by PN orders as:
that it is easier to implement the boundary condition of
hTT =hTT +hTT +hTT +hTT +... . (3.16)
no incoming gravitational radiation for a wave equation ij (4)ij (5)ij (6)ij (7)ij
like Eq. (3.7) than for a system of first-order differential It is important that only a finite number of terms from
equations like Eqs. (3.5) and (3.6). Inserting the 3.5PN- the near-zone expansion [Eq. (3.14)] contribute to a spe-
accurate interaction Hamiltonian [Eq. (3.3)], one gets cific PN order due to the increasing number of time
derivatives therein. Therefore, one obtains
(cid:20)
(cid:3)hTijT =δkTlTij 2B(4)kl+2B(6)kl−8πGH(m2a)tterhTklT hTT =2δTTij∆−1B , (3.17)
(4)ij kl (4)kl
+(cid:0)φ hTT (cid:1) −2∂ (cid:16)Vk φ (cid:17)(cid:21), (3.9) hT(5T)ij =χ˙(4)ij, (3.18)
(2) kl,m ,m ∂t (3) (2),l ...
1 (cid:16) (cid:17) hT(7T)ij =Π˙1ij +Π˙2ij +Π¨3ij +Π4ij +Qij, (3.19)
πijTT = h˙TT−δTTij Vk φ −πkl , (3.10)
can 2 ij kl (3) (2),l (5)matter where
with the definition: χ(4)ij =−2δkTlTijL1B(4)kl, (3.20)
B =Bˆ +π˙ij . (3.11) Π1ij =−2δkTlTijL1B(6)kl, (3.21)
(6)ij (6)ij (5)matter (cid:16) (cid:17)
Π =8πGδTTijL hTT Hmatter , (3.22)
One can get alternative expressions for B and 2ij kl 1 (4)kl (2)
√ (4)ij (cid:16) (cid:17)
B(6)ij in terms of Tij = γTij by comparing the wave Π =2δTTijL Vk φ , (3.23)
equation for hTT with the Einstein equations (see [1]), 3ij kl 1 (3) (2),l
ij
e.g., Π4ij =−2δkTlTijL3B(4)kl, (3.24)
(cid:16) (cid:17)
1 Q =−8πGδTTij∆−1 hTT Hmatter . (3.25)
B =−8πGT − φ φ . (3.12) ij kl (5)kl (2)
(4)ij (4)ij 8 (2),i (2),j
NoticethattheapplicationofL toatotaldivergencelike
1
ThisshouldagreewithEq.(3.4)aftertheTT-projection. (φ hTT ) leads to a vanishing result. It will become
(2) kl,m ,m
apparent in the next section that hTT is not needed in
(6)ij
the present paper (but it contributes to the conservative
B. Near-zone expansion
3PN Hamiltonian). The definitions P = Π˙ , P =
... 1ij 1ij 2ij
Attheconsideredorder, aspectliketaileffectsplayno Π˙2ij, P3ij =Π¨3ij, and Rij = Π4ij were used in [2].
An application of the operator L obviously leads to
role(seee.g.[56]). Wemaythereforesolvethewaveequa- 1
tion for hTT by an order-by-order evaluation of the re- a field depending on time only (i.e., not depending on
ij x). This allows an easy calculation of the (regularized)
tardedsolution. Further,thefieldsolutionisonlyneeded
TT-projections in Eqs. (3.20) – (3.23) by means of the
in the near-zone.
formula
In order to discuss the near-zone expansion, we write
the wave equation for hTT schematically as: 2
ij δTTijA (t)= ASTF(t), (3.26)
kl kl 5 ij
(cid:3)hTT =−8πGδTTijS . (3.13)
ij kl kl valid for an arbitrary x-independent function Akl(t) (see
[2]). Here STF denotes the symmetric trace-free part,
The near-zone expansion of the retarded solution to this
equation corresponds to a series in c−1 entering through 1 1
the retarded time tret =t−c−1|x−x(cid:48)|, reading ASijTF = 2(Aij +Aji)− 3δijAkk. (3.27)
(cid:20) Further, hTT is a function of time only, hTT =0. As
hTT =−8πGδTTij L S −L S˙ (5)ij (5)ij,k
ij kl 0 kl 1 kl a consequence of these simplifications, we finally have
(3.14)
... (cid:21) 1 (cid:90)
+L2S¨kl−L3Skl+... , χ(4)ij = 5π d3xB(S4T)iFj, (3.28)
5
1 (cid:90)
Π = d3xBSTF , (3.29) In order to distinguish the nondynamical matter vari-
1ij 5π (6)ij ables from the dynamical ones, we attach a prime to
4G(cid:90) their object label as in, e.g., P or P , and also talk
Π =− d3xhTT Hmatter, (3.30) 1(cid:48) a(cid:48)
2ij 5 (4)ij (2) of primed and unprimed variables for short. Further, we
1 (cid:90) (cid:16) (cid:17)STF introduceanexplicittimederivative∂tex,whichonlyacts
Π3ij =−5π d3x V(i3)φ(2),j , (3.31) on the primed variables (The partial and ordinary time
derivatives act on both primed and unprimed variables
1 (cid:90)
Π = δTTij d3x(cid:48)|x−x(cid:48)|2B (x(cid:48),t), (3.32) here). A superscript a→a(cid:48) is attached to a field to de-
4ij 12π kl (4)kl notethatitssolutionshouldbeexpressedintermsofthe
Q = 1hTT δTTijφ , (3.33) primedvariables. Thisdenotesanexchangeofall object
ij 2 (5)kl kl (2) labels by labels with a prime, not just of label a. Thus
hTT and hTT in Eqs. (3.34) and (3.35) should better
wherethePN-expandedHamiltonconstraintintheform (5)ij (7)ij
∆φ = −16πGHmatter was used to arrive at the last be denoted by hTT a→a(cid:48) and hTT a→a(cid:48) from now on.
(2) (2) (5)ij (7)ij
equation. After the equations of motion have been obtained from
theHamiltonian,onemayidentifyprimedandunprimed
variables(e.g.,theobjects1and1(cid:48)),whichingeneralre-
C. Radiation-reaction Hamiltonians quires another application of regularization techniques.
The formulas for the radiation-reaction Hamiltonians
Thedissipationthroughemissionofgravitationalradi- Eqs. (3.34) and (3.35) can be simplified further. First,
ationentersthePN-expansionviahTT andhTT ,which Eq. (3.34) may be written as:
(5)ij (7)ij
are antisymmetric under time reversal. The parts of
1 (cid:90)
the Hamiltonian linear in hTT or hTT thus give the Hint = hTT a→a(cid:48) d3xBSTF , (3.37)
(5)ij (7)ij 2.5PN 16πG (5)ij (4)ij
radiation-reaction Hamiltonians at the considered order.
NoticethatHfielddoesnotcontributetothematterequa-
where the x-independent hTT was pulled in front of
tions of motion, so we only need to consider Hint. The (5)ij
the integral and B is contracted with the symmet-
radiation-reaction Hamiltonians are thus given by: (4)ij
ric trace-free hTT . As explained previously, hTT must
1 (cid:90) (5)ij (5)ij
Hint = d3xB hTT , (3.34) be replaced by hTT a→a(cid:48). The remaining integral in Eq.
2.5PN 16πG (4)ij (5)ij (5)ij
1 (cid:90) (cid:20) (3.37) is identical up to a prefactor to the definition of
H3in.5tPN = 16πG d3x B(4)ijhT(7T)ij +V(i3)φ(2),jh˙T(5T)ij χ(4)ij, cf. (3.28). Finally we obtain, inserting Eq. (3.18),
+(cid:16)B −4πGHmatterhTT (cid:17)hTT (cid:21) Hint = 5 χ˙a→a(cid:48)χ , (3.38)
(6)ij (2) (4)ij (5)ij 2.5PN 16G (4)ij (4)ij
1 d (cid:90)
− d3xhTT πij , (3.35) which is a well-known result (see [2] and references
16πGdt (5)ij (5)matter therein). The problem was reduced to the calculation
where we used hTT = 0, with Eqs. (3.11) and (3.10). ofχ(4)ij via(3.28). Rememberthatχ˙a(4→)iaj(cid:48) inthisHamil-
(5)ij,k
tonian is explicitly time-dependent.
Equation (3.10) reads explicitly:
We proceed with a simplification of the individual
πijTT = 1h˙TT . (3.36) parts of Eq. (3.35). Analogous to the simplification of
(6)can 2 (5)ij Hint given in the last paragraph we have
2.5PN
The last term in Eq. (3.35) corresponds to a canonical 1 (cid:90) 5
transformation and could be dropped, but we keep it for d3xhTT a→a(cid:48)B = χ˙a→a(cid:48)Π , (3.39)
16πG (5)ij (6)ij 16G (4)ij 1ij
now.
1 (cid:90) 5
One has to be aware of a subtlety here. The matter d3xh˙TT a→a(cid:48)Vi φ =− χ¨a→a(cid:48)Π ,
variables entering the Hamiltonian via the solution for 16πG (5)ij (3) (2),j 16G (4)ij 3ij
hTT play a special role as they may not be treated as (3.40)
ij
dynamical (i.e., phase space) variables. Otherwise, the
matter equations of motion resulting from the Hamil- where Eqs. (3.29) and (3.31) were used. We may further
tonian would in general be wrong (at the conservative write
level one can use a Routhian to avoid this problem, 1(cid:90) 5
see [8]). Instead these nondynamical matter variables −4 d3xhT(5T)ija→a(cid:48)hT(4T)ija→a(cid:48)H(m2a)tter = 16Gχ˙a(4→)iaj(cid:48)Π(cid:101)2ij,
entering through hTT are treated as functions depend-
ij (3.41)
ing explicitly on time only. This introduces an explicit
with the definition
time-dependence into the radiation-reaction Hamiltoni-
ans, which is a very natural description of a dissipative 4G(cid:90)
system via canonical methods. Π(cid:101)2ij =− 5 d3xhT(4T)ija→a(cid:48)H(m2a)tter. (3.42)
6
The notation Π(cid:101)2ij was chosen because of the similarity This agrees with [2] (with misprints corrected in [11]).
to Π , cf. Equation (3.30). If the self-interaction con- It should be noted that no time derivatives are present
2ij
tributions to the integral in (3.30) vanish, then Π(cid:101)2ij can in Eq. (3.3), so all time derivatives in Eqs. (3.38) and
be obtained from Π by a relabeling of objects only. (3.54) are introduced by above insertions. Indeed, all
2ij
these time derivatives should be understood as abbrevi-
Forthespin-dependentpartofΠ(cid:101)2ij, thiswillturnoutto
be possible. The integral over B hTT in Eq. (3.35) ations and be performed before the equations of motions
(4)ij (7)ij are derived from the Hamiltonians. However, for time
splits into the following five parts, cf. Equation (3.19),
derivatives of primed variables it is irrelevant at which
1 (cid:90) 5 stage they are eliminated (These are actually all time
d3xΠ˙a→a(cid:48)B = Π˙a→a(cid:48)χ , (3.43) derivativesexcepttheoneactingonO ). Oneshouldbe
16πG 1ij (4)ij 16G 1ij (4)ij ij
aware that an insertion of equations of motion leads to a
1 (cid:90) 5
d3xΠ˙a→a(cid:48)B = Π˙a→a(cid:48)χ , (3.44) recombination of PN orders, e.g., inserting the 1PN con-
16πG 2ij (4)ij 16G 2ij (4)ij servative part of the equations of motion leads to 3.5PN
1 (cid:90) d3xΠ¨a→a(cid:48)B = 5 Π¨a→a(cid:48)χ , (3.45) contributions from H2in.5tPN, cf. Equation (3.38). Further,
16πG 3ij (4)ij 16G 3ij (4)ij oneshouldnoticethatΠ(cid:101)2ij,R(cid:48),R(cid:48)(cid:48),Q(cid:48)ij,andQ(cid:48)i(cid:48)j depend
1 (cid:90) d3x.Π..a→a(cid:48)B =(∂ex)3(R(cid:48)+R(cid:48)(cid:48)), (3.46) onbothprimedandunprimedvariablesbyvirtueoftheir
16πG 4ij (4)ij t definitions.
1 (cid:90)
d3xQa→a(cid:48)B =χ˙a→a(cid:48)(Q(cid:48) +Q(cid:48)(cid:48)). (3.47)
16πG ij (4)ij (4)ij ij ij
Notice that here Π , Π , and Π are independent of
1ij 2ij 3ij
x. The relations Eqs. (3.33) and (3.18) were used in the
last integral. The last two integrals were each split into IV. CALCULATION OF THE HAMILTONIANS
two parts using Eq. (3.12) and the following definitions:
1(cid:90) Uptoformal3.5PNorder,theinteractionHamiltonian
R(cid:48) =− d3xT Πa→a(cid:48), (3.48)
2 (4)ij 4ij is given by Eqs. (3.38) and (3.54). The quantities enter-
1 (cid:90) ing these expressions must be calculated by solving the
R(cid:48)(cid:48) =−128πG d3xφ(2),iφ(2),jΠa4i→ja(cid:48), (3.49) integrals appearing in their definitions [see Eqs. (3.17),
(3.28–3.32),(3.42),(3.48–3.51],and(3.53)). Theleading-
1(cid:90)
Q(cid:48) =− d3xT δTTklφa→a(cid:48), (3.50) order source terms in the pole-dipole case entering these
ij 4 (4)kl ij (2) integrals read
1 (cid:90)
Q(cid:48)(cid:48) =− d3xφ φ δTTklφa→a(cid:48). (3.51)
ij 256πG (2),k (2),l ij (2)
(cid:88)
Hmatter = m δ , (4.1)
The fact that the explicit time derivative ∂ex only acts (2) a a
t
a
on primed variables was used in (3.46) to pull it in front
(cid:20) (cid:21)
(cid:88) 1
of the whole expression. Finally, it holds that T(4)ij = m PaiPajδa+Pa(iSa(j))(k)∂kδa , (4.2)
a
1 (cid:90) a
− d3xhTT a→a(cid:48)πij =−χ˙a→a(cid:48)O , 1
16πG (5)ij (5)matter (4)ij ij B(4)ij =−8πGT(4)ij − 8φ(2),iφ(2),j, (4.3)
(3.52)
with the definition
Oij = 161πG(cid:90) d3xπ(i5j)matter. (3.53) smeaes[s1e]sfaonrdmδoare=deδt(axil−s. zˆHa)e.reφ, (m2)ais(ap=rop1o,r2t.io.n.)alatroe tthhee
Newtonian potential of point-masses, namely:
Summing up the contributions from Eqs. (3.39) –
(3.41), (3.43) – (3.47), and the total time derivative of
Eq. (3.52), one gets: φ =−16πG∆−1Hmatter =4G(cid:88)ma , (4.4)
(2) (2) r
a
a
5 (cid:104)
Hint = χ (Π˙a→a(cid:48) +Π˙a→a(cid:48) +Π¨a→a(cid:48))
3.5PN 16G (4)ij 1ij 2ij 3ij
(cid:105)
+χ˙a(4→)iaj(cid:48)(Π1ij +Π(cid:101)2ij)−χ¨a(4→)iaj(cid:48)Π3ij where ra = |x−zˆa|. Notice that φ(2) is independent of
the spins. The expression for B was derived in [1]:
(6)ij
+χ˙a→a(cid:48)(Q(cid:48) +Q(cid:48)(cid:48))+(∂ex)3(R(cid:48)+R(cid:48)(cid:48))
(4)ij ij ij t
d(cid:104) (cid:105)
− χ˙a→a(cid:48)O .
dt (4)ij ij
(3.54)
7
(cid:88)(cid:20) P2 5 P2 1
B =16πG a P P δ + P P φ δ + a P S δ − P P P S δ
(6)ij 4m3 ai aj a 8m ai aj (2) a 4m3 ai a(j)(k) a,k 4m3 al aj ak a(l)(i) a,k
a a a a a
5 (cid:0) (cid:1) 1 1
+ P S φ δ + P S φ δ − P S φ δ
8m ai a(j)(k) (2) a ,k 2m ai a(k)(j) (2),k a 8m ak a(k)(i) (2),j a
a a a (4.5)
1 (cid:16) (cid:17) (cid:21)
+ S Vj +Vk δ
2 a(k)(i) (3),k (3),j a
1 3 5 (cid:16) (cid:17) 1
+ φ φ + φ φ + φ φ φ +2π˜jk π˜k −π˜i +2π˜ij Vk + π˜ij π˜k .
2 1(4) (2),ij 8 2(4) (2),ij 64 (2) (2),i (2),j (3) (3),i (3),k (3),k (3) 2 (3) (3),k
The field quantities entering Eq. (4.5) are equal to: been computed in [14] and reads:
φ1(4) =2G(cid:88)(cid:20)mP2ar + PaiSma(i)(j) (cid:18)r1 (cid:19) (cid:21), (4.6a) hT(4T)isjpin =G(cid:88)a PanSmaa(k)(l)(cid:20)(4δk(iδj)n∂l(cid:21)−2δijδkn∂l)r1a
a a a a a ,j +(δ ∂ ∂ ∂ −2δ ∂ ∂ ∂ )r , (4.8)
φ =−2G2(cid:88)(cid:88)mamb , (4.6b) kn i j l k(i j) n l a
2(4) r r
ab a
a b(cid:54)=a where we use the superscript “spin” to denote the spin-
(cid:20) (cid:18) (cid:19) (cid:21)
π˜i =G(cid:88) 2Pai +S 1 , (4.6c) dependent part of a quantity from now on. In order to
(3) r a(i)(j) r obtain the spin contributions to the radiation-reaction
a a a ,j Hamiltonian up to formal 3.5PN order, we also need to
(cid:20) (cid:18) (cid:19) (cid:21)
Vi =G(cid:88) 2Pai − 1P r +S 1 , computethespinpartofhTT andhTT . hTTspin iseasy
(3) r 4 aj a,ij a(i)(j) r (5)ij (7)ij (5)ij
a a a ,j to compute. From Eqs. (3.28), (4.3), and (4.2), we have
(4.6d)
π˜(i3j) =G(cid:88)a (cid:20)2Pai(cid:18)r1a(cid:19),j +2Paj(cid:18)r1a(cid:19),i χs(4p)inij =−85G(cid:88)a (cid:20)PaiSma(aj)(k) (cid:90) d3x∂kδa(cid:21)STF=0,
(cid:18) (cid:19) (4.9)
1 1
−δ P − P r (4.6e) and thus also hTTspin = 0 [see Eq. (3.18)]. There is no
ij ak r 2 ak a,ijk (5)ij
a ,k spin contribution to the 2.5PN hTT, which is the rea-
(cid:18) 1 (cid:19) (cid:18) 1 (cid:19) (cid:21) ij
son why the leading-order source terms (4.2) are not
−S −S ,
a(k)(i) ra ,kj a(k)(j) ra ,ki sufficient to derive the leading-order radiation-reaction
Hamiltonian. hTTspin would be more difficult to derive,
(6)ij
where r = |zˆ −zˆ |. Notice that for nonspinning sys- butitisnotneededinourcalculationoftheleading-order
ab a b
radiation-reactionHamiltonianwithspins,sowewillnot
tems the result in [2] is reproduced. Further notice that
π˜i does not depend on spin. Finally, the spin correc- discuss it in the present paper.
(3),i Analogous to Eq. (3.19), we decompose the solution
tion to the field momentum is given by:
for hTTspin into several parts,
(7)ij
πij =−(cid:88)4πGP P S δ , (4.7) hTTspin =Π˙spin+Π˙spin+Π¨spin+.Π..spin, (4.10)
(5)matter m2 ak a(i a(j))(k) a (7)ij 1ij 2ij 3ij 4ij
a a
where the following definitions are used:
to the required order. 1 (cid:90)
Πspin = d3xBSTFspin, (4.11)
1ij 5π (6)ij
4G(cid:90)
Πspin =− d3xhTTspinHmatter, (4.12)
2ij 5 (4)ij (2)
A. Spin-dependent part of hTijT Πspin =− 1 (cid:90) d3x(cid:16)Vispinφ (cid:17)STF, (4.13)
3ij 5π (3) (2),j
The explicit solutions for the point-mass, i.e., spin- 2G (cid:90)
Πspin =− δTTij d3x(cid:48)|x−x(cid:48)|2Tspin(x(cid:48),t), (4.14)
independent, contributions to hTijT can be found in [2, 4ij 3 kl (4)kl
8, 11, 57] (but notice that [2] contains some misprints).
The spin part of hTT , arising from the spin-dependent and obviously Qspin = 0, cf. Equations (3.29) – (3.33)
(4)ij ij
source terms in Eq. (4.2) via Eqs. (4.3) and (3.17), has and (4.3). These integrals yield the results:
8
4G2 (cid:88)(cid:88)(cid:26) 1 (cid:104)
Πspin = 3(n ·P )nk (nj S +ni S )−3P (nj S +ni S )
1ij 5 r2 ab b ab ab a(i)(k) ab a(j)(k) bk ab a(i)(k) ab a(j)(k)
a b(cid:54)=a ab
(cid:105) m 1 (cid:104)
−3nk (P S +P S )+4(3ni nj −δ )nk P S + b P (nj S +ni S )
ab bj a(i)(k) bi a(j)(k) ab ab ij ab bl a(k)(l) m r2 ak ab a(i)(k) ab a(j)(k)
a ab
(cid:105) S (cid:104)
+(4δ −6ni nj )nk P S +4nk (P S +P S ) − a(k)(l) (3ni nj −δ )S
ij ab ab ab al a(k)(l) ab aj a(i)(k) ai a(j)(k) r3 ab ab ij b(k)(l)
ab
(cid:105)(cid:27)
+3nk (nj S +ni S )+3(δ −5ni nj )nk nn S , (4.15)
ab ab b(i)(l) ab b(j)(l) ij ab ab ab ab b(n)(l)
Πspin =−4G2 (cid:88)(cid:88) mb 1 (cid:104)−2P (ni S +nj S )+nk (P S +P S )
2ij 5 m r2 ak ab a(j)(k) ab a(i)(k) ab ai a(j)(k) aj a(i)(k)
a b(cid:54)=a a ab (4.16)
(cid:105)
+3(n ·P )nk (ni S +nj S )+(δ +3ni nj )nk P S ,
ab a ab ab a(j)(k) ab a(i)(k) ij ab ab ab al a(k)(l)
Πspin = 4G2 (cid:88)(cid:88)mbnk (nj S +ni S ), (4.17)
3ij 5 r ab ab a(i)(k) ab a(j)(k)
ab
a b(cid:54)=a
Πs4pijin = 41G5 (cid:88)mra (cid:104)Pak(njaSa(i)(k)+niaSa(j)(k))−2nka(PajSa(i)(k)+PaiSa(j)(k)+δijPalSa(k)(l))(cid:105), (4.18)
a
a
where n =(x−zˆ )/r and n =(zˆ −zˆ )/r . Notice spin-orbit and spin(1)-spin(2) equations of motion [28–
a a a ab a b ab
that it holds 30], provided in this paper by Eqs. (5.2) and (5.3) later
on].
4G
Πspin+Πspin+Π˙spin =− Ispin, (4.19) The spin part of the formal 3.5PN order interaction
1ij 2ij 3ij 5 ij
Hamiltonian Eq. (3.54) can be written as:
at the considered PN order, where I is a multipole mo-
ij 5 (cid:104)
ment of the far-zone expansion of hTT and can be ex- Hintspin = (Π˙spin+Π˙spin+Π¨spin)a→a(cid:48)χ
pressed as a double time derivativeiojf a very compact 3.5PN 16G 1ij 2ij 3ij (4)ij
(cid:105)
expression, see Eqs. (6.15) and (6.18) in [1]. +χ˙a→a(cid:48)(Πspin+Π(cid:101)spin)−χ¨a→a(cid:48)Πspin
(4)ij 1ij 2ij (4)ij 3ij
+χ˙a→a(cid:48)Q(cid:48)spin+(∂ex)3(R(cid:48)spin+R(cid:48)(cid:48)spin)
(4)ij ij t
B. Derivation of spin contributions to 2.5PN and d(cid:104) (cid:105)
− χ˙a→a(cid:48)Ospin , (4.22)
3.5PN interaction Hamiltonians dt (4)ij ij
When taking into account the fact that Eq. (4.9) tells where we used χspin = 0 and Q(cid:48)(cid:48)spin = 0. The latter
(4)ij ij
us that χspin = 0, we immediately see that the formal is trivial from Eq. (3.51), as only the spin-independent
(4)ij
2.5PN order interaction Hamiltonian Eq. (3.38), potential φ appears there. Πspin, Πspin, Πspin, and
(2) 1ij 2ij 3ij
Πspin were already derived in Sec. IVA. The missing
5 4ij
H2in.5tPN = 16Gχ˙a(4→)iaj(cid:48)χ(4)ij, (4.20) quantities Π(cid:101)s2pijin, Q(cid:48)isjpin, R(cid:48)spin, R(cid:48)(cid:48)spin, and Oisjpin can be
obtained from:
hasonlythewell-knownpoint-masscontribution[57,58]:
4G(cid:90)
Π(cid:101)spin =− d3xhTTspina→a(cid:48)Hmatter. (4.23)
4G(cid:88)(cid:20) 2 2ij 5 (4)ij (2)
χ(4)ij = 15 a ma(P2aδij −3PaiPaj) (cid:21) (4.21) Q(cid:48)isjpin =−14(cid:90) d3xT(s4p)iknlδiTjTklφa(2→)a(cid:48), (4.24)
−G(cid:88)mamb(δ −3ni nj ) , 1(cid:90)
r ij ab ab R(cid:48)spin =− d3x(T Πa→a(cid:48))spin, (4.25)
b(cid:54)=a ab 2 (4)ij 4ij
1 1 (cid:90)
but no direct spin contribution. However, indirect spin- R(cid:48)(cid:48)spin =− d3xφ φ Πspina→a(cid:48), (4.26)
816πG (2),i (2),j 4ij
contributions arise from Eq. (4.20) via the time deriva-
1 (cid:90)
tive therein and first appear at the formal 3.5PN level Ospin = d3xπij spin , (4.27)
[after taking into account the leading-order conservative ij 16πG (5)matter
9
1(cid:90)
using Eqs. (3.42), (3.48) – (3.51), and (3.53). We also R(cid:48)spin =− d3xTspinΠspina→a(cid:48). (4.31)
split R(cid:48)spin into three parts, 3 2 (4)ij 4ij
R(cid:48)spin =R(cid:48)spin+R(cid:48)spin+R(cid:48)spin, (4.28) Here, PM denotes the point-mass parts of a function.
1 2 3
1(cid:90) The quantities entering above integrals will be all given
R(cid:48)spin =− d3xTspinΠPMa→a(cid:48), (4.29) in the present paper, except for ΠPM, which can be read
1 2 (4)ij 4ij 4ij
1(cid:90) from Eq. (36) in [2] using Rij = ∂t3Π4ij. The results of
R(cid:48)spin =− d3xTPMΠspina→a(cid:48), (4.30) the above integrations read as follows:
2 2 (4)ij 4ij
Π(cid:101)s2pijin =−4G52 (cid:88)mma r21 (cid:104)2Pa(cid:48)k(niaa(cid:48)Sa(cid:48)(j)(k)+njaa(cid:48)Sa(cid:48)(i)(k))−nkaa(cid:48)(Pa(cid:48)iSa(cid:48)(j)(k)+Pa(cid:48)jSa(cid:48)(i)(k))
a,a(cid:48) a(cid:48) aa(cid:48) (4.32)
(cid:105)
−3(n ·P )nk (ni S +nj S )−(δ +3ni nj )nk P S ,
aa(cid:48) a(cid:48) aa(cid:48) aa(cid:48) a(cid:48)(j)(k) aa(cid:48) a(cid:48)(i)(k) ij aa(cid:48) aa(cid:48) aa(cid:48) a(cid:48)l a(cid:48)(k)(l)
Q(cid:48)spin = G(cid:88)ma(cid:48) 1 (cid:104)2P (ni S +nj S )−nk (P S +P S )
ij 4 m r2 ak aa(cid:48) a(j)(k) aa(cid:48) a(i)(k) aa(cid:48) ai a(j)(k) aj a(i)(k)
a,a(cid:48) a aa(cid:48) (4.33)
(cid:105)
−3(n ·P )nk (ni S +nj S )−(δ +3ni nj )nk P S ,
aa(cid:48) a aa(cid:48) aa(cid:48) a(j)(k) aa(cid:48) a(i)(k) ij aa(cid:48) aa(cid:48) aa(cid:48) al a(k)(l)
R(cid:48)spin = G (cid:88)S (cid:18) 4ra(cid:48)a (cid:104)P2 ni P −(n ·P )P P −2(P ·P )ni P (cid:105)
1 15 a(i)(j) m m a(cid:48) a(cid:48)a aj a(cid:48)a a(cid:48) a(cid:48)i aj a(cid:48) a a(cid:48)a a(cid:48)j
a(cid:48) a
a,a(cid:48)
+ G (cid:88) ma(cid:48)mb(cid:48)(cid:26)17ni P − 2ra(cid:48)a(cid:104)17(n ·P )ni nj +7n iP (cid:105) (4.34)
7 m a(cid:48)b(cid:48) aj r a(cid:48)b(cid:48) a a(cid:48)b(cid:48) a(cid:48)a a(cid:48)a aj
a a(cid:48)b(cid:48)
b(cid:48)(cid:54)=a(cid:48)
6r2 (cid:104) (cid:105) 8r (cid:104) (cid:105)(cid:27)(cid:19)
+ a(cid:48)a ni P +2(n ·P )ni nj + a(cid:48)a r2 ni P −r2 ni P ,
r2 a(cid:48)b(cid:48) aj a(cid:48)a a a(cid:48)b(cid:48) a(cid:48)a r3 a(cid:48)a a(cid:48)a aj b(cid:48)a a(cid:48)a aj
a(cid:48)b(cid:48) a(cid:48)b(cid:48)
R2(cid:48)spin = 41G5 (cid:88)mraam(cid:48) Sa(cid:48)(i)(j)(cid:104)P2aniaa(cid:48)Pa(cid:48)j −2(Pa(cid:48) ·Pa)niaa(cid:48)Paj +(naa(cid:48) ·Pa)Pa(cid:48)iPaj(cid:105), (4.35)
a(cid:48) a
a,a(cid:48)
(cid:20) (cid:21)
4G(cid:88) (cid:88) 1 3
R(cid:48)spin = S P P S −(P ·P )S −P P S , (4.36)
3 15 m m a(i)(j) 2 a(cid:48)k ai a(cid:48)(k)(j) a(cid:48) a a(cid:48)(i)(j) a(cid:48)i ak a(cid:48)(k)(j)
a(cid:48) a
a a(cid:48)(cid:54)=a
R(cid:48)(cid:48)spin = 21G52 (cid:88)(cid:88)mmambrra(cid:48)aSa(cid:48)(i)(j)(cid:104)na(cid:48)aiPa(cid:48)j −2(nab·Pa(cid:48))nia(cid:48)anjab−(na(cid:48)a·nab)nabiPa(cid:48)j(cid:105), (4.37)
a(cid:48) ab
a,a(cid:48)b(cid:54)=a
Ospin =(cid:88) 1 P (P S +P S ). (4.38) Noticethatinthisexpressionnoregularizationisneeded
ij 8m2 ak ai a(k)(j) aj a(k)(i) for taking x = zˆ , as primed and unprimed objects are
a a a
not identified yet. In contrast to that there may be con-
The term in Eq. (4.34) containing 17nia(cid:48)b(cid:48)Paj actually tributions from Hadamard regularization in Eq. (4.39).
cancelsifthesumsovera(cid:48) andb(cid:48) areperformedandmay However, for the spin-dependent part, no such contri-
therefore be dropped. butions appear (in contrast to the nonspinning case in
NoticethatΠs2pijin,Π(cid:101)s2pijin,andQ(cid:48)isjpinaregivenbyalmost [2]), which explains the great similarity between Πs2pijin
identical expressions, cf. Equations (4.16), (4.32), and and Π(cid:101)spin. Further, insertion of Eq. (4.4) into Eq. (4.24)
(4.33). This is not accidental, but due to similarities of 2ij
leads to:
their defining integrals. With the source mass density
given by Eq. (4.1), we obtain from Eq. (4.12): Q(cid:48)spin =−1(cid:90) d3xhTTspin(Hmatter)a→a(cid:48),
ij 4 (4)ij (2)
4G(cid:88) (cid:12)
Πs2pijin =− 5 a ma hT(4T)ijspin(cid:12)(cid:12)x=zˆa . (4.39) =−14(cid:88)a(cid:48) ma(cid:48) hT(4T)ijspin(cid:12)(cid:12)(cid:12)x=zˆa(cid:48) , (4.41)
Similarly, Eq. (4.23) leads to
after performing several partial integrations and using
Π(cid:101)s2pijin =−45G(cid:88)a ma (hT(4T)ijspin)a→a(cid:48)(cid:12)(cid:12)(cid:12)x=zˆa . (4.40) Eneqesd.e(d4..3)Thaendsim(3i.l1a7ri)t.y tHoerEeqsa.ls(o4.n23o)roergu(l4a.r4i0za)tiiosnobis-
10
vious. The difference is simply an overall factor and a WesubstituteEqs.(4.20)and(4.22)intoEq.(5.1)(For
mutual exchange of primed and unprimed variables. the point-mass part of Hint this was already done in
3.5PN
[2]). Afterthat,weneedtoeliminatethetimederivatives
in Eq. (5.1) using the leading-order spin-orbit, spin(1)-
V. ENERGY LOSS OF A BINARY SYSTEM spin(2),andNewtonianequationsofmotionderivedfrom
thecorrespondingHamiltonians[see,e.g.,Eqs.(7.28)and
A. Derivation of the energy loss from the (7.29) in [14]),
Hamiltonian
The instantaneous (near-zone) energy loss of a two- zˆ˙i = pi1(cid:48) − G nj1(cid:48)2(cid:48) (cid:0)3m S +4m S (cid:1) ,
body system due to gravitational radiation can be writ- 1(cid:48) m 2m r2 2(cid:48) 1(cid:48)(j)(i) 1(cid:48) 2(cid:48)(j)(i)
1(cid:48) 1(cid:48) 1(cid:48)2(cid:48)
ten in the form (see, e.g., [2, 11]): (5.2a)
Li≤n3st.5PN =−∂tex(H2in.5tPN+H3in.5tPN). (5.1) zˆ˙2i(cid:48) =(1(cid:10)2), (5.2b)
Notice that this energy loss is gauge-dependent in con-
trast to the energy flux at infinity.
Gm m Gni (cid:40)9m (cid:20) (cid:21) (cid:20) (cid:21)
p˙i =− 1(cid:48) 2(cid:48)ni + 1(cid:48)2(cid:48) 2(cid:48) (p ×S )·n −6 (p ×S )·n
1(cid:48) r2 1(cid:48)2(cid:48) r3 2m 1(cid:48) 1(cid:48) 1(cid:48)2(cid:48) 2(cid:48) 1(cid:48) 1(cid:48)2(cid:48)
1(cid:48)2(cid:48) 1(cid:48)2(cid:48) 1(cid:48)
(cid:20) (cid:21)
15 3
+ (n ·S )(n ·S )− (S ·S )+6 (p ×S )·n
r 1(cid:48)2(cid:48) 1(cid:48) 1(cid:48)2(cid:48) 2(cid:48) r 1(cid:48) 2(cid:48) 1(cid:48) 2(cid:48) 1(cid:48)2(cid:48)
1(cid:48)2(cid:48) 1(cid:48)2(cid:48)
(cid:20) (cid:21)(cid:41)
9m
− 1(cid:48) (p ×S )·n (5.3a)
2m 2(cid:48) 2(cid:48) 1(cid:48)2(cid:48)
2(cid:48)
(cid:40)
G 3m 3
+ − 2(cid:48) (p ×S )i+2(p ×S )i− (n ·S )Si −2(p ×S )i
r3 2m 1(cid:48) 1(cid:48) 2(cid:48) 1(cid:48) r 1(cid:48)2(cid:48) 2(cid:48) 1(cid:48) 1(cid:48) 2(cid:48)
1(cid:48)2(cid:48) 1(cid:48) 1(cid:48)2(cid:48)
(cid:41)
3m 3
+ 1(cid:48) (p ×S )i− (n ·S )Si ,
2m 2(cid:48) 2(cid:48) r 1(cid:48)2(cid:48) 1(cid:48) 2(cid:48)
2(cid:48) 1(cid:48)2(cid:48)
p˙i =(1(cid:10)2). (5.3b) pi =(1(cid:10)2), (5.4b)
2(cid:48) 2
toexpresstheparticlemomentap intermsoftheparti-
a
Note that because the 2.5PN order Hamiltonian does clecoordinatevelocitiesva,whichcanbeeasilyobtained
not have spin contributions as we discussed in Sec. IV, from Eq. (5.2). Here, r =r12, and n=n12. Note we do
we do not include the 1PN point-mass terms because not include the 1PN point-mass terms in this expression
substituting them into the 2.5PN Hamiltonian only pro- for the reason described above.
duces point-mass terms at 3.5PN order, while substi- To put the energy loss into a more convenient form,
tuting them into the 3.5PN Hamiltonian only produces we rewrite the individual masses m1,m2 into the total
4.5PN terms which is beyond the scope of this paper. mass of the system M ≡ m1 + m2, the reduced mass
µ≡m m /M, and the symmetric mass-ratio parameter
1 2
Atthispoint, wenolongerneedtodistinguishthedif- η ≡µ/M using the relations (assuming m ≥m ):
1 2
ference between the primedand unprimed variables. Us-
ingtheHadamardregularizationmethod, weremovethe µ (cid:16) (cid:112) (cid:17)
m = 1+ 1−4η , (5.5a)
singularitiesproducedbythelimitzˆ →zˆ andzˆ →zˆ 1 2η
1(cid:48) 1 2(cid:48) 2
and obtain an expression of the energy loss in terms of µ (cid:16) (cid:112) (cid:17)
m = 1− 1−4η . (5.5b)
zˆ and p . By realizing that zˆ˙ ≡v , we may use 2 2η
1(2) 1(2) a a
We also transform the individual coordinate velocities
of each particle into the center of mass frame using the
pi =m vi − Gnj (cid:0)3m S +4m S (cid:1) , (5.4a) relations:
1 1 1 2 r2 2 1(j)(i) 1 2(j)(i)