A skeleton approximate solution of the Einstein field equations for multiple black-hole systems

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by  G. Faye

A skeleton approximate solution of the Einstein field equations
for multiple black-hole systems
Guillaume Faye,1,∗ Piotr Jaranowski,2,† and Gerhard Scha¨fer1,‡
1Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at, Max-Wien-Platz 1, 07743 Jena, Germany
2Institute of Theoretical Physics, University of Bial ystok, Lipowa 41, 15-424 Bial ystok, Poland
(Dated: February 7, 2008)
An approximate analytical and non-linear solution of the Einstein field equations is derived for
a system of multiple non-rotating black holes. The associated space-time has the same asymptotic
structure as the Brill-Lindquist initial data solution for multiple black holes. The system admits
an Arnowitt-Deser-Misner (ADM) Hamiltonian that can particularly evolve the Brill-Lindquist so-
lution over finite time intervals. The gravitational field of this model may properly be referred
to as a skeleton approximate solution of the Einstein field equations. The approximation is based
on a conformally flat truncation, which excludes gravitational radiation, as well as a removal of
4 some additional gravitational field energy. After these two simplifications, only source terms pro-
0 portionaltoDiracdeltadistributionsremainintheconstraintequations. TheskeletonHamiltonian
0 is exact in thetest-body limit, it leads to theEinsteinian dynamics up to the first post-Newtonian
2 approximation, and in the time-symmetric limit it gives the energy of the Brill-Lindquist solution
n exactly. Theskeletonmodelforbinarysystemsmayberegardedasakindofanalyticalcounterpart
a to thenumerical treatment of orbiting Misner-Lindquist binary black holes proposed by Gourgoul-
J hon, Grandcl´ement, and Bonazzola, even if they actually treat the corotating case. Along circular
0 orbits,thetwo-black-holeskeletonsolution isquasi-stationaryanditfulfillstheimportantproperty
2 ofequalityofKomarandADMmasses. Explicitcalculationsforthedeterminationofthelaststable
circular orbit of the binary system are performed up to the tenthpost-Newtonian order within the

2 skeleton model.
v
8 PACSnumbers: 04.25.-g,04.20.Fy,04.25.Nx
1
0
1 I. INTRODUCTION sentially perform two simplifications: on the one hand,
1 we adopt the often-used assumption of conformally flat
3 space metric; on the other hand, we attribute the non-
The description of the motion of binary black holes
0 linear gravitational field energy to point-like sources,
within the Einsteinian theory of gravity is quite a chal-
c/ lenging problem and is likely to be finally solvable by whicharemodeledasDiracdeltadistributions. Wesolve
q the resulting truncated Einstein equations analytically
numerical means only. However, it seems that the full
- and get what we call, in reference to the point-like char-
r success will occur in a far future. It has turned out in
g acter of the source support, the skeleton solution. The
the past that analytical developments did always influ-
: calculations are carried out for bodies in a mathe-
v ence numerical investigations, hence further progress on N
matically sound way by working from the beginning in
i theanalyticalsideiscertainlydesirable. Inthispaperwe
X (d+1)-dimensionalspace-timeandtakingthelimitd 3
shallpresentananalyticalsolutionoftruncatedEinstein →
r in the final expressions, so that the Dirac delta distribu-
a equations for systems of non-spinning point-like objects,
tions modeling the sources are consistently handled [7].
which shows many aspects of past numerical models.
Our space metric and extrinsic curvature may be
Most of the numerical computations of quasi-
regarded as some generalization of the multiple black
stationaryinitialdataforblack-holebinariesarebasedon
hole Brill-Lindquist initial value solution [5] to arbitrary
the assumption that the space metric is conformally flat
black-hole momenta preserving its original asymptotic
[1, 2, 3]. They often assume as well the existence of an
structure. Moreover,the energy function can be used as
approximatehelicalKillingvectorwhichpermitstoelab-
generator of the dynamical evolution of the system. We
orate some manageable formulations (also see [4]). Re-
shallrefertoitastheskeletonHamiltonianassociatedto
cently, the time-symmetric initial data of Brill-Lindquist
the full Arnowitt-Deser-Misner (ADM) Hamiltonian. It
[5] were generalized by adding a non-conformally flat
agrees with the exact Hamiltonian (i) in the test body
contributionthatincorporatespieces ofinformationpro-
limit, (ii) at the first post-Newtonian (1PN) approxima-
vided by perturbative post-Newtonian calculations [6].
tion, i.e., at the first order in powers of 1/c2 (c denoting
The present paper does not follow this line, but rather
the speedoflight)beyondthe Newtoniandynamics,and
keeps the spirit of former numerical simulations. We es-
(iii) inthelimitofspecialrelativityinabsenceofgravity.
Theskeletongravitationalfieldisgivenexplicitly,aswell
as the equations of motion.
As a first application, we shall investigate the loca-
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] tion of the last stable (circular) orbit (LSO) for binary
‡Electronicaddress: [email protected] systems. In order to reach a proper accuracy, the last-
2
stable-orbit parameters are worked out up to the tenth c4
H = ddx N +Ni
post-Newtonian (10PN) approximationwithin the skele- 16πG H Ji
ton model. The comparison of second and third post- Z
(cid:0) (cid:1)
Newtonian results derived from the skeleton model with
+ ddx ∂ (∂ γ ∂ γ ) , (2.1)
the corresponding quantities computed in the full Ein- i j ij − i jj !
steintheorydeliversaquantitativemeasureforthemiss- Z
ing terms in the skeleton approach. where ∂ ∂/∂xi. The super-Hamiltonian density
i
≡ H
appearing under the integral sign is defined by
1 1
II. DESCRIPTION OF THE BLACK-HOLE R√γ+ πiπj (πi)2
DYNAMICS IN ADM FORMALISM H≡− √γ (cid:18) j i− d−1 i (cid:19)
16πG N γijp p 21
+ m2 + Ai Aj δ , (2.2a)
We are interested in a system of black holes inter- c2 A c2 A
acting gravitationally. Its descriptiNon will be achieved AX=1(cid:18) (cid:19)
within the ADM Hamiltonian frameworkusing a (d+1)
while the supermomentum density is given by
i
splitting of space-time. The spatial indices are denoted J
byLatinlettersandvaryfrom1to d,whereasthespace- N
16πG
ctihmaret,in(dxic=esxair,etG=reexk0/acn)d, tvhaerymfortoimon0otfothde. IAntha bgilvaecnk Ji ≡−2∂jπji+πkl∂iγkl− c3 A=1pAiδA. (2.2b)
X
hole is described by a curve xi (t), for A = 1, ..., .
A N InEq.(2.2a)Rdenotesthespacecurvatureofthehyper-
The conjugate momentum of the position xi is called
A surface t=const and γ is the determinant of the matrix
p . The constantmass parameter of the Ath black hole
Ai
γ . As the lapse andshift functions areLagrangemulti-
is denotedbym . The Hamiltonianvariablesassociated ij
A
pliers,thevariationoftheHamiltonian(2.1)withrespect
tothegravitationalfieldaretakentobe(i)thespacemet-
to them leads to the constraint equations
ric γ induced by the full space-time metric g on the
ij µν
hypersurfacet=const,and(ii)itsconjugatemomentum
=0, =0. (2.3)
(c3/16πG)πij where G is the Newtonian gravitational H Ji
constant. Theothercomponentsofgµν areparametrized Before solving Eqs. (2.3), we must first fix the coor-
by the lapse and shift functions: N (g0ig0jγij g00)12 dinate system. It is convenient to work in the so-called
≡ −
and N g , with γij standing for the inverse of γ . It ADM transverse trace-free (ADMTT) gauge [16, 17] de-
i 0i ij
is also u≡seful to introduce the shift vector Ni γijN . fined by the two conditions
j
≡
The spatial indices will be raised and lowered with the
help of γij and γij, respectively. ∂ γ 1δ γ =0, πii =0. (2.4)
We now assume that the “center-of-field” x (t) of ob- j ij − d ij kk
A (cid:18) (cid:19)
ject A can be modeled in the fictitious space-time asso-
In this gauge, the metric decomposes into its trace γ ,
ciated with the flat metric by means of the Dirac distri- kk
bution δ δ(x x ). This prescription is known to which may be parametrized as dΨ4/(d−2) (in order to
A ≡ − A get a simple expressionfor the curvature density), and a
lead to the Brill-Lindquist Hamiltonian when the black
transverse trace-free part hTT. We thus have
hole velocities are set to zero, provided the mass param- ij
eters m , A = 1, ..., , are identified with the Brill-
LindquiAst masses [8, 9].NIt also leads to the correct 2PN γij =Ψd−42δij +hTijT, ∂jhTijT =0, hTiiT =0, (2.5a)
dynamics [10, 11] for compact binaries, which can itself
πii =0. (2.5b)
be recovered by means of the extended-body approach
[12]. Moreover, the singularities generated by the dis-
The momentum πij needs itself to be split into a trans-
tributions δA have shown to be curable by dimensional verse trace-free contribution πij , and a rest π˜ij,
regularization at the 3PN order [7]. Similarly, we shall TT
seeinthe presentworkthat,foranappropriatechoiceof πij =π˜ij +πij . (2.6)
the spacedimensiond, theδ -type sourcesdonotgener- TT
A
ate any ambiguity, so that the 3-dimensional quantities These two terms are uniquely defined assuming they
can be obtained by dimensional regularization. Finally, decay as 1/(xixi)d−21 at spatial infinity and are given
this point-particle approachallowsto bypass delicate in-
as a combination of derivatives of the Poisson integral
vestigations about the behavior of the gravitationalfield ∆−1∂ πij, and the Poisson integral of the latter quan-
j
near the objects. tity, namely ∆−2∂ πij.
j
Inanasymptoticallyflatspace-timeofdimensiond+1, We can then try to compute Ψ and π˜ij from the two
theHamiltonianof pointmassesprovidedbytheADM constraintequations(2.3)supplementedbytheboundary
N
formalism reads [13, 14, 15] conditions at spatial infinity resulting from the asymp-
totic flatness of space-time; notably Ψ 1 (xkxk)d−22,
− ∼
3
π˜ij∼1/(xkxk)d−21. It is actually more convenient to hTijT to be zero on Σ0 is always possible, though it is not
solve the Hamiltonian constraint for the quantity φ physically realistic in generalfor isolatedself-gravitating
≡
4(d−1)(Ψ 1) rather than Ψ to get rid off the spurious systems. Our attitude in this instance will be to accept
codn−s2tant a−t spatial infinity. We thus have the resulting error. In return, we shall be able to deter-
mine the fieldγ without assumingzerovelocitiesofthe
ij
d 2 bodies at initial time.
Ψ=1+ − φ.
4(d 1) We thus demand here that
−
After solving the constraint equations (2.3), we insert hTijT =0, (3.1)
the ensuingexpressionsforφandπ˜ij into the right-hand
and the problem reduces to solve the constraints =0,
side of Eq. (2.1). We get a reduced Hamiltonian depend- H
=0underthiscondition. Thespacemetricγ isthen
ing on xi , p , hTT, and πij only: Ji ij
A Ai ij TT proportional to δ , which implies that in the supermo-
ij
mentum the term πkl∂ γ vanishes [cf. Eq. (2.2b)].
Hred =Hred[xiA,pAi,hTijT,πTijT] The fieldJmiomentum πij reilaktles to πi through the iden-
j
c4 tities
= ddx ∂ (∂ γ ∂ γ )
i j ij i jj
16πcG4 Z − πij =γjkπik =Ψ−d−42πji. (3.2)
= ddx ∆φ. (2.7)
−16πG In particular,πi is symmetric and trace-free(STF) as is
Z j
πij. The determinant γ appearing in is merely
ThisHamiltoniancontainsthefullinformationaboutthe H
evolution of the matter (xiA,pAi) and field (hTijT,πTijT) γ =Ψd4−d2 . (3.3)
variables[14, 17]. Asthe changeofvariables(γ ,πij)
ij
(hTT,πij ) is not canonical, the Poissonbrackets of hT→T Finally,the 3-curvaturedensityR√γ takesaremarkably
ij TT ij simple form [22]:
and πij take a form involving a transverse trace-free
TT
projection operator [14, 17]. R√γ = Ψ∆φ. (3.4)
−
Collecting these relations together with the constraint
III. SKELETON HAMILTONIAN equations (2.3), we find:
The reduced Hamiltonian Hred provides the equations ∆φ=−Ψ−3dd−−22πijπji
of motion of the bodies together with the evolution
ecqouupatleiodnistoisfhthoepeglNreasvsittoatsioolnvaeltfiheelmd.aAnasltyhteicyalalyre,bsutrtotnhgelyy − 16cπ2G N mAδAΨ−1 1+Ψ−d−42mp22Ac2 12 ,
A=1 (cid:18) A (cid:19)
can be solved perturbatively using, notably, the post- X (3.5a)
Newtonian iterative scheme [17, 18, 19]. The resulting
N
dynamics is highly reliable as long as the typical speeds ∂ πj = 8πG p δ , (3.5b)
remainsmallerthan,say,about0.3c. Whenbinaryblack j i − c3 Ai A
A=1
holes reach their last stable orbits, the reliability of the X
approximation breaks down since the system enters a with p2 p p .
strongfieldregime. Theonlywaytounderstandprecisely InthAe≡systAeimA(i3.5)thesecondequationdecouplesfrom
this crucial stage is to perform numerical simulations in the first one and is solved by searching for a particular
full general relativity. Our purpose here is to search for solutionproportionalto the symmetric trace-freepartof
a simple non-perturbative model applicable to arbitrary the derivative of a certain vector potential V ; namely,
i
strong fields, eventually leaving the Einsteinian theory
2
but always staying close enough to it, in order to give πi (part) =STF(2∂ V ) ∂ V +∂ V δ ∂ V . (3.6)
an element of comparison with numerical investigations. j i j ≡ i j j i−d ij k k
The space metric obtained in our approach should also
The equation for the divergence ∂ V is obtained by ap-
k k
be usable as initial condition.
plying the operator ∂ ∂ ∂ simultaneously on the
ij i j
We shall adopt here a Wilson-Mathews-type prescrip- two sides of (3.6), henc≡e 2∆∂ V = d/(d 1)∂ πi.
tion [20, 21], setting the non-conformallyflatpartof γ , k k − ij j
ij This yields immediately ∂ πj = ∆V + (d 1)/(d
namely hTT, equal to zero. As hTT is of order 1/c4, the j i i 2 − −
ij ij 1)∆−1∂ πj, and from the constraint (3.5b) it follows
differencebetweenourγij andtheEinsteinianspacemet- ijk k
ricshowsupatthe 2PNlevel. Forsphericallysymmetric that:
space-times the difference is zero. In the case where the N
8πG d 2
presentmodelisusedto fixinitialCauchydata,the con- ∆V = p δ − p ∂ ∆−1δ .
dition of a conformally flat space metric needs to hold i − c3 A=1(cid:18) Ai A− 2(d−1) Aj ij A(cid:19)
X
on only one hypersurface, say Σ : t = t . Choosing (3.7)
0 0
4
We now notice that the d-dimensional Laplacian of the The integration of the Hamiltonian constraint (3.5a)
mth power of rA ≡[(xi−xiA)(xi−xiA)]12 satisfies is much more involved, principally because of the ap-
pearance of φ in the source and the presence of a non-
∆rm =m(m 2+d)rm−2 (3.8) compact support term. On the contrary, the compact
A − A
support term is harmless. It is made of a sum of Dirac
in the sense of functions, and that, in the sense of distri-
distributions δ that formally determine the boundary
A
butions,
conditionsnearthe blackholes. Bysubtractingfromthe
d conformal factor the 1/rd−2 terms entering φ, we get an
∆r2−d = 4π2 δ . (3.9) object that satisfies an eqAuation whose source is entirely
A −Γ d−22 A free of δA’s (this is the essence of paper [24]). By con-
trast, we shall keep here the constraint equation (3.5a)
As a result we may choose: (cid:0) (cid:1)
under its original form in order to determine the depen-
GΓ(d−2) N 4p (d 2)p denceofthecoefficientof1/rAd−2 onthepositionsxiA and
Vi = c3 2πd2−22 A=1 rAd−A2i − (d−−1)(4−Ajd)∂ijrA4−d!, moTmoenstoalvpeAit.he Hamiltonian constraint (3.5a) itera-
X
(3.10) tively, we have to face the delicate computation of
whichisnothingbutthe1PNvectorpotentialV(i3) ofRef. ∆−1 Ψ2d−−32dπijπji . The expansion of the prefactor
[7]Ovuerrifsyeicnogndπijsim=pSlTifiFca(2ti∂oinV(cj3o))n+sisOts(1in/cs5t)a.ting that the Ψ2d−−32(cid:16)d = 1+ d(cid:17)−2 φ 2d−−32d in powers of φ is known to
4(d−1)
homogeneous solution of Eq. (3.5b), πi (hom) πi give rise t(cid:16)o poles when(cid:17) d 3 at the level correspond-
j ≡ j − →
πi (part), is precisely zero, which implies πi TT = 0. ing to the 3PN approximation [7]. In full general rela-
j j tivity, these poles cancel in the Hamiltonian with simi-
When this prescription is imposed on one unique hyper-
lar quantities coming from hTT, but such a cancellation
surface, it merely represents a particular choice of initial ij
cannot occur in our case. In particular, the 3PN ap-
data. The resultingfieldmomentumπi =STF(2∂ V )is
j i j proximation of the Wilson-Mathews model does not ex-
identical to the rescaled extrinsic curvature proposed by
ist for binary black-holes. We must therefore treat the
BowenandYorkintheirtime-asymmetricformulationof
π2 term differently. Let us first note that this term
the Cauchy problem [23] and adopted e.g. in the punc-
enters the space metric at the 3PN order only, since
turemethod[24]orvariouscalculationsofthelaststable
πiπj = (1/c6), but contributes to the 1PN Hamil-
circular orbit [25, 26, 27]. Although it is a mathemati- j i O
tonian [because of the global factor c4/(16πG) in Eq.
calsolutionofthe constraintequationsfora conformally
(2.7)]. To be consistent with the accuracy of the for-
flatmetric,itis notrealisticinthe sensethatitdoesnot
merassumptionhTT =0whichamountstoneglectsome
result from an evolution process in absence of incoming ij
2PN corrections in the Hamiltonian, we have to keep
radiation. It does not coincide either with the Wilson-
Mathewsmomentum(πi) derivedfromthecondition at least the leading term in the post-Newtonian expan-
j WM 2−3d
h˙TT =0: sion of 1+ d−2 φ d−2 πiπj, modulo a possible total
ij 4(d−1) j i
spacede(cid:16)rivative. Now(cid:17),thetensordensityπi isitselfpro-
1 j
(πi) =(1+φ)πi (part)+(d 1)π˜ij + , (3.11) portional to the symmetric trace-free part of the space
j WM j − (5) O c6
(cid:18) (cid:19) derivative ∂iVj, so that
where the quantity π˜(i5j), defined as the 1/c5 part of πijπji =2STF(πij)STF(∂iVj)=2πij∂iVj.
the solution [matching the form (3.6)] of the equation
(d 1)∂ π˜ij = ∂ (φπj(part)), is explicitly given in pa- Asaconsequence,thelatterexpressioncanbesplitintoa
− j (5) − j i “skeleton” term 2V ∂ πi with compact support, and a
per [18] for d = 3. The difference (πij)WM−πij (part) is “flesh”term∂i(2−Vjπijj)iwhjichtakesintoaccountthefield
onlyoforder1/c5,whichcorrespondstothe2PNapprox- between the point particles. The flesh term contributes
imation. On the Hamiltonian level, it is of 3PN order to the Hamiltonian through the quantity:
only. One might then argue that the Wilson-Mathews
hc˙oTnTdi=tio0n, ihsTiijnT c=on0tr,avdailcitdiofnortoaltlhteimcoens,diit.ieo.npπaritTicTul=arl0y. 16cπ4G ddxΨ2d−−32d∂i(2Vjπij)
ij j Z
However, it is not the case because the momentum con-
straint equation is linear in πij and decouples from any = 32cπ4G3dd−12 ddx Ψ4(d1−−2d)Vjπij∂iφ .
otherfieldequationwhenthespacemetricisconformally − Z !
flat,sothatthequantityobtainedbyaddinganarbitrary
We are allowed to regard it as negligible in our scheme
homogeneous solution to πi still satisfies the constraint.
j andkeeptheskeletontermaloneinthedecompositionof
As our πij differs from (πij)WM by such a homogeneous πiπj. Ourthirdsimplificationconsitsthusinperforming
solution (i.e. a transverse-trace free object), there is no j i
the substitution
contradiction between the two approaches.
5
Ψ2d−−32dπijπji →−2Ψ2d−−32dVj∂iπij and the conformal factor reads
= 16cπ3GΨ2d−−32d N pAjVjδA. (3.12) Ψ=1+Ad N rαd−A2 , (3.14)
AX=1 AX=1 A
After the replacement (3.12) is made, the right-hand
where we have posed, in order to lighten the notation,
side ofEq.(3.5a)takesthe form f (x)δ , wherethe
A A A
f ’s are some unknown functions. For d belonging to
aAn appropriate range of values, fPA is regular at x=xA, A Gd−2Γ d−22 .
and ∆φ = AfA(xA)δA. We see that in our model the d ≡ c2d−1 π(cid:0)d−22 (cid:1)
source of the Hamiltonian constraint reduces to some
P
“skeleton” made of a linear combination of Dirac dis- It is also useful to introduce the (regularized)value of Ψ
tributions. For this reason, all quantities computed in at the particle position x :
A
the presentapproximationwill be referredto as skeleton
quantities. Defining the quantities α by the relation α
N A Ψ Ψ(x=x )=1+A B ,
A ≡ A d rd−2
16πGα f (x ), A=1, ..., , BX6=A AB
− c2 A ≡ A A N
the function φ can be written as where rAB ≡[(xiA−xiB)(xiA−xiB)]21.
Aftermakingthereplacement(3.12)ontheright-hand
4GΓ(d−2) N α side of Eq. (3.5a), inserting there expressions (3.13) and
φ= 2 A , (3.13) (3.14), we arrive at
c2 πd−22 A=1rAd−2
X
c2 ∆φ= N α δ = N m 1+A αB −1 1+ 1+A αC −d−42 p2A 12
−16πG A=1 A A A=1( A(cid:18) dB6=ArAd−B2(cid:19) " (cid:18) dC6=ArAd−C2(cid:19) m2Ac2#
X X X X
2−3d
αB d−2 pAiVAi
+ 1+A δ , (3.15)
(cid:18) dB6=ArAd−B2(cid:19) c ) A
X
with V V (x=x ). Making use of Eq. (3.10), we see that
Ai i A
≡
A 3d 2
p V = d − (p p )+(d 2)(ni p )(nj p ) r2−d. (3.16)
Ai Ai 2c d 2 Ai Bi − AB Ai AB Bj AB
B6=A(cid:18) − (cid:19)
X
Because of the linear independence of the distributions δ , the Hamiltonian constraint (3.15) is finally equivalent to
A
a system of algebraic equations which reads
N
1
2
m p2/(m2c2) p V /c
αA = 1+AdPBA6=A rαAd−BB2 1+ 1+AdA C6=AA rαAd−CC2 d−42 + 1+Ad ABi 6=AAi rαAd−BB2 3dd−−22, A=1, ..., N. (3.17)
 (cid:16) P (cid:17)  (cid:16) P (cid:17)
Solvingtheabovesystemprovidesinprinciplethevalues which shows that the ADM mass M = H/c2 is just
ADM
ofallthe α ’s. These quantities representsomeeffective the sum of the effective masses α . The Hamilto-
A A
N
masses, as suggested by the approximate equality α = nian (3.18) depends only on the matter variables: H =
A
m + (1/c2). Theyarefunctionsofxi xi (forA=1) H(xi xi,p ),sothatthesystemisautomaticallycon-
A O A− 1 6 A− 1 Ai
and p [defined implicitly by Eqs. (3.17)] . servative. The space metric is deduced from Eqs. (2.5a),
Ai
The skeleton Hamiltonian reads (3.1), and (3.14):
4
c4 N N α d−2
H = ddx ∆φ=c2 α , (3.18) γ = 1+A A δ . (3.19)
−16πG A ij d rd−2! ij
Z A=1 A=1 A
X X
6
By construction, H and γij coincide with their coun- IV. SKELETON GRAVITATIONAL FIELD
terparts in the Einsteinian theory up to the 1PN order.
Thetest-bodylimitisachievedbytakingthemassofone
IntheADMformalism,theevolutionofthespacemet-
blackhole,sayoflabel1,muchlargerthanthe massesof
ricin(d+1)dimensionsisgivenby[cf.Eq.(4.70)inRef.
its companion: m m , where A = 2, ..., . To
1 ≫ A N [15]]1
get the test-body Hamiltonian exactly, one has to let
formally m go to infinity (m + ) in the Hamil-
1 1 → ∞ 1 N
tonian (3.18) while keeping m /rd−2, A = 2, ..., , ∂ γ = (2π πkγ )+2D N , (4.1)
1 1A N c t ij √γ ij − k ij (i j)
constant as well as all linear momenta p . It follows
Ai
that particularly the flesh term ∂ (2V πi) does not con-
i j j
tribute to this approximation. In absence of gravity wheretheoperatorDidenotesthespacecovariantderiva-
(G 0) our Hamitlonian has the form of the exact tive. Its explicit action on the shift function Ni after
spec→ial-relativistic Hamitlonian for the system of free symmetrization is
N
point particles. When p = 0 for all A, we recover the
Ai
Brill-Lindquist solution, for which Eqs. (3.17) reduce to 1
D N =γ ∂ Nk+ Nk∂ γ . (4.2)
α (1+A α /rd−2)=m . Asaconsequenceour (i j) k(i j) 2 k ij
A d B6=A B AB A
model also generalizes the Brill-Lindquist data.
P
Let us observe that the particle masses mA cannot be For the conformally flat 3-metric γij = Ψd−42δij, we
equated to the ADM masses computed at the spatial in- have γ 1γ δ =0, hence
finities of the black hole sheets when one or several ij − d kk ij
N
momenta are non-zero. When such an identification is
1
performed,itleadsautomaticallytothesamerelationbe- ∂ γ γ δ =0.
t ij kk ij
tweenα andm asinthezeromomentumcase,whereas − d
A A (cid:18) (cid:19)
the correct dynamics up to the 1PN order is known to
be the one that derives from the Hamiltonian (3.18) de- After replacingin the above equation∂ γ and ∂ γ by
t ij t kk
termined by the set of Eqs. (3.17) with p = 0. The theexpressionsfollowingfromEq.(4.1)anddroppingall
Ai
actual dependence of the effective masses on6 the m ’s contributions proportionalto πk =0, we arrive at:
A k
must take the kinetic energy of the system into account.
Twhitihs rinesdpiceacttetsoaearcehlaotitvheerbso.oTstheofistshuee ibslaccokmphloilceatsehdeebtys STF(∂iNj)=−Ψ−d2−d2Nπij. (4.3)
the fact that the Bowen-York data for one single black
holedonotresultfromaboostedSchwarzschildfield[28]. The evolution equation for the field momenta πij is
This point, being still rather unclear, needs to be inves- knowninddimensionsaswell[cf.Eq.(4.71)inRef.[15]].
tigated in detail. We do not intend to discuss it here. It reads
1 We use here and below the usual notation for symmetrization:
2D(iNj)≡DiNj+DjNi.
1 1 N 1 1
∂ πij = √γ N Rij γijR DiDjN +γijD DmN + πijπk 2πi πkj + γij πklπ (πk)2
c t − " (cid:18) − 2 (cid:19)− m # √γ" k− k 2 (cid:18) kl− 2 k (cid:19)#
N −1
πkjD Ni+πkiD Nj D (πijNk) + 8πGN pAkpAlγikγjl 1+ pAmpAnγmn 2 δ . (4.4)
− k k − k c4 m m2c2 A
A=1 A (cid:18) A (cid:19)
(cid:2) (cid:3) X
In the ADM gauge, the trace (with respect to flat metric δ ) of (4.4) reduces to
ij
Ψ2 d−2 2NR−(d−1)Ψ−d−42DiDiN − 4−2 dΨ−2(dd−+22)Nπijπji−2πklDkNl
(cid:18) (cid:19)
+ 8cπ4GNΨ−d−82 N mp2A 1+ mp22Ac2Ψ−d−42 −12 δA =0. (4.5)
A=1 A (cid:18) A (cid:19)
X
Making now use of Eqs. (3.4), (3.5a), (4.3), and of the two useful relations D D N = γ ∂ (DkN)+ 1∂ γ DkN
(i j) k(i j) 2 k ij
and D D N =2∂ N∂ lnΨ+∆N, Eq. (4.5) can be rewritten as
i i i i
7
−(d−1)∆(NΨ)+ 3d4−2Ψ−3dd−−22Nπijπji+ 4πc2G(d−2)Ψ−1N N mA 1+ mp22Ac2Ψ−d−42 21 δA
A=1 (cid:18) A (cid:19)
X
+ 8πc4GΨ−dd−+22N N mp2A 1+ mp22Ac2Ψ−d−42 −12 δA =0. (4.6)
A=1 A (cid:18) A (cid:19)
X
The formofthe latter equationsuggeststosolveit forthe auxiliaryfunctionχ NΨratherthanfor the lapseN [9].
≡
At this stage, we apply to the right-hand side of Eq. (4.6) the replacement (3.12) which amounts to deleting a flesh
term in the product Ψ−3dd−−22πijπji. Since the Newtonian potential comes from the 1/c2 part of the lapse, the term
Ψ−1Ψ−3dd−−22πijπji = O(1/c6) is indeed a 2PN quantity and may be neglected consistently with our approximation.
Finally, we obtain a Poisson equation for χ with the source term being some linear combination of the Dirac deltas:
∆χ= 4πGd−2 N χ 3d−2pAiVAiΨ−4(dd−−21)
c2 d 1 A d 2 c A

− A=1 −
X
+m Ψ−2 1+ p2A Ψ−d−42 −12 1+ d p2A Ψ−d−42 δ . (4.7)
A A m2c2 A d 2m2c2 A A
(cid:18) A (cid:19) (cid:18) − A (cid:19)!
FcoonllfoowrminagltfhaectsoarmineSaergc.umIIeIn,twaesminaythweridteerivation of the d−d 1∆(∂iNi)=−∂ij Ψ−d2−d2Nπij . (4.10b)
(cid:16) (cid:17)
The source term of the second equation is the
N
β
χ=1−AdAX=1rAd−A2 , (4.8) nsuammeloyf a−∂riat∂iojn(Ψal−df2u−d2nNcti)oπnij ,of prAlu,s Aa =dis1t,ri.b.u.t,ioNn,,
sothat∆χ= 4πG(d−2) N β δ . Wenextreplacethe −∂i Ψ−d2−d2N(cid:0)∂jπij . The Po(cid:1)isson integral of the term
c2 (d−1) A=1 A A with non-compact support is not explicitly known, but
left-handsideofEq. (4.7)byvirtueofthelatterrelation, that(cid:0)of the distribu(cid:1)tional term is easy to compute. Now,
P
and equate the coefficients of δA. We are led to the new “flesh” term does not contribute at the Newto-
nianlevel. Itisthereforeatleastpost-Newtonianandwill
β = 1 A βB 3d−2pAiVAiΨ−4(dd−−21) eventuallyplayaroleinthedynamicsonlyatthe2PNor-
A (cid:18) − dB6=ArAd−B2(cid:19) d−2 c A der;itwillbeconsistentlyneglectedinthepresentmodel.
X Thus,thesubstitution(3.12)willbesupplementedbythe
+m Ψ−2 1+ p2A Ψ−d−42 −12 replacement:
A A m2c2 A
(cid:18) A (cid:19) ∂j Ψ−d2−d2Nπji →Ψ−d2−d2N∂jπji
×(cid:18)1+ d−d 2mp2A2Ac2Ψ−Ad−42(cid:19)!, A=1, ...(,4N.9). (cid:16) (cid:17)=−8πc3GΨ−d2−d2N N pAiδA. (4.11)
A=1
X
This system of equations for unknowns provides Taking Eq. (4.11) into account, Eq. (4.10b) becomes
N N
in principle the values of the monopoles β entering the
A
ethxepnregssivioenn boyf χ, Eq. (4.8). The lapse function itself is ∆(∂iNi)= 8πc3Gd d 1∂i N χAΨA−3dd−−22pAiδA .
− (cid:18)A=1 (cid:19)
χ X
(4.12)
N = .
Ψ The solution of (4.12) can be obtained with the help of
formula (3.9). It reads
For =2 and d=3, we recover the result of Ref. [9] in
N
the time-symmetric limit p =0 (with A=1,2).
Ai 2G d Γ(d−2)
InordertosolveEq.(4.3)fortheshiftfunctionNi,we ∂ Ni = 2
take the first and second order space derivatives of both i − c3 d−1 πd−22
sides. We obtain:
N
χ Ψ−3dd−−22p ∂ 1 . (4.13)
∆Ni+ d−d 2∂ijNj =−2∂j Ψ−d2−d2Nπij , (4.10a) ×AX=1 A A Ai i rAd−2!
(cid:16) (cid:17)
8
ThePoissonequation(4.10a)afterinsertingtheaboveex- where the skeleton Hamiltonian H expressedin terms of
pressionfor∂ Nj andperformingthe substitution(4.11) the quantities Ψ equals
j A
takes the form
2c4r
12
H = (Ψ +Ψ 2). (5.4)
2Gd 2Γ(d−2) G 1 2−
∆Ni = − 2
c3 d−1 πd−22 Using the theorem of implicit differentiation, we put
Eqs. (5.3) in a form apparently more convenient for nu-
N
χ Ψ−3dd−−22p ∂ 1 mericalintegration. We startby differentiatingEq.(5.2)
× A A Aj ij rd−2! withrespecttothepositions. Wefind(hereB =1,2and
A=1 A
X
A=B):
N 6
+ 16πG χ Ψ−3dd−−22p δ . (4.14) ∂Ψ
c3 A A Ai A ∂ Ψ B =η˜i ζ ∂ Ψ , (5.5a)
A=1 1i B ≡ ∂xi A1− A 1i A
X 1
By virtue of the formulas (3.8) and (3.9), the solution of ∂Ψ
∂ Ψ B =η˜i ζ ∂ Ψ , (5.5b)
Eq. (4.14) is simply 2i B ≡ ∂xi A2− A 2i A
2
Ni = cG3Γπ(dd−−2222)AN=1χAΨA−3dd−−22 where η˜i GmAni12 1+ p2A 12
X A1 ≡− 2c2r2 Ψ m2c2Ψ4
d 2 p 12 A(cid:18) A A(cid:19)
− p ∂ r4−d 4 Ai . (4.15)
×(cid:18)(4−d)(d−1) Aj ij A − rAd−2(cid:19) + G∂1i(p2Ac3jΨVA7j/r12), (5.6a)
A
η˜i η˜i , (5.6b)
V. BLACK-HOLE BINARY EQUATIONS OF A2 ≡− A1
MOTION
∂Ψ (xi ,p ,Ψ )
ζ B C Ci C
A
≡− ∂Ψ
For a black-hole binary, the number of coupled equa- A
tionscomposingthesystem(3.17)thatgivestheeffective Gm 1+3p2/(m2c2Ψ4)
= A A A A
massesreducestotwo,andthesum C6=AαC/rAd−C2 con- 2c2r12Ψ2A[1+p2A/(m2Ac2Ψ4A)]12
tains only one term. It is now convenient to regard the
two quantities ΨA as the unknowns:P + 7GpAjVAj
2c3r Ψ8
12 A
α α
C B
ΨA =1+Ad rd−2 =1+Adrd−2, =χB −1. (5.6c)
CX6=A AC 12 χA
A=1,2, B =A. (5.1)
Equality (5.6b)follows fromthe factthat the right-hand
6
side of relation (5.2) depends on xi and xi only through
Let us restrict ourselves to the space dimension d = 3. 1 2
The constant Ad reads then A3 =G/(2c2), and the sys- theblack-holeseparationr12 ≡[(xi1−xi2)(xi1−xi2)]12,and
that we have ∂ r = ∂ r . Solving Eqs. (5.5a) with
tem(3.17)consistsoftwoequationsforthetwounknowns 1i 12 − 2i 12
respect to ∂ Ψ and Eqs.(5.5b) with respect to ∂ Ψ ,
Ψ : 1i B 2i B
B
we find
ΨB =1+ 2rG12mc2AΨA (cid:18)1+ m2Apc2A2Ψ4A(cid:19)21 + 2Gr1p2Aci3VΨA7Ai , ∂1iΨB = η˜Ai11−−ζζ1Aζη˜2Bi 1, (5.7a)
∂ Ψ = ∂ Ψ . (5.7b)
2i B 1i B
B =1,2, A=B. (5.2) −
6
It remains to evaluate the momentum derivatives
It is clearlypossible to decouple the two equations (5.2). ∂ Ψ ∂Ψ /∂p and ∂ Ψ ∂Ψ /∂p . The sys-
In Appendix A we show that each ΨB is a root of a tepm1i oBf e≡quatioBns fo1ri∂p1iΨBp2iis gBiv≡en byB 2i
polynomial of order 200.
The equations of motion of a black-hole binary read ∂ Ψ =θ˜i ζ ∂ Ψ , B =1,2, A=B, (5.8)
p1i B A1− A p1i A 6
∂H where we have posed:
x˙i = , (5.3a)
A ∂p
Ai G p δ
θ˜i 1i A1
p˙Ai =−∂∂xHiA, (5.3b) A1 ≡ 2mAc4r12Ψ5A 1+ m2pc2A2Ψ4 12
(cid:18) A A(cid:19)
9
+ G∂p1i(pAjVAj). (5.9) adiabatically in time while the frequency increases ac-
2c3r Ψ7 cordingto the relativisticversionof the Kepler law. The
12 A
processends as soon as the first instable orbit is reached
The coefficients ζA are identical to those appearing in before the coalescence phase. The point where this oc-
∂1iΨB becausetheyrepresentthe resultofthe differenti- curs changes according to the model (see Refs. [29, 30]
ationofthe right-handsideofrelation(5.2)with respect and references therein for some comparisons).
toΨA atxiA andpAi constant. ThesolutionofEqs.(5.8) For binary black holes the skeleton Hamiltonian reads
with respect to ∂p1iΨB reads H = (α1 +α2)c2, where α1 and α2 are solutions of the
coupled system of equations (3.17). It is impossible to
∂ Ψ = θ˜Ai1−ζAθ˜Bi 1, (5.10) solve this system analytically, but it is not difficult to
p1i B 1 ζ ζ solve it perturbatively within the post-Newtonian set-
1 2
− ting,treatingε 1/c2asasmallparameter. AtthenPN
fromwhichwededuce∂p2iΨB byexchangingtheparticle approximation,≡we searchthe post-Newtonianexpansion
labels. of the coefficients α in the form
A
It is convenient to introduce the rescaled variables
n+1
2c2r 2c4r α =m + α(i−1)εi+ (εn+2), A=1,2. (6.1)
ηi 12 η˜i , θi 12 θ˜i , A=1,2. A A A O
A ≡ G AA AB ≡ G AB i=1
X
(5.11)
Note that we need to carry the series up to the order
Making use of Eqs. (5.4), (5.7), (5.10), and (5.11), the n+1inεto getannPN-accuratedynamicswithrespect
equations of motion (5.3) can be written in the form to the exact skeleton evolution (which first differs from
the exact Einstein’s evolution at the 2PN order). We
1 now substitute (6.1) into Eqs. (3.17) and expand their
x˙i = (1 ζ )θi +(1 ζ )θi , (5.12a)
1 1 ζ ζ − 2 11 − 1 21 right-hand sides up to the order n+1. By identifying
1 2
− (cid:16) (cid:17) the coefficients of different powers εi, i = 1, ..., n+1,
p˙1i =−1 cζ21ζ2 (1−ζ2)η1i −(1−ζ1)η2i wdiespolbaytahinerae tshyestfiemrstotfharelgeeebqruaiactieoqnusa:tions for α(Ai). We
− (cid:16) (cid:17)
2c4
(Ψ +Ψ 2)ni α(0) p p Gm
− G 1 2− 12 1 = 1i 1i 2,
m 2m2 − 2r
c2 1 1 12
= (1 ζ )ζ ηi (1 ζ )ζ ηi
1−ζ1ζ2(cid:16) − 1 2 1− − 2 1 2(cid:17) α(11) = (p1ip1i)2 + Gm2 5p1ip1i + 7p1ip2i
c 1 + 1 ∂ (p V ), (5.12b) m1 − 8m41 2r12 (cid:18)− 2m21 2m1m2
− Ψ7 Ψ7 1i 1j 1j
(cid:18) 1 2(cid:19) ni nj p p α(0) G2m2
+ 12 12 1i 2j 2 + 2,
wdehfienrietitohnes(la5s.6tae)q–u(5a.l6itby).foSlilmowilsarfreoqmuarteiolantsiofonrsx˙(i5.a2n)dapn˙d 2m1m2 − m2 ! 4r122
2 2i
hold with the role oflabels 1 and 2 exchanged. Knowing
the positions and momenta at a given time t, we can α(12) = (p1ip1i)3 + Gm2 9(p1ip1i)2 5p1ip1iα(20)
solve the algebraic equations (5.2) numerically, calculate m1 16m61 2r12 8m41 − 2m21m2
ηi , θi and ζ by means of Eqs. (5.6), (5.9), (5.11),
A AB A
and determine xiA(t+dt) as well as pAi(t+dt) from the α(1) G2m2 15p p 49p p
evolution equations (5.12). 2 + 2 1i 1i 1i 2i
− m2 ! 2r122 (cid:18) 4m21 − 4m1m2
VI. LAST STABLE CIRCULAR ORBIT 7ni nj p p α(0) G3m3
12 12 1i 2j + 2 2 ;
− 4m1m2 m2 !− 8r132
As an example of application of our skeleton Hamil-
tonian, we shall discuss in this section the parameters similar equalities hold with labels 1 and 2 exchanged. It
of the last stable (circular) orbit (LSO) in the relative isstraightforwardtosolveupthissystemto therequired
motion of two inspiraling compact objects. This class of order.
systems is of primary importance, being one of the most For convenience, we shall consider instead of H the
promising sources of detectable gravitational waves for dimensionlessHamiltonian(afterrulingoutthe constant
ground-interferometry experiments as LIGO (Laser In- rest-mass contribution)
terferometer Gravitational Wave Observatory), VIRGO,
GEO600, and TAMA300. Because of gravitational-wave H (m +m )c2
Hˆ − 1 2 ,
emission, the separation between the bodies decreases ≡ µc2
10
µ m m /(m +m ) denoting the reduced mass of the of the equality M =M is given in Appendix B for
1 2 1 2 ADM K
≡
system. At the nPN level, we have an arbitrary number of space dimensions.
Equation (6.7) is solved perturbatively for r = r(j).
2 n+1α(i−1) By plugging the result into Eq. (6.6), we find the gauge
Hˆ = A εi+ εn+2 . (6.2)
µ O invariantlinkbetweentheenergyE andtheangularmo-
AX=1Xi=1 (cid:0) (cid:1) mentum j in the case of circular motion:
We restrict our study to circular orbits in the center-
E Hˆ(r(j),j)
of-mass reference frame. Within the ADM gauge, the =
ε ε
frame shift is achieved by imposing the condition
1 1
p +p =0, (6.3) = (9+ν)ε
1i 2i − 2j2 − 8j4
whereas circularity requires that 1
81+41ν 5ν2 ε2+ (ε3). (6.8)
− 16j6 − O
ni p =ni p =0. (6.4)
12 1i 12 2i (cid:0) (cid:1)
Itisconvenienttorewriteformula(6.8)intermsofthe
The form of Hˆ becomes particularly simple when use is dimensionless variable
made of the reduced relative position and momentum:
2
dE 3
ri xi1−xi2 , pi p1i = p2i . x≡(cid:18)√ε dj (cid:19) . (6.9)
≡ G(m +m ) ≡ µ − µ
1 2
Theparameterxisnowtreatedasasmallquantityofthe
When both relations (6.3) and (6.4) hold, the Hamil- order ε [cf. Eq. (6.8)]. The definition (6.9) establishes
tonian (6.2) can be expressed with the help of ri and the link between x and j. We invert it order by order
p only. Notice also that it depends on the masses m to obtain j as a function of x. Inserting j = j(x) into
i 1
and m exclusively through the symmetric mass ratio Eq. (6.8) leads to a gauge invariant expression of the
2
ν m m /(m +m )2. energyE dependingontheparameterx. Thecalculation
1 2 1 2
≡
Inthenextstep,weintroducetheconserved(reduced) isperformedatthe10PNapproximationinordertocome
angular momentum j along the cirular orbit of radius r: reasonablyclose to the exactresult. We display here the
explicit formula with the 3PN-accuracy. It reads
j2 x 3 ν 27 29 17
p2 = . (6.5) E = + + x2+ + ν ν2 x3
r2 − 2 8 24 16 16 − 48
(cid:18) (cid:19) (cid:18) (cid:19)
675 8585 7985 1115
After eliminating the linear momentum pi by means of + + ν ν2+ ν3 x4
Eq. (6.5), the dimensionless energy E Hˆ(r,j) of the 128 384 − 192 10368
≡ (cid:18) (cid:19)
binary system becomes a function of the variables r and 11
j: + e xi+ (x12). (6.10)
i
O
i=5
E j2 1 1 j2 j4 ε X
= + (3+ν)+ ( 1+3ν) The higher order post-Newtonian coefficients e (for i =
ε 2r2 − r r2 − r3 4r4 − 2 i
(cid:18) (cid:19) 5, ..., 11) can be found in Appendix C.
1 j2 5j4
We want to compare the result of Eq. (6.10) with the
+ (1+ν)+ (5+ν)+ (1 4ν)
− 2r3 r4 4r5 − relation E = E (x) of the Wilson-Mathews model
(cid:18) WM WM
j6 ε2 (rememberthatinthismodelthepost-Newtonianexpan-
+ (1 5ν+5ν2) + (ε3). (6.6) sion terminates at the 2PN order)2:
8r6 − 2 O
(cid:19)
x 3 ν
The coordinate radius r of the circular orbit and the E = + + x2
WM
angular momentum j are related through one of the − 2 8 24
(cid:18) (cid:19)
canonical equations of motion, namely 27 39 17
+ ν ν2 x3+ (x4). (6.11)
16 − 16 − 48 O
∂Hˆ(r,j) (cid:18) (cid:19)
=0, (6.7)
∂r The 2PN coefficient proportional to ν2 differs from that
of the post-Newtonian computations carried out in gen-
a relationthat may be regardedas expressing the quasi-
eral relativity. Indeed, the energy for circular orbits in
stationarity of the configuration. Remarkably, it can be
shown in this case, i.e., for circular motion, that the
ADM mass of the system equates the so-called Komar
massM ,i.e.,themass-likequantityproportionaltothe
K 2 This relation is obtained by taking n = j in Eq. (C4) of Ref.
monopole part of g at spatial infinity satisfying the re-
00 [32] and replacing there the angular momentum j by jWM(x)
lation g00 = 1+4A3MK/r+ (1/r2) [31]. The proof accordingtoEq.(B1)ofRef.[29].
− O