Three-body hamiltonian in general relativity

Volume 123, number 7

PHYSICS LETTERS A

THREE-BODY HAMILTONIAN IN GENERAL RELATIVITY

24 August 1987

*

Gerhard SCHAFER Groupe d’Astrophysique Relatjvjste, CNRS, DARC, Observatoire de Paris, Section deMeudon, 92195 Meudon Principal Cedex, France Received Ii May 1987; revised manuscript received 15 June 1987; accepted for publication 18 June 1987 Communicated by J.P. Vigier

We present the complete hamiltonian for a three-body system interacting gravitationally according to Einstein’s theory up to the second post-newtonian approximation. The hamiltonian is obtained in the ADM gauge.

Although a formal expression for the n-body hamiltonian, respectively lagrangian, at the second post-newtonian approximation of general relativity has been obtained by Ohta et a!. [1] some time ago, it has not yet been possible to compute it in closed form for a general value of n. In ref. [1] is indicated that the case n = 2 is calculable in closed form. However, this work was vitiated by a calculational error and by an unclear treatment of the effects of different coordinate conditions. Both topics have been clarified later by Damour and the author [21.The first fully correct calculation of the two-body lagrangian has been performed in the de Donder coordinate-gauge condition [3]. The compatibility ofthis result with the general results of ref. [1] has also been proven in ref. [2]. This led to the knowledge of the two-body hamiltonian in the ADM gauge [4J up to the second post-newtonian approximation. Today the two-body dynamics is known in the de Donder gauge [3] and in the ADM gauge [5] up to the second and a half post-newtonian approximation. In the meantime there has been some criticism [61 against the lagrangian given in ref. [31 insofar as there are no terms present which are quadratic in the accelerations of the bodies. With regard to ref. [7], where the effects of coordinate transformations and the use of the equations of motion on the lagrangian level are discussed, this has led to the opinion that the lagrangian in ref. [3] does not belong to the de Donder gauge. It has been overlooked there, however, that a coordinate transformation by which a double zero term is eliminated is indeed the identity, so that no coordinate transformations are involved by the eliminations of double zero terms with the aid of the equations of motion. This is different for linearly occurring accelerations. Their eliminations involve coordinate transformations if no auxiliary functions are introduced. The utilization of the latter is precisely the crucial point in the “method of the double zero” developed by Barker and O’Connell [8] (for identity transformations, see also ref. [9]) for eliminating linearly occurring accelerations on the lagrangian level. On the other hand, from the point of view of singular lagrangian systems (for a recent review see, e.g. ref. [10]) truncated lagrangians with acceleration terms occurring quadratically as well as (truncated) lagrangians with terms linear in the accelerations are singular. The difference is only that the former are in general less singular than the latter [11]. Up to the second post-newtonianapproximation, in the ADM gauge, the (only) difficult potential term which is not explicitly known for systems with more than two bodies reads [1]: U(TT)=_thJd3xf~f~L

(1)

~ Supported by the Deutsche Forschungsgemeinschaft (DFG).

336

0375-960 11871$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 123, number 7

PHYSICS LETTERS A

24 August 1987

with 2 ~ ~ f75 T =f~—~G a by-a

r

ii 2 2 mamblô,JI——--——+--i--na~nab L \rarb rarab rab

+_~_1_~_](ln(ra+rb+rab)_-~_+_4_na.nab) X ÔX rab 2r 0~,

_(n~n~b+n~n~b)]

rab

and f

C’ 2 V

a a

1 1 L L \ mamb-------~—J-ankra-1-rb-1-rab,,

V

~

a by-a

(IXa VXj,

where ra = I X Xa I, rab = Xa Xb I, rana = X Xa, ra,,nab= Xa Xb, ma is the mass-parameter of the ath body, Ti’ means the transverse-traceless part of the affixed tensor, and the velocity oflight is set equal to one. The expression (1) is wetl defined. This is even the case if in one of the two f’~-functionsin (1) the prescription a ~ b is dropped. The value of U~’~ is not changed hereby. In this letter we are interested in (1) for the case of three bodies. In general, the unique decomposition U(TT) = ~ + U~r)+ U~1~)holds, where ~ ~ and ~ are the four-, three-, and two-point correlation functions contained in ~ respectively. By a quite lengthy, but nearly straightforward calculation the integrations in (1) can be performed taking into account several times partial integrations (in order to prevent that the integrals become meaningless the technique of a4ialytic continuation has been applied (appendix A of ref. [2]), see also ref. [12]) and making use of a modified version of the divergent relation (AS. 11) of ref. [13]. The modification consists in the replacement of —

—

—

_~_Jd3xr;4r~r~1by

—

_~Jd3xr;2(Vr)(Vr)+r~1(r~,2+r~2)

with b ~ a, c ~ a, b, and the involved integral has been evaluated by the author taking advantage of the Feynman trick in momentum space [14]: (2)

rhcrab2r~2=rbc1(rab2+r~2)_~~_fd3xra2(VrbI)(Vrc1).

By the way, eq. (2) also solves one of the two unperformed integrals in the expression X~0~ of ref. [13], see also ref. [2]. The result fOr the three-point correlation function in (I) reads: m~mbm~ [18r~br~~ 60r~br~24r~brac(rab + r~)+ 6Orabracr~c

U~J~ =

—

—

a by-acy-a.b

—

rcbrcarba

+ 56r~br~~ 72rabr~+ 354 + 6r~b].

(3)

—

After several substitutions expression (3) is in perfect agreement with expression (A. 10) of ref. [15] forthe contribution o1~f,~ to the metric component g00 if one recognizes that in general ~ = ~>~~m~g(x= x~)holds and that U~f~ = ~m~g~)(x=x~) is valid provided g~~)(x) belongs to a two-body system (the prime at the suMmation sign indicatesthat body c has to be different from the related bodies in gET) (x); these are two bodies each time). Therefore, we get the two-point correlation function from (3) by dividing that expression by a factor 2 and letting one body coalesce with one of the related other bodies. Hereby the expression (3) seems to become divergent and a regularization procedure has to be applied. However, consistent with —

—

337

Volume 123, number 7

PHYSICS LETTERS A

24 August 1987

(1) this is not the case and without any regularization we find independently and consistent with refs. [3] and [2], but different from ref. [1] (4)

m~m~

a by-a

~ab

It may be worth noting that the derivations ofthe three- and two-point correlation functions just indicated are neither mentioned nor done in ref. [15]. It is evident that the knowledge of U(~ for a four-body system would be sufficient to get U(TT) for a nbody system with n>4. ~ is the most difficult four-point correlation function at the second post-newtonian approximation. It is as important for point-particle models as for continuously distributed matter. Within the full continuous case a correlation function related to U~T~is, e.g. expression (5.12) in ref. [16]. Gathering results of refs. [13,1,2] and eq. (3) of this letter, we obtain for the three-body hamiltonian in the ADM gauge: H—~ma+~--~G~ a

—

a ma

~G ~ a by-a

mamb (6 Pa rab \ ma /

+

~

aba

mambi~(P~) rab a

Pa ~Pb

—

(flab ~Pa

mamb

ma

)( ~ab ~Pb ~ + ~G~~

r

2\~

/

2\2

~m~(~-P-~) +i~G>~~ mamb [lo~-~-) a ma a by-a Tab ma

j~

+

2

2

2

P~(1tabPb) mam,,

2

2

2

rabrac

~

PaPb

21J’aPbJ

mamb

3

mambmC

a by-a cy-a 2

—11

(Pa~Pb)(1’ab~Pa)(1’abPb) —

/

mamb

mamb

(flabPa)2(lIabPb)2 2

—

mamb

2

mamb

+~G2~~ ~ mambmc[l 8Pa a by- a cy-a

—14

(flabPa)(flabPb)

mamb

2 ~ + ~G

a by-a cy-a

ma

Tab Tac

+ 14

+l4Pb m,,

~p~)+ 2 (flab ~Pb)( “ac ~p~)+ 5~abfl mbmC

l

ma mbm~

a by-acy-a.b (rab+rbc+rca)

+

m~

~‘-“~

ma /

+ 1G2 2

~

2

~ m~mb(Pa+Pb2PaPb a by-a rab \ma mb mamb

(nab + n ~)( n~,,+ fl~b)(8 Pa,PcJ mamc

mambmc

a by-a cy-a,b (Tab +rbc+rca)rab

338

3Pa~Pb(’Sab~Pa)(~tabPb) mamb — 4P~(flab~Pc) m~

—

—16

PajPc, + 3 Pa,Pbj mamc mamb

( 8Pa~Pc(flabPa)(h’abPc)

\

P~(1abPa)2

2 —

mb m~

mbmC

/

2 ~

+l7~”’~

ma mb

mbm~

4(hlabPc)(1acPcY~+jG2>~

+ ~G

50PaPb

m,,

(ttabPb)(h1ab~Pc) +flab•flac (flab’Pb)(tac~Pc)

ma rn,,m~(2 (“a~~ Pa ) (“ac rab mamc

(1tacPc)2

2(flabj’b)

m~

mamc

Volume 123, number 7

~,m~rnbmc -~G3~ a by-acy-b rabr&

—thG3

PHYSICS LETTERS A

jG3~

~m~rnbmc

a by-acy-a

rabrac

~G3~ a

24 August 1987

~ by-ccy-a,b

m~mbmC r~bra~r,,~

Ea by-acy-a.b ~ rn~rnbm~ [l8r~br~~—60r~br~ —24r~brac(Tab + r,,~)+60rabracr~+ 56r~br,,~ rabracrbc

—72rabr~c+35r~c+6r~b1—~G3 ~ a by-a

m~rn~

(5)

rab

where a, b, c run over 1, 2, 3. The two-body hamiltonian is a special case of (5): here a, b, c run over 1, 2 only. By the applic~ationof the coordinate transformation (32) of ref. [2] to the lagrangian which corresponds to our hamiltoniati (5) we might get the explicit three-body lagrangian in the de Donder gauge. As this lagrangian turns out to be singular the associated hamiltonian structure, however, will be more complicated than the one we presented in this letter. The author thanks T. Damour for helpful discussions.

References [1] T. Ohta, H. bkamura, T. Kimuraand K. Hiida, Prog. Theor. Phys. 51(1974)1220, 1598. [2] T. Damour aId 0. Schafer, Gen. Rd. Gray. 17 (1985) 879. [3] T. Damour, C.R. Acad. Sci. 29411(1982)1355. [4] R. Arnowilt, $. DeserandC.W. Misner, Phys. Rev. 120 (1960) 313. [5] G. Schafer, A~in.Phys. (NY) 161 (1985) 81. 6] L.P. Grishchi~kand S.M. Kopejkin, in: Relativity in celestial mechanics and astronomy, eds. J. Kovalevsky and V.A. Brumberg (Reidel, Dor~recht,1986) p. 19. [71G. SchAfer, P~ys.Lett. A 100 (1984) 128. [8] B.M. Barker and R.F. O’Connell, Phys. Lett. A 78 (1980) 231; Can. J. Phys. 58 (1980) 1659; Gen. Rd. Gray. 18 (1986) 1055. [9] B.M. Barker ~nd R.F. O’Connell, Phys. Rev. D 29 (1984) 2721. [10] K. Sunderme~’er,Constrained dynamics (Springer, Berlin, 1982). [11) X. Jaén, J. Lh?sa and A. Molina, Phys. Rev. D 34 (1986) 2302. [12] T. Damour, in: Gravitational radiation, eds. N. Deruelle and T. Piran (North-Holland, Amsterdam, 1983) p. 59. [131T. Ohta, H. c~kamura,T. Kimura and K. Hiida, Prog. Theor. Phys. 50 (1973) 492. [14] R.P. Feynman, Phys. Rev. 76 (1949) 749. [15] T. Ohta, T. Klmura and K. Hiida, Nuovo Cimento B 27 (1975) 103. [16] S.M. Kopejkin, Astron. Zh. 62 (1985) 889 [in Russian].

339