On the conditions for static balance in the post-post-Newtonian approximation

Volume 63A, number 3

PHYSICS LETTERS

14 November 1977

ON THE CONDITIONS FOR STATIC BALANCE IN THE POST-POST-NEWTONIAN APPROXIMATION T. KIMURA Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima-ken 725, Japan

and

T. OHTA Department of Physics, Miyagi University of Education, Sendai 980, Japan Received 16 August 1977 2 (a = 1, 2) obtained by Barker and It is shown that the alternate condition for static balance ea = ±(4irGm tm 2)”down in the post-post-Newtonian apO’Connell from the post-Newtonian analysis of the two-body problem is broken proximation.

Recently, from a post-Newtonian analysis of the two-body problem Barker and O’Connell [1] have found an alternate condition for static balance~1 ea

=

±(4irGm

1/2, 1m2) in addition to the usual condition

(1)

ea

=

±(4irG)’I2ma,

(2)

ma and ea (a = 1, 2) denote the masses and charges of ath particle. They posed a question whether the condition (1) will hold beyond the post-Newtonian approximation. It is the aim of this note to show that the condition (1) does not hold in the post-post-Newtonian approximation, while the usual one (2) does. Before we enter into the main problem of this note, we touch upon the gauge dependence of the static potential in the post-Newtonian approximation from which Barker and O’Connell obtained the conditions (1) and (2). As shown by Hiida and Okamura [2, 3] the static gravitational potential in the order of G 2 is where

~

=~~~G212(

2 21 +m2)~l(1 —x)G 2

r12

+m2)

(3)

r12

where r12 = 1z1 z21, za being the coordinate of ath particle, and the parameter x depends on the gauge. The gauge x = 1 has been used2bystatic Einstein et at [4]potential while x gives = 3 was by Feynman [5] and DeWitt [6] (1) Inand the (2) latgravitational rise used to zero. At first sight, the conditions ter case (i.e. x = 3), the G do not hold provided the gauge x = 3 is adopted. However, it is seen that the careful employment of the canonical formalism of Arnowitt et al. [7] leads to the static potential between two particles in the post-Newtonian approximation*2, —

,

*1

We use the units c

=

.

l6irG = 1 and denote Newton’s gravitational constant by G. The charge ea(4ITY”2 corresponds to ea of

2 Barker and O’Connell. A comma in a subscript denotes a partial derivative and repetition of indices implies summation. 1’ Cf. ref. [3] where the gauge parameter z is related with x by z — 1 f(x — 1). We describe the distribution of the point particles by good ~ functions introduced by Infeld [81. In other words, we ignore terms of the self-energy type.

193

Volume 63A, number 3

PHYSICS LETTERS

1

UPN

— —

TG G

m1m2(m1

+

m2) 2

12

+.~(l_X)G

1 +m2)

ji

2 + m2) \ e~e~(m1 r 12

~

—

~

14 November 1977

+

(1

2 \ ~e1m2r2 + e2m1 12

(4) (m1e22+m2e~))

The first term on the right hand side of eq. (4) is the static potential used by Barker and O’Connell. Since the second term vanishes provided either of two conditions (1) and (2) is satisfied, the conditions (1) and (2) are independent of the gauge parameter x. Therefore we shall adopt the gauge x = 1 and calculate the static potential in the post-post-Newtonian approximation by means of the canonical formalism of Arnowitt et al. [7]. The calculation is carried out by extending our previous method [9] so as to include the effect of point charges. In the coordinate system withy (1 x) = 0 the metric tensor becomes ~-

—

~ (i,j1,2,3), (5) where h~Tis the transverse-traceless part of h~ 1= g~1 and hT = ~ h11,11. By putting 1 4), + ~ we obtain (l + the and expanding 0 in power series of (v/c), 0 = + 0 + 0 + (h,~T starts from the order of (v/c) required Hamiltonian in the order of (v/c)8 (2) (4) (6) —

—

...

HPPN

=

—

fd3x ~hT

=

—

fd3x

(6)

~,

by solving the constraint equations G 01 T0~= 0. In HPPN the static potential part which contains no h~Tis given by —

UPPN(l, 2)= _-fd3x ~ (8)

=~_fd3x{2os~os + q~S~ç5s — 8 (2) (6) (4) (4)

(ØS (4)

—

~

4~S oS)iLciL} (2) (2)

=G[_~~ mlm2(m1+m2+3m1m2)3(1)m1m2(e1+e2)

~ ji +~G

2~)+ ~ j~\ m TG ~4i~j 1m2e1e2 ~

\ e~e~(m~ +m r 12

/

i(1\2

~

(7)

e1e2(e~+r3e~ 3e1e2) —

12

by successively solving the following equation =

—

~

—

~(1

+~

—

~ma~(x

—

Za),

(8)

where =

[—e1/4irr1

—

e2/4irr2],~,

in which ra = IXZaI. On the other hand, the static potential due to h~’Tis

194

(9)

PHYSICS LEUERS

Volume 63A, number 3

+f

u~T~~(l, 2) =~fd3xh~’~h~ ++fd3x h~Te~euI~

14 November 1977

fd3x(h~Tos),~ (2)”

=_+fd3xh~h~ =

where

—G (Gmim2

(10) 2~_~_

—

(~_~)eie2)

h~T= —4G {Gmim

2.)ln(ri 2

—

(~_)eie2}[(a2+

~

+

r 2 + ri2)].

(11)

T~~(l, 2) vanish provided either of (1) and (2) is satistransverse-traceless partthe h~Tand theofpotential fied.The The sum of the first and last twothen terms UPPN(l, U~ 2) given by (7) also vanishes provided either of (1) and (2) is satisfied, while the second and the third terms cancel with each other only when the usual condition (2) is satisfied. We, therefore, conclude that the alternate condition (1) does not hold for static balance in the postpost-Newtonian approximation.

References [1] B.M. Barker and R.F. O’Connell, Phys. Lett. 61A (1977) 297. [2] K. Hiida and H. Okamura, Prog. Theor. Phys. 47 (1972) 1743. [3] T. Kimura and T. Toiya, Prog. Theor. Phys. 48 (1972) 316. [4] A. Einstein, L. Infeld and B. Hoffmann, Ann. Math. 39 (1938) 66. [5] R.P. Feynman, Acta Phys. Pol. 24 (1963) 700. [6] B.S. DeWitt, Phys. Rev. 162 (1967) 1243. [7] R. Arnowitt, S. Deser and C.W. Misner, in: Gravitation, an introduction to current research, ed. L. Witten (JohnWiley and Sons Inc., New York, 1962). [8] L. Infeld and J. Plebanski, Motion and Relativity (Pergamon Press, Oxford, 1960). [9] T. Ohta, H. Okamura, T. Kimura and K. Hiida, Prog. Theor. Phys. 51(1974)1598.

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