482

Astrophysics, Vol. 43, No. 4, 2000

0571-7256/00/4304-0482$22.00 ©2000 Kluwer Academic/Plenum Publishers

VARIANT OF THE BIMETRIC THEORY OF GRAVITATION. III.GRAVITATIONAL RADIATION

R. M. Avagyan and A. H. Yeranyan UDC: 52.423

Gravitational radiation in a variant of the bimetric theory of gravitation is investigated in the case of slowmotions and weak fields. Questions of the propagation velocity, polarization, and generation of a weakgravitational wave are considered. The PetersMatthews coefficients and the dipole emission coefficient aredetermined.

1. Introduction

In [1] we proposed a variant of the bimetric theory of gravitation as a new alternative to the general theory of

relativity (GTR). As we showed in [2], the predictions of the proposed theory and the GTR coincide in the post-Newtonian

approximation. It therefore becomes necessary to consider, within the framework of this theory, those phenomena for which

a difference between the predictions of the theories may prove to be significant. One such phenomenon may be

gravitational radiation, and the present work is devoted to an investigation of it within the framework of the proposed

theory.

2. Gravitational Radiation

A consistent study of the problem of gravitational radiation is possible in the case of weak waves because of the

strong nonlinearity of the field equations. Because of the extremely low intensity of gravitational radiation, it is sufficient

for our purposes to use linearized field equations. In the quasi-Cartesian coordinate system ( ),,,,diag 11

11

11

10

---- ---=g ccccik

where γik is the background metric while c

0 and c

1 are cosmological coupling constants [2, 4], we represent the metric in

the form

( ),

0

ikikik hgg += (1)

where

( )( ) ,11,1,1,diag

0

---=ikg (2)

Yerevan State University, Armenia. Translated from Astrofizika, Vol. 43, No. 4, pp. 653-658, October-December,2000. Original article submitted February 4, 2000; accepted for publication August 11, 2000.

483

with |hik| << 1 in all of space, including a region occupied by a source. In this case, to within quantities of first order in

hik, the contravariant metric tensor is

( ),

0ikikik hgg -= (3)

while the determinant of the tensor gik is

( )( ) ,1

0

hgg += (4)

where h = hi

i all indices are raised and lowered using the metric

( )0

ikg .

In this coordinate system the linearized field equations [1, 2] have the form

( ),

8

2

1

2

14,

0

iklnlnik

lilk,

lkli,ik T

c

Gga

p=ú

û

ùêë

éy-y+y÷

ø

öçè

æ-+Y (5)

,0, =Ymmka (6)

where

( ),

2

1 0

hgh ikikik -=Y (7)

,ll,

x

ff

¶

¶º

( )

kiik

xxg

¶¶

¶-=

20

is dAlemberts operator, a is one of the dimensionless parameters of the theory, ( )0

ikT is

obtained from the tensor Tik by the substitution

( )0

ikik gg ® in it and by replacing the covariant derivative by an ordinary

derivative. In this approximation, the covariant conservation law 0; =kkiT takes the form

( ).0,

0

=kkiT (8)

We first consider Eq. (6). It should hold in all of space, for any a, and for an arbitrary weak source. For an isolated source

at infinity we have 0, ®Y kki Because of the condition that the solution be regular in the rest of space, it is found that

in all space we have

.0, =Y kki (9)

Generally speaking, Eq. (6) has a solution in the form of plane waves, representing an external field unrelated to the field

of the isolated system. We can therefore discard such solutions with no loss of generality [5].

With allowance for (9), Eq. (5) takes the form

( ).

16 0

4 ikik Tc

Gp=Y (10)

In the weak-field approximation, the tensor ( )0

ikT does not depend on hik (or Ψ

ik). If Ψ

ik is a solution of the system (8) and

(10), therefore, another solution will be

( ),,

0

,,mmikikkiikik g x+x-x-y=y¢ (11)

where the four-vector ξm satisfies the equation

.0=xm (12)

484

The transformation (11) is a gauge transformation and is not related to a coordinate transformation. Observable physical

quantities do not vary in such a transformation. The condition

,1, <<x nm (13)

guaranteeing that the field is weak, must be imposed on ξm. Using this arbitrariness, we choose ξm so that the components

Ψ0α and the trace a

aY+Y=Y 00 vanish (the Greek indices take the values 1, 2, 3). The conditions of this kind for the

field Ψik are called the TT calibration. The corresponding expressions for ξm were given, for example, in [6].

Outside the source we have the equation

,0=Yik (14)

from which it follows that a graviton propagates at the speed of light. This result distinguishes this version of the theory

of gravitation from most other alternative bimetric theories, in which the propagation velocity of gravitational waves

depends on the cosmological coupling constants [4]. In Rosens theory [7], for example, this velocity is 01 cc .

Eardley and Lee [8] developed a Lorentz-invariant scheme E(2) of classification of the polarization of gravitational

waves in metric theories of gravitation. According to it, the class of a theory is determined by the kind of linearized field

equations for a plane wave in a vacuum. These equations have the same form for the present theory and the GTR. This

fact means that the two theories belong to the same E(2) class N2 with NeumannPenrose parameters ψ

2 = ψ

3 = Ψ

22 = 0,

ψ4

≠ 0, i.e., in this theory (as in the GTR) a physical gravitational field has spin 2 and helicity ±2.

3. Generation of Gravitational Waves

We now consider the emission of gravitational waves by slowly moving sources, particularly the multipole nature

of such emission. The importance of the latter is due to the fact that by analyzing the variation of the orbital period of

a binary pulsar due to gravitational radiation, one can obtain information about the multipole nature of the radiation. For

this we write the solution of Eq. (10) in the form

( ) ,4

4 R

VdT

c

GcRt

ki

ki

¢-=Y

-ò (15)

where R is the distance of the observation point from the integration element. Using (8), we can show that, far from the

source and in the case of low velocities [6, 9], we have

( ) ,,2

02

2

04

VdxxrcRttRc

G¢¢¢¢-r

¶

¶-=Y baab ò

r

(16)

where ρ is the mass density of the radiating matter and R0 is the distance of the observation point from some element within

the mass distribution.

To calculate the intensity of the gravitational radiation we use the covariant, differential conservation law [3]

,08

4

=úúû

ù

êêë

é÷÷ø

öççè

æ+

p-

g|k

ikikikLL TS

G

ct

g(17)

where ikLLt is the tensor generalization of the LandauLifshitz pseudotensor [3], an expression for S ik can be found in [2],

and a vertical bar denotes a covariant derivative with respect to the background metric γik.

In a quasi-Cartesian coordinate system we have

485

( ) .08

4

=úúû

ù

êêë

é

÷÷ø

öççè

æ+

p--

k,

ikikikLL TS

G

ctg (18)

Integrating (18) over some large volume and assuming that there is no mass flux through the surface bounding the

integration volume, we obtain

( ) ( ) ,88

500

40

aaaòò ÷

÷ø

öççè

æ

p---=÷

÷ø

öççè

æ+

p--

¶

¶dfS

G

cctgdVTS

G

ctg

tkk

LLkkk

LL (19)

where the integration on the right side is carried out over the two-dimensional surface bounding the integration region on

the left side. Since for k = 0 the left side of Eq. (19) represents the energy loss by the system per unit time, the energy

flux of gravitational radiation through an area element dfα will be

( ) .8

05

0a

aa÷÷ø

öççè

æ

p--= dfS

G

cctgdI LL (20)

Choosing a sphere as the integration surface and using ,20 W-= aa dnRdf where dΩ is an element of solid angle and

0R

xn aa = is a unit vector (nαnα =1), we obtain

( ) .8

05

020 ÷

÷ø

öççè

æ

p---=

Waa

a SG

cctRng

d

dILL (21)

At large distances from the radiating system and in the case of weak fields in the TT calibration (h0i

= h = 0, 0=aba,h ), we

have

,32

00

40

,,LL hhnG

ct et

etaa

p= (22)

( ) ,00

0,,hhanS et

etaa -= (23)

where the delayed nature of hik = hik(t - R

0/c) (nαhαβ

= 0) is also taken into account.

Substituting (22) and (23) into (21) and with allowance for g ≈ 1 in the approximation under consideration, we obtain

( ) .4

1

8 0000

20

5

úû

ùêë

é+

p=

W etet

etet

,,,, hhahhG

Rc

d

dI(24)

Equation (24) differs from the corresponding GTR equation given in [8] by the second term inside the brackets. However,

this term, being a total time derivative, vanishes in averaging over a time interval exceeding the wave period. An analogous

situation occurs in the GTR if in calculating the radiation intensity one uses, for example, the energymomentum

pseudotensor given by Sahakian [5]. This term will therefore be omitted in subsequent calculations.

Since we have hαβ = Ψαβ in the TT calibration, from (16) we obtain

,2

1

3

2

04 etab

ettb

eaab ÷

ø

öçè

æ--= DPPPP

Rc

Gh &&

(25)

where dots denote a time derivative,

( )VdgrxxD

0¢÷÷

ø

öççè

æ¢-¢¢r= abbaab ò 23 (26)

is the systems quadrupole moment, and

486

ab

ab

ab +d= nnP (27)

is the projection operator, which satisfies the conditions ,2=aaP and .a

bsb

as = PPP

The radiation intensity in all directions, i.e., the energy loss by the system per unit time, will be

.45 5

abab==- DD

c

GI

dt

dE&&&&&& (28)

The expression for the radiation intensity thus coincides with the analogous one in the GTR, i.e., dipole and monopole

radiation are absent in the variant of the theory under consideration. This result distinguishes this variant of the theory

from the known alternative theories [4].

For a binary system of bodies with masses m1 and m

2 we obtain, using the procedure presented in [4, 9],

( ) ,111215

8 22265

223

úûù

êëé u-u=

rr

RRRc

MmGI (29)

where m = m1 + m

2, ,

21

21

mm

mmM

+= and υ is the relative velocity of the bodies. From a comparison with the general equation

for the radiation intensity in metric theories [4] we obtain the PetersMatthews coefficients k1 and k

2 and the dipole radiation

coefficient kD,

;0;11;12 21 === Dkkk (30)

The theory under consideration thus does not differ from the GTR in questions related to gravitational radiation.

The authors thank participants of a seminar in the Department of Theoretical Physics of Yerevan State University

for useful discussions.

REFERENCES

1. R. M. Avagyan (Avakian) and L. Sh. Grigorian, Astrophys. Space Sci., 146, 183 (1988).2. R. M. Avagyan and A. H. Yeranyan, Astrofizika, 43, 303 (2000).3. R. M. Avagyan and A. H. Yeranyan, Astrofizika, 43, 493 (2000).4. C. M. Will, Theory and Experiment in Gravitational Physics, Cambridge Univ. Press, Cambridge, Eng.New York

(1981).5. G. S. Sahakyan, SpaceTime and Gravitation [in Russian], Erev. Gos. Univ., Yerevan (1985).6. A. A. Logunov and M. A. Mestvirishvili, Relativistic Theory of Gravitation [in Russian], Nauka, Moscow (1989).7. N. Rosen, Ann. Phys., 84, 455 (1974).8. D. M. Eardley and D. L. Lee, Phys. Rev., D8, 3308 (1973).9. L. D. Landau and E. M. Lifshitz, Field Theory [in Russian], Nauka, Moscow (1988) [The Classical Theory of Fields,

4th ed., Pergamon Press, New York (1976)].