General Relativity and Gravitation, Vol. 8, No. 8 (1977), pp. 617-621

E x t r e m a l i t y o f M a s s in t h e B i m e t r i c T h e o r y o f G r a v i t a t i o n 1

ITZHAK GOLDMAN and NATHAN ROSEN

Department o f Physics, Teehnion-Israel Institute o f Technology, Haifa, Israel

Abs t rac t

In the bimetric theory of gravitation, the static spherically symmetric case involving matter characterized by density and pressure is considered. It is found that the condition that the mass be stationary under small variations of the field variables (including the density) for a fixed number of baryons leads to the field equations and to the equilibrium condition. If one considers only solutions of the field equations, then the mass for a fixed baryon number is stationary (one can expect it to be extremal in most cases) if the equilibrium condition holds.

In the past [1 ] extremali ty properties of the mass of a static spherically symmetric configuration were proved in the framework of the general theory of relativity. It was shown that, given a fixed number of baryons, the momentar i ly static, spherically symmetric configurations of cold matter which extremize the mass are those which satisfy the equilibrium equation. The purpose o f this paper is to point out that one has a somewhat similar situation in the bimetric theory of gravitation [2, 3].

In the framework of the bimetric theory, let us consider the static spheri- cally symmetric case. With spherical polar coordinates (r, 0, q0, in the case of

matter characterized by mass density p(r) and pressure p(r ) , the Riemannian line element takes the form [3]

dS 2 = e2r 2 - eZr + r2 d~2 2) (1)

and the field equations can be reduced to

V2~b = 4rr•(p + 3p) (2)

V 2 ~ = -4~rK(p - p) (3) 1 Dedicated to AchiUe Papapetrou on the occasion of his retirement.

617 This journal is ct~pyrighted by Plenum. Each article is available for $7.50 from Plenum Publishing Corporation. 227 West 17th Street, New York, N.Y. 100[ 1,

618

with

and

GOLDMAN AND ROSEN

= e r162 (4)

V 2 r = r + (2/r) r (5)

Furthermore the energy-momentum equation

T~; v = 0 (6)

gives in this case the equilibrium condition

p ' + (p + p) q~' = 0 (7)

Let us assume that we are dealing with cold matter so that there exists an equa- tion of state as a relation between p and P. Equations (2), (3), and (7), together with the equation of state and the boundary conditions, serve to determine the field variables.

If we assume that the matter is in the form of a sphere of radius R, then we have for r ~> R

r = -M/r, t~ = M'/r (S)

where

and

fo R M = 4zr r + 3p)r2dr (9)

fo R M'= 4zt K(p - p)r2dr (10)

Here M is the primary mass, or simply the mass, which determines the motion of a slowly moving test particle, and M' is the secondary mass, which influences the mot ion of a high-speed particle and that of a light ray.

It has been shown [4] that the mass M, as given by (9), is equal to the total energy (including that of the gravitational field) and can be written

f o ~ ~ 1 ] M = 4Ir Kp + ~ (r _ 3q/2 _ 6q~'~') r2dr (11)

For the sake of simplicity, let us assume that the matter is made up of par- ticles of one kind, taken to be baryons, and let n be the baryon number density. In the general case the conservation of baryon number is given by

[(-g)l/2nutt], u = 0 (12)

where u u is the velocity 4-vector. One can also write this

nu~t;u + n,uu u = 0 (13)

EXTREMALITY OF MASS 619

On the other hand, if one takes T uv for matter characterized by O, P and u u, namely,

Tt~V = (p + p)uUu v _ pg~V (14)

then one finds that multiplying (6) by u** gives the relation

(0 + P)UU;u + P, uu ~ = 0 (15)

For (13) to hold, it is therefore necessary that

dn do - ( 1 6 )

n p + p

If we take into account the equation of state of our cold matter, we see that one can express p and p as functions o f n (or n and p as functions of p). From (12) we also see that the baryon number (which is conserved) is given by

N= f (-g)a/2nu~ (17)

or, in the present case (with u ~ = e-0),

N= 4rr e3q~ n(r)r2dr (18)

It follows that the variation of N is given by

or, if one uses (16),

~ o o

8N = 47r Jo ea~ r2(Sn + 3n6 4) dr (19)

We shall assume that the baryon number is fixed, so that we shall have as a constraint

8N=0 (21)

Let us now consider the variation 8M, where M is given by equation (11) and where we vary 0, ~, and ~ arbitrarily. With the help of (4) we get

fo { 8M= 4rr KSp+Kp(SC)+38t))

+ - - [ (4 / - 3 ~ ' ) 8 ~ ' - 3((a'+ r r2dr (22) 16a

620 G O LD MA N AND ROSEN

Integrating by parts and noting that 5~ = 6 ~b = 0 at infinity, we obtain

fo~176 [ 1 ] 6M=47r Kfp+ K R + I ~ ( 3 V 2 ~ - V 2 ~ b ) 8~b

+3 Kp+l--~(V2%+V2ff) 6~ r2dr (23)

Let us now impose the condition that M is extremal for fixed N, i.e., that

6M = 0 (24)

subject to the constraint (21). As usual, let us then make use of a Lagrangian multiplier k and replace (24) by the condition

6M- X6N = 0 (25)

o r

] [ 1 ] )me3~ 8/3+ Kp+ (3V2~ - V2~) 6~ p + p ~ n

+3 Kp+l- -~(V ~ + V 2 ~ ) - X e a ~ n 6~b r2dr=O (26)

Since the variations 6p, 84~, and 6q; are arbitrary, we must take their coeffi- cients to vanish. From the coefficient of ~p we get

n = (l/X) (p +p)e 4~ (27)

The coefficient of 64) gives

V2~ - 3V 2 ~ = 16rr•p (28)

while from that of 8 ~ we get, using (27),

V 2 c~ + V 2 ~ = 16nKp (29)

We see immediately that (28) and (29) give the field equations (2) and (3). As for (27), differentiating it gives

t t t n p + p q~, - + ( 3 0 )

n p + p

and, if we make use of (16), we get equation (7), the equilibrium condition. It has been shown therefore that in the static, spherically symmetric case the

mass, for a fixed baryon number, is stationary for the states in which the field equations and the equilibrium condition are satisfied.

Returning to equation (27), we can readily evaluate the Lagrangian multi- plier X. On the surface of the matter sphere (r = R), where p = 0, let us write

E X T R E M A L I T Y O F M A S S 621

P = Po, n = no. Since we have here q5 = -M/R, then

1 _ no eM/R (31) 3, Po

Let us consider briefly another approach, one that is closer to what has been done in the general relativity case [1 ]. We assume that the field equations (2) and (3) are satisfied. Then equation (23) can be written

L ~

6M= 41r {K6p + 3K(p +p)6~}r~dr (32)

In that case the condition (25) gives

L ~ ( Xne3r I 47r K P+P ][~p+3(p+p)6~]r2dr=O (33)

and we get again equation (27) and hence the equilibrium condition. We see that, if we compare various solutions of the static spherically sym-

metric field equations, the mass is stationary for small variations with a fixed number of baryons if the equilibrium condition holds. One can expect that in most cases the mass will actually be extremal.

References

1. Harrison, B. K., Thorne, K. S., Wakano, M., and Wheeler, J. A. (1965). Gravitation Theory and Gravitational Collapse, Chap. 3. University of Chicago Press.

2. Rosen, N. (1973). Gen. Rel. Gray., 4, 435. 3. Rosen, N. (1974).Ann. Phys. N. Y., 84,455. 4. Goldman, I. (1976). Gen. ReL Gray., 7, 681.