L F ~ T T : E R E A L N U O V O C I M ] ~ N T O V O L . 10, N . 5 1 Giugno 1 9 7 4

On Rosen's Bimetric Theory of Gravitation (*).

H. YILMAZ

Perception Technology Corporation - Winchester, Mass.

(ricevuto il 15 Febbraio 1974)

The purpose of this note is to show that the field equations of Rosen's bimetric theory of gravitation (1) are soluble and the solutions are the same as in the author's theory of gravitation (~). However, Rosen's conservation laws are not consistent with these solutions except in the case of a flat space. Other conceptual difficulties connected with the theory, such as Rosen's second metric, 7~, and his propagation equations will be briefly discussed. I t will be shown that the difficulties arise either from the second- order terms of the solutions or from their first-order time dependence. The theory is therefore completely defensible as a static, firsVorder theory and predicts the red shift, light deflection and time delay correctly.

The field equations of Rosen's theory are

(i)

(2)

v 1 v = - - 8 7 g ~ T ; ,

Here 7 ~ is a fiat metric, gz, is the curved-space metric and la implies ordinary partial derivative a~. The conservation laws are given by

(3) T~:,=0,

where the semicolon denotes covariant derivative. The quant i ty x is related to the determinants of 7 ~ and ga~ as ~ = (g/7)�89 ~ / ~ . The theory so provided allows arbitrary co-ordinate transformations and, to narrow down the reference frame to a reasonable extent, RosE~ imposes the co-ordinate condition

(4) a~(ug,~)=o

which, as far as we are concerned, completes the formulation of the theory.

(*) Supported by Advanced Research Projects Agency, Department of Defense. (1) N. RosE~: (Ten. Rel. Gray., 6, 435 (1973); Ann. o/Phys. , 22, 1 (1963); Phys. Rev., 57, 147 (1940). (:) H. YILMAZ: Phys. Rev., l l l , 1417 (1958); Phys. Rev. Lett., 20, 1399 (1971); Phys. Today, 26, 15 (1973); LeSt. Nuovo Cimento, 6, 181 (1973); 7, 337 (1973); Ann. o] Phys., 81, 179 (1973); Nuovo Cimento, 10 B, 79 (1972).

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2 0 2 H. YILMAZ

We shall first show that if we ignore the conservation laws (3) of Ros~N, the remaining three of his equations are exactly soluble. We shall assume the following form:

(5) ~ , = [~ exp [Ale + 2 B ~ ] ] ~ ,

where ~ is the Lorentz metric, I is the uni t matrix and ~ = ~(x) is a local-field tensor. v The quant i ty e is the trace of e~, namely e~. A and B are two constants to be deter-

mined from the field equations (1). For A = - - B = 2 this form will be seen to v satisfy (1) and (2). Then it will be clear that (4) is also satisfied if we impose on eg

the gauge condition

(~) ~ , e ~ = o .

To prove these assertions we construct N~ and N from (5) and insert them into (1). We obtain

(7) (A + B)~ ~ u ~ ~ 8 ~ T ~

The first term cannot accommodate arbitrarily moving matter, au~u ~, hence in the case of (r matter (no pressures or Maxwell stresses) on has A + B = O .

v If one assumes eg is the analogue of the Newtonian field, then one gets B = - - 2. The solutions of (1) and (2) are therefore given by

(8) = ,~ exp [ 2 ( e - - 2~)2,

where for convenience we used e = ~ Tr ~. The determinant and inverse of this expression are (z), respectively, - - g = exp [4e] and g ' ~ = (~-l)m. We see that using (6) we can satisfy (4) as follows:

(9)

(10)

y-1 = ~ oxp [-- 2(e - - 2~)] , x/---~ = exp [2e],

a , ( ,g" ' ) = a , ( ~ / : ~ g " ' ) = a , [~ exp [4r = ~[exp [4~]] "~ a~e~ = o.

The solutions of Rosen's equations (1), (2) and (4) are, therefore, the same as in the author 's theory of gravitation (2). Consequently, if nothing went wrong with the con- servation equations (3) or with tho second metric y ~ of ROSEN, the theory Would cor- respond to a viable theory of gravitation with respect to all known experiments (~), would lead to a <~ gray hole ~ concept (3) for collapsed matter instead of (, black holes ~ and would have legitimate gravitational waves (4). For example, the metric gu = - - oxp [-- 4Q], g2~ = - - exp [4Q], where Q 1 2 = e l ( t - z) = - e 2 ( t - z) is a rigorous solu- t ion of the above equations and represents a plane gravitational wave (4). Unfortunately, however, these hopes are destroyed, either by the conservation laws (3), or by the second metric 7 ~ of ROSEN, as will be shown in the subsequent two paragraphs.

To show that the conservation laws (3) are incompatible with the solutions of the other three equations of ROSEN we now compute, using (8), the divergence-free Einstein tensor

v 1 ~ R 0; ; v ~ 0 (11) G~= R , - - ~ , ,

(8) R . E. CLAI'P: Phys. Rev. D, 7, 345 (1973). ( ') R . A . B o ~ E ~ a n d R . PAVELLE: ~pring Meeting, Washington, D.C., 1974, Bull. Am. Phys. Not., (in press) .

O N R O S E N ' S B I M ] g T R I C T H E O R Y O F G R A V I T A T I O N 9 , 0 3

y where R~ is taken as the negative of the usual definition. I t is known from the work of the author that (8) yields (~)

(12) R~-- �89 ~ R = - - 8~(T~ -[- t ~ / 4 ~ ) ,

v where t~ is, up to a divergence, given by

(13) t , - - 2 ( ~ ~--�89 + ~,~ ~ - - ~ ~

From (3) and (11) we therefore have

(14) t~;,= o .

This consequence of (3) and (11), however, contradicts the solution (8) because, at least in first and second order in %

which implies the vanishing of the field gradients, namely

The exponential solution (8) then reduces to a constant metric, hence represents a fiat space. If this computation appears a little complicated, it is perhaps easier to check the t ruth of the conclusion in the simple static case, where only ~(x , y, z) is present. In this ease ~0~ ~, hence (8) yields

(17) goo = exp [-- 2q], --gii = exp [2(p] ,

p then tg is now the simple Newtonian stress energy tensor (2)

y y (18) t i t= - - ~ v 8 ~ + �89 b ~ v S ~ .

I t is then explicitly found that

(19) t~;v = a~ ~a~ 8~q~ = - - exp [-- 2~] 4 = a a ~ = 0 ,

(20) O~v = 0 , ~ = constant .

y Thus under the conscrvation laws T#;~= 0 of ROSEN even the unique static me- tric (17) is reduced to a completely fiat space.

We next turn to a difficulty connected with Rosen's fiat metric ya~. For the purpose of the argument we pretend not to know about the above difficulty with the field equa- tions. The solution (8) is then not necessarily fiat, hence usable, as a curved space-time geometry. We are interested in the d'Alembert operator of this curved space-time geometry. Using (4) with ~ = (g/y)�89 = ~ /~g , y = - 1, we find

(21) ~ > = ( ~ / - - - 5 ) - ~ , v ' ~ r = g~%~e, ( 2 2 ) ~ ~ ~ g v a ~ g = 4ztTg.

2 0 4 H. YILMAZ

On the other hand, by using (7) with A + B = 0, B = - - 2, Rosen's field equations (1) become

(23) ~ ~ ~ v y %~% = 4Jr~T~.

One finds that, in order to be interpretable as a propagation law, Rosen's field equa- tions (23) must yield

(24) g ~ = ~ - - l y ~ : e x p [-- 2~] y ~ .

However, the flat metric ~ = exp [2r reduces to a constant only in the static limit (17) of (8) and only for the special part ~" as

(25) ?,is= __ 1 , ~,oo= exp [4~o] .

Therefore, if the field is time independent, the two equations (22) and I24) become equivalent and the metric (17) is a legitimate solution. However, in view of (24) this would be the only legitimate curved-space solution allowed by Rosen's equations and, as explained in the previous paragraph, it would still be in conflict with his conservation laws (3) and the flat space condition yoo= 1.

In the author 's theory of gravitation to which Rosen's theory bears a resemblance the above mentioned difficulties do not arise. This is because (3) is rejected in favor of the identi ty (11), hence (14) does not apply. As for our propagation laws we have (22) instead of Rosen's (23) and we avoid the conflict with the curved-space d'Alembertian (21) and the constraint (24), because we do not have a second, flat-space metric 7~. According to our theory the physical observation is a local process and this is interpreted as the local unobservability of the absolute values of ~%. We arc then able to use the potential differences ~ = ~ ( x ) - ~(x'), where x' is the position of the observer. Locally (that is when x--->x') the metric behaves as g a ~ 7 ~ without its gradients becoming zero and without getting into the difficulties of (24) and (25). ROSEN attempts to obtain a local Lorentz invarianee by introducing a second, flat metric y ~ (hence the name bimetrie theory) and this necessarily conflicts with the propagation equation of the curved geo- metry as it must.

To close with a cheerful note we may, however, point out that something interesting may still be salvaged from Rosen's theory. To see this we note again that in the absence of (3) ~he static solution (17) is rigorously valid. The conservation laws (3), on the other

v hand, contain first- and second-order terms of which only the second-order part t~ causes the difficulty. Under the assumption the geodesic equations of motion to be separately postulated, the first-order part of the static metric (17), namely

(25) goo = 1 - 2~o, --g,:~ = 1 + 2q~, c~~ = 0 ,

is therefore a defensible solution. (Static form of linearized Einstein solution.) In this case one may ignore the yoo = exp [4~0] part of (25) by saying that wc are interested only in the spaeial part of (24) which yields the proper curved-space Poisson equation. Then the theory predicts the three first-order effects, namely red shift, light bending and time delay, correctly.