Abstra.c t. We discuss the field equat.ions which stem from a variational 1)rinciIde containing t.ho quadratic terms 3iRp, HPY . He showed that the Hilbcrt,-Eiiist,cin Lagmngian, R, anppleinenttid by teriiis proport,ional to RpVR'ly anc RZ, leads--in the linear approxiaiation-to stat,ic sphericalsymmetric solutions which are a sun1 of the Newtonian and t.he Yultawa potentials. They t,encl t o a finite valne at t.he origin T = 0 and represent; the static analogues of the short-range as well the long-rangc, parts of the gravitational potential glrv.
The question of the gravlton rest mass is briefly discussed and then it is shown that the Sciama-Dicke formulation of Mach's princ~ple admits, In the linear approxlmatlon. thc. calculation of the graviton rest mass. One finds that the value of the graviton rest mass depends on the cosmological model adopted, the mean matter density In the universe, the speed of light, and the constant of gravitation. The value obtained for an infinite, stationary universe is 7.6 X lo-" g. The value for evolutionary cosmological models is found to depend critically on the mass and "radius" of the universe, both null and non-null values occurring only for certain values of these parameters. Problems that arise as a consequence of the linear approximation are pointed out.Over the past few y e a r s , considerable interest has been manifested in m a s s i v e graviton theories of gravitation stemming f r o m the investigations of Ogievetsky and Polubarinov' and Freund etal.' This, in turn, h a s r a i s e d questions a s to the magnitude of the upper limit to the graviton r e s t m a s s . Goldhaber and Nieto, a f t e r noting that the question of the graviton r e s t m a s s i s yet to be settled, have recently shown that observations on the intergalactic scale set an upper limit of 2 X g on the graviton r e s t m a s s . 3 Enlploying Mach's principle a s formulated by S~i a m a~'~ and Dicke,' it can be shown that this upper limit, if certain assunlptions about the m a s s of the universe a n d / o r the exactitude of the linear approximation a r e granted, i s about five o r d e r s of magnitude g r e a t e r than the l a r g e s t value theoretically to be expected-assuming, of course, the validity of this statement of Mach's principle.According to Sciama and Dicke, Mach's principle implies the relation That i s , the r e s t energy (inc2) of a neutral body is equal to i t s total gravitational potential energy a r i s i n g f r o m the gravitational inertial interaction with other m a s s e s ( m i ) located elsewhere (at d i stances I . , ) in the universe. G i s the constant of gravitation, and c the speed of light, and this r elationship i s a Newtonian approximation in Minkows k i space-time that can merely be regarded a s accurate to o r d e r of tnagnitude since cosmological distances a r e involved. (Sciama obtains a n equivalent expression by considering the inertial r e a ction of bodies to accelerations, presumably induced by their interaction with other m a t t e r in the universe.') If we make the usual simplifying cosmological assumptions of homogeneous and isotropic m a t t e r (including energy in the f o r m of radiation and s t r e s s e s ) density p at z e r o p r e s s u r e i n the universe, then, with the deletion of r i l , Eq. (1) becomes where 0 i s the ( a r b i t r a r y ) location of an o b s e r v e r and R his event horizon, i.e., his observed "radius" of the universe. The integral in Eq. (2) i s readily recognized a s the gravitational potential of the m a t t e r in the observable universe a t the ...
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