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A Metric Theory of Gravitation Without Minimal Coupling of Matter and Gravitational Field H

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A Metric Theory of Gravitation Without Minimal Coupling of Matter and Gravitational Field H A Metric Theory of Gravitation without Minimal Coupling of Matter and Gravitational Field H. Goenner Institut für Theoretische Physik der Universität Göttingen (Z. Naturforsch. 31a, 1451-1456 [1976] ; received October 20, 1976) A metric theory of gravitation is suggested which reduces to Einstein's theory in the case of vanishing matter. If matter is present, in the Lagrangian formulation of the theory the principle of minimal coupling is given up by directly linking the matter variables to the curvature tensor. The theory contains a free parameter of dimension length. It is considered not to be a universal constant but a length characteristic for the mass of the material system described. Results diverging from those of General Relativity are to be expected for regions with high cur- vature i. e. especially for gravitational collapse and dense phases of the cosmos. An exact, static and spherically symmetric solution with constant matter density is discussed; it indicates that, possibly, gravitational collapse is avoided. 1. Introduction duces to the expressions of special relativity. This procedure is considered an expression of the prin- Einstein's theory of general relativity represents 2 ciple of minimal coupling . The gravitational field both a theory of gravitation and a framework for described by gxx is coupled to the matter variables macroscopic theories of matter. In the Lagrangian A u via gxi and gx^v only, not via second derivatives. formulation the field equations derive from a varia- In the opinion of many relativists the principle of tional principle with scalar Lagrangian density: minimal coupling is a necessary consequence of the £ = £F + 2x£M (1) strong equivalence principle by the following line of reasoning. The strong equivalence principle states where Z = 8.TG/C4 is the coupling constant and* that, in a local inertial system, the physical laws £F=V~gR\ (2) hold as formulated in special relativity theory. The describes the gravitational field while local inertial system is identified with the geodesic xoordinates. As far as the equivalence principle is A A £M = V-gLM (gx;i, u , u .a,...) (3) supported by, for example, the free fall experiments is the matter part of the Lagrangian. uA denotes the of Eötvös, Renner, Dicke and Braginsky 3'4 the prin- variables of matter and matter fields (A is a global ciple of minimal coupling is a sine qua non con- index). By variation of the action integral with dition for any theory describing the interaction of regard to the metric and the variables uA the field matter and gravitation. equations of gravitation and of matter follow. As direct consequence of minimal coupling we The important point, here, is the prescription for may take the equation for the matter tensor obtaining Ly from the Lagrangian of the corre- sponding material system in special relativity: Taß;ß = 0 (4) Replace, in the special relativistic Lagrangian following from Einsteins field equations uAi uA,u,...) the Minkowski metric rjxi by the Riemannian metric gyK and the partial derivative -y.T'ß (5) 3/3:ra by the covariant derivative with regard to and the contracted Bianchi identities. In a local in- . This prescription guarantees that, in each event, ertial system, it coincides with the special relativistic a coordinate system (geodesic coordinates) can be conservation laws for energy and momentum found in which the general relativistic theory re- Taß,ß = 0. Replacements of the type suggested, for example, by Rastall5 of Eq. (4) Reprint requests to Dr. H. Goenner, Institut für Theoreti- sche Physik, Bunsenstr. 9, D-3400 Göttingen. Tt.^XK^sf* (6) * Greek indices run from 0 to 3, Latin indices from 1 to 3. a violate the formulation of the equivalence principle Sign (gxx) — —2. For the definition of R ßY& and R0° the sign conventions of Eisenhart1 are used. given above 6. 1452 H. Goenner • A Metrie Theory of Gravitation Nevertheless, some doubts seem to be permitted relativistic conservation law T^3ß = 0 ought to fol- as to this line of arguments. The high-precision mea- low. surements backing the equivalence principle have Of course, these assumptions cannot determine all been made outside of matter. In fact, with a few the Lagrangian of the theory. Detailed investiga- exceptions4'7'8, it is rarely touched upon what tions will have to show which if any of the various meaning Einstein's thought experiment with the possibilities is most preferable. We now replace the freely falling elevator has within macroscopic mat- Lagrangian density (1) by ter. Also, in the derivation of Eq. (4) from kinetic £ = £ + 2xA(fr )£ (7) theory9 [without use of the field equations (5)] it F M M is tacitly assumed that the principle of minimal where, as in Eqs. (2), (3) coupling holds. In view of the many theories of gravitation de- £f = V~-g R°o = V~gR vised which contain additional geometrical objects A A (scalar fields, tetrads etc.) or even leave the frame- £.v = V-g Lm (g-A, u , u ;a<...) work of Riemannian geometry, it many be worth to while A(R1ßri) is a scalar function of the (fourteen) investigate theories of the type of General Relativity scalar invariants of the curvature tensor satisfying in which the principle of minimal coupling is given up, provided that those results of Einstein's theory >4(0) =1. (7) supported strongly by observation and experiment A simple choice for A makes it a function of the are regained. Ricci scalar R alone: A class of such theories is suggested in Sect. 2 and narrowed to one single theory whose field A(R*ßyd)=A(R) . (8) equations are then derived. In Sect. 3 the case of an The variation of the Lagrangian (7) with respect to ideal fluid matter distribution is studied. In Sect. 4 g, ß leads to the point particle approach to matter is dealt with, a briefly. The Newtonian approximation of the theory d£ = S£f + 2X (SA £m + A d£M) is treated in Sect. 5 while Sect. 6 is devoted to spherically symmetric, static gravitational fields. An where, with Eq. (8) exact solution for constant energy density is dis- £m SA = V - g [ - A' LM R*FI cussed. In a concluding section we give an outlook on further possibilities of breaking the principle of minimal coupling. + divergence terms (9) and A : = dA/'dR. Leaving aside the divergence 2. A Theory of Gravitation without Minimal terms, the field equations become: Coupling -y.[AT*fi + 2 A' LmR*P (10) We set the following guideposts for the construc- tion of a theory without minimal coupling: Here, T^ : = ( — 2/V — g) d£Mfdguß is the energy- 1) The field equations ought to be second order momentum tensor of Einstein's theory. partial differential equations for gaß and to follow In order to obtain second order equations we put from a variational principle. 2 2) The variables of matter (matter fields) are A = l+el0 R, £=±1; (8) coupled directly to the curvature tensor. This is to /0 is a constant of dimension length which we inter- be done in a way independent of any specific mate- pret to characterize the mass of the matter distribu- rial system (universal coupling). tion (see below and Section 5). The sign factor £ 3) If Ly = 0, i. e. if no matter is present, the field is left undetermined, at this stage. To this extent /0 equations ought to coincide with the Einstein vacu- cannot be fitted arbitrarily to observational data. um field equations. With (8) we arrive at the field equations 4) If R2ßys= -0, i.e. if no permanent gravitational fields are present, then, in each event, the special G*P=-x6*fi (11) 1453 H. Goenner • A Metrie Theory of Gravitation x with the new energy-momentum tensor of matter: where a is the radius and rs = 2GMc~ the Schwarz- 2 schild radius of the gravitating object. We now put: <9^:=(1 +£l0 R)T*P (12) 2 2 k = rs. (16) + 2 el Lm R*-2e I LM, X ,X (R^ - ST' g*) • 2 Thus, for ordinary stars and white dwarfs, but not For notational convenience we introduce / : = f /0 x. for neutron stars and collapsing objects the results From Eq. (11) of the new theory should be the same as in General <9*^ = 0 (13) Relativity. which, if expressed in terms of the energy-momen- tum tensor and the matter Lagrangian of Einstein's 3. Gravitating Ideal Fluid theory, reads as In order to apply the preceeding theory we start 2 Aß from the special relativistic Lagrangian for an T ;FI= - [\og{\ + el0 (14) ideal fluid. The fluid is described by its rest mass -L [\og(L+El 2R)], g>0. M 0 ß density o, its internal energy density e(Q), its total There is no unusual ambiguity in this definition of energy density F(q) with F = o[c2 + e ((?)] = : Q y the new energy-momentum tensor The obvious and its 4-velocity ua. Upon these matter variables choice is to take the r.h.s. of the field equations the following constraints are imposed which directly results from the variation c2 = r\,.i u* ux, (17 a) (-2/V-g)d(C-£F)/dgaß 0 =({?»*).*• (17b) except for the dropping of divergence terms. How- 10 ever, the name may appear strange as <9I/? now With Ray we add the constraint equations to the contains the Ricci tensor explieitely. This reflects matter Lagrangian: (18) the violation of minimal coupling. In the present Lm=-F theory of gravitation it is even more difficult to by means of Lagrangian multipliers , /2: separate out material sources and gravitational field L.\i + /i(r] ).
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