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The Principle of General Covariance and the Principle of Equivalence: Two Distinct Concepts

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The Principle of General Covariance and the Principle of Equivalence: Two Distinct Concepts Revista Brasileira de Física, Vol. 10, NP 1, 1980 The Principle of General Covariance and the Principle of Equivalence: Two Distinct Concepts H. V. FAGUNDES* Instituto de Física Tedrica, São Paulo, SP Recebido em 11 de Junho de 1979 It is shown how to construct a theory with general covarian- ce but without the equivalence principle. Such a theory is in disagree- ment with experiment, but it serves to illustrate the independence ofthe former principle from the latter one. Mostramos como construi r uma teori a com covari ância gera l mas sem o principio de equivalência. Tal teoria está em desacordo com a ex- periência, mas serve para i lustrar a independência do primeiro princípio com relação ao segundo. 1. INTRODUCTION Some texbooks on the theory of general relativi ty (GRT) in- troduce the equivalence princi ple and the general covariance princi ple almost in the same breath (see, for example, ~insteinl,p.56; pauli2, pp.143-144; Tonnelat3, p.262; Wha t happened2 y3 s that soon after Einstein's discovery of the theory of special relativ ty, two problems presented themselves: (1) how to incorporate gravi ty in the new theory, and (2) how to general ize covariance in inertial frames of reference into covariance in accelera- ted frames. As is well known, Einstein coupled and solved these two pro- * Work supported by FINEP, under Contract 522/CT. blems simultaneously, precisely through a ctose asso&a.tion of the two principles under discussion. But then we feel that the student may fail to appreciate the import of general covariance per se, seeing i t perhaps as a mere conse- quence of the equivalence princi ple. Therefore, we make some conjectures based on the assumption of general covariance without equi va lence. Thi s serves to i i lustra te the independence of the former concept from the latter one5. In this spirit we discuss in Sec.2, the law of motion of a particle under the influence of a force, and in Sec. 3 we present the e- quations of electromagnetism as well as a hypothetical set of equationsfor the gravitational field. We conclude remembering the pedagogical nature of this paper. 2. THE MOTION OF A MASSIVE PARTICLE UNDER THE INFLUENCE OF A FORCE In a Lorentz system of reference (x') with metric the equation of motion of a particle of inertial mass 40, under the in- f luence of a 4-force FV is where s is the particle's proper time (c-1) and is its bvelocity. FP is the net externa1 force appl ied on the particle, for example the electromagnetic force as usual, but here also the gravi- tational force. ' If we now go to an arbitrary system (x"), with the metric invariant becomes according to the general rule for covariant tensors6. The 4-velocit~ is def i ned as i n Eq .(h), and u' and F" are given by and Equat ion (3), however, takes a different look, because we have to replace du" by the covariant differential The equation of motion is therefore In the absence of forces (3'' = 0) ~q .(l3) becomes the geode- sic equation In the absence of matter, Eq. (14) is the same as in GRT. Thus, for instance, the questions discussed by Adler et aZ.7, Ser. 4.2, about space and time in a uniformly rotating system remain valid here.An interesting fact in this connection is that, on the one hand, space-time is flat in the rotating system (t,r,g,z), since curvature is an invari- ant. On the other hand, the 3-spatial geometry is curved. The reason is that the defini tion of distances demands the concept of s imu 1 tanei ty, and this is obtained by dt* = 0, where t* is a non-integrable time-coor- dina te7. For completeness, let us calculate the Gaussian curvature of a plane z = const. in the rotating frame. The line element dk is given - 3~~ < O . (I- u2r2) We observe here that our theory predicts a rotational red shift (see Ref.7, p.126), but no gravitational red shift, since gravity does not influence the metric. This is why we say that the theory iscon- tradicted by experiment. 3. THE ELECTROMAGNETIC AND GRAVITATIONAL FORCES AND FIELDS The equat ions of electromagnetism are expressed i n th e sarne way as in GRT~". Thus the force on a particle of charge q in a field F" is Maxwel I's equations and the continuity equation are v where I lv indicates the absolute derivative wi th respect to 3: , jUis the b-current densi ty, and *FVv is the tensor dual to FUV But of course there is a difference with GRT: our g 's are ?Jv not influenced by the presence of the f ield FUV, Now we must face the problem of dealing with gravitation. We shall consider only the simplest situation, i .e., we assume the gravi ty field to be derived frorn a scalar potential $(x). We generalize Poisson's equation v2g = 4a G (I!)) P i nto where the 4-scalar p(x) is the densi ty of gravitational mass. Eq. (20) is formaliy similar to the equation for Brans and Dicke's scalar f ield7, but there are two important differences: our coii- pl ing constant G is the usual gravi tational constant, and our source can- not be the energy-momentum tensor, because this tensor is related to i - nertial rnass. Finally, the law of force on a particle of gravitational mass is a general ization of the Newtonian law which goes into (22) the expression for the motion of a particle under the action of grav 4. CONCLUSION We emphaslze that we have no intention of invalidating the equivalence principle. Our purpose should rather be viewed as a pedago- gical effort to clarify the independence of the general covariance prin- clple. REFERENCES AND NOTES 1. A. Einstein, The Meaning of Rezativity, 5 th ed. Princeton Univ.Press, Princeton 1955. 2. W. Pauli, The Theory of ReZativity, Pergamon Press, London, 1958. 3. M. A. Tonnela t, Les Principes de Za ~héorieBZectromagnétique et de Za ~ezativité,Masson , Par i s 1959. 4. C. Mdl ler, The Theory of ReZativity, 2nd ed., Clarendon Press, Oxford, 1972. 5. S. Wei nberg argues that general covariance fol lows f rom equivalence: see his Gravitation and CosmoZogy, John Wiley, New York, 1972. We do not deny this, but rather affirm that general covariance would still be pos- sible without equivalence. 6. For this and other general matters see, for example, L. Landau and E. Lifshitz, CZassicaZ meory of Fiel&, English translation by M. Hamer- mesh, Addison-Wesley, Cambridge, 1951. 7. R. Adler, M. Bazin and M. Schiffer, Introduction to General Rezativi- ty, 2nd ed. McGraw-Hill Kogakusha, Tokyo, 1975. 8. E. Kreysz ig, Intro&ction to EfferentiaZ Geometry and Riemannian Geometry, Univ. of Toronto Press , Toronto, 1975. .
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