Towards a Relativistic Theory of Gravity
The need for a relativistic generalization of Newton’s theory of gravity is phenomenologically motivated, and a possible generalization based on the introduction of a space–time metric more general than the Minkowski one is suggested. The crucial role of
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Towards a Relativistic Theory of Gravity
The equations of Newton’s gravitational theory, which provide the theoretical foundations of Kepler’s celestial mechanics and seem to describe so well the gravitational force at all macroscopical scales, are not compatible, unfortunately, with the principles of Einstein’s special relativity. The Newton equations, in fact, predict for the gravitational effects an infinite speed of propagation in any medium. Also, they do not take into account the possible transformation properties of the gravitational field of forces from one reference frame to another. The Newtonian theory defines indeed the forces generated by static matter sources, but gives us no hint about the forces produced by moving sources. Hence, the theory may describe the gravitational field of a mass M through the static potential φ(r) = −GM/r only in the non-relativistic approximation, i.e. in the regime where the modulus of the potential energy mφ of a test mass m is negligible with respect to its rest energy mc2 , namely: GM 1. rc2
(2.1)
A correct description of gravity in the relativistic regime thus require an appropriate generalization of Newton’s theory. Which kind of generalization? A natural answer seems to be suggested by the close formal analogy existing between the Newton force among static masses and the Coulomb electrostatic force among electric charges. In the same way as the Coulomb potential corresponds to the fourth component of the electromagnetic vector potential, the Newton potential might correspond to the component of a four-vector, and the relativistic gravitational interaction might be represented by an appropriate vector field, in close analogy with the electromagnetic theory. Such an attractive speculation, however, has to be immediately discarded for a very simple reason: vector-like interactions produce repulsive static interactions between sources of the same sign, while—as is well known—the static gravitational interaction between masses of the same sign is attractive. Another simple (and formally consistent) possibility is based on the assumption that the Newton potential may be treated as an invariant under a general change M. Gasperini, Theory of Gravitational Interactions, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-88-470-2691-9_2, © Springer-Verlag Italia 2013
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Towards a Relativistic Theory of Gravity
of frame, i.e. that gravity may be correctly described by a relativistic scalar field. However, also this hypothesis has to be discarded on the ground of phenomenological results, even if the reasons, this time, are more subtle. In view of our subsequent applications it is worthwhile to recall here one of these reasons, concerning the precession of planetary orbits. Let us consider the motion of a relativistic test body of mass m, interacting with a central (i.e. radially oriented) field of forces described by the scalar potential U (r) = −GM/r. The dynamics of the problem is controlled by the relativistic Lagrangian v2 (2.2) L = −mc2 1 − 2 − mU, c
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