An improved theory of gravitation. Part I

Abstract

A theory of gravitation is developed from assumptions that differ as little as possible from those of special relativity and the Newtonian theory of gravitation. As in special relativity, one assumes the existence of preferred coordinate systems (Newtonian charts) in which the nondiagonal components of the metric vanish, and in which the spatial, diagonal components are equal. The metric is determined by a single real function, the gravitational potential, which is assumed, as in the Newtonian theory, to be arbitrary to the extent of an additive constant. A uniqueness theorem is proved for Newtonian charts, and the functional dependence of the metric on the gravitational potential is determined (apart from two constants, which are later fixed by requiring that the equations of motion of a particle have the correct, nonrelativistic limit, and that the potential due to a fixed particle have the Newtonian form at great distances). By a simple change in the units of space and time, the geometry is made Minkowskian. A similar change in the units of mass makes the theory formally similar to special relativity. Particle dynamics is developed. The red shift and the deflection of light by a star are calculated, and agree with the Einstein results. The combination of the assumptions that the potentials due to particles are additive and that the potential due to a fixed particle is not proportional to 1/r, is shown to lead to difficulties. The weight of a simple system is found to be proportional to its total energy, including its gravitational interaction energy. Continuous, static mass distributions are considered. A field equation is derived for the static gravitational potential, and an expression for the energy density of the static gravitational field. The field equation is modified by assuming that the gravitational energy density is itself a source of the gravitational potential. The potential due to a static, spherically symmetric body is calculated, and the perihelion advance of a planet is found to be 11/12 of the Einstein value, in good agreement with the results of Dicke.