A theory of gravitation

Abstract

A theory of gravitation is developed -which is a natural generalization of the Newtonian theory and of special relativity. Space-time is Riemannian, but its metric is determined by a scalar functionΦ, the analogue of the Newtonian potential. The functional dependence of the metric tensor onΦ is deduced, apart from two constant parameters, from the hypothesis that no measurement in an isolated region of space-time can determine whether the potential in that region isΦ orΦ +K, whereK is any constant. Test particles are assumed to travel along geodesies, and one of the unknown parameters in the metric tensor is fixed by requiring that the motions of test particles should have the correct classical limit. The usual expression for the gravitational red-shift then follows. The action of a system of gravitationally interacting particles is found. A field equation is obtained forΦ, and an energy-momentum tensor for the whole system. The remaining undetermined parameter in the metric tensor is fixed by requiring that in the static caseΦ depend linearly on the energy density of the sources. New units of mass, length, and time are introduced, in terms of which many equations of the theory become identical in form with the corresponding special relativistic equations. Conservation laws for energy and momentum hold when these quantities are expressed in the new units. The principle of equivalence is discussed.