Semigenerally covariant equations of motion— I. Derivation

Abstract

The equation of motion of a particle of finite mass m moving in an external gravitational field is derived. As an external gravitational field we choose the field obtained from the whole field by putting m = 0 in the later, where m here is the Infeld inertial mass. We use special coordinates, those of Fermi, in which the Christoffel symbols of the external field vanish along the world line of the particle and the calculations being made at an (arbitrary) assigned point q of the world line at which the velocity of the particle equals zero (using a Lorentz transformation). This is an extension of the well known method of Infeld and Schild for deriving the equation of motion of a test particle in terms of an external field. Here we go to the second approximation in the (finite) mass. The equation of motion so obtained looks like the classical equation of Dirac, based on Maxwell-Lorentz theory, for a charge moving in an external electromagnetic field. In our case the mass replaces the charge, the absolute derivative replaces the proper-time derivative in flat space, and the radiation reaction forces appear under two guises, the x term with a different coefficient, and a “tail” term whose contribution we have not estimated. The equation so obtained is a semigenerally covariant, in the sense that it is invariant under the infinite continuous group of all coordinate transformations not including the parameter m.