Semigenerally covariant equations of motion—II the significance of the “tail” and the relation to other equations of motion

Abstract

The semigenerally covariant equation of motion of a particle of finite mass moving in an “external” gravitational field, obtained in the previous paper, is further discussed. At first we decompose it into four scalar equations along four orthonormal vectors moving with the particle. It is shown that if the particle moves along a geodesic line in the external field, then the “tail” vanishes. The vanishing of the “tail,” on the other hand, implies a motion that tends to be geodetic, without the appearance of any runaway solution. The “tail” term is then determined for the case of slow motion (to an accuracy up to the seventh order in c −1) of two finite mass particles. It is shown that its structure is very similar to that obtained for the (test) charged particle moving in a static gravitational field, i.e., it depends on the velocity of the particle and the Riemann tensor of the “external” field. We then show that both the Einstein-Infeld-Hoffmann and the Havas-Goldberg equations of motion can be obtained as limiting cases of the semigenerally covariant equation when one expands the latter in c −1 and G, respectively.