On the gravitational influence of direct particle fields

Abstract

The problem of the contribution of direct particle interaction of the Fokker type to the gravitational equations is solved. It is shown that the usual procedure for obtaining the gravitational equations, of making small variations of geometry, δg_ik+δg_ik replacing g_ik in finite regions, with δg_ik=0 on their boundaries, and of requiring that the action be stationary for such variations, can be carried through with the aid of Green functions. This procedure, due to Hilbert, serves to define the energy tensor T_ik associated with each of the fields. That for the C-field turns out exactly the same as we have used in the macroscopic form of the theory. That for the electromagnetic field turns out to have some new features. These are terms containing the vector potential and its derivative when world-lines are broken, although these terms vanish when there is charge conservation. The terms in the field F_ik are identical with the usual tensor if the field is calculated from retarded potentials. In former work no decision has been made on the form the tensor should take when the potentials are ½ (retarded + advanced). Wheeler & Feynman showed that alternative choices are possible and that a decision cannot be made from electromagnetic considerations alone. Our analysis leads to a unique result, the Frenkel tensor.