Electromagnetism and Gravitation: an Action-at-a-Distance Confluence

It is shown how the distinct action-at-a-distance theories of electrodynamics (Wheeler-Feynman theory) and of gravitation (Einstein-Infeld-Hoffmann theory) may be brought together in the construction of a joint electromagnetic-gravitational Hamiltonian for the classical motions of interacting point particles. This is accomplished through a general scheme for finding a Hamiltonian for the Wheeler-Feynman theory proceeding by powers of $e^2$, in which the terms (involving only particle position and conjugate momentum variables) are all a species of functional of whatever arbitrary unperturbed (electrodynamically) Hamiltonian is stipulated. When the latter is in fact taken to be that of purely gravitating particles according to Einstein, Infeld, and Hoffmann, a well-defined joint Hamiltonian is produced. The lowest order electrodynamic-gravitational contribution to the Hamiltonian of a pair of particles is worked out explicitly as an example; it arises from the coupling of Newtonian gravitation to an electromagnetic interaction term of order $\frac{1}{c^4}$, which is the next in line after the Coulomb ($\frac{1}{c^0}$) and Darwin-Breit ($\frac{1}{c^2}$) interactions.