Interaction Energy in Geometrostatics

Abstract

The "geometrodynamical" description of particles by means of topological features of empty space-time is applied here to the case of $N$ charged masses which are momentarily at rest. These particles are represented by Einstein-Rosen bridges in a mainfold which satisfies the time-symmetric initial value equations of gravitation and electromagnetism. Invariant definitions are given for the total mass energy of the system, and for the "bare mass" of each Einstein-Rosen bridge. These masses characterize various asymptotically Schwarzschildian regions of the manifold and are, therefore, conserved in time. The total mass of the system differs from the sum of the bare masses by contributions from the gravitational and electrostatic interaction energies. It is shown that the interaction energy is always negative, and that it reduces to the classical expression in the limit of large separation between the masses. The shape of the minimal surface associated with each Einstein-Rosen bridge, another invariant feature of the "particle," is discussed. The minimal surfaces are also used to characterize manifolds which can be interpreted as a closed universe containing $N+1$ "particles."