The two-body problem in geometrodynamics

The problem of two interacting masses is investigated within the framework of geometrodynamics. It is assumed that the space-time continuum is free of all real sources of mass or charge; particles are identified with multiply connected regions of empty space. Particular attention is focused on an asymptotically flat space containing a “handle” or “wormhole.” When the two “mouths” of the wormhole are well separated, they seem to appear as two centers of gravitational attraction of equal mass. To simplify the problem, it is assumed that the metric is invariant under rotations about the axis of symmetry, and symmetric with respect to the time t = 0 of maximum separation of the two mouths. Analytic initial value data for this case have been obtained by Misner; these contain two arbitrary parameters, which are uniquely determined when the mass of the two mouths and their initial separation have been specified. We treat a particular case in which the ratio of mass to initial separation is approximately one-half. To determine a unique solution of the remaining (dynamic) field equations, the coordinate conditions g0α = −δ0α are imposed; then the set of second order equations is transformed into a quasilinear first order system and the difference scheme of Friedrichs used to obtain a numerical solution. Its behavior agrees qualitatively with that of the one-body problem, and can be interpreted as a mutual attraction and pinching-off of the two mouths of the wormhole.

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