Initial Value Problems on Noneuclidean Topologies.

Einstein's equations of general relativity, studied as a Cauchy problem are presented. The equations are split into space-plus-time form, and the dynamical and kinematical variables are distinguished. The initial value equations are specialized to maximal, conformally flat initial spacelike hypersurfaces. Time symmetric solutions of the initial value equations are investigated. When the symmetry conditions are applied to the extrinsic curvature tensor, a simple form for the vacuum solution of the momentum constraint is obtained, representing an object with linear momentum. It is argued that the resulting three-geometry consists of two physically identical, asymptotically flat regions connected by an extremal area throat. Solutions including angular momentum and electric fields are also presented. The solutions found are purely geometric models that may represent moving, (classically) spinning, or electrically charged particles.