Two black holes with axisymmetric parallel spins: initial data

The extrinsic curvature on an initial spacelike hypersurface of two black holes with axisymmetric parallel spins is derived in terms of an analytic infinite series. This curvature constitutes an important piece of the initial data required to solve completely the Cauchy problem of a head-on collision of two holes with spin. The extrinsic curvature is constructed so that the resulting spacetime will have the topology of two Einstein-Rosen bridges; a physical equivalence of the top and bottom sheets of the initial hypersurface is also built in. It is shown that one may freely specify the spins of the two black holes. The authors display some sample numerical evaluations indicating that the series, for reasonable configurations, converges rapidly and hence is amenable to numerical work. They then pose the approximate numerical problem to obtain the rest of the initial data.