Maximally slicing a black hole with minimal distortion.

Abstract

Equations are derived which determine the maximal hypersurfaces of the analytically extended Kerr-Newman spacetime. An analytic solution is obtained in the charged, nonrotating case for the asymptotically flat maximal hypersurfaces of the Reissner-Nordstr\"om spacetime using spatial coordinates which minimize the coordinate distortion. The slices tend asymptotically in time to a limiting hypersurface lying between the inner and outer horizons, while covering the domain of outer communication of the black hole. The coordinate lines are drawn down the black hole if coordinate symmetry is maintained across the throat. The equation for the limiting hypersurface in the Kerr geometry is solved numerically. An apparently unique solution exists for all rotating black holes with specific angular momentum \ensuremath{\Vert}a\ensuremath{\Vert}<M, where M is the black-hole mass. An excellent analytic approximation is derived for the value of the Boyer-Lindquist coordinate r on the hypersurface. Implications for gravitational-collapse calculations are discussed.