Axisymmetric constant mean curvature slices in the Kerr spacetime

Abstract

Recently, there have been efforts to solve Einstein’s equation in the context of a conformal compactification of spacetime. Of particular importance in this regard are the so-called constant mean curvature (CMC) foliations, characterized by spatial hyperboloidal hypersurfaces with a constant extrinsic mean curvature K. However, although of interest for general spacetimes, CMC slices are known explicitly only for the spherically symmetric Schwarzschild metric. This work is devoted to numerically determining axisymmetric CMC slices within the Kerr solution. We construct such slices outside the black hole horizon through an appropriate coordinate transformation in which an unknown auxiliary function A is involved. The condition K = const throughout the slice leads to a nonlinear partial differential equation for the function A, which is solved with a pseudo-spectral method. The results exhibit exponential convergence, as is to be expected in a pseudo-spectral scheme for analytic solutions. As a by-product, we identify CMC slices of the Schwarzschild solution which are not spherically symmetric.