Axisymmetric constant mean curvature slices in the Kerr space-time

Abstract

Recently, there have been efforts to solve Einstein's equation in the context of a conformal compactification of space-time. Of particular importance in this regard are the so called CMC-foliations, characterized by spatial hyperboloidal hypersurfaces with a constant extrinsic mean curvature K. However, although of interest for general space-times, 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.

14 pages, 4 figures, Version published in CQG

References in corpus (6)

• Binary-black-hole initial data with nearly-extremal spins
• Regularity of the Einstein Equations at Future Null Infinity
• Gravitational perturbations of Schwarzschild spacetime at null infinity and the hyperboloidal initial value problem
• Tetrad formalism for numerical relativity on conformally compactified constant mean curvature hypersurfaces
• Black hole initial data on hyperboloidal slices
• The general spherically symmetric constant mean curvature foliations of the Schwarzschild solution

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• Evolution of scalar fields surrounding black holes on compactified constant mean curvature hypersurfaces
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• Hyperboloidal neutron star and black hole initial data in spherical symmetry