Excised black hole spacetimes: quasi-local horizon formalism applied to the Kerr example

TL;DR

This paper develops a numerical method for computing excised initial data of black hole spacetimes using the isolated horizon formalism, validating it against the Kerr solution and analyzing boundary conditions.

Contribution

It introduces a new approach combining Dirac gauge and constrained formalism to accurately model black hole horizons without boundary conditions on the horizon.

Findings

Good agreement with Kerr solution in Kerr-Schild coordinates

Validation of the isolated horizon multipolar analysis

Assessment of conformally flat approximation limitations

Abstract

We present a numerical work aiming at the computation of excised initial data for black hole spacetimes in full general relativity, using Dirac gauge in the context of a constrained formalism for the Einstein equations. Introducing the isolated horizon formalism for black hole excision, we especially solve the non-conformally flat part of the equations, and assess the boundary condition problem for this part. In the stationary single black hole case, we present and justify a no-boundary treatment on the black hole horizon. We compare the data obtained with the well-known analytic Kerr solution in Kerr-Schild coordinates, and assess the widely used conformally flat approximation for simulating axisymmetric black hole spacetimes. Our method shows good concordance on physical and geometrical issues, with the particular application of the isolated horizon multipolar analysis to confirm that the solution obtained is indeed the Kerr spacetime. Finally, we discuss a previous suggestion in the literature for the boundary conditions for the conformal geometry on the horizon.