Horizon Boundary Condition for Black Hole Spacetimes

TL;DR

This paper details a numerical scheme for black hole spacetime evolution that avoids singularities by excising the interior region, using horizon locking coordinates and causal differencing to improve stability and accuracy.

Contribution

It introduces a horizon boundary condition scheme with specific coordinate and differencing methods, enabling long-term stable evolutions of black holes in numerical relativity.

Findings

Black holes can be evolved accurately beyond t=1000 M.

The scheme effectively avoids singularities in numerical simulations.

Different shift conditions influence the horizon locking coordinate stability.

Abstract

It was recently shown that spacetime singularities in numerical relativity could be avoided by excising a region inside the apparent horizon in numerical evolutions. In this paper we report on the details of the implementation of this scheme. The scheme is based on using (1)~a horizon locking coordinate which locks the coordinate system to the geometry, and (2)~a finite differencing scheme which respects the causal structure of the spacetime. We show that the horizon locking coordinate can be affected by a number of shift conditions, such as a ``distance freezing'' shift, an ``area freezing'' shift, an ``expansion freezing'' shift, or the minimal distortion shift. The causal differencing scheme is illustrated with the evolution of scalar fields, and its use in evolving the Einstein equations is studied. We compare the results of numerical evolutions with and without the use of this horizon boundary condition scheme for spherical black hole spacetimes. With the boundary condition a black hole can be evolved accurately well beyond $t=1000 M$, where $M$ is the black hole mass.