Numerical Evolution of Black Holes with a Hyperbolic Formulation of General Relativity

TL;DR

This paper presents a novel numerical code that employs a hyperbolic formulation of Einstein's equations to simulate black hole spacetimes, successfully evolving a Schwarzschild black hole in spherical symmetry with stable, second-order accurate methods.

Contribution

First implementation of a hyperbolic formulation for evolving a black hole spacetime numerically, demonstrating stable evolution with excision and causal differencing in spherical symmetry.

Findings

Successful evolution of a Schwarzschild black hole using hyperbolic formulation

Implementation of a causal differencing method for stability

Achieved second-order accuracy in spherical symmetry

Abstract

We describe a numerical code that solves Einstein's equations for a Schwarzschild black hole in spherical symmetry, using a hyperbolic formulation introduced by Choquet-Bruhat and York. This is the first time this formulation has been used to evolve a numerical spacetime containing a black hole. We excise the hole from the computational grid in order to avoid the central singularity. We describe in detail a causal differencing method that should allow one to stably evolve a hyperbolic system of equations in three spatial dimensions with an arbitrary shift vector, to second-order accuracy in both space and time. We demonstrate the success of this method in the spherically symmetric case.