Three dimensional numerical relativity: the evolution of black holes

TL;DR

This paper introduces a new 3D numerical relativity code for evolving black hole spacetimes, demonstrating techniques to improve accuracy and stability in simulations without symmetry assumptions.

Contribution

The paper presents a novel 3D numerical code for Einstein equations, including advanced techniques for black hole evolution and handling singularities, with performance on parallel supercomputers.

Findings

Successful evolution of 3D black hole spacetimes up to t=50M

Implementation of special conformal factor treatment improves accuracy

Apparent horizon mass conserved within 5% during evolution

Abstract

We report on a new 3D numerical code designed to solve the Einstein equations for general vacuum spacetimes. This code is based on the standard 3+1 approach using cartesian coordinates. We discuss the numerical techniques used in developing this code, and its performance on massively parallel and vector supercomputers. As a test case, we present evolutions for the first 3D black hole spacetimes. We identify a number of difficulties in evolving 3D black holes and suggest approaches to overcome them. We show how special treatment of the conformal factor can lead to more accurate evolution, and discuss techniques we developed to handle black hole spacetimes in the absence of symmetries. Many different slicing conditions are tested, including geodesic, maximal, and various algebraic conditions on the lapse. With current resolutions, limited by computer memory sizes, we show that with certain lapse conditions we can evolve the black hole to about $t=50M$, where $M$ is the black hole mass. Comparisons are made with results obtained by evolving spherical initial black hole data sets with a 1D spherically symmetric code. We also demonstrate that an ``apparent horizon locking shift'' can be used to prevent the development of large gradients in the metric functions that result from singularity avoiding time slicings. We compute the mass of the apparent horizon in these spacetimes, and find that in many cases it can be conserved to within about 5\% throughout the evolution with our techniques and current resolution.