Constrained evolution in axisymmetry and the gravitational collapse of prolate Brill waves

TL;DR

This paper investigates numerical schemes for Einstein's equations in axisymmetric vacuum spacetimes, identifies issues with existing methods, proposes a new stable scheme, and applies it to study gravitational collapse of prolate Brill waves leading to black hole formation.

Contribution

The paper introduces a new stable numerical scheme for axisymmetric Einstein equations that overcomes issues with indefinite elliptic equations in previous methods.

Findings

Existing schemes have indefinite elliptic equations causing solver failures.

The new scheme avoids these issues and is numerically stable.

Prolate Brill waves collapse to black holes rather than naked singularities.

Abstract

This paper is concerned with the Einstein equations in axisymmetric vacuum spacetimes. We consider numerical evolution schemes that solve the constraint equations as well as elliptic gauge conditions at each time step. We examine two such schemes that have been proposed in the literature and show that some of their elliptic equations are indefinite, thus potentially admitting nonunique solutions and causing numerical solvers based on classical relaxation methods to fail. A new scheme is then presented that does not suffer from these problems. We use our numerical implementation to study the gravitational collapse of Brill waves. A highly prolate wave is shown to form a black hole rather than a naked singularity.