Schwarzschild Tests of the Wahlquist-Estabrook-Buchman-Bardeen Tetrad Formulation for Numerical Relativity

TL;DR

This paper adapts the WEBB tetrad formulation of Einstein's equations to spherical symmetry, testing its accuracy and stability in simulating Schwarzschild black holes, and compares different gauge choices for stable evolution.

Contribution

It extends the WEBB tetrad formulation to spherical symmetry and evaluates its stability and accuracy in black hole simulations, highlighting gauge-dependent stability.

Findings

Stable evolution achieved with Nester gauge under certain conditions

Lorentz gauge did not yield stable evolution in tests

Boundary conditions and gauge choices significantly affect stability

Abstract

A first order symmetric hyperbolic tetrad formulation of the Einstein equations developed by Estabrook and Wahlquist and put into a form suitable for numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted to explicit spherical symmetry and tested for accuracy and stability in the evolution of spherically symmetric black holes (the Schwarzschild geometry). The lapse and shift which specify the evolution of the coordinates relative to the tetrad congruence are reset at frequent time intervals to keep the constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the spatial coordinate satisfying a kind of minimal rate of strain condition. By arranging through initial conditions that the constant-time hypersurfaces are asymptotically hyperbolic, we simplify the boundary value problem and improve stability of the evolution. Results are obtained for both tetrad gauges (``Nester'' and ``Lorentz'') of the WEBB formalism using finite difference numerical methods. We are able to obtain stable unconstrained evolution with the Nester gauge for certain initial conditions, but not with the Lorentz gauge.