A hyperbolic tetrad formulation of the Einstein equations for numerical relativity

TL;DR

This paper reformulates the Einstein equations using a hyperbolic tetrad approach suitable for numerical relativity, simplifying previous equations and demonstrating their hyperbolic nature under various gauge conditions.

Contribution

It introduces a simplified, hyperbolic tetrad formulation of Einstein's equations that is adaptable to different gauge choices for numerical simulations.

Findings

The equations form a first-order symmetrizable hyperbolic system.

The formulation is applicable under multiple gauge conditions.

Inclusion of lapse and shift enables general coordinate evolution.

Abstract

The tetrad-based equations for vacuum gravity published by Estabrook, Robinson, and Wahlquist are simplified and adapted for numerical relativity. We show that the evolution equations as partial differential equations for the Ricci rotation coefficients constitute a rather simple first-order symmetrizable hyperbolic system, not only for the Nester gauge condition on the acceleration and angular velocity of the tetrad frames considered by Estabrook et al., but also for the Lorentz gauge condition of van Putten and Eardley, and for a fixed gauge condition. We introduce a lapse function and a shift vector to allow general coordinate evolution relative to the timelike congruence defined by the tetrad vector field.