Dynamical Gauge Conditions for the Einstein Evolution Equations

TL;DR

This paper develops extended symmetric hyperbolic formulations of Einstein's equations by incorporating gauge degrees of freedom as dynamical variables, allowing for adjustable characteristic speeds and improved numerical stability.

Contribution

It introduces new hyperbolic systems that include gauge fields as dynamical variables with adjustable parameters for causality and stability optimization.

Findings

Characteristic speeds can be made causal by parameter adjustment.

The systems generalize known gauge conditions like K-driver and Gamma-driver.

Parameters allow for stability tuning in numerical simulations.

Abstract

The Einstein evolution equations have been written in a number of symmetric hyperbolic forms when the gauge fields--the densitized lapse and the shift--are taken to be fixed functions of the coordinates. Extended systems of evolution equations are constructed here by adding the gauge degrees of freedom to the set of dynamical fields, thus forming symmetric hyperbolic systems for the combined evolution of the gravitational and the gauge fields. The associated characteristic speeds can be made causal (i.e. less than or equal to the speed of light) by adjusting 14 free parameters in these new systems. And 21 additional free parameters are available, for example to optimize the stability of numerical evolutions. The gauge evolution equations in these systems are generalizations of the ``K-driver'' and ``Gamma-driver'' conditions that have been used with some success in numerical black hole evolutions.