Exploiting gauge and constraint freedom in hyperbolic formulations of Einstein's equations

TL;DR

This paper introduces new hyperbolic formulations of Einstein's equations with flexible gauge conditions, aiming to improve numerical relativity simulations by providing systems with adjustable parameters and controlled characteristic speeds.

Contribution

It presents two new multi-parameter hyperbolic formulations of Einstein's equations that incorporate general gauge conditions, expanding the tools for 3D numerical relativity.

Findings

Formulations include 30 and 34 variables with algebraic gauge conditions.

Systems have free parameters allowing customization for numerical stability.

Conditions identified to avoid superluminal characteristic speeds.

Abstract

We present new many-parameter families of strongly and symmetric hyperbolic formulations of Einstein's equations that include quite general algebraic and live gauge conditions for the lapse. The first system that we present has 30 variables and incorporates an algebraic relationship between the lapse and the determinant of the three metric that generalizes the densitized lapse prescription. The second system has 34 variables and uses a family of live gauges that generalizes the Bona-Masso slicing conditions. These systems have free parameters even after imposing hyperbolicity and are expected to be useful in 3D numerical evolutions. We discuss under what conditions there are no superluminal characteristic speeds.