First-order symmetric-hyperbolic Einstein equations with arbitrary fixed gauge

TL;DR

This paper introduces a new family of variables that transform the 3+1 Einstein equations into a first-order symmetric-hyperbolic form for any fixed gauge, enhancing the mathematical structure of Einstein's equations.

Contribution

It presents a one-parameter family of variable transformations that achieve symmetric-hyperbolic form for Einstein equations under arbitrary fixed gauges.

Findings

Recast Einstein equations into symmetric-hyperbolic form

Defined a new lapse function based on metric determinant

Ensured hyperbolicity without gauge-fixing constraints

Abstract

We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an arbitrary factor times a power of the determinant of the 3-metric; under certain assumptions, the exponent can be chosen arbitrarily, but positive, with no implication of gauge-fixing.