New first-order formulation for the Einstein equations

TL;DR

This paper introduces a novel first-order formulation of Einstein's equations that reduces the number of unknowns and maintains hyperbolicity, facilitating more efficient numerical simulations in general relativity.

Contribution

A new first-order formulation of Einstein's equations with fewer unknowns, based on 3+1 decomposition, and symmetric hyperbolic structure in linearized form.

Findings

Fewer unknowns than previous formulations

Symmetric hyperbolic system in linearized case

Potential for improved numerical stability

Abstract

We derive a new first-order formulation for Einstein's equations which involves fewer unknowns than other first-order formulations that have been proposed. The new formulation is based on the 3+1 decomposition with arbitrary lapse and shift. In the reduction to first order form only 8 particular combinations of the 18 first derivatives of the spatial metric are introduced. In the case of linearization about Minkowski space, the new formulation consists of symmetric hyperbolic system in 14 unknowns, namely the components of the extrinsic curvature perturbation and the 8 new variables, from whose solution the metric perturbation can be computed by integration.