A Conformal Hyperbolic Formulation of the Einstein Equations

TL;DR

This paper introduces a new hyperbolic formulation of Einstein's equations that separates conformal and non-conformal degrees of freedom, ensuring well-posedness and physical characteristic speeds for numerical relativity.

Contribution

It presents a conformal hyperbolic formulation with a two-parameter family of systems, fixed by trace-free conditions, improving the mathematical structure of Einstein evolution equations.

Findings

Two-parameter family of hyperbolic systems for non-conformal degrees.

System can be made consistently trace-free.

Constructs conformal hyperbolic system with physical speeds.

Abstract

We propose a re-formulation of the Einstein evolution equations that cleanly separates the conformal degrees of freedom and the non-conformal degrees of freedom with the latter satisfying a first order strongly hyperbolic system. The conformal degrees of freedom are taken to be determined by the choice of slicing and the initial data, and are regarded as given functions (along with the lapse and the shift) in the hyperbolic part of the evolution. We find that there is a two parameter family of hyperbolic systems for the non-conformal degrees of freedom for a given set of trace free variables. The two parameters are uniquely fixed if we require the system to be ``consistently trace-free'', i.e., the time derivatives of the trace free variables remains trace-free to the principal part, even in the presence of constraint violations due to numerical truncation error. We show that by forming linear combinations of the trace free variables a conformal hyperbolic system with only physical characteristic speeds can also be constructed.