Hyperbolicity of Hamiltonian formulations in General Relativity

TL;DR

This paper classifies the hyperbolicity of Hamiltonian formulations of Einstein's equations, identifying a symmetric hyperbolic form compatible with common gauge conditions, crucial for well-posedness and numerical relativity.

Contribution

It develops tools to analyze hyperbolicity in Hamiltonian Einstein equations and finds a symmetric hyperbolic formulation compatible with standard gauge choices.

Findings

Classified a large class of Hamiltonian Einstein formulations by hyperbolicity.

Identified a symmetric hyperbolic Hamiltonian formulation with puncture-like gauge conditions.

Provided tools for assessing well-posedness in numerical relativity models.

Abstract

Well-posedness of the initial (boundary) value problem is an essential property, both of meaningful physical models and of numerical applications. To prove well-posedness of wave-type equations their level of hyperbolicity is an essential ingredient. We develop helpful tools and classify a large class of Hamiltonian versions of Einstein's equations with live gauge conditions with respect to their hyperbolicity. Finally we find a symmetric hyperbolic Hamiltonian formulation that allows for gauge conditions which are similar to the puncture gauge.