Hyperbolic formulations of General Relativity with Hamiltonian structure

TL;DR

This paper develops hyperbolic formulations of General Relativity with Hamiltonian structure, enabling stable evolution systems compatible with popular gauge conditions, and introduces tools applicable to various order formulations.

Contribution

It presents a method to derive symmetric hyperbolic Hamiltonian formulations of GR, including gauge conditions like the puncture gauge, with new tools for analyzing hyperbolicity.

Findings

Hamiltonian structure can simplify hyperbolicity proofs

Established symmetric hyperbolic formulations with puncture gauge

Tools applicable to different order forms of equations

Abstract

With the aim of deriving symmetric hyperbolic free-evolution systems for GR that possess Hamiltonian structure and allow for the popular puncture gauge condition we analyze the hyperbolicity of Hamiltonian systems. We develop helpful tools which are applicable to either the first order in time, second order in space or the fully second order form of the equations of motion. For toy models we find that the Hamiltonian structure can simplify the proof of symmetric hyperbolicity. In GR we use a special structure of the principal part to prove symmetric hyperbolicity of a formulation that includes gauge conditions which are very similar to the puncture gauge.