Hyperbolicity of Physical Theories with Application to General Relativity

TL;DR

This paper analyzes the hyperbolic structure of physical theories, especially general relativity, focusing on gauge freedoms and formulating a family of wave-like gauge conditions to improve understanding and numerical stability.

Contribution

It introduces a five-parameter family of gauge conditions for general relativity, extending harmonic gauge formulations and clarifying the role of gauge choices in hyperbolic formulations.

Findings

Identifies the standard gauge freedom in constrained Hamiltonian theories.

Characterizes gauge choices compatible with wave-like formulations.

Develops a generalized family of gauge conditions for Einstein's equations.

Abstract

We consider gauge theories from the free evolution point of view, in which initial data satisfying constraints of a theory are given. Because the constraints are compatible with the field equations they remain so. We study a model constrained Hamiltonian theory and identify a particular structure in the equations of motion which we call the standard gauge freedom. The pure gauge subsystem of this model theory is identified and the manner in which the gauge variables couple to the field equations is presented. We demonstrate that the set of gauge choices that can be coupled to the field equations to obtain a, properly defined, wave-like formulation is exactly the set of wave-like pure gauges. Consequently we analyze a parametrized family of formulations of general relativity. The generalization of the harmonic gauge formulation to a five parameter family of gauge conditions is obtained.