Geometrical Hyperbolic Systems for General Relativity and Gauge Theories

TL;DR

This paper develops a gauge covariant first order symmetric hyperbolic formulation for Einstein's and Maxwell's theories, decoupling gauge and physical variables, and clarifying the propagation of radiative gravitational and electromagnetic fields.

Contribution

It introduces a novel hyperbolic system formulation that isolates gauge degrees of freedom and describes the gauge-invariant propagation of physical fields in general relativity and electromagnetism.

Findings

Gauge variables are decoupled from dynamical variables.

Radiative variables propagate along the light cone in a gauge-invariant manner.

The formulation explicitly relates gravitational fields to the Riemann tensor components.

Abstract

The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom [the spatial shift vector $\beta^{i}(t,x^{j})$ and the spatial scalar potential $\phi(t,x^{j})$, respectively] are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of $(t,x^{j})$ from subsidiary equations that are not part of the evolution equations. Propagation of certain (``radiative'') dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by $(1)$ taking a further time derivative of the equation of motion of the canonical momentum, and $(2)$ adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier $\beta^{i}$) or of the Gauss's law constraint of electromagnetism (Lagrange multiplier $\phi$). General relativity also requires a harmonic time slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first order symmetric form. The dynamically propagating gravity fields straightforwardly determine the ``electric'' or ``tidal'' parts of the Riemann tensor.