Differential Forms and Wave Equations for General Relativity

TL;DR

This paper develops a differential forms-based wave equation for vacuum general relativity, providing a gauge-covariant, causal formulation that parallels tensor methods but uses differential forms for clarity and simplicity.

Contribution

It introduces a new wave equation formulated with differential forms, offering an alternative to tensor-index methods for describing gravitational propagation in general relativity.

Findings

Formulates a wave equation using differential forms for vacuum GR.

Provides a gauge-covariant, causal system separating physical and gauge degrees.

Connects the differential forms approach to the ADM gravitational momentum.

Abstract

Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and York (AACY) have cast the 3+1 evolution equations of general relativity in gauge-covariant and causal ``first-order symmetric hyperbolic form,'' thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's 2-form, which in the ``time-gauge'' is built linearly from the ``extrinsic curvature 1-form.'' The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt-Deser-Misner gravitational momentum.