Gauge Symmetry and Gravito-Electromagnetism

TL;DR

This paper develops a tensor framework for perturbative Einstein gravity, establishing an analogy with electromagnetism to describe gravitational fields and forces, highlighting both its usefulness and limitations.

Contribution

It introduces a gauge-invariant tensor description of gravito-electromagnetism, enabling Maxwell-like equations in perturbative gravity on flat backgrounds.

Findings

Maxwell-like form of linearized gravitational equations on flat backgrounds

Restrictions on solutions prevent propagating vector gravitational fields

Gravito-electromagnetic analogy useful for stationary phenomena

Abstract

A tensor description of perturbative Einsteinian gravity about an arbitrary background spacetime is developed. By analogy with the covariant laws of electromagnetism in spacetime, gravito-electromagnetic potentials and fields are defined to emulate electromagnetic gauge transformations under substitutions belonging to the gauge symmetry group of perturbative gravitation. These definitions have the advantage that on a flat background, with the aid of a covariantly constant timelike vector field, a subset of the linearised gravitational field equations can be written in a form that is fully analogous to Maxwell's equations (without awkward factors of 4 and extraneous tensor fields). It is shown how the remaining equations in the perturbed gravitational system restrict the time dependence of solutions to these equations and thereby prohibit the existence of propagating vector fields. The induced gravito-electromagnetic Lorentz force on a test particle is evaluated in terms of these fields together with the torque on a small gyroscope. It is concluded that the analogy of perturbative gravity to Maxwell's description of electromagnetism can be valuable for (quasi-)stationary gravitational phenomena but that the analogy has its limitations.