Gauge Symmetry and Gravito-electromagnetism

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 are defined to emulate the gauge transformations of electromagnetic potentials under substitutions belonging to the gauge symmetry group of perturbative gravitation. These definitions have the advantage that on a flat background a subset of the perturbative gravitational field equations can be written in a form that is fully analogous to Maxwell’s equations (without awkward factors of 4) with the aid of a covariantly constant timelike vector field. 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.