Gravitational self-force in nonvacuum spacetimes

Abstract

The gravitational self-force has thus far been formulated in background spacetimes for which the metric is a solution to the Einstein field equations in vacuum. While this formulation is sufficient to describe the motion of a small object around a black hole, other applications require a more general formulation that allows for a nonvacuum background spacetime. We provide a foundation for such extensions, and carry out a concrete formulation of the gravitational self-force in two specific cases. In the first we consider a particle of mass $m$ and scalar charge $q$ moving in a background spacetime that contains a background scalar field. In the second we consider a particle of mass $m$ and electric charge $e$ moving in an electrovac spacetime. The self-force incorporates all couplings between the gravitational perturbations and those of the scalar or electromagnetic fields. It is expressed as a sum of local terms involving tensors defined in the background spacetime and evaluated at the current position of the particle, as well as tail integrals that depend on the past history of the particle. Because such an expression is rarely a useful starting point for an explicit evaluation of the self-force, we also provide covariant expressions for the singular potentials, expressed as local expansions near the world line; these can be involved in the construction of effective extended sources for the regular potentials, or in the computation of regularization parameters when the self-force is computed as a sum over spherical-harmonic modes.Comment: 20 pages, no figure

Self-force as a cosmic censor in the Kerr overspinning problem

“…There is work in progress at Southampton to directly (numerically) evaluate δL const c via Eq. (8). Second, it may be possible to replace some (or all) of the numerical input for our analysis with analytical arguments.…”

“…A complete analysis would require a calculation of both electromagnetic and gravitational self-forces in the coupled problem, which remains a difficult challenge despite recent progress [8][9][10].…”

Inspirals into a charged black hole

“…Radiation reaction on the smaller binary component occurs through a mechanism called the self-force [41][42][43][44]. Self-force studies in charged black hole spacetimes have been developed through scalar field toy models [45,46] and the groundwork is being laid for more realistic charged self-force scenarios [47][48][49]. Leading-order self-force effects are accessible through an adiabatic approximation [50].…”

Self-force and fluid resonances

“…In this situation the black hole is charged, the spacetime is filled with an electrostatic field, and the coupling between gravitational and electromagnetic perturbations creates complications in the formulation of the self-force. In this case an adequate formulation of the self-force for background spacetimes containing either a scalar or electromagnetic field was provided by Zimmerman and Poisson [28,29] and Linz, Friedman, and Wiseman [30]. In a related development, the self-force was extended to scalar-tensor theories of gravity by Zimmerman [31].…”