A matched expansion approach to practical self-force calculations

TL;DR

The paper presents a practical method combining local covariant Taylor series and mode sum expansions to compute the gravitational self-force on particles in curved spacetime, especially near black holes.

Contribution

It introduces a matched expansion technique for self-force calculations that can be applied to generic geometries and specific cases like black hole orbits.

Findings

The method shows good convergence properties for the series expansions.

It enables calculation of the gravitational self-force for small-mass particles orbiting black holes.

Technical challenges remain, but the approach is promising for practical implementations.

Abstract

We discuss a practical method to compute the self-force on a particle moving through a curved spacetime. This method involves two expansions to calculate the self-force, one arising from the particle's immediate past and the other from the more distant past. The expansion in the immediate past is a covariant Taylor series and can be carried out for all geometries. The more distant expansion is a mode sum, and may be carried out in those cases where the wave equation for the field mediating the self-force admits a mode expansion of the solution. In particular, this method can be used to calculate the gravitational self-force for a particle of mass mu orbiting a black hole of mass M to order mu^2, provided mu/M << 1. We discuss how to use these two expansions to construct a full self-force, and in particular investigate criteria for matching the two expansions. As with all methods of computing self-forces for particles moving in black hole spacetimes, one encounters considerable technical difficulty in applying this method; nevertheless, it appears that the convergence of each series is good enough that a practical implementation may be plausible.