Regularization of static self-forces

TL;DR

This paper demonstrates the equivalence of various regularization methods for computing the static self-force in curved spacetimes, establishing connections between Hadamard-based and Green's function-based approaches, and providing explicit parameters for spherically symmetric cases.

Contribution

It proves the equivalence of different regularization techniques for static self-force calculations and derives local Green's function expansions for general static spacetimes.

Findings

All regularization methods yield the same static self-force.

Local Green's function expansions agree up to high order, possibly all orders.

Provides regularization parameters for static particles in spherically symmetric spacetimes.

Abstract

Various regularization methods have been used to compute the self-force acting on a static particle in a static, curved spacetime. Many of these are based on Hadamard's two-point function in three dimensions. On the other hand, the regularization method that enjoys the best justification is that of Detweiler and Whiting, which is based on a four-dimensional Green's function. We establish the connection between these methods and find that they are all equivalent, in the sense that they all lead to the same static self-force. For general static spacetimes, we compute local expansions of the Green's functions on which the various regularization methods are based. We find that these agree up to a certain high order, and conjecture that they might be equal to all orders. We show that this equivalence is exact in the case of ultrastatic spacetimes. Finally, our computations are exploited to provide regularization parameters for a static particle in a general static and spherically-symmetric spacetime.