Gravitational self force by mode sum regularization

TL;DR

This paper introduces a practical mode-sum regularization scheme to compute the local gravitational self-force on a particle in black hole spacetime, applicable to both weak and strong fields, with detailed derivations and example cases.

Contribution

It extends the mode-sum regularization method from scalar models to the gravitational case, providing a systematic way to calculate the self-force in black hole spacetimes.

Findings

Derived regularization parameters for static particles in flat spacetime.

Derived regularization parameters for particles at radial turning points in Schwarzschild spacetime.

Established an analytic foundation for applying the scheme to general cases.

Abstract

We propose a practical scheme for calculating the local gravitational self-force experienced by a test mass particle moving in a black hole spacetime. The method---equally effective for either weak or strong field orbits---employs the {\em mode-sum regularization scheme} previously developed for a scalar toy model. The starting point for the calculation, in this approach, is the formal expression for the regularized self-force derived by Mino et al. (and, independently, by Quinn and Wald), which involves a worldline integral over the tail part of the retarded Green's function. This force is decomposed into multipole (tensor harmonic) modes, whose sum is subjected to a carefully designed regularization procedure. This procedure involves an analytic derivation of certain ``regularization parameters'' by means of a local analysis of the Green's function. This manuscript contains the following main parts: (1) Introduction of the mode sum scheme as applied to the gravitational case. (2) Two simple cases studied: the test case of a static particle in flat spacetime, and the case of a particle at a turning point of a radial geodesic in Schwarzschild spacetime. In both cases we derive all necessary regularization parameters. (3) An Analytic foundation is set for applying the scheme in more general cases. (In this paper, the mode sum scheme is formulated within the harmonic gauge. The implementation of the scheme in other gauges shall be discussed in a separate, forthcoming paper.)