Gravitational self force by mode sum regularization

TL;DR

This work develops a practical mode-sum regularization framework for computing the local gravitational self-force on a small mass in Schwarzschild spacetime, extending prior scalar-field results to gravity within the harmonic gauge. It starts from the MSTQW tail-term expression and decomposes the metric perturbation into tensor harmonics, yielding decoupled multipole equations with inter-mode couplings. The core contribution is the mode-sum regularization, where the divergent sum over bare force modes is regularized by analytically computed parameters $A_\alpha$, $B_\alpha$, $C_\alpha$, and $D_\alpha$ derived from local Green's-function analysis, together with a tilde-summation procedure to handle oscillations. The authors demonstrate the method on two simple tests—a static particle in flat space and a turning-point radial geodesic in Schwarzschild—deriving explicit regularization parameters and verifying consistency with independent numerical results, thereby establishing a path toward more general orbits and Kerr spacetime extensions.

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.)