Low multipole contributions to the gravitational self-force

TL;DR

This paper computes the monopole and dipole contributions to the gravitational self-force on a particle orbiting a Schwarzschild black hole, highlighting their significance alongside higher modes and providing analytical and numerical results.

Contribution

It provides the first detailed calculation of low multipole (monopole and dipole) self-force contributions in the Lorenz gauge, including a numerical approach for the even-parity dipole case.

Findings

Dipole self-force affects orbital dynamics at Newtonian level.

Closed-form results obtained for monopole and odd-parity dipole modes.

Numerical method developed for even-parity dipole mode.

Abstract

We calculate the unregularized monopole and dipole contributions to the self-force acting on a particle of small mass in a circular orbit around a Schwarzschild black hole. From a self-force point of view, these non-radiating modes are as important as the radiating modes with l greater than 2. In fact, we demonstrate how the dipole self-force contributes to the dynamics even at the Newtonian level. The self-acceleration of a particle is an inherently gauge-dependent concept, but the Lorenz gauge is often preferred because of its hyperbolic wave operator. Our results are in the Lorenz gauge and are also obtained in closed form, except for the even-parity dipole case where we formulate and implement a numerical approach.