Gravitational self-force on a particle in eccentric orbit around a Schwarzschild black hole

TL;DR

This paper introduces a numerical method to compute the gravitational self-force on a particle in eccentric orbit around a Schwarzschild black hole, separating dissipative and conservative effects in the self-force.

Contribution

The authors develop a time-domain numerical code in Lorenz gauge for calculating the self-force, including the first computation of the conservative component in eccentric strong-field orbits.

Findings

Validated the dissipative self-force against energy and angular momentum fluxes.

First calculation of conservative self-force in eccentric strong-field orbits.

Demonstrated the impact of the conservative self-force on orbital phase evolution.

Abstract

We present a numerical code for calculating the local gravitational self-force acting on a pointlike particle in a generic (bound) geodesic orbit around a Schwarzschild black hole. The calculation is carried out in the Lorenz gauge: For a given geodesic orbit, we decompose the Lorenz-gauge metric perturbation equations (sourced by the delta-function particle) into tensorial harmonics, and solve for each harmonic using numerical evolution in the time domain (in 1+1 dimensions). The physical self-force along the orbit is then obtained via mode-sum regularization. The total self-force contains a dissipative piece as well as a conservative piece, and we describe a simple method for disentangling these two pieces in a time-domain framework. The dissipative component is responsible for the loss of orbital energy and angular momentum through gravitational radiation; as a test of our code we demonstrate that the work done by the dissipative component of the computed force is precisely balanced by the asymptotic fluxes of energy and angular momentum, which we extract independently from the wave-zone numerical solutions. The conservative piece of the self force does not affect the time-averaged rate of energy and angular-momentum loss, but it influences the evolution of the orbital phases; this piece is calculated here for the first time in eccentric strong-field orbits. As a first concrete application of our code we recently reported the value of the shift in the location and frequency of the innermost stable circular orbit due to the conservative self-force [Phys. Rev. Lett.\ {\bf 102}, 191101 (2009)]. Here we provide full details of this analysis, and discuss future applications.