Gravitational perturbations and metric reconstruction: Method of extended homogeneous solutions applied to eccentric orbits on a Schwarzschild black hole

TL;DR

This paper develops and applies a numerical method to accurately compute gravitational perturbations caused by a small mass in eccentric orbit around a Schwarzschild black hole, enabling precise metric reconstruction and flux calculations.

Contribution

It demonstrates the successful application of the extended homogeneous solutions method to handle singular source terms in gravitational perturbation equations for eccentric orbits.

Findings

Accurate mode calculations for specific orbital parameters.

Precise energy and angular momentum fluxes at infinity and the horizon.

Effective metric reconstruction from master functions in the presence of singular sources.

Abstract

We calculate the gravitational perturbations produced by a small mass in eccentric orbit about a much more massive Schwarzschild black hole and use the numerically computed perturbations to solve for the metric. The calculations are initially made in the frequency domain and provide Fourier-harmonic modes for the gauge-invariant master functions that satisfy inhomogeneous versions of the Regge-Wheeler and Zerilli equations. These gravitational master equations have specific singular sources containing both delta function and derivative-of-delta function terms. We demonstrate in this paper successful application of the method of extended homogeneous solutions, developed recently by Barack, Ori, and Sago, to handle source terms of this type. The method allows transformation back to the time domain, with exponential convergence of the partial mode sums that represent the field. This rapid convergence holds even in the region of $r$ traversed by the point mass and includes the time-dependent location of the point mass itself. We present numerical results of mode calculations for certain orbital parameters, including highly accurate energy and angular momentum fluxes at infinity and at the black hole event horizon. We then address the issue of reconstructing the metric perturbation amplitudes from the master functions, the latter being weak solutions of a particular form to the wave equations. The spherical harmonic amplitudes that represent the metric in Regge-Wheeler gauge can themselves be viewed as weak solutions. They are in general a combination of (1) two differentiable solutions that adjoin at the instantaneous location of the point mass (a result that has order of continuity $C^{-1}$ typically) and (2) (in some cases) a delta function distribution term with a computable time-dependent amplitude.