Gravitational and electromagnetic perturbations of a charged black hole in a general gauge condition

TL;DR

This paper derives coupled equations for gravitational and electromagnetic perturbations of a charged black hole without gauge restrictions, including sources, and solves them numerically to explore correlations useful for gravitational wave detection.

Contribution

It presents a gauge-independent derivation of coupled perturbation equations for Reissner-Nordström black holes, including sources, and provides a numerical solution framework.

Findings

Coupled gravitational and electromagnetic signals are obtained.

The analysis includes source terms for a charged fluid.

A basis of spin-weighted harmonics is used for numerical solutions.

Abstract

We derive a set of coupled equations for the gravitational and electromagnetic perturbation in the Reissner-Nordstr\"om geometry using the Newman Penrose formalism. We show that the information of the physical gravitational signal is contained in the Weyl scalar function $\Psi_4$, as is well known, but for the electromagnetic signal the information is encoded in the function $\chi$ which relates the perturbations of the radiative Maxwell scalars $\varphi_2$ and the Weyl scalar $\Psi_3$. In deriving the perturbation equations we do not impose any gauge condition and our analysis contains as a limiting case the results obtained previously for instance in Chandrashekhar's book. In our analysis, we also include the sources for the perturbations and focus on a dust-like charged fluid distribution falling radially into the black hole. Finally, by writing the functions on a basis of spin weighted spherical harmonics and the Reissner-Nordstr\"om spacetime in Kerr-Schild type coordinates a hyperbolic system of coupled partial differential equations is presented and numerically solved. In this way, we solve completely a system which generates a gravitational signal as well as an electromagnetic/gravitational one, which sets the basis to find correlations between them and thus facilitating the gravitational wave detection via the electromagnetic signal.