Signatures of the sources in the gravitational waves of a perturbed Schwarzschild black hole

TL;DR

This paper derives the perturbation equations for gravitational waves in Schwarzschild black holes with matter sources and applies them to simulate gravitational wave emission caused by surrounding dust-like matter.

Contribution

It provides the explicit form of the perturbation equation for the Weyl scalar with matter sources in Schwarzschild spacetime, enabling numerical analysis of gravitational wave emission.

Findings

Derived the perturbation equation with matter sources for Schwarzschild black holes.

Applied the equation to simulate gravitational wave emission from dust-like matter.

Demonstrated the use of spherical harmonic decomposition for simplified numerical analysis.

Abstract

The explicit form of perturbation equation for the $\Psi_4$ Weyl scalar, containing the matter source terms, is derived for general type D spacetimes. It is described in detail the particular case of the Schwarzschild spacetime using in-going penetrating coordinates. As a practical application, we focused on the emission of gravitational waves when a black hole is perturbed by a surrounding dust-like fluid matter. The symmetries of the spacetime and the simplicity of the matter source allow, by means of a spherical harmonic decomposition, to study the problem by means of a one dimensional numerical code.