Second order perturbations of a Schwarzschild black hole

TL;DR

This paper develops a second order perturbation formalism for Schwarzschild black holes, reducing Einstein's equations to a wave equation with a quadratic source, aiding the validation of first order astrophysical models.

Contribution

It introduces a second order perturbation framework for Schwarzschild black holes, extending first order analyses with a wave equation including a quadratic source term.

Findings

Reduction of Einstein equations to a single wave equation with source

Formalism to assess the validity of first order perturbation results

Framework applicable to astrophysical scenarios involving black holes

Abstract

We study the even-parity $\ell=2$ perturbations of a Schwarzschild black hole to second order. The Einstein equations can be reduced to a single linear wave equation with a potential and a source term. The source term is quadratic in terms of the first order perturbations. This provides a formalism to address the validity of many first order calculations of interest in astrophysics.