Quadratic perturbations of the Schwarzschild black hole: The algebraically special sector

TL;DR

This paper analyzes quadratic algebraically special perturbations of Schwarzschild black holes, deriving analytical solutions that reveal exponential growth at horizons and quadratic corrections to mass and spin, extending linear perturbation theory.

Contribution

It provides the first analytical solutions for quadratic ASPs of Schwarzschild black holes, extending linear perturbation results to the nonlinear regime.

Findings

Quadratic ASPs exhibit exponential growth at horizons.

Explicit analytical expressions for quadratic ASP profiles are derived.

Quadratic zero modes relate to corrections in black hole mass and spin.

Abstract

We investigate quadratic algebraically special perturbations (ASPs) of the Schwarzschild black hole. Their dynamics are derived from the expansion up to second order in perturbation of the most general algebraically special twisting vacuum solution of general relativity. Following this strategy, we present analytical expressions for the axial-axial, polar-polar and polar-axial source terms entering in the dynamical equations. We show that these complicated inhomogeneous equations can be solved analytically and we present explicit expressions for the profiles of the quadratic ASPs. As expected, they exhibit exponential growth both at the past and future horizons even in the non-linear regime. We further use this result to analyze the quadratic zero modes and their interpretation in terms of quadratic corrections to mass and spin of the Schwarzschild black hole. The present work provides a direct extension beyond the linear regime of the original work by Couch and Newman.