A complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole

TL;DR

This paper develops a comprehensive gauge-invariant formalism for analyzing second-order perturbations of Schwarzschild black holes, facilitating studies of non-linear gravitational effects, mode coupling, and stability.

Contribution

It introduces a general gauge-invariant framework for second-order perturbations, including the construction of Zerilli and Regge-Wheeler equations with any first-order modes.

Findings

Derived second-order master equations for perturbations.

Reconstructed the perturbed metric from master scalars.

Computed energy radiated at null infinity.

Abstract

Using recently developed efficient symbolic manipulations tools, we present a general gauge-invariant formalism to study arbitrary radiative $(l\geq 2)$ second-order perturbations of a Schwarzschild black hole. In particular, we construct the second order Zerilli and Regge-Wheeler equations under the presence of any two first-order modes, reconstruct the perturbed metric in terms of the master scalars, and compute the radiated energy at null infinity. The results of this paper enable systematic studies of generic second order perturbations of the Schwarzschild spacetime. In particular, studies of mode-mode coupling and non-linear effects in gravitational radiation, the second-order stability of the Schwarzschild spacetime, or the geometry of the black hole horizon.