Non-modal linear stability of the Schwarzschild black hole

TL;DR

This paper proves that solutions to the linearized vacuum Einstein equations around a Schwarzschild black hole can be characterized by two gauge-invariant scalar fields, which remain bounded outside the event horizon, indicating stability.

Contribution

It introduces a gauge-invariant scalar field parameterization of linearized solutions around Schwarzschild black holes, demonstrating their boundedness and non-modal stability.

Findings

Solutions are parameterized by two scalar fields.

Scalar fields are gauge-invariant and encode perturbation information.

Fields are pointwise bounded outside the horizon.

Abstract

A proof is given that the space $\L$ of solutions of the linearized vacuum Einstein's equation around a Schwarzschild black hole is parameterized by two scalar fields which are gauge invariant combinations of perturbed algebraic and differential invariants of the Weyl tensor, and encode the information on the odd (-) and even (+) sectors $\L_{\pm}$ These fields measure the distortion of the geometry caused by a generic perturbation, and are shown to be pointwise bounded on the outer region $r \geq 2M$.