A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited

TL;DR

This paper offers a new proof of the linear stability of Schwarzschild black holes under gravitational perturbations, utilizing a novel geometric gauge and transport equations to simplify and strengthen the stability analysis.

Contribution

It introduces a new geometric gauge and leverages transport equations to prove linear stability without future gauge normalizations, simplifying previous methods.

Findings

Proves linear stability of Schwarzschild spacetime.

Uses a new geometric gauge to control perturbations.

Eliminates the need for future gauge normalizations.

Abstract

We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of \cite{benomio_kerr}, specialised to the $|a|=0$ case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin $\pm 2$ Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an initial-data gauge normalisation suffices to establish both orbital and asymptotic stability for all the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of \cite{DHR}, which requires a future normalised (double-null) gauge to establish asymptotic stability for the full system.