A new gauge for gravitational perturbations of Kerr spacetimes I: The linearised theory

TL;DR

This paper introduces a novel geometric gauge and null frame framework for analyzing gravitational perturbations of Kerr black holes, aiming to simplify and improve the understanding of their stability properties.

Contribution

It develops a new geometric formulation of linearized Einstein equations around Kerr, tailored to outgoing principal null geodesics, facilitating the analysis of stability.

Findings

Derived linearized Einstein equations in the new framework.

Identified stabilizing properties of red-shifted transport equations.

Set the stage for future nonlinear stability proofs.

Abstract

We propose a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range $|a|<M$. Central to our framework is a new formulation of nonlinear gravitational perturbations of Kerr, whose two novel ingredients are the choice of a geometric gauge and non-integrable null frames both tailored to the outgoing principal null geodesics of Kerr. The vacuum Einstein equations for the perturbations are formulated in our gauge as a system of equations for the connection coefficients and curvature components relative to the chosen frames. When renormalised with respect to Kerr, the null structure equations with the form of outgoing transport equations do not possess any derivatives of renormalised connection coefficients on the right hand side. In this work, we derive the linearised vacuum Einstein equations around Kerr in the new framework. Our new framework is designed to effectively capture the stabilising properties of the red-shifted transport equations, thereby isolating one of the crucial structures of the problem. Such a feature is suggestive of future simplifications in the analysis. As an illustration, our companion work \cite{benomio_schwarzschild} employs the system of linearised gravity and its enhanced red-shifted transport equations, specialised to the $|a|=0$ case, to produce a new simplified proof of linear stability of the Schwarzschild solution. The full linear stability analysis in the full sub-extremal range $|a|<M$ is deferred to future work. As already apparent from \cite{benomio_schwarzschild}, our framework will allow to combine the new structure in the transport equations with the elliptic part of the system to establish a linear orbital stability result without loss of derivatives, indicating that the framework may be well suited to address nonlinear stability in the full sub-extremal range $|a|<M$.