A New Gauge for Gravitational Perturbations of Kerr Spacetimes I: The Linearised Theory

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. The system of linearised gravity exhibits two key structural properties. The first is well-known and consists of the exact decoupling of two gauge invariant linearised quantities in the system, satisfying two decoupled spin \(\pm \, 2\) Teukolsky equations. The second is a new, gauge dependent structure in the outgoing transport equations for the linearised connection coefficients inherited from the nonlinear system: Unlike previous works, these equations do not contain any derivatives of linearised connection coefficients on the right hand side and induce a new hierarchy of only outgoing transport equations including all gauge dependent linearised quantities. 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 (Benomio in A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited, 2022) 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 (Benomio 2022), 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 \(|a|<M\).