Perturbations of Kerr with approximate Killing spinors -- Wave version
Thomas Bäckdahl
TL;DR
This work proposes a covariant framework for non-linear Kerr perturbations that avoids relying on a fixed Kerr background by employing an approximate Killing spinor to encode deviations. It constructs a Kerr-adapted set of variables, including an approximate Killing spinor and derived small and non-small quantities, and derives a comprehensive first-order symmetric hyperbolic evolution system in both covariant and GHP/space-spinor formalisms. The framework yields approximate Killing vectors and local expressions for mass and angular-momentum-like constants, providing a path toward decay estimates and stability analysis in subsequent work. The approach is designed to be background-free with respect to Kerr parameters, enabling gauge-robust tracking of the spacetime’s proximity to Kerr and facilitating applications to Kerr stability and self-force problems.
Abstract
In this paper we develop a new framework for non-linear perturbations of the Kerr spacetime. This is based on a characterization of the Kerr spacetime in terms of a Killing spinor. On the perturbed spacetime, one can construct an approximation of the Killing spinor. Based on this, a number of quantities are constructed measuring the deviation from Kerr. Evolution equations for these quantities are derived. Approximations of the Killing vectors, the mass and angular momentum parameters etc., are constructed along with a full set of equations for their derivatives. In this setting, we don't need a reference background solution. Instead, we covariantly construct the relevant structures on the perturbed spacetime itself. This can eliminate many issues and allow for a cleaner analysis.