Algebraically special perturbations of the Kerr black hole: a metric formulation

TL;DR

This work develops a metric-based treatment of algebraically special perturbations (ASLP) of the Kerr black hole by exploiting the most general twisting Petrov type II vacuum solution space and linearizing it about Kerr. It derives two coupled PDEs for the first-order ASP perturbations $(f_1,F_1)$, and introduces a slow-rotation resolution scheme that separates angular and radial/time dependence, enabling an exact analytic Kerr ASP solution up to cubic order in the spin for a selected mode. The study further furnishes complete stationary zero-mode analyses, providing explicit perturbations that realize mass and spin shifts, as well as mappings to Kerr-NUT and the spinning C-metric through concrete coordinate changes. These metric-only ASP results offer a rare, tractable testbed for hidden symmetries and potential non-linear extensions of Kerr perturbations, without recourse to the Teukolsky Weyl perturbation framework. Overall, the paper demonstrates a concrete, fully metric formulation of Kerr ASP and opens avenues for deeper geometric and symmetry-based insights into black hole perturbations.

Abstract

Perturbations of the Kerr black hole are notoriously difficult to describe in the metric formalism and are usually studied in terms of perturbations of the Weyl scalars. In this work, we focus on the algebraically special linear perturbations (ASLP) of the Kerr geometry and show how one can describe this subsector of the perturbations solely using the metric formulation. To that end, we consider the most general twisting algebraically special solution space of vacuum General Relativity. By linearizing around the Kerr solution, we obtain two coupled partial differential wave equations describing the dynamics of the Kerr ASLP. We provide an algorithm to solve them analytically in the small spin approximation up to third order, providing the first exact solution of this kind in the metric formulation. Then, we use this framework to study the stationary zero modes of the Kerr geometry. We present the exact analytical form of the shifts in mass and spin together with the required change of coordinates needed to identify them. Finally, we also provide for the first time closed expressions for the solution-generating perturbations generating the NUT and acceleration charges, thus deforming the Kerr solution to the linearized Kerr-NUT and spinning C-metric. These results provide a first concrete and rare example of perturbations of the Kerr black hole which can be treated entirely in the metric formulation. They can serve as a useful testbed to search for hidden symmetries of the Kerr perturbations.

Figures (1)

• Figure 1: Solutions $\omega$ to equation \ref{['eq:det-chandra']} for $\ell = 3$ and different values of $m$. The mass $M$ is set to 1, and the spin $a$ spans the interval $\interval{0}{0.5}$, with a dot every 0.05.