Gravitational Waves on Kerr Black Holes II: Metric Reconstruction with Cosmological Constant

TL;DR

The paper extends linearized gravitational perturbation theory to Kerr--(A)dS spacetimes with a cosmological constant $\Lambda$, deriving a separable Teukolsky equation for the Weyl scalars $\psi_0$ and $\psi_4$ on a Petrov type D background. It provides explicit metric reconstruction in ingoing and outgoing radiation gauges via Hertz potentials, connecting Weyl-mode data to a full metric perturbation through a detailed operator framework and mode inversion. The analysis leverages angular and radial ODEs governed by generalized Heun functions, defines Teukolsky--Starobinsky constants/identities, and establishes consistency across Weyl scalars, Hertz potentials, and reconstructed metrics, with a holographic interpretation for AdS boundaries. The work generalizes Kerr results (Paper I) to nonflat asymptotics, clarifying how holographic data and boundary conditions influence perturbations, and supplies explicit formulas and mode-level inversions that are essential for numerical and holographic studies of black hole perturbations in a de Sitter or anti-de Sitter background.

Abstract

In this second paper of our series started with \cite{Berens2024}, we investigate linearized gravitational perturbations of a rotating Kerr black hole in a non-asymptotically flat spacetime with (anti-)de Sitter boundary conditions. Here, we explicitly write down the metric components (in both ingoing and outgoing radiation gauge) of the perturbations that correspond to a given mode of either Weyl scalar. We provide formulas involving Hertz potentials (intermediate quantities with a holographic interpretation) as well as some that involve only the separated radial and angular modes. We expect these analytic results to prove useful in numerical studies of black hole perturbation theory in the context of the holographic correspondence.

Figures (1)

• Figure 1: The allowed region (blue) in the $(a,\Lambda)$ plane (shown in units of $M$).