Kerr-AdS and its Near-horizon Geometry: Perturbations and the Kerr/CFT Correspondence

TL;DR

This work analyzes linear perturbations of spin $s$ fields on Kerr-$AdS$ and its near-horizon geometry $NHEK$-$AdS$ using the Teukolsky/Newman–Penrose framework. It derives decoupled master equations, separates them into radial and angular parts, and solves the angular sector numerically while obtaining exact hypergeometric solutions for the radial part in $NHEK$-$AdS$; stability is then probed under outgoing boundary conditions. The results show no linear instabilities in either axisymmetric or non-axisymmetric sectors, aligning with the Durkee–Reall conjecture, and reveal that the Kerr/CFT fall-off conditions are violated by general linear perturbations in $NHEK$-$AdS$, with axisymmetric modes ($m=0$) providing a consistent truncation. The analysis also maps the radial equation to the massive charged scalar equation in $AdS_2$ with a homogeneous electric field, illustrating a deep structural link between perturbations in near-horizon AdS geometries and lower-dimensional effective dynamics. These findings clarify the relationship between near-horizon stability and full black hole stability, and they highlight subtleties in applying Kerr/CFT boundary conditions to dynamical perturbations.

Abstract

We investigate linear perturbations of spin-s fields in the Kerr-AdS black hole and in its near-horizon geometry (NHEK-AdS), using the Teukolsky master equation and the Hertz potential. In the NHEK-AdS geometry we solve the associated angular equation numerically and the radial equation exactly. Having these explicit solutions at hand, we search for linear mode instabilities. We do not find any (non-)axisymmetric instabilities with outgoing boundary conditions. This is in agreement with a recent conjecture relating the linearized stability properties of the full geometry with those of its near-horizon geometry. Moreover, we find that the asymptotic behaviour of the metric perturbations in NHEK-AdS violates the fall-off conditions imposed in the formulation of the Kerr/CFT correspondence (the only exception being the axisymmetric sector of perturbations).

Figures (4)

• Figure 1: $\eta^2$, defined in \ref{['radialeq_eta definition']}, for $|s|=2$ and $l=3$. a)$\eta^2$ vs $r_+/\ell\,$ for $|m|=0,1,2,3$, and b)$\eta^2$ vs $m$ for $r_+/\ell=0\,$ (black points) and $r_+/\ell=0.55 \simeq 1/\sqrt{3}\,$ (red points).
• Figure 2: $\eta^2$ for $|s|=2$ and $l=16$. a)$\eta^2$ vs $r_+/\ell\,$ for the representative $|m|$ cases (from top to bottom these are $|m|=0,10,12,13,16$), and b)$\eta^2$ vs $m$ for $r_+/\ell=0\,$ (black points) and $r_+/\ell=0.55 \simeq 1/\sqrt{3}\,$ (red points).
• Figure 3: $\eta^2$ as a function of $-l\leq m\leq l$ and $r_+/\ell\,$, for $l=3$ ( left) and $l=16$ ( right). The red points (curve segments) have $\eta^2>0$ while the blue points (curve segments) have $\eta^2<0$. In Section \ref{['sec:PhysInterp']} we will conclude that the red dots describe normal modes ($\eta\in\mathbb{R}$), while blue dots describe traveling waves ($\eta\in\mathbb{I}$). The green dots correspond to modes on which we cannot impose outgoing boundary conditions.
• Figure 4: Critical value $(r_+/\ell)_c\,$ for $l\leq16$. In Section \ref{['sec:PhysInterp']} we conclude that no traveling waves exist for $(r_+/\ell)_c<r_+/\ell<\sup\{r_+/\ell\}$.