Boundary Conditions for Kerr-AdS Perturbations

TL;DR

This paper establishes physically relevant boundary conditions for Kerr-AdS perturbations using the Teukolsky formalism, enabling detailed studies of stability, quasinormal modes, and holographic stress-energy tensor behavior in the context of AdS/CFT.

Contribution

It introduces two Robin boundary conditions for Teukolsky variables that ensure asymptotically global AdS perturbations and connects them to the Kodama-Ishisbashi formalism, facilitating stability analysis.

Findings

Derived Robin boundary conditions for Kerr-AdS perturbations.

Mapped Teukolsky variables to Kodama-Ishisbashi formalism.

Reproduced the quasinormal mode spectrum of AdS-Schwarzschild.

Abstract

The Teukolsky master equation and its associated spin-weighted spheroidal harmonic decomposition simplify considerably the study of linear gravitational perturbations of the Kerr(-AdS) black hole. However, the formulation of the problem is not complete before we assign the physically relevant boundary conditions. We find a set of two Robin boundary conditions (BCs) that must be imposed on the Teukolsky master variables to get perturbations that are asymptotically global AdS, i.e. that asymptotes to the Einstein Static Universe. In the context of the AdS/CFT correspondence, these BCs allow a non-zero expectation value for the CFT stress-energy tensor while keeping fixed the boundary metric. When the rotation vanishes, we also find the gauge invariant differential map between the Teukolsky and the Kodama-Ishisbashi (Regge-Wheeler-Zerilli) formalisms. One of our Robin BCs maps to the scalar sector and the other to the vector sector of the Kodama-Ishisbashi decomposition. The Robin BCs on the Teukolsky variables will allow for a quantitative study of instability timescales and quasinormal mode spectrum of the Kerr-AdS black hole. As a warm-up for this programme, we use the Teukolsky formalism to recover the quasinormal mode spectrum of global AdS-Schwarzschild, complementing previous analysis in the literature.