Boundary Conditions for Kerr-AdS Perturbations

TL;DR

This work formalizes physically well-posed boundary conditions for Kerr–AdS perturbations by deriving two Robin boundary conditions on the Teukolsky master variables that preserve asymptotically global AdS geometry. It builds a bridge between the Teukolsky and Kodama–Ishibashi formalisms in the non-rotating limit, showing a one-to-one mapping of Teukolsky BCs to KI scalar and vector sectors. The authors apply these BCs to recover global AdS normal modes and compute AdS–Schwarzschild quasinormal modes, including hydrodynamic limits, thereby enabling precise studies of stability, QNM spectra, and holographic relaxation. Overall, the framework provides a robust, AdS/CFT-consistent approach to Kerr–AdS perturbations and their boundary dynamics.

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.

Figures (6)

• Figure 1: Left panel: Hydrodynamic QNM (which ends in the black point) and the first four microscopic QNM curves (that start at the vector normal modes of AdS pinpointed as red dots) of the $\ell=2$ harmonic of the vector QNM spectrum of GAdSBH. Right panel: Imaginary part of the hydrodynamic vector QNM as a function of the horizon radius in AdS units. For large $r_+/L$, the data approaches the black curve which is the analytical prediction \ref{['hydroV']}. The green dots in the hydrodynamic and in the lowest-lying ($p=0$) microscopic QNM curves are exactly the values taken from Table 2 of Cardoso:2001bb. See text for detailed discussion of these plots.
• Figure 2: Lowest-lying ($p=0$) microscopic vector QNM of the $\ell=2$ harmonic of GAdSBH. The green dots are exactly the values taken from Table 2 of Cardoso:2001bb. The blue (magenta) dots are obtained solving numerically the Teukolsky (KI) equations. (See text for detailed discussion of these plots).
• Figure 3: Lowest-lying ($p=0$) microscopic vector QNMs of the first 8 harmonics of the GAdSBH. From bottom to top we have: $\ell=2,3,\cdots,10$. The red dots give the vector normal mode frequencies \ref{['VnormalM']}. The green dots are exactly taken from Table 2 of Cardoso:2001bb.
• Figure 4: Hydrodynamic QNM (which ends at the black point) and the first three microscopic QNM curves (that start at the vector normal modes of AdS pinpointed as red dots) of the $\ell=2$ harmonic of the scalar QNM spectrum of GAdSBH. The green dots have exactly the values taken from Table 1 and 2 of Michalogiorgakis:2006jc (we have added the $r_+=0.5$ green points in the $p\geq 1$ curves). (See text for detailed discussion of this plot).
• Figure 5: Hydrodynamic scalar QNM of the $\ell = 2$ harmonic of GAdSBH. The green dots are exactly the values taken from taken from Table 1 and 2 of Michalogiorgakis:2006jc. The blue (magenta) dots are obtained solving numerically the Teukolsky (KI) equations. (See text for detailed discussion of these plots).