Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS

TL;DR

The paper advances the understanding of holographic thermalization in AdS spacetimes by solving the linearized gravitational perturbations of Kerr-AdS and Myers–Perry–AdS black holes in D=4 and D=5 under global AdS boundary conditions. It develops a Teukolsky-type master equation, imposes Robin BCs that preserve the conformal boundary, and employs three numerical methods, including a novel Newton-Raphson approach, to map the full QNM spectrum and superradiant instabilities. The work uncovers detailed onset curves for scalar and vector perturbations, identifies maximal growth rates, and argues for new single-KVF black hole phases via a perturbative thermodynamic construction. It further validates the hydrodynamic limit through AdS/CFT by matching long-wavelength QNMs with CFT3/4 hydrodynamics, reinforcing the η/s=1/(4π) correspondence in rotating backgrounds. Finally, it lays out open problems, notably the nonlinear existence and endpoint of the single-KVF/Hairy branches and the extension to higher dimensions and extremal limits.

Abstract

Black holes in anti-de Sitter (AdS) backgrounds play a pivotal role in the gauge/gravity duality where they determine, among other things, the approach to equilibrium of the dual field theory. We undertake a detailed analysis of perturbed Kerr-AdS black holes in four- and five-dimensional spacetimes, including the computation of its quasinormal modes, hydrodynamic modes and superradiantly unstable modes. Our results shed light on the possibility of new black hole phases with a single Killing field, possible new holographic phenomena and phases in the presence of a rotating chemical potential, and close a crucial gap in our understanding of linearized perturbations of black holes in anti-de Sitter scenarios.

Figures (19)

• Figure 1: Left panel: Allowable region for $a/L$ and $r_+/L$: the vertical dashed line is given by $r_+/L=1/\sqrt{3}$, the dashed dotted lines indicate extremality, and the horizontal solid lines indicate $|a|=L$. Right panel: Allowable region for $\Omega_h L$ and $R_+/L$: the horizontal dashed line marks the onset of superradiance, the dashed dotted lines indicate extremality.
• Figure 2: Imaginary part of the QNM frequency as a function of the rotation parameter $a/L$, for fixed horizon radius $r_+/L=0.005$, for scalar ( Right Panel) and vector modes ( Left Panel). This is for $\ell=2$ modes with no radial overtone. These are an example of the QNM spectrum in the regime $a/L<r_+/L \ll 1$ where the analytical matching analysis is valid and its approximated results can be used to both test our numerical code (valid in any regime), and estimate more precisely the regime of validity of the analytical approximation. The red dots are the exact results from our numerical code. The green curve is the numerical solution of the matching transcendental equation \ref{['FreqMatchingl2']}, while the dashed black curve is the approximated analytical solution \ref{['SnormalM']} or \ref{['VnormalM']} of \ref{['FreqMatchingl2']}. In both figures there is a critical rotation where ${\rm Im}(\tilde{\omega} L)=0$ and ${\rm Re}(\tilde{\omega})-m\Omega_H\simeq 0$ to within $0.01\%$. For lower rotations the QNMs are damped and with ${\rm Re}(\tilde{\omega})-m\Omega_H>0$, while for higher rotations we have unstable superradiant modes with ${\rm Re}(\tilde{\omega})-m\Omega_H<0$.
• Figure 3: The onset of superradiance for the first $\ell=m=2,3,4,5$scalar modes of the Kerr-AdS BH. The left panel shows the OC in the phase diagram described by the gauge invariant parameters $(R_+/L,\Omega_h L)$ (the inset plot zooms out the main plot to show an enlarged view of the parameter space). Regular Kerr-AdS BHs exist in the blue shaded area all the way up to the black curve where extremality is attained. In the right panel we show the value of the angular eigenvalue $\lambda$ as a function of the areal radius $R_+/L$ as we move along the OC. In both plots, the larger black points on the left with $R_+/L=0$ are fixed by the properties \ref{['wAdSscalar']} of scalar normal modes of global AdS.
• Figure 4: Superradiant modes and QNMs for the $\ell=m=2$ scalar harmonic. The left panel plots the imaginary part Im$(\omega)$ of the frequencies while the right panel shows the real part of the superradiant factor, i.e. $(\text{Re}(\omega L)-m\Omega_h L)/(4\pi T_h)$, as a function of the horizon radius $r_+/L$ and rotation $a/L$ parameters. The blue curve is the superradiant OC with Im$(\omega)=0$ and Re$(\varpi)=0$. The large red point signals the Kerr-AdS BH that is most unstable to scalar superradiance described by \ref{['scalar:MaxInstab']}.. The black curves have constant radius $r_+/L=0.1;\, 0.2;\, 0.3;\, 0.4;\, 0.445;\, 0.5;\, 0.6;\, 0.7;\, 0.8$. These plots are discussed in more detail in the text.
• Figure 5: Imaginary ( left panel) and real ( right panel) part of the angular eigenvalues of the superradiant modes and QNMs of the $\ell=m=2$scalar harmonic whose frequencies are shown in Fig. \ref{['Fig:ScalarFreq']}. The color coding of the lines/points is the same as Fig. \ref{['Fig:ScalarFreq']}.