Quasinormal modes of Reissner-Nordström-anti-de Sitter black holes: scalar, electromagnetic and gravitational perturbations

TL;DR

This work computes quasinormal modes for scalar, electromagnetic, and gravitational perturbations of Reissner-Nordström–anti-de Sitter black holes, extending known Schwarzschild–AdS results to charged backgrounds. Using two independent frequency-domain methods, the authors reveal near-isospectral behavior for large black holes, quantify how QNM frequencies depend on charge with universal fits for Q̄<1/3, and identify purely damped modes whose damping grows toward extremality. A key finding is the potential marginal instability of extremal RN–AdS black holes if the amplitudes of these modes do not vanish, with important implications for AdS/CFT thermalization timescales. The results provide practical, accurate fitting formulas and deepen understanding of stability and spectral properties of charged AdS black holes in the context of holography.

Abstract

We study scalar, electromagnetic and gravitational perturbations of a Reissner-Nordström-anti-de Sitter (RN-AdS) spacetime, and compute its quasinormal modes (QNM's). We confirm and extend results previously found for Schwarzschild-anti-de Sitter (S-AdS) black holes. For ``large'' black holes, whose horizon is much larger than the AdS radius, different classes of perturbations are almost exactly {\it isospectral}; this isospectrality is broken when the black hole's horizon radius is comparable to the AdS radius. We provide very accurate fitting formulas for the QNM's, which are valid for black holes of any size and charge $Q<Q_{ext}/3$. Electromagnetic and axial perturbations of large black holes are characterized by the existence of pure-imaginary (purely damped) modes. The damping of these modes tends to infinity as the black hole charge approaches the extremal value; if the corresponding mode amplitude does not tend to zero in the same limit, this implies that {\it extremally charged RN-AdS black holes are marginally unstable}. This result is relevant in view of the AdS/CFT conjecture, since, according to it, the AdS QNM's give the timescales for approach to equilibrium in the corresponding conformal field theory.

Figures (5)

• Figure 1: Plot of the S-AdS QNM frequencies for the fundamental scalar mode with $l=0$ (continuous line), for the first non-purely damped axial mode with $l=2$ (dashed line) and for the fundamental polar mode for $l=2$ (dotted line). All calculations are started at $r_+=100$ (top right of the diagram). The step in $r_+$ is reduced by $\Delta r_+=0.1$ and the calculation is iterated until our numerical code fails to converge (bottom left). Corresponding to selected values of the horizon radius (namely, $r_+=100,~50,~10,~5,~1,~0.8,~0.6,~0.4$, whose S-AdS frequencies are tabulated in Table I of Horowitz and Hubeny), we "switch on" the charge and follow the modes in the complex plane. The resulting trajectories are the "tails" departing from the continuous scalar QNM line. Axial and polar perturbations are almost isospectral both in the large and in the small black hole limit.
• Figure 2: The top two panel shows the characteristic "wiggling" of the real part of the QNM frequency, $\omega_R$, as a function of the normalized charge $\bar{Q}=Q/Q_{ext}$, for selected values of the horizon radius $r_+$; the bottom two panels show the imaginary part of the QNM frequency, $\omega_I(\bar{Q})$, for the same black holes. Scalar perturbations are denoted by solid lines, axial perturbations by dashed lines, and polar perturbations by dotted lines. The isospectrality of different kinds of perturbations, which holds in the large black hole limit, is clearly lost as the black hole "size" becomes comparable to the AdS radius, $r_+\sim 1$. Notice that $\omega_I"(\bar{Q})=0$ when $\omega_R'(\bar{Q})=0$ (a prime denoting differentiation with respect to $\bar{Q}$).
• Figure 3: Imaginary part of the purely damped mode reducing to pure axial gravitational perturbations with $l=2$ in the zero-charge limit. Starting from S-AdS results, we track the modes for selected values of the horizon radius $r_+$ (indicated to the right of each curve). The calculation is terminated when our root finder fails to converge. The dashed line indicates the last mode we can find before the numerical method breaks down as we approach the "algebraically special" ($\omega_I=2$, $r_+=1$) frequency.
• Figure 4: Imaginary part of the purely damped mode reducing to pure electromagnetic perturbations with $l=1$ (left panel) and $l=2$ (right panel) in the zero-charge limit. Continuous, dashed and dotted lines refer, respectively, to black holes having horizon radius $r_+=100,~50$ and $10$ (top to bottom in the plots). For $l=1$, we also plot the pure imaginary mode with $r_+=5$ (dash-dotted line).
• Figure 5: Real (left panel) and imaginary parts (right panel) of the small-black hole QNM frequencies for modes reducing to pure electromagnetic perturbations with $l=1$ in the zero-charge limit. The size of the black hole horizon radius corresponding to each curve is indicated in the inlay.