Quasinormal behavior of massless scalar field perturbation in Reissner-Nordström anti-de Sitter spacetimes

We present a comprehensive study of the massless scalar field perturbation in the Reissner-Nordström anti-de Sitter (RNAdS) space-time and compute its quasinormal modes (QNM). For the lowest lying mode, we confirm and extend the dependence of the QNM frequencies on the black hole charge obtained in previous works. In near extreme limit, under the scalar perturbation we find that the imaginary part of the frequency tends to zero, which is consistent with the previous conjecture based on electromagnetic and axial gravitational perturbations. For the extreme value of the charge, the asymptotic field decay is dominated by a power-law tail, which shows that the extreme black hole can still be stable to scalar perturbations. We also study the higher overtones for the RNAdS black hole and find large variations of QNM frequencies with the overtone number and black hole charge. The nontrivial dependence of frequencies on the angular index $\ell$ is also discussed.

Figure 1

Illustration of the filtering process. (top) Data obtained from the characteristic algorithm. (bottom) The ${\chi }^2$-fitting made in the characteristic data subtracted from the Horowitz-Hubeny NOM solution. The results obtained from this fitting are compatible with the Horowitz-Hubeny OM results. In the graphs $r_{+}=100$, $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.41$, $R=1$, $\ell =0$, and $n=0$.

Figure 2

Semilog graphs of the scalar field in the event horizon, showing the transition from oscillatory to nonoscillatory asymptotic decay. In the graphs, $r_{+}=100$, $\ell =0$, and $R=1$.

Figure 3

Graph of ${\omega }_R(\mathrm{O}\mathrm{M})$ with $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$, using Horowitz-Hubeny method and characteristic integration results. For this graph, $r_{+}=100$, $R=1$, $\ell =0$, and $n=0$.

Figure 4

Graph of ${\omega }_I$ (OM and NOM) with $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$, showing that ${\omega }_I(\mathrm{N}\mathrm{O}\mathrm{M})$ tends to zero as $Q$ tends to $Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$. In the graph, $r_{+}=100$, $R=1$, $\ell =0$, and $n=0$.

Figure 5

Log-log graphs of the scalar field in the event horizon. (left) Extreme case for different values of $\ell$, showing power-law tails. If $\ell =0$, $\Psi \propto v^{-2}$, but if $\ell >0$, $\Psi \propto v^{-1}$. (right) Approach to the extreme limit, showing transition from exponential decay to power-law decay ($\ell =1$). For all curves, $r_{+}=100$, and $R=1$.

Figure 6

Log-log graphs of the function $V(r^{*})$, from the near extreme limit approaching the extreme limit. At a certain point, if $\delta >0$, there is a transition from power-law decay to exponential decay, indicated for each graph. For the curves, $r_{+}=100$, $R=1$, and $\ell =0,1$ (left and right figures, respectively).

Figure 7

Graphs of ${\omega }_R(\mathrm{O}\mathrm{M})$ and ${\omega }_I(\mathrm{O}\mathrm{M})$ versus $n$, for $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.1$ (represented by triangles) and $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.4$ (represented by bullets). The continuous lines indicate the linear fits. With the increase of Q, ${\omega }_R(\mathrm{O}\mathrm{M})$ increases slower with $n$, while $|{\omega }_I(\mathrm{O}\mathrm{M})|$ increases faster with $n$. For the curves, $r_{+}=100$, $R=1$ and $\ell =0,2,4,6,8,10$.

Figure 8

Graphs of ${\omega }_R(\mathrm{O}\mathrm{M})$ with $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$, showing the “wiggle behavior.” Values of $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$ in graphs indicating when nonoscillating solution dominates. For the curves, $r_{+}=100$, and $\ell =0$.

Figure 9

Graphs of ${\omega }_I$ (OM and NOM solutions) with $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$ for several values of $n$. The continuous lines indicate the oscillatory solution, dotted lines indicate nonoscillatory solutions. For the curves, $r_{+}=100$, $R=1$, and $\ell =0$.

Figure 10

Graphs of ${\omega }_I(\mathrm{O}\mathrm{M})$ with $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$, showing the increase of $|\mathrm{I}\mathrm{m}\mathrm{(}\mathrm{\omega }\mathrm{)}\mathrm{|}$ with $n$. For the curves, $r_{+}=100$, $R=1$, $\ell =0$, and $n=0$.

Figure 11

Graphs of ${\omega }_R(\mathrm{O}\mathrm{M})$ with $\ell$, for several values of $n$. The bullets indicate the numerical results and the lines indicate the quadratic fittings. For the curves, $r_{+}=100$, $R=1$, and $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.1$.

Figure 12

Graphs of ${\omega }_I(\mathrm{O}\mathrm{M})$ with $\ell$, for several values of $n$. The bullets indicate the numerical results and the lines indicate the quadratic fittings. For the curves, $r_{+}=100$, $R=1$, and $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.1$.

Figure 13

Graphs of ${\omega }_R(\mathrm{O}\mathrm{M})$ with $\ell$, for several values of $n$. The bullets indicate the numerical results and the lines indicate the quadratic fittings. For the curves, $r_{+}=100$, $R=1$, and $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.4$.

Figure 14

Graphs of ${\omega }_I(\mathrm{O}\mathrm{M})$ with $\ell$, for several values of $n$. The bullets indicate the numerical results and the lines indicate the quadratic fittings. For the curves, $r_{+}=100$, $R=1$, and $Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.4$.