Excited hairy black holes: Dynamical construction and level transitions

We study the dynamics of unstable Reissner-Nordström anti–de Sitter black holes under charged scalar field perturbations in spherical symmetry. We unravel their general behavior and approach to the final equilibrium state. In the first part of this work, we present a numerical analysis of massive charged scalar field quasinormal modes. We identify the known mode families—superradiant modes, zero-damped modes, AdS modes, and the near-horizon mode—and we track their migration under variation of the black hole and field parameters. We show that the zero-damped modes become superradiantly unstable for large RNAdS with large gauge coupling; the leading unstable mode is identified with the near-horizon condensation instability. In the second part, we present results of numerical simulations of perturbed large RNAdS, showing the nonlinear development of these unstable modes. For generic initial conditions, charge and mass are transferred from the black hole to the scalar field, until an equilibrium solution with a scalar condensate is reached. We use results from the linear analysis, however, to select special initial data corresponding to an unstable overtone mode. We find that these data evolve to produce a new equilibrium state—an excited hairy black hole with the scalar condensate in an overtone configuration. This state is, however, unstable, and the black hole eventually decays to the generic end state. Nevertheless, this demonstrates the potential relevance of overtone modes as transients in black hole dynamics.

Figure 1

Apparent horizon area of a black hole as it undergoes a series of transitions through metastable excited hairy black hole states. Initial data chosen to consist primarily of $n=2$ overtone, with subleading $n=1$ overtone.

Figure 2

Continued fraction values for RNAdS with $r_{+}=0.1$, $\alpha =0.8$, $q=4$, and $m=0$. We plot the logarithm of the difference between the left hand side and right hand side of (43). Darker colors correspond to higher values. Quasinormal modes (minima in the plot) are marked by black crosses. We see that nonzero $q$ breaks the symmetry between positive and negative real part. Two branches of modes are present, a vertical branch of zero-damped modes, and a more horizontal branch AdS modes. The dashed vertical line corresponds to the superradiant bound frequency; modes satisfying $0<\mathrm{Re}\omega <qQ/r_{+}$ are unstable.

Figure 3

Spacing $\delta$ between two zero-damped modes versus surface gravity $\kappa$. Here, we take $q=0$ and $r_{+}=1$. We see that $\delta \to \kappa$ as $\kappa \to 0$.

Figure 4

Quasinormal frequencies of RNAdS for $r_{+}=0.1$, $\alpha =0.8$, $m=0$, and $0\le q\le 12$. Dashed gray curves track the quasinormal frequencies under variation of $q$. The two mostly-horizontal trajectories are zero-damped modes, and the others are AdS modes. The AdS modes become unstable when $\mathrm{Re}\omega$ drops below the superradiant bound frequency (indicated by dashed vertical lines).

Figure 5

Continued fraction values for RNAdS with $r_{+}=5.0$ and $\alpha =0.95$, and field parameters $q=2$, $m^2=-1.49^2$. We see two diagonal branches, and one unstable mode. Not visible due to resolution is the set of zero-damped modes.

Figure 6

Quasinormal modes for $\alpha =0.995$, $r_{+}=5$, $q=0$ and $m^2=-1.49^2$. One mode in unstable here: this corresponds to the near-horizon mode.

Figure 7

Tracking of the first four AdS modes (${\mathrm{AdS}}_i$) and the first zero-damped mode (${\mathrm{ZDM}}_0$) for $r_{+}$ varied between 0.1 and 5 by steps of 0.05, for $\alpha =0.8$, $q=0$, and $m=0$. On each trajectory, the arrow indicates the direction of increasing $r_{+}$. The trajectories approach a fixed point for large black holes, which arises as a consequence of a scaling symmetry: when $r_{+}\gg L$, the modes are invariant under the transformation $r_{+}\to \lambda r_{+}$, $r_{-}\to \lambda r_{-}$, $\omega \to \lambda \omega$. This can be derived from (35). We also find that whereas for small RNAdS, the AdS modes are longest lived, for large RNAdS the zero-damped modes are longest lived.

Figure 8

Quasinormal frequencies for a large black hole with $r_{+}=5$, $\alpha =0.95$, $m=0$, and $q$ varying between 0 and 10 by steps of 0.5. In the main figure we plot trajectories for the first two zero-damped modes and the first two AdS modes, and the arrow indicates the direction of increasing $q$. We see that the zero-damped modes become unstable for large $q$. The AdS mode frequencies pass through a kink in their migration. The inset shows additional zero-damped mode trajectories (gray dashed), and illustrates how the second AdS mode slots itself in between two of them.

Figure 9

Normalized area of the apparent horizon as a function of time, for initial black hole with $r_{+}=100$ and $\alpha =0.9$. Different colors correspond to different gauge coupling $q$.

Figure 10

Normalized charge contained in the black hole and in the scalar field as a function of time, for initial black hole with $r_{+}=100$ and $\alpha =0.9$. Solid curves denote scalar field charge outside the apparent horizon, and dashed curves denote charge within the black hole apparent horizon.

Figure 11

End state radial profile of the scalar field, for initial black hole with $r_{+}=100$ and $\alpha =0.9$. For smaller gauge coupling $q$ the field has support closer to the horizon, whereas for larger $q$ the support is away from the black hole.

Figure 12

Spectrogram of $\mathrm{Re}{\phi }_3(v)$ for evolution starting from $r_{+}=100$, $\alpha =0.9$, and $q=24$. Initially there are nine unstable modes; in the end, the final state is in the fundamental mode. Dashed lines correspond to the quasinormal frequencies of Table 1, solid to the superradiant bound frequency, $-q{\mathrm{\Phi }}_{\mathrm{AH}}(v)$.

Figure 13

Radial profiles of initial data for initial values $r_{+}=100$, $\alpha =0.9$, and $q=11$. Two curves correspond to single-mode data, and the third to mixed-mode data.

Figure 14

Normalized area of the apparent horizon for initial $r_{+}=100$ and $\alpha =0.9$. Dashed curves correspond to $q=8$, solid to $q=11$. Excited hairy black hole solutions occur at the temporary plateaus.

Figure 15

Charge transfer as a function of time for initial data with $r_{+}=100$, $\alpha =0.9$, and $q=11$. Solid curves are the (normalized) charge of the scalar field, dashed curves the charge of the black hole, that is the charge within the apparent horizon.

Figure 16

Spectrogram of ${\phi }_3(v)$ for mixed mode initial data, with $r_{+}=100$, $\alpha =0.9$, $q=11$, and $a_{\mathrm{Mix}}=0.999$. This shows a cascade through hairy black holes with $n=2\to 1\to 0$. Dashed lines indicate values of the initial mode frequencies (calculated using the linear analysis), solid corresponds to the superradiant bound $-q{\mathrm{\Phi }}_{\mathrm{AH}}(v)$.

Figure 17

Imaginary part of the NH mode as a function of the number of terms used in the continued fraction $N$, for different values of the extremality parameter $a$. We can observe that convergence is more difficult when $a$ gets closer to 1, as the equations used break down.