Superradiant instability and black resonators in AdS

Rapidly rotating Myers-Perry-AdS black holes are unstable against rotational superradiance. From the onset of the instability, cohomogeneity-1 black resonators are constructed in five-dimensional asymptotically AdS space. By using the cohomogeneity-1 metric, perturbations of the cohomogeneity-1 black resonators are also studied. Copyright T. Ishii. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 01-11-2020 Accepted 17-11-2020 Published 13-08-2021 Check for updates doi:10.21468/SciPostPhysProc.4.008


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In asymptotically AdS space, rotational gravitational superradiance leads to superradiant insta-12 bility to rotating AdS black holes [1,2]. From the onset of the instability, solutions called black 13 resonators branches off [3]. They are connected to geons in the limit of the zero horizon size.

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The range of the radial coordinate r is from either the origin r = 0 (for geons) or some horizon 30 radius r = r h (for black holes) to asymptotic infinity r → ∞. The metric is assumed to be 31 asymptotically AdS 5 in r → ∞ where h approaches a constant and f , g, α, β → 1.

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The Myers-Perry AdS black hole with equal angular momenta is a solution to the Einstein 33 equations within the metric ansatz (1). It is given by where the angular velocity Ω is The horizon radius r h is given by the largest real root of g(r h ) = 0. The isometry group of This perturbation is SU(2)-invariant, but the terms with σ 2 ± have the U(1)-charges ±2, respec-45 tively, and hence it is not invariant under the U(1) shift of χ: χ → χ +const. This perturbation 46 results in a decoupled perturbation equation for δα, where g and β are given in (3). For fixed r h , this equation can be solved by a nontrivial normal 48 modes δα at a critical value of Ω that corresponds to the onset of the superradiant instability.

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The Myers-Perry black hole with equal angular momenta is unstable above that frequency, and 50 the U(1) isometry is broken when this instability turns up.

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The Einstein equations under the metric ansatz (1)  These boundary conditions correspond to the rotating frame at infinity, for which we use (τ, χ). 56 We can switch to the non-rotating frame at infinity (t, ψ) by In the non-rotating frame, periodic time dependence explicitly appears in the metric as whereσ a are the invariant 1-forms for the non-rotating frame at infinity by replacing χ in σ a

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(2) with ψ. The τ-translation isometry R is interpreted to be a helical Killing vector The (E, J) phase diagram is shown in Fig. 1. The entropy S is shown by the color map.

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The black resonators branch off from the onset of the superradiant instability. Remarkably, 62 they can be found even in the region where no regular Myers-Perry solutions exist. While the 63 metric in the non-rotating frame at infinity is time dependent (9), the entropy is not. The zero This gives the energy-momentum tensor consistent with the metric ansatz.
Thus, the Killing vector is asymptotically spacelike K 2 > 0 because Ω > 1. It has been shown 83 that solutions with such a property should be unstable [8]. In the case of the cohomogeneity-1 84 black resonators, we can consider perturbations breaking the SU(2) isometry of the metric.

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For simplicity, we consider only the massless scalar field perturbation in the black resonator 86 and geon backgrounds. To find the onset of instability, it is convenient to consider the eigen- The scalar field can be decomposed to modes by using the Wigner D-matrices as where the indices of and summation over ( j, m) are suppressed. Using this, we obtain coupled 91 equations for φ k with the following structure, where 93 L k = (1 + r 2 )g d 2 d r 2 + 1 + r 2 2 and Note that φ k vanishes if |k| > j. is "double-stepping," i.e. the mode with k is coupled to those with k ± 2. 98 We solve the coupled equations (14) and identify the instability of φ k in the black resonator for modes with j = 9/2, j = 5, and j = 11/2. For j = 5, there are actually two onset curves corresponding to even and odd parity modes, but they almost coincide. In the inset, we zoom 104 into the region near the geon where it is easier to see the difference between these modes.

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We studied the superradiant instability of the five-dimensional Myers-Perry black hole with 107 equal angular momenta for SU(2)-invariant metric perturbations. We then constructed black 108 resonators that branch off from the onset of the instability and are given by cohomogeneity-1 109 metric. In addition to gravitational black resonators in Einstein gravity, we obtained photonic 110 black resonators in Einstein-Maxwell theory. By using the cohomogeneity-1 black resonator 111 as the background, we studied the perturbation of the black resonators and identified their 112 instability. 113