Abstract
We study the quasinormal modes (QNMs) of a charged scalar field on a Reissner-Nordstr\"om-anti-de Sitter (RN-${\mathrm{AdS}}_{5}$) black hole in the small radius limit by using the isomonodromic method. We also derive the low-temperature expansion of the fundamental QNM frequency. Finally, we provide numerical evidence that instabilities appear in the small radius limit for large values of the charge of the scalar field.
Highlights
The energy extraction from a rotating uncharged black hole can be realized by the scattering of a wave of frequency ω and axial quantum number m that satisfies the superradiance condition ω < mΩ, where Ω is the angular velocity of the black hole
We study the quasinormal modes (QNMs) of a charged scalar field on a Reissner-Nordström-anti-de Sitter (RN-AdS5) black hole in the small radius limit by using the isomonodromic method
We study the occurrence of instabilities in RN-AdS5 by computing the QNMs frequencies of a charged scalar field scattering on a RN-AdS5 black hole
Summary
The energy extraction from a rotating uncharged black hole can be realized by the scattering of a wave of frequency ω and axial quantum number m that satisfies the superradiance condition ω < mΩ, where Ω is the angular velocity of the black hole. By imposing an asymptotic AdS spacetime or a mirrorlike boundary condition, Reissner-Nordström black holes can develop such instabilities, as it has been shown in four dimensions [5,6] and d ≥ 4 dimensions in the small outer horizon radius, rþ, limit [7]. As argued in [14], in the small rþ limit the first correction to the (fundamental) QNMs frequencies receives contributions from all the intermediate channels of the c 1⁄4 1 Virasoro conformal blocks expansion These contributions can be resummed by using the generating function for the Catalan numbers, allowing us to obtain an asymptotic expression for the frequencies [14]. We show that at high temperature the QNMs are given by the zeros of the PVI tau function, and compute an asymptotic expression in the small rþ limit for the l 1⁄4 0 modes in Sec. III A. In Appendix we review the Nekrasov expansion of the PVI tau function
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