Abstract
An analytic expression for the scalar quasinormal modes of the generic, spinning Kerr-$\mathrm{AdS_5}$ black holes was previously proposed by the authors in ref. 1, in terms of transcendental equations involving the Painlev\'e VI (PVI) $\tau$ function. In this work, we carry out a numerical investigation of the modes for generic rotation parameters, comparing implementations of expansions for the PVI $\tau$ function both in terms of conformal blocks (Nekrasov functions) and Fredholm determinants. We compare the results with standard numerical methods for the subcase of Schwarzschild black holes. We then derive asymptotic formulas for the angular eigenvalues and the quasinormal modes in the small black hole limit for generic scalar mass and discuss, both numerically and analytically, the appearance of superradiant modes.
Highlights
The quasinormal fluctuations of black holes play an important role in general relativity
An analytic expression for the scalar quasinormal modes of generic, spinning Kerr-AdS5 black holes was previously proposed by the authors [J
For the Heun equation related to the Kerr–de Sitter and Kerr–anti–de Sitter black holes, the solution for the scattering problem has been given in terms of transcendental equations involving the Painleve VI (PVI) τ function
Summary
The quasinormal fluctuations of black holes play an important role in general relativity. There have been many studies of quasinormal modes of various types of perturbations on several background solutions in AdS spacetime, and we refer to Ref. For the Heun equation related to the Kerr–de Sitter and Kerr–anti–de Sitter black holes, the solution for the scattering problem has been given in terms of transcendental equations involving the PVI τ function. III, we give approximate expressions for the monodromy parameters in terms of the isomonodromy time t0 Applying these results to the angular equation, we obtain an approximate expression for the separation constant for slow rotation or near rotating black holes. In Appendix B, we give an explicit parametrization of the monodromy matrices given the monodromy parameters
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