Abstract
We study rotating global AdS solutions in five-dimensional Einstein gravity coupled to a multiplet complex scalar within a cohomogeneity-1 ansatz. The onset of the gravitational and scalar field superradiant instabilities of the Myers-Perry-AdS black hole mark bifurcation points to black resonators and hairy Myers-Perry-AdS black holes, respectively. These solutions are subject to the other (gravitational or scalar) instability, and result in hairy black resonators which contain both gravitational and scalar hair. The hairy black resonators have smooth zero-horizon limits that we call graviboson stars. In the hairy black resonator and graviboson solutions, multiple scalar components with different frequencies are excited, and hence these are multioscillating solutions. The phase structure of the solutions are examined in the microcanonical ensemble, i.e. at fixed energy and angular momenta. It is found that the entropy of the hairy black resonator is never the largest among them. We also find that hairy black holes with higher scalar wavenumbers are entropically dominant and occupy more of phase space than those of lower wavenumbers.
Highlights
Work on this problem involved studying perturbations of the Kerr-AdS and Myers-Perry-AdS black holes, where quasi-normal spectra were obtained in [10, 11]
We have studied asymptotically global AdS solutions of Einstein gravity coupled to a (2j + 1) complex scalar multiplet within a cohomogeneity-1 ansatz
The following solutions are available within our ansatz: Myers-Perry-AdS black holes, black resonators, black holes with scalar hair, and hairy black resonators
Summary
Our metric ansatz will contain deformations of an S3 whose perturbations can naturally be written in terms of Wigner D-matrices Dmj k(θ, φ, χ). Where σi (i = 1, 2, 3) are 1-forms defined by σ1 = − sin χdθ + cos χ sin θdφ , σ2 = cos χdθ + sin χ sin θdφ , σ3 = dχ + cos θdφ These satisfy the SU(2) Maurer-Cartan equation dσi = (1/2) ijkσj ∧ σk. The Wigner D-matrices satisfy a convenient formula for summation, j (Dmj k )∗Dmj k = δk k This can be proved using the ladder operators. Dk depends on the index j, we suppress it for notational simplicity because we will generally keep j fixed once the content of the scalar field is specified. In this notation, eq (2.12) is written as.
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