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A (charged) rotating black hole may be unstable against a (charged) massivescalar field perturbation due to the existence of superradiance modes. Thestability property depends on the parameters of the system. In this paper, thesuperradiant stable parameter space is studied for the four-dimensionalextremal Kerr and Kerr-Newman black holes under massive and charged massivescalar perturbation. For the extremal Kerr case, it is found that when theangular frequency and proper mass of the scalar perturbation satisfy theinequality $\omega<\mu/\sqrt{3}$, the extremal Kerr black hole and scalarperturbation system is superradiantly stable. For the Kerr-Newman black holecase, when the angular frequency of the scalar perturbation satisfies$\omega<qQ/M$ and the product of the mass-to-charge ratios of the black holeand scalar perturbation satisfies $\frac{\mu}{q}\frac{M}{Q} > \frac{\sqrt{3k^2+2} }{ \sqrt{k^2+2} },~k=\frac{a}{M}$, the extremal Kerr-Newman black holeis superradiantly stable under charged massive scalar perturbation.

Extremal rotating black holes, scalar perturbation and superradiant stability