The large-mass limit of cloudy black holes

Abstract: The interplay between black holes and fundamental fields has attracted much attention over the years from both physicists and mathematicians. In this paper we study {\it analytically} a physical system which is composed of massive scalar fields linearly coupled to a rapidly-rotating Kerr black hole. Using simple arguments, we first show that the coupled black-hole-scalar-field system may possess stationary bound-state resonances (stationary scalar `clouds') in the bounded regime $1<\mu/m\Omega_{\text{H}}<\sqrt{2}$, where $\mu$ and $m$ are respectively the mass and azimuthal harmonic index of the field, and $\Omega_{\text{H}}$ is the angular velocity of the black-hole horizon. We then show explicitly that these two bounds on the dimensionless ratio $\mu/m\Omega_{\text{H}}$ can be saturated in the asymptotic $m\to\infty$ limit. In particular, we derive a remarkably simple analytical formula for the resonance mass spectrum of the stationary bound-state scalar clouds in the regime $M\mu\gg1$ of large field masses: $\mu_n = \sqrt{2}m \Omega_{\text{H}} \big[1-{{\pi({\cal R}+n)} \over {m|\ln\tau|}}\big]$, where $\tau$ is the dimensionless temperature of the rapidly-rotating (near-extremal) black hole, ${\cal R}<1$ is a constant, and $n=0,1,2,...$ is the resonance parameter. In addition, it is shown that, contrary to the flat-space intuition, the effective lengths of the scalar field configurations in the curved black-hole spacetime approach a {\it finite} asymptotic value in the large mass $M\mu\gg1$ limit. In particular, we prove that in the large mass limit, the characteristic length scale of the scalar clouds scales linearly with the black-hole temperature.