The large-mass limit of cloudy black holes

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.