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 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 \gg 1 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 finite asymptotic value in the large mass M \mu \gg 1 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 .