Quasi-bound states of massive scalar fields in the Kerr black-hole spacetime: Beyond the hydrogenic approximation

Rotating black holes can support quasi-stationary (unstable) bound-state resonances of massive scalar fields in their exterior regions. These spatially regular scalar configurations are characterized by instability timescales which are much longer than the timescale $M$ set by the geometric size (mass) of the central black hole. It is well-known that, in the small-mass limit $\alpha\equiv M\mu\ll1$ (here $\mu$ is the mass of the scalar field), these quasi-stationary scalar resonances are characterized by the familiar hydrogenic oscillation spectrum: $\omega_{\text{R}}/\mu=1-\alpha^2/2{\bar n}^2_0$, where the integer $\bar n_0(l,n;\alpha\to0)=l+n+1$ is the principal quantum number of the bound-state resonance (here the integers $l=1,2,3,...$ and $n=0,1,2,...$ are the spheroidal harmonic index and the resonance parameter of the field mode, respectively). As it depends only on the principal resonance parameter $\bar n_0$, this small-mass ($\alpha\ll1$) hydrogenic spectrum is obviously degenerate. In this paper we go beyond the small-mass approximation and analyze the quasi-stationary bound-state resonances of massive scalar fields in rapidly-spinning Kerr black-hole spacetimes in the regime $\alpha=O(1)$. In particular, we derive the non-hydrogenic (and, in general, non-degenerate) resonance oscillation spectrum ${{\omega_{\text{R}}}/{\mu}}=\sqrt{1-(\alpha/{\bar n})^2}$, where $\bar n(l,n;\alpha)=\sqrt{(l+1/2)^2-2m\alpha+2\alpha^2}+1/2+n$ is the generalized principal quantum number of the quasi-stationary resonances. This analytically derived formula for the characteristic oscillation frequencies of the composed black-hole-massive-scalar-field system is shown to agree with direct numerical computations of the quasi-stationary bound-state resonances.Comment: 7 page

Marginally stable resonant modes of the polytropic hydrodynamic vortex

The polytropic hydrodynamic vortex describes an effective $(2+1)$-dimensional acoustic spacetime with an inner reflecting boundary at $r=r_{\text{c}}$. This physical system, like the spinning Kerr black hole, possesses an ergoregion of radius $r_{\text{e}}$ and an inner non-pointlike curvature singularity of radius $r_{\text{s}}$. Interestingly, the fundamental ratio $r_{\text{e}}/r_{\text{s}}$ which characterizes the effective geometry is determined solely by the dimensionless polytropic index $N_{\text{p}}$ of the circulating fluid. It has recently been proved that, in the $N_{\text{p}}=0$ case, the effective acoustic spacetime is characterized by an {\it infinite} countable set of reflecting surface radii, $\{r_{\text{c}}(N_{\text{p}};n)\}^{n=\infty}_{n=1}$, that can support static (marginally-stable) sound modes. In the present paper we use {\it analytical} techniques in order to explore the physical properties of the polytropic hydrodynamic vortex in the $N_{\text{p}}>0$ regime. In particular, we prove that in this physical regime, the effective acoustic spacetime is characterized by a {\it finite} discrete set of reflecting surface radii, $\{r_{\text{c}}(N_{\text{p}},m;n)\}^{n=N_{\text{max}}}_{n=1}$, that can support the marginally-stable static sound modes (here $m$ is the azimuthal harmonic index of the acoustic perturbation field). Interestingly, it is proved analytically that the dimensionless outermost supporting radius $r^{\text{max}}_{\text{c}}/r_{\text{e}}$, which marks the onset of superradiant instabilities in the polytropic hydrodynamic vortex, increases monotonically with increasing values of the integer harmonic index $m$ and decreasing values of the dimensionless polytropic index $N_{\text{p}}$.Comment: 13 page

Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis

The spheroidal harmonics $S_{lm}(\theta;c)$ have attracted the attention of both physicists and mathematicians over the years. These special functions play a central role in the mathematical description of diverse physical phenomena, including black-hole perturbation theory and wave scattering by nonspherical objects. The asymptotic eigenvalues $\{A_{lm}(c)\}$ of these functions have been determined by many authors. However, it should be emphasized that all previous asymptotic analyzes were restricted either to the regime $m\to\infty$ with a fixed value of $c$, or to the complementary regime $|c|\to\infty$ with a fixed value of $m$. A fuller understanding of the asymptotic behavior of the eigenvalue spectrum requires an analysis which is asymptotically uniform in both $m$ and $c$. In this paper we analyze the asymptotic eigenvalue spectrum of these important functions in the double limit $m\to\infty$ and $|c|\to\infty$ with a fixed $m/c$ ratio.Comment: 5 page

Self-gravitating ring of matter in orbit around a black hole: The innermost stable circular orbit

We study analytically a black-hole-ring system which is composed of a stationary axisymmetric ring of particles in orbit around a perturbed Kerr black hole of mass $M$. In particular, we calculate the shift in the orbital frequency of the innermost stable circular orbit (ISCO) due to the finite mass $m$ of the orbiting ring. It is shown that for thin rings of half-thickness $r\ll M$, the dominant finite-mass correction to the characteristic ISCO frequency stems from the self-gravitational potential energy of the ring (a term in the energy budget of the system which is quadratic in the mass $m$ of the ring). This dominant correction to the ISCO frequency is of order $O(\mu\ln(M/r))$, where $\mu\equiv m/M$ is the dimensionless mass of the ring. We show that the ISCO frequency increases (as compared to the ISCO frequency of an orbiting test-ring) due to the finite-mass effects of the self-gravitating ring.Comment: 11 page