On the number of light rings in curved spacetimes of ultra-compact objects

In a very interesting paper, Cunha, Berti, and Herdeiro have recently claimed that ultra-compact objects, self-gravitating horizonless solutions of the Einstein field equations which have a light ring, must possess at least {\it two} (and, in general, an even number of) light rings, of which the inner one is {\it stable}. In the present compact paper we explicitly prove that, while this intriguing theorem is generally true, there is an important exception in the presence of degenerate light rings which, in the spherically symmetric static case, are characterized by the simple dimensionless relation $8\pi r^2_{\gamma}(\rho+p_{\text{T}})=1$ [here $r_{\gamma}$ is the radius of the light ring and $\{\rho,p_{\text{T}}\}$ are respectively the energy density and tangential pressure of the matter fields]. Ultra-compact objects which belong to this unique family can have an {\it odd} number of light rings. As a concrete example, we show that spherically symmetric constant density stars with dimensionless compactness $M/R=1/3$ possess only {\it one} light ring which, interestingly, is shown to be {\it unstable}.Comment: 5 page

The Hawking paradox and the Bekenstein resolution in higher-dimensional spacetimes

The black-hole information puzzle has attracted much attention over the years from both physicists and mathematicians. One of the most intriguing suggestions to resolve the information paradox is due to Bekenstein, who has stressed the fact that the low-energy part of the semi-classical black-hole emission spectrum is partly blocked by the curvature potential that surrounds the black hole. As explicitly shown by Bekenstein, this fact implies that the grey-body emission spectrum of a (3+1)-dimensional black hole is considerably less entropic than the corresponding radiation spectrum of a perfectly thermal black-body emitter. Using standard ideas from quantum information theory, it was shown by Bekenstein that, in principle, the filtered Hawking radiation emitted by a (3+1)-dimensional Schwarzschild black hole may carry with it a substantial amount of information, the information which was suspected to be lost. It is of physical interest to test the general validity of the "information leak" scenario suggested by Bekenstein as a possible resolution to the Hawking information puzzle. In the present paper we analyze the semi-classical entropy emission properties of higher-dimensional black holes. In particular, we provide evidence that the characteristic Hawking quanta of $(D+1)$-dimensional Schwarzschild black holes in the large $D\gg1$ regime are almost unaffected by the spacetime curvature outside the black-hole horizon. This fact implies that, in the large-$D$ regime, the Hawking black-hole radiation spectra are almost purely thermal, thus suggesting that the emitted quanta cannot carry the amount of information which is required in order to resolve the information paradox. Our analysis therefore suggests that the elegant information leak scenario suggested by Bekenstein cannot provide a generic resolution to the intriguing Hawking information paradox.Comment: 6 page

Strong cosmic censorship in charged black-hole spacetimes: As strong as ever

It is proved that dynamically formed Reissner-Nordstr\"om-de Sitter (RNdS) black holes, which have recently been claimed to provide counter-examples to the Penrose strong cosmic censorship conjecture, are characterized by unstable (singular) inner Cauchy horizons. The proof is based on analytical techniques which explicitly reveal the fact that {\it charged} massive scalar fields in the charged RNdS black-hole spacetime are characterized, in the large-coupling regime, by quasinormal resonant frequencies with $\Im\omega^{\text{min}}<{1\over2}\kappa_+$, where $\kappa_+$ is the surface gravity of the black-hole event horizon. This result implies that the corresponding relaxation rate $\psi\sim e^{-\Im\omega^{\text{min}}t}$ of the collapsed charged fields is slow enough to guarantee, through the mass-inflation mechanism, the instability of the dynamically formed inner Cauchy horizons. Our results reveal the physically important fact that, taking into account the unavoidable presence of {\it charged} matter fields in dynamically formed {\it charged} spacetimes, non-asymptotically flat RNdS black holes are globally hyperbolic and therefore respect the fundamental strong cosmic censorship conjecture.Comment: 7 page

Upper bound on the center-of-mass energy of the collisional Penrose process

Following the interesting work of Ba\~nados, Silk, and West [Phys. Rev. Lett. {\bf 103}, 111102 (2009)], it is repeatedly stated in the physics literature that the center-of-mass energy, ${\cal E}_{\text{c.m}}$, of two colliding particles in a maximally rotating black-hole spacetime can grow unboundedly. For this extreme scenario to happen, the particles have to collide at the black-hole horizon. In this paper we show that Thorne's famous hoop conjecture precludes this extreme scenario from occurring in realistic black-hole spacetimes. In particular, it is shown that a new (and larger) horizon is formed {\it before} the infalling particles reach the horizon of the original black hole. As a consequence, the center-of-mass energy of the collisional Penrose process is {\it bounded} from above by the simple scaling relation ${\cal E}^{\text{max}}_{\text{c.m}}/2\mu\propto(M/\mu)^{1/4}$, where $M$ and $\mu$ are respectively the mass of the central black hole and the proper mass of the colliding particles.Comment: 5 page

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