On the instability regime of the rotating Kerr spacetime to massive scalar perturbations

The instability of rotating Kerr black holes due to massive scalar perturbations is investigated. It is well known that a bosonic field impinging on a Kerr black hole can be amplified as it scatters off the hole. This superradiant scattering occurs for frequencies in the range $\omega<m\Omega$, where $\Omega$ is the angular frequency of the black hole and $m$ is the azimuthal harmonic index of the mode. If the incident field has a non-zero rest mass, $\mu$, then the mass term effectively works as a mirror, reflecting the scattered wave back towards the black hole. The wave may bounce back and forth between the black hole and some turning point amplifying itself each time. This may lead to a dynamical instability of the system, a phenomena known as a "black-hole bomb". In this work we provide a bound on the instability regime of rotating Kerr spacetimes. In particular, we show that Kerr black holes are stable to massive perturbations in the regime $\mu\geq\sqrt{2}m\Omega$.Comment: 5 pages. arXiv admin note: text overlap with arXiv:0910.073

Quantum-Gravity Fluctuations and the Black-Hole Temperature

Bekenstein has put forward the idea that, in a quantum theory of gravity, a black hole should have a discrete energy spectrum with concomitant discrete line emission. The quantized black-hole radiation spectrum is expected to be very different from Hawking's semi-classical prediction of a thermal black-hole radiation spectrum. One naturally wonders: Is it possible to reconcile the {\it discrete} quantum spectrum suggested by Bekenstein with the {\it continuous} semi-classical spectrum suggested by Hawking ? In order to address this fundamental question, in this essay we shall consider the zero-point quantum-gravity fluctuations of the black-hole spacetime. In a quantum theory of gravity, these spacetime fluctuations are closely related to the characteristic gravitational resonances of the corresponding black-hole spacetime. Assuming that the energy of the black-hole radiation stems from these zero-point quantum-gravity fluctuations of the black-hole spacetime, we derive the effective temperature of the quantized black-hole radiation spectrum. Remarkably, it is shown that this characteristic temperature of the {\it discrete} (quantized) black-hole radiation agrees with the well-known Hawking temperature of the {\it continuous} (semi-classical) black-hole spectrum.Comment: 6 page

Upper bound on the energies of the emitted Hawking quanta

Using Thorne's hoop conjecture, it is argued that the energies of the Hawking quanta emitted from canonical Schwarzschild black holes are bounded from above by the simple quantum relation ${\cal E}<{\cal E}_{\text{max}}={\hbar^{2/3}}/M^{1/3}$. In particular, it is shown that, due to non-linear (self-gravity) effects of the tunneling quanta, higher energy field modes are re-absorbed (rather than escape to infinity) by the black hole.Comment: 5 page

Bekenstein's generalized second law of thermodynamics: The role of the hoop conjecture

Bekenstein's generalized second law (GSL) of thermodynamics asserts that the sum of black-hole entropy, $S_{\text{BH}}=Ac^3/4\hbar G$ (here $A$ is the black-hole surface area), and the ordinary entropy of matter and radiation fields in the black-hole exterior region never decreases. We here re-analyze an intriguing gedanken experiment which was designed by Bekenstein to challenge the GSL. In this historical gedanken experiment an entropy-bearing box is lowered into a charged Reissner-Nordstr\"om black hole. For the GSL to work, the resulting increase in the black-hole surface area (entropy) must compensate for the loss of the box's entropy. We show that if the box can be lowered adiabatically all the way down to the black-hole horizon, as previously assumed in the literature, then for near-extremal black holes the resulting increase in black-hole surface-area (due to the assimilation of the box by the black hole) may become too small to compensate for the loss of the box's entropy. In order to resolve this apparent violation of the GSL, we here suggest to use a generalized version of the hoop conjecture. In particular, assuming that a physical system of mass $M$ and electric charge $Q$ forms a black hole if its circumference radius $r_{\text{c}}$ is equal to (or smaller than) the corresponding Reissner-Nordstr\"om black-hole radius $r_{\text{RN}}=M+\sqrt{M^2-Q^2}$, we prove that a new (and larger) horizon is already formed before the entropy-bearing box reaches the horizon of the original near-extremal black hole. This result, which seems to have been overlooked in previous analyzes of the composed black-hole-box system, ensures the validity of Bekenstein's GSL in this famous gedanken experiment.Comment: In memory of Prof. Jacob D. Bekenstein (1947--2015), whose seminal papers have inspired my researc