# Shahar Hod

## Contact Details

NameShahar Hod |
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## Pubs By Year |
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## Pub CategoriesGeneral Relativity and Quantum Cosmology (50) High Energy Physics - Theory (49) High Energy Astrophysical Phenomena (47) Mathematical Physics (2) Mathematics - Mathematical Physics (2) Nonlinear Sciences - Exactly Solvable and Integrable Systems (1) |

## Publications Authored By Shahar Hod

The physical properties of bound-state charged massive scalar field configurations linearly coupled to a spherically symmetric charged reflecting shell are studied {\it analytically}. To that end, we solve the Klein-Gordon wave equation for a static scalar field of proper mass $\mu$, charge coupling constant $q$, and spherical harmonic index $l$ in the background of a charged shell of radius $R$ and electric charge $Q$. It is proved that the dimensionless inequality $\mu R<\sqrt{(qQ)^2-(l+1/2)^2}$ provides an upper bound on the regime of existence of the composed charged-spherical-shell-charged-massive-scalar-field configurations. Read More

One of the most remarkable predictions of the general theory of relativity is the existence of black-hole "photonspheres", compact null hypersurfaces on which massless particles can orbit the central black hole. We prove that every spherically-symmetric asymptotically flat black-hole spacetime is characterized by a photonsphere whose radius is bounded from above by $r_{\gamma} \leq 3M$, where $M$ is the total ADM mass of the black-hole spacetime. It is shown that hairy black-hole configurations conform to this upper bound. Read More

The magnetically charged SU(2) Reissner-Nordstr\"om black-hole solutions of the coupled nonlinear Einstein-Yang-Mills field equations are known to be characterized by infinite spectra of unstable (imaginary) resonances $\{\omega_n(r_+,r_-)\}_{n=0}^{n=\infty}$ (here $r_{\pm}$ are the black-hole horizon radii). Based on direct {\it numerical} computations of the black-hole instability spectra, it has recently been observed that the excited instability eigenvalues of the magnetically charged black holes exhibit a simple universal behavior. In particular, it was shown that the numerically computed instability eigenvalues of the magnetically charged black holes are characterized by the small frequency universal relation $\omega_n(r_+-r_-)=\lambda_n$, where $\{\lambda_n\}$ are dimensionless constants which are independent of the black-hole parameters. Read More

We present evidence for the existence of a quantum lower bound on the Bekenstein-Hawking temperature of black holes. The suggested bound is supported by a gedanken experiment in which a charged particle is dropped into a Kerr black hole. It is proved that the temperature of the final Kerr-Newman black-hole configuration is bounded from below by the relation $T_{\text{BH}}\times r_{\text{H}}>(\hbar/r_{\text{H}})^2$, where $r_{\text{H}}$ is the horizon radius of the black hole. Read More

It is proved that spherically symmetric compact reflecting objects cannot support static bound-state configurations made of scalar fields whose self-interaction potential $V(\psi^2)$ is a monotonically increasing function of its argument. Our theorem rules out, in particular, the existence of massive scalar hair outside the surface of a spherically symmetric compact reflecting star. Read More

The intriguing superradiant amplification phenomenon allows an orbiting scalar field to extract rotational energy from a spinning Kerr black hole. Interestingly, the energy extraction rate can grow exponentially in time if the black-hole-field system is placed inside a reflecting mirror which prevents the field from radiating its energy to infinity. This composed Kerr-black-hole-scalar-field-mirror system, first designed by Press and Teukolsky, has attracted the attention of physicists over the last four decades. Read More

We study analytically the Klein-Gordon wave equation for stationary massive scalar fields linearly coupled to spinning Kerr black holes. In particular, using the WKB approximation, we derive a compact formula for the discrete spectrum of scalar field masses which characterize the stationary composed Kerr-black-hole-massive-scalar-field configurations in the large-coupling regime $M\mu\gg1$ (here $M$ and $\mu$ are respectively the mass of the central black hole and the proper mass of the scalar field). We confirm our analytically derived formula for the Kerr-scalar-field mass spectrum with numerical data that recently appeared in the literature. Read More

Charged rotating Kerr-Newman black holes are known to be superradiantly unstable to perturbations of charged massive bosonic fields whose proper frequencies lie in the bounded regime $0 < \omega < \text{min} \{\omega_{\text{c}} \equiv m \Omega_{\text{H}} + q\Phi_{\text{H}},\mu\}$ [here $\{\Omega_{\text{H}}, \Phi_{\text{H}}\}$ are respectively the angular velocity and electric potential of the Kerr-Newman black hole, and $\{m,q,\mu\}$ are respectively the azimuthal harmonic index, the charge coupling constant, and the proper mass of the field]. In this paper we study analytically the complex resonance spectrum which characterizes the dynamics of linearized charged massive scalar fields in a near-extremal Kerr-Newman black-hole spacetime. Interestingly, it is shown that near the critical frequency $\omega_{\text{c}}$ for superradiant amplification and in the eikonal large-mass regime, the superradiant instability growth rates of the explosive scalar fields are characterized by a non-trivial (non-monotonic) dependence on the dimensionless charge-to-mass ratio $q/\mu$. Read More

Following the interesting work of Ba\~nados, Silk, and West [Phys. Rev. Lett. Read More

We determine the characteristic timescales associated with the linearized
relaxation dynamics of the composed
Reissner-Nordstr\"om-black-hole-charged-massive-scalar-field system. To that
end, the quasinormal resonant frequencies
$\{\omega_n(\mu,q,M,Q)\}_{n=0}^{n=\infty}$ which characterize the dynamics of a
charged scalar field of mass $\mu$ and charge coupling constant $q$ in the
charged Reissner-Nordstr\"om black-hole spacetime of mass $M$ and electric
charge $Q$ are determined {\it analytically} in the eikonal regime $1\ll
M\mu

The seminal works of Bekenstein and Hawking have revealed that black holes have a well-defined thermodynamic description. In particular, it is often stated in the physical literature that black holes, like mundane physical systems, obey the first law of thermodynamics: $\Delta S=\Delta E/T_{\text{BH}}$, where $T_{\text{BH}}$ is the Bekenstein-Hawking temperature of the black hole. In the present work we test the regime of validity of the thermodynamic description of gravity. Read More

It has recently been suggested [S. B. Giddings, Phys. Read More

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. Read More

Bekenstein and Mayo have revealed an interesting property of evaporating $(3+1)$-dimensional Schwarzschild black holes: their entropy emission rates $\dot S_{\text{Sch}}$ are related to their energy emission rates $P$ by the simple relation $\dot S_{\text{Sch}}=C_{\text{Sch}}\times (P/\hbar)^{1/2}$. Remembering that $(1+1)$-dimensional perfect black-body emitters are characterized by the same functional relation, $\dot S^{1+1}=C^{1+1}\times(P/\hbar)^{1/2}$, Bekenstein and Mayo have concluded that, in their entropy emission properties, $(3+1)$-dimensional Schwarzschild black holes behave effectively as $(1+1)$-dimensional entropy emitters. One naturally wonders whether all black holes behave as simple $(1+1)$-dimensional entropy emitters? In order to address this interesting question, we shall study in this paper the entropy emission properties of Reissner-Nordstr\"om black holes. Read More

Spinning Kerr black holes are known to be superradiantly unstable to massive scalar perturbations. We here prove that the instability regime of the composed Kerr-black-hole-massive-scalar-field system is bounded from above by the dimensionless inequality $M\mu < m \cdot \sqrt{{{2(1+\gamma) (1-\sqrt{1-\gamma^2}) - \gamma^2} \over {4\gamma^2}}}$, where $\{\mu,m\}$ are respectively the proper mass and azimuthal harmonic index of the scalar field and $\gamma\equiv r_-/r_+$ is the dimensionless ratio between the horizon radii of the black hole. It is further shown that this {\it analytically} derived upper bound on the superradiant instability regime of the spinning Kerr black hole agrees with recent {\it numerical} computations of the instability resonance spectrum. Read More

The well-known superradiant amplification mechanism allows a charged scalar field of proper mass $\mu$ and electric charge $q$ to extract the Coulomb energy of a charged Reissner-Nordstr\"om black hole. The rate of energy extraction can grow exponentially in time if the system is placed inside a reflecting cavity which prevents the charged scalar field from escaping to infinity. This composed black-hole-charged-scalar-field-mirror system is known as the {\it charged black-hole bomb}. Read More

It has recently been shown that the Hawking evaporation process of $(3+1)$-dimensional Schwarzschild black holes is characterized by the dimensionless ratio $\eta\equiv\tau_{\text{gap}}/\tau_{\text{emission}}\gg1$, where $\tau_{\text{gap}}$ is the characteristic time gap between the emissions of successive Hawking quanta and $\tau_{\text{emission}}$ is the characteristic timescale required for an individual Hawking quantum to be emitted from the Schwarzschild black hole. This strong inequality implies that the Hawking cascade of gravitons from a $(3+1)$-dimensional Schwarzschild black hole is extremely {\it sparse}. In the present paper we explore the semi-classical Hawking evaporation rates of {\it higher}-dimensional Schwarzschild black holes. Read More

More than three decades ago, Detweiler provided an analytical formula for the gravitational resonant frequencies of rapidly-rotating Kerr black holes. In the present work we shall discuss an important discrepancy between the famous {\it analytical} prediction of Detweiler and the recent {\it numerical} results of Zimmerman et. al. Read More

The holographic principle has taught us that, as far as their entropy content is concerned, black holes in $(3+1)$-dimensional curved spacetimes behave as ordinary thermodynamic systems in flat $(2+1)$-dimensional spacetimes. In this essay we point out that the opposite behavior can also be observed in black-hole physics. To show this we study the quantum Hawking evaporation of near-extremal Reissner-Nordstr\"om black holes. Read More

The coupled gravitational-electromagnetic quasinormal resonances of charged rotating Kerr-Newman black holes are explored. In particular, using the recently published numerical data of Dias, Godazgar, and Santos [Phys. Rev. Read More

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. Read More

A well-established phenomenon in general relativity is the dragging of inertial frames by a spinning object. In particular, due to the dragging of inertial frames by a ring orbiting a central black hole, the angular-velocity of the black-hole horizon in the composed black-hole-ring system is no longer related to the black-hole angular-momentum by the simple Kerr-like (vacuum) relation $\Omega^{\text{Kerr}}_{\text{H}}(J_{\text{H}})=J_{\text{H}}/2M^2R_{\text{H}}$. Will has performed a perturbative treatment of the composed black-hole-ring system in the regime of slowly rotating black holes and found the explicit relation $\Omega^{\text{BH-ring}}_{\text{H}}(J_{\text{H}}=0,J_{\text{R}},R)=2J_{\text{R}}/R^3$ for the angular-velocity of a central black hole with zero angular-momentum. Read More

It has recently been shown by Yang. et. al. Read More

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,. Read More

The quasinormal resonance spectrum $\{\omega_n(\mu,q,M,Q)\}_{n=0}^{n=\infty}$ of charged massive scalar fields in the charged Reissner-Nordstr\"om black-hole spacetime is studied {\it analytically} in the large-coupling regime $qQ\gg M\mu$ (here $\{\mu, q\}$ are respectively the mass and charge coupling constant of the field, and $\{M,Q\}$ are respectively the mass and electric charge of the black hole). This physical system provides a striking illustration for the validity of the universal relaxation bound $\tau \times T \geq \hbar/\pi$ in black-hole physics (here $\tau\equiv 1/\Im\omega_0$ is the characteristic relaxation time of the composed black-hole-scalar-field system, and $T$ is the Bekenstein-Hawking temperature of the black hole). In particular, it is shown that the relaxation dynamics of charged massive scalar fields in the charged Reissner-Nordstr\"om black-hole spacetime may {\it saturate} this quantum time-times-temperature inequality. Read More

It is shown that rapidly-rotating Kerr black holes are characterized by the dimensionless ratio $\tau_{\text{gap}}/\tau_{\text{emission}}=O(1)$, where $\tau_{\text{gap}}$ is the average time gap between the emission of successive Hawking quanta and $\tau_{\text{emission}}$ is the characteristic timescale required for an individual Hawking quantum to be emitted from the black hole. This relation implies that the Hawking cascade from rapidly-rotating black holes has an almost continuous character. Our results correct some inaccurate claims that recently appeared in the literature regarding the nature of the Hawking black-hole evaporation process. Read More

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. Read More

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. Read More

It is well-known that the SU(2) Reissner-Nordstr\"om black-hole solutions of the Einstein-Yang-Mills theory are characterized by an infinite set of unstable (imaginary) eigenvalues $\{\omega_n(T_{\text{BH}})\}_{n=0}^{n=\infty}$ (here $T_{\text{BH}}$ is the black-hole temperature). In this paper we analyze the excited instability spectrum of these magnetically charged black holes. The numerical results suggest the existence of a universal behavior for these black-hole excited eigenvalues. Read More

It is shown that highly-charged Reissner-Nordstr\"om black holes in the charge interval $8/9<{(Q/M)}^2<1$ are stable to charged scalar perturbations. Read More

It was first pointed out by Press and Teukolsky that a system composed of a spinning Kerr black hole surrounded by a reflecting mirror may develop instabilities. The physical mechanism responsible for the development of these exponentially growing instabilities is the superradiant amplification of bosonic fields confined between the black hole and the mirror. A remarkable feature of this composed black-hole-mirror-field system is the existence of a critical mirror radius, $r^{\text{stat}}_{\text{m}}$, which supports {\it stationary} (marginally-stable) field configurations. Read More

Static spherically-symmetric matter distributions whose energy-momentum tensor is characterized by a non-negative trace are studied analytically within the framework of general relativity. We prove that such field configurations are necessarily highly relativistic objects. In particular, for matter fields with $T\geq\alpha\cdot\rho\geq0$ (here $T$ and $\rho$ are respectively the trace of the energy-momentum tensor and the energy density of the fields, and $\alpha$ is a non-negative constant), we obtain the lower bound $\text{max}_r\{2m(r)/r\}>(2+2\alpha)/(3+2\alpha)$ on the compactness (mass-to-radius ratio) of regular field configurations. Read More

The elegant `no short hair' theorem states that, if a spherically-symmetric static black hole has hair, then this hair must extend beyond 3/2 the horizon radius. In the present paper we provide evidence for the failure of this theorem beyond the regime of spherically-symmetric static black holes. In particular, we show that rotating black holes can support extremely short-range stationary scalar configurations (linearized scalar `clouds') in their exterior regions. Read More

It is well known that the U(1) Reissner-Nordstr\"om black hole is stable within the framework of the Einstein-Maxwell theory. However, the SU(2) Reissner-Nordstr\"om black-hole solution of the coupled Einstein-Yang-Mills equations is known to be unstable. In fact, this magnetically charged black hole is characterized by an infinite set of unstable (growing in time) perturbation modes. Read More

Recent numerical studies of the coupled Einstein-Klein-Gordon system in a cavity have provided compelling evidence that {\it confined} scalar fields generically collapse to form black holes. Motivated by this intriguing discovery, we here use analytical tools in order to study the characteristic resonance spectra of the confined fields. These discrete resonant frequencies are expected to dominate the late-time dynamics of the coupled black-hole-field-cage system. Read More

In a recent paper (arXiv:1410.0694) Zilh\~ao, Cardoso, Herdeiro, Lehner, and Sperhake have studied the nonlinear stability of Kerr-Newman black holes. We show that their numerical results for the time evolutions of the spacetime deformations of near-extremal Kerr-Newman black holes are described extremely well by a {\it universal} formula for the quasinormal resonances of the black holes. Read More

It is shown that Kerr-Newman black holes can support linear charged scalar fields in their exterior regions. To that end, we solve analytically the Klein-Gordon wave equation for a stationary charged massive scalar field in the background of a near-extremal Kerr-Newman black hole. In particular, we derive a simple analytical formula which describes the physical properties of these stationary bound-state resonances of the charged massive scalar fields in the Kerr-Newman black-hole spacetime. Read More

The hydrodynamic vortex, an effective spacetime geometry for propagating sound waves, is studied analytically. In contrast with the familiar Kerr black-hole spacetime, the hydrodynamic vortex model is described by an effective acoustic geometry which has no horizons. However, this acoustic spacetime possesses an ergoregion, a property which it shares with the rotating Kerr spacetime. Read More

A co-rotating bosonic field interacting with a spinning Kerr black hole can extract rotational energy and angular momentum from the hole. This intriguing phenomenon is known as superradiant scattering. As pointed out by Press and Teukolsky, the black-hole-field system can be made unstable (explosive) by placing a reflecting mirror around the black hole which prevents the extracted energy from escaping to infinity. Read More

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). Read More

The late-time dynamics of massive spin-2 fields in flat and curved spacetimes are studied analytically. We find that the time evolutions of the massive fields are characterized by oscillatory power-law decaying tails at asymptotically late times. In a flat spacetime the decaying exponent depends on the multipole number and the parity of the mode. Read More

The physical properties of an axisymmetric black-hole-ring system are studied analytically within the framework of general relativity to second order in the dimensionless mass ratio $\mu\equiv m/M$. In particular, we analyze the asymptotic behaviors of the binding-energy and the total angular-momentum of the two-body system in the vicinity of the light ring at $R=3M$, where the circular orbit becomes null. We find that both quantities diverge {\it quadratically} in $\mu(1-3M/R)^{-1}$ at the light ring. Read More

The Klein-Gordon equation for a massive scalar field in the background of a rapidly-rotating Kerr black hole is studied analytically. In particular, we derive a simple formula for the stationary (marginally-stable) resonances of the field in the black-hole spacetime. The analytically derived formula is shown to agree with direct numerical computations of the resonances. Read More

We prove the existence of a unique family of non-oscillatory (purely-imaginary) polar quasinormal resonances of rapidly-rotating Kerr black holes. These purely imaginary resonances can be expressed in the compact form: w_n=-i2*pi*T_{BH}*(l+1+n), where T_{BH} is the black-hole temperature, l is the spheroidal harmonic index of the mode, and n=0,1,2,.. Read More

A simple black-hole-ring system is proposed as a toy model for the two-body problem in general relativity. This toy-model yields the fractional shift $\Delta\Omega_{\text{isco}}/\Omega_{\text{isco}}={{29}\over{81\sqrt{2}}}\eta$ in the Schwarzschild ISCO (innermost stable circular orbit) frequency, where $\eta\equiv m/M_{\text{ir}}\ll 1$ is the dimensionless ratio between the mass of the particle and the irreducible mass of the black hole. Our model suggests that the second-order spin-orbit interaction between the black hole and the orbiting particle (the dragging of inertial frames) is the main element determining the observed value of the ISCO shift. Read More

A charged scalar field impinging upon a charged Reissner-Nordstrom black hole
can be amplified as it scatters off the hole, a phenomenon known as
superradiant scattering. This scattering process in the superradiant regime
w

The phenomenon of wave tails has attracted much attention over the years from both physicists and mathematicians. However, our understanding of this fascinating phenomenon is not complete yet. In particular, most former studies of the tail phenomenon have focused on scattering potentials which approach zero asymptotically ($x\to\infty$) faster than $x^{-2}$. Read More

The fundamental role played by black holes in many areas of physics makes it highly important to explore the nature of their stability. The stability of charged Reissner-Nordstr\"om black holes to {\it neutral} (gravitational and electromagnetic) perturbations was established almost four decades ago. However, the stability of these charged black holes under {\it charged} perturbations has remained an open question due to the complexity introduced by the well-known phenomena of superradiant scattering: A charged scalar field impinging on a charged Reissner-Nordstr\"om black hole can be {\it amplified} as it scatters off the hole. Read More

**Authors:**Shahar Hod

Prolate spin-weighted spheroidal harmonics play a key role in black-hole perturbation theory. In particular, the highly damped quasinormal resonances of rotating Kerr black holes are closely related to the asymptotic eigenvalues of these important functions. We here present a novel and compact derivation of the asymptotic eigenvalues of the prolate spin-weighted spheroidal harmonics. Read More

Spin-weighted spheroidal harmonics play a central role in the mathematical description of diverse physical phenomena, including black-hole perturbation theory and wave scattering. We present a novel and compact derivation of the asymptotic eigenvalues of these important functions. Our analysis is based on a simple trick which transforms the corresponding spin-weighted spheroidal angular equation into a Schr\"odinger-like wave equation which is amenable to a standard WKB analysis. Read More