Area spectrum of rotating black holes via the new interpretation of quasinormal modes
Elias C. Vagenas
备注：
v1: 4 pages, REVTeX, no figures; v2: clarifications and comments added, comments on last result modified, Abstract slightly changed; v3: comments and references added;
Motivated by the recent work on a new physical interpretation of quasinormal modes by Maggiore, we utilize this new proposal to the interesting case of Kerr black hole. In particular, by modifying Hod's idea, the resulting black hole horizon area is quantized and the resulting area quantum is in full agreement with Bekenstein's result. Furthermore, in an attempt to show that the area spectrum is equally spaced, we follow Kunstatter's method. We propose a new interpretation as a result of Maggiore's idea, for the frequency that appears in the adiabatic invariant of a black hole. The derived area spectrum is similar to that of the quantum-corrected Kerr black hole but it is not equally spaced.
A Bohr-Sommerfeld equation is derived for the highly-damped quasinormal mode frequencies omega(n>>1) of rotating black holes. It may be written as 2\int_C(p_r+ip_0)dr=(n+1/2)h, where p_r is the canonical momentum conjugate to the radial coordinate r along null geodesics of energy hbar*omega and angular momentum hbar*m, p_0=O(omega^0), and the contour C connects two complex turning points of p_r. The solutions are omega(n) = - m*omega_0 - i(phi + n*delta), where {omega_0,delta}>0 are functions of the black-hole parameters alone. Some physical implications are discussed.
Motivated by novel results in the theory of black-hole quantization, we study {\it analytically} the quasinormal modes (QNM) of ({\it rotating}) Kerr black holes. The black-hole oscillation frequencies tend to the asymptotic value $ω_n=mΩ+i2πT_{BH}n$ in the $n \to \infty$ limit. This simple formula is in agreement with Bohr's correspondence principle. Possible implications of this result to the area spectrum of quantum black holes are discussed.
The study of the quasinormal modes (QNMs) of the 2+1 dimensional rotating draining bathtub acoustic black hole, the closest analogue found so far to the Kerr black hole, is performed. Both the real and imaginary parts of the quasinormal (QN) frequencies as a function of the rotation parameter B are found through a full non-linear numerical analysis. Since there is no change in sign in the imaginary part of the frequency as B is increased we conclude that the 2+1 dimensional rotating draining bathtub acoustic black hole is stable against small perturbations.
Following Bekenstein's suggestion that the horizon area of a black hole should be quantized, the discrete spectrum of the horizon area has been investigated in various ways. By considering the quasinormal mode of a black hole, we obtain the transition frequency of the black hole, analogous to the case of a hydrogen atom, in the semiclassical limit. According to Bohr's correspondence principle, this transition frequency at large quantum number is equal to classical oscillation frequency. For the corresponding classical system of periodic motion with this oscillation frequency, an action variabl…
In this paper we study the shadow of rotating Kaluza-Klein (KK) black holes and the connection between the shadow radius and the real part of quasi-normal modes (QNMs) in the eiokonal limit. In addition we have explored the quasi-periodic oscillations (QPOs) in the rotating KK black hole.
We provide semi-analytic expressions for quasinormal mode frequencies of slowly rotating Kerr black holes to the quadratic order in the rotation parameter. We apply the parametrized black hole quasinormal mode ringdown formalism to the Chandrasekhar-Detweiler equation and the Sasaki-Nakamura equation instead of the Teukolsky equation, and compare our result with the previous numerical calculations.
We study analytically the fundamental resonances of near-extremal, slowly rotating Kerr-Newman black holes. We find a simple analytic expression for these black-hole quasinormal frequencies in terms of the black-hole physical parameters: omega=m Omega-2i pi T(l+1+n), where T and Omega are the temperature and angular velocity of the black hole. The mode parameters l and m are the spheroidal harmonic index and the azimuthal harmonic index of a co-rotating mode, respectively. This analytical formula is valid in the regime Im omega << Re omega <<1/M, where M is the black-hole mass.
We compute the quasinormal modes (QNMs) of a massive scalar field in the background of a rotating three-dimensional Hořava AdS black hole, and we analyze the effect of the breaking of the Lorentz invariance on the QNMs. Imposing on the horizon that there are only ingoing waves and at infinity the Dirichlet boundary conditions and the Neumann boundary condition, we calculate the oscillatory and the decay modes of the QNMs. We find that the propagation of the scalar field is stable in this background and employing the holographic principle we find the different times of the perturbed system to r…
We study the holographic duals of bosonic and fermionic fields and compute their Quasinormal Frequencies (QNF) by exploiting that they are massive charged fields in $\mathrm{AdS_2 \times \mathcal{K}}$. In particular, we assess the stability of the scalar and fermionic operators in Myers-Perry's black holes with equal angular momenta. We compute the central charge and the Hawking-entropy and introduce extremality parameters to determine the properties of the dual CFT. Then, we briefly discuss how detection of Spin-2 fields can pave the way for more deep understanding of the holographic principl…
We demonstrate the separability of the massive vector (Proca) field equation in general Kerr-NUT-(A)dS black hole spacetimes in any number of dimensions, filling a long-standing gap in the literature. The obtained separated equations are studied in more detail for the four-dimensional Kerr geometry and the corresponding quasinormal modes are calculated. Two of the three independent polarizations of the Proca field are shown to emerge from the separation ansatz and the results are found in a perfect agreement with those of the recent numerical study where the full coupled partial differential e…
A conjectured connection to quantum gravity has led to a renewed interest in highly damped black hole quasinormal modes (QNMs). In this paper we present simple derivations (based on the WKB approximation) of conditions that determine the asymptotic QNMs for both Schwarzschild and Reissner-Nordstrom black holes. This confirms recent results obtained by Motl and Neitzke, but our analysis fills several gaps left by their discussion. We study the Reissner-Nordstrom results in some detail, and show that, in contrast to the asymptotic QNMs of a Schwarzschild black hole, the Reissner-Nordstrom QNMs a…
In this work we discuss charged rotating black holes in $AdS_5 \times S^5$ that degenerate to extremal black holes with zero entropy. These black holes have scaling properties between charge and angular momentum similar to those of Fermi surface operators in a subsector of $\mathcal{N}=4$ SYM. We add a massless uncharged scalar to the five dimensional supergravity theory, such that it still forms a consistent truncation of the type IIB ten dimensional supergravity and analyze its quasinormal modes. Separating the equation of motion to a radial and angular part, we proceed to solve the radial e…
We study the main properties of general linear perturbations of rotating black holes in asymptotically flat higher-dimensional spacetimes. In particular, we determine the quasinormal mode (QNM) spectrum of singly spinning and equal angular momenta Myers-Perry black holes (MP BHs). Emphasis is also given to the timescale of the ultraspinning and bar-mode instabilities in these two families of MP BHs. For the bar-mode instabilities in the singly spinning MP BH, we find excellent agreement with our linear analysis and the non-linear time evolution of Shibata and Yoshino for d=6,7 spacetime dimens…
The rotating linear dilatonic black hole is an asymptotically non-flat solution to Einstein-Maxwell-Dilaton-Axion gravity theory due to the existence of non-trivial matter fields. We have analytically studied the wave equation of scalar field in this background and shown that the radial wave equation can be solved in terms of hypergeometric function. By determining the ingoing and the outgoing fluxes at the asymptotic infinity, we have found the analytical expressions for reflection coefficient and greybody factor for certain scalar modes. In the high frequency regime, we obtain the Hawking te…
The pure quadratic term of New Massive Gravity in three dimensions admits asymptotically locally flat, rotating black holes. These black holes are characterized by their mass and angular momentum, as well as by a hair of gravitational origin. As in the Myers-Perry solution in dimensions greater than five, there is no upper bound on the angular momentum. We show that, remarkably, the equation for a massless scalar field on this background can be solved in an analytic manner and that the quasinormal frequencies can be found in a closed form. The spectrum is obtained requiring ingoing boundary co…
In this work, we explore the connection between the critical curves ("shadows") and the quasinormal mode frequencies (in the eikonal limit) of Kerr black holes. This mapping has been previously established for non-rotating black holes. We show that, the shadow seen by an distant observer at a given inclination angle, can be mapped to a family of quasinormal modes with $m/(\ell+1/2)$ bounded within certain range, where $m$ is is azimuthal node number and $\ell$ is the angular node number. We discuss the possibility of testing such relation with space-borne gravitational wave detectors and the n…
In this work we consider the evolution of a massive scalar field in cylindrically symmetric space-times. Quasinormal modes have been calculated for static and rotating cosmic cylinders. We found unstable modes in some cases. Rotating as well as static cosmic strings, i.e., without regular interior solutions, do not display quasinormal oscillation modes. We conclude that rotating cosmic cylinder space-times that present closed time-like curves are unstable against scalar perturbations.
Numerical studies of the coupled Einstein-Klein-Gordon system have recently revealed that confined scalar fields generically collapse to form caged black holes. In the light of this finding, we analytically study the characteristic resonance spectra of the confined scalar fields in rotating linear dilaton black hole geometry. Confining mirrors (cage) are assumed to be placed in the near-horizon region of a caged rotating linear dilaton black hole $\frac{r_{m}-r_{2}}{r_{2}}\ll 1$ ($r_{m}$ is the radius of the cage and $r_{2}$ represents the event horizon). The radial part of the Klein-Gordon eq…
Scalar quasinormal modes of rotating black holes in parity-violating gravity
Tokyo Metropolitan University（东京都立大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；College of Humanities and Sciences, Nihon University（日本大学文理学部）；Chulalongkorn University（朱拉隆功大学）；The University of Tokyo（东京大学）
Respecting the angular momentum conservation of torque-free systems, it is natural to consider rotating solutions of massive objects. Besides that, motivated by the realistic astrophysical black holes that rotate, we use the Newman-Janis formalism to construct a regular rotating black hole. We start with a nonlinearly charged regular static black hole in the framework of the standard general relativity and then obtain the associated rotating solution through such a formalism. We investigate the geometrical properties of the metric by studying the boundary of ergosphere. We also analyze thermod…
Many properties of black holes can be studied using acoustic analogues in the laboratory through the propagation of sound waves. We investigate in detail sound wave propagation in a rotating acoustic (2+1)-dimensional black hole, which corresponds to the ``draining bathtub'' fluid flow. We compute the quasinormal mode frequencies of this system and discuss late-time power-law tails. Due to the presence of an ergoregion, waves in a rotating acoustic black hole can be superradiantly amplified. We compute superradiant reflection coefficients and instability timescales for the acoustic black hole …
Black holes provide a natural laboratory to study particle physics and astrophysics. When black holes are surrounded by matter fields, there will be plenty of phenomena which can have observational consequences, from which we can learn about the matter fields as well as black hole spacetime. In this work, we investigate the massive scalar field in the vicinity of a newly proposed rotating regular black hole inspired by quantum gravity. We will especially investigate how this non-singular spactime will affect the superradiance instability and quasinormal modes of the scalar filed. We derive the…
It has been recently found that for the near extremal Kerr black holes appearing of Zero Damped Modes (accompanied by qusinormal mode branching) signifies about inapplicability of the regime of small perturbations and the onset of turbulence. Here we show that this phenomena is not limited by Kerr or Kerr-Newman solutions only, but also takes place for rotating dilatonic black holes for which we have found Zero Damped Modes both numerically and analytically. We have also shown that, contrary to recent claims, there is no instability of a charged massive scalar field in the background of the ro…
Quasinormal modes of rotating black holes beyond general relativity in the WKB approximation
Observatoire de Paris, Universite PSL（巴黎天文台，巴黎文理研究大学）；CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL（里斯本大学，技术研究所物理系与理论研究中心）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理学研究所（爱因斯坦研究所））；Johns Hopkins University（约翰斯·霍普金斯大学）
We study the quasinormal modes of massive Dirac spinors in the background of rotating black holes. In particular, we consider the Kerr geometry as well as the five dimensional Myers-Perry spacetime with equal angular momenta. We decouple the equations using the standard methods from the literature. In the five dimensional Myers-Perry black hole the angular equation is solved analytically. Using the continued fraction method, we calculate the spectrum of quasinormal modes for the ground modes and first excited modes. We analyze, in a systematic way, its dependence on the different parameters of…
Motivated by the recent work about a new physical interpretation of quasinormal modes by Maggiore, we investigate the quantization of near-extremal Schwarzschild-de Sitter black holes in the four dimensional spacetime. Following Kunstatter's method, we derive the area and entropy spectrum of near-extremal Schwarzschild-de Sitter black holes which differs from Setare's result. Furthermore, we find that the derived a universal area spectrum is $2πn$ which is equally spaced.
Quasinormal modes of rotating black holes with ultralight scalar fields
Beijing Institute of Mathematical Sciences and Applications（北京数学与交叉科学研究院）；St John’s College, Cambridge（剑桥大学圣约翰学院）；DAMTP, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心应用数学与理论物理系）
Astrophysical black holes are generally surrounded by accretion disks, galactic matter and the omnipresent cosmic microwave background radiation, thus allowing for the concurrent propagation of both gravitational and sound waves. Recently, acoustic black holes were embedded in Schwarzschild spacetime allowing for the coexistence of event and acoustic horizons. Here, we obtain a class of perturbative solutions to the field equations of the relativistic Gross-Pitaevskii and Yang-Mills theories, which describe sound waves propagating on a curved slowly-rotating acoustic black hole, akin to Lense-…
This note addresses quasinormal mode frequencies of four-dimensional asymptotically de Sitter rotating black holes. The main motivation is that Mathematica 12.1 has implemented a new family of special functions: Heun functions. Using the fact that Teukolsky's equations for Kerr-de Sitter black holes are mapped to Heun's equations, we are able to compute their quasinormal mode frequencies by the Heun function. In this approach, Mathematica normally evaluates these frequencies to arbitrary numerical precision in a few seconds. We further discuss an application to asymptotically flat rotating bla…
We compute numerically the quasinormal modes of Kerr-Newman black holes in the scalar case, for which the perturbation equations are separable. Then we study different approximations to decouple electromagnetic and gravitational perturbations of the Kerr-Newman metric, computing the corresponding quasinormal modes. Our results suggest that the Teukolsky-like equation derived by Dudley and Finley gives a good approximation to the dynamics of a rotating charged black hole for Q<M/2. Though insufficient to deal with Kerr-Newman based models of elementary particles, the Dudley-Finley equation shou…
A certain identification of points in a planar Schwarzschild-anti de Sitter (AdS) black hole generates a four-dimensional static black string. In turn, a rotating black string can be obtained from a static one by means of an `illegitimate coordinate transformation', a local boost in the compact direction. On the basis of the gauge/gravity duality, these black strings are dual to rotating thermal states of a strongly interacting conformal field theory (CFT) that lives on a cylinder. In this work, we obtain the complete quasinormal mode (QNM) spectrum of the gravitational perturbations of rotati…
Based on the geometric-optics correspondence between the parameters of a quasinormal mode and the conserved quantities along geodesics, we propose an equation to calculate the typical shadow radius for asymptotically flat and rotating black holes when viewed from the equatorial plane given by \begin{equation}\notag \bar{R}_s=\frac{\sqrt{2}}{2}\left(\sqrt{\frac{ r_0^{+}}{f'(r)|_{r_0^{+}}}}+\sqrt{\frac{ r_0^{-}}{f'(r)|_{r_0^{-}}}}\right), \end{equation} with $r_0^{\pm}$ being the radius of circular null geodesics for the corresponding mode. Furthermore we have explicitly related the shadow radiu…
The analogy between sound wave propagation and light waves led to the study of acoustic holes, the acoustic analogues of black holes. Many black hole features have their counterparts in acoustic holes. The Kerr metric, the rotating metric for black holes in general relativity, has as analogue the draining bathtub metric, a metric for a rotating acoustic hole. Here we report on the progress that has been made in the understanding of features, such as quasinormal modes and tails, superresonance, and instabilities when the hole is surrounded by a reflected mirror, in the draining bathtub metric. …
It has recently been shown by Yang. et. al. [Phys. Rev. D {\bf 87}, 041502(R) (2013)] that rotating Kerr black holes are characterized by two distinct sets of quasinormal resonances. These two families of quasinormal resonances display qualitatively different asymptotic behaviors in the extremal ($a/M\to 1$) black-hole limit: The zero-damping modes (ZDMs) are characterized by relaxation times which tend to infinity in the extremal black-hole limit ($\Imω\to 0$ as $a/M\to 1$), whereas the damped modes (DMs) are characterized by non-zero damping rates ($\Imω\to$ finite-values as $a/M\to 1$). In …
Black-hole quasinormal modes (QNM) have been the subject of much recent attention, with the hope that these oscillation frequencies may shed some light on the elusive theory of quantum gravity. We compare numerical results for the QNM spectrum of the (rotating) Kerr black hole with an {\it exact} formula Re$ω\to T_{BH}\ln 3+Ωm$, which is based on Bohr's correspondence principle. We find a close agreement between the two. Possible implications of this result to the area spectrum of quantum black holes are discussed.
