We study the stability of quasinormal modes (QNM) in asymptotically flat black hole spacetimes by means of a pseudospectrum analysis. The construction of the Schwarzschild QNM pseudospectrum reveals the following: (i) the stability of the slowest-decaying QNM under perturbations respecting the asymptotic structure, reassessing the instability of the fundamental QNM discussed by Nollert [H. P. Nollert, About the Significance of Quasinormal Modes of Black Holes, Phys. Rev. D 53, 4397 (1996)] as an "infrared" effect; (ii) the instability of all overtones under small-scale ("ultraviolet") perturbations of sufficiently high frequency, which migrate towards universal QNM branches along pseudospectra boundaries, shedding light on Nollert's pioneer work and Nollert and Price's analysis [H. P. Nollert and R. H. Price, Quantifying Excitations of Quasinormal Mode Systems, J. Math. Phys. (N.Y.) 40, 980 (1999)]. Methodologically, a compactified hyperboloidal approach to QNMs is adopted to cast QNMs in terms of the spectral problem of a non-self-adjoint operator. In this setting, spectral (in)stability is naturally addressed through the pseudospectrum notion that we construct numerically via Chebyshev spectral methods and foster in gravitational physics. After illustrating the approach with the Pöschl-Teller potential, we address the Schwarzschild black hole case, where QNM (in)stabilities are physically relevant in the context of black hole spectroscopy in gravitational-wave physics and, conceivably, as probes into fundamental high-frequency spacetime fluctuations at the Planck scale.
Recent investigations of the pseudospectrum in black hole spacetimes have shown that quasinormal mode frequencies suffer from spectral instabilities. This phenomenon may severely affect gravitational-wave spectroscopy and limit precision tests of general relativity. We extend the pseudospectrum analysis to horizonless exotic compact objects which possess a reflective surface arbitrarily close to the Schwarzschild radius, and find that their quasinormal modes also suffer from an overall spectral instability. Even though all the modes themselves decay monotonically, the pseudospectrum contours o…
Black hole spectroscopy is a powerful tool to probe the Kerr nature of astrophysical compact objects and their environment. The observation of multiple ringdown modes in gravitational waveforms could soon lead to high-precision gravitational spectroscopy, so it is critical to understand if the quasinormal mode spectrum is stable against perturbations. It was recently shown that the pseudospectrum can shed light on the spectral stability of black hole quasinormal modes. We study the pseudospectrum of Reissner-Nordström spacetimes and we find a spectral instability of scalar and gravitoelectric …
The merger phase of binary black hole coalescences is a transient between an initial oscillating regime (inspiral) and a late exponentially damped phase (ringdown). In spite of the non-linear character of Einstein equations, the merger dynamics presents a surprisingly simple behaviour consistent with effective linearity. On the other hand, energy loss through the event horizon and by scattering to infinity renders the system non-conservative. Hence, the infinitesimal generator of the (effective) linear dynamics is a non-selfadjoint operator. Qualitative features of transients in linear dynamic…
Pseudospectrum and (in)stability of black hole total transmission modes
University of Science and Technology of China（中国科学技术大学）；Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院）；Peng Huanwu Center for Fundamental Theory（彭桓武理论研究中心）
Recent work applying the notion of pseudospectrum to gravitational physics showed that the quasinormal mode spectrum of black holes is unstable, with the possible exception of the longest-lived (fundamental) mode. The fundamental mode dominates the expected signal in gravitational wave astronomy, and there is no reason why it should have privileged status. We compute the quasinormal mode spectrum of two model problems where the Schwarzschild potential is perturbed by a small "bump" consisting of either a Pöschl-Teller potential or a Gaussian, and we show that the fundamental mode is destabiliz…
Recent studies based on the notion of black hole pseudospectrum indicated substantial instability of the fundamental and high-overtone quasinormal modes. Besides its theoretical novelty, the details about the migration of the quasinormal mode spectrum due to specific perturbations may furnish valuable information on the properties of associated gravitational waves in a more realistic context. This work generalizes the matrix method for black hole quasinormal modes to cope with a specific class of perturbations to the metric featured by discontinuity, which is known to be intimately connected w…
We use the Fock-Ivanenko formalism to obtain the Dirac equation which describes the interaction of a massless 1/2-spin neutral fermion with a gravitational field around a Schwarzschild black hole (BH). We obtain approximated analytical solutions for the eigenvalues of the energy (quasi-normal frequencies) and their corresponding eigenstates (quasi-normal states). The interesting result is that all the excited states [and their supersymmetric (SUSY) partners] have a purely imaginary frequency, which can be expressed in terms of the Hawking temperature. Furthermore, as one expects for SUSY Hamil…
We determine the black hole quasi-normal mode spectrum for tensor perturbations in topologically massive AdS-gravity. In the special case of chiral gravity quasi-normal modes are absent despite of the presence of a horizon. In the process we uncover a simple algebraic structure in the quasi normal modes spectrum: the tower of QNM's consists of descendents of a "chiral highest weight'' QNM which in turn satisfies a first order equation.
We generalize the one-to one correspondence between quasi normal modes in 3- dimensional anti deSitter black holes and the poles of the retarded correlators in the boundary conformal field theory to include logarithmic operators in the latter. This analysis is carried out explicitly for the logarithmic mode in chiral topologically massive gravity.
Following the monodromy technique performed by Motl and Neitzke, we consider the analytic determination of the highly damped (asymptotic) quasi-normal modes of small Schwarzschild-de Sitter (SdS) black holes. We comment the result as compared to the recent numerical data of Konoplya and Zhidenko.
A novel method, based on superpotentials is proposed for obtaining the quasi-normal modes of anti-de Sitter black holes. This is inspired by the case of the three-dimensional BTZ black hole, where the quasi-normal modes can be obtained exactly and are proportional to the surface gravity. Using this approach, the quasi-normal modes of the five dimensional Schwarzschild anti-deSitter black hole are computed numerically. The modes again seem to be proportional to the surface gravity for very small and very large black holes. They reflect the well-known instability of small black holes in anti-deS…
We discuss a method of calculating analytically the asymptotic form of quasi-normal frequencies for large AdS black holes in five dimensions. In this case, the wave equation reduces to a Heun equation. We show that the Heun equation may be approximated by a Hypergeometric equation at large frequencies. Thus we obtain the asymptotic form of quasi-normal frequencies in agreement with numerical results. We also present a simple monodromy argument that leads to the same results. We include a comparison with the three-dimensional case in which exact expressions are derived.
We discuss the quasi-normal oscillations of black holes which are sourced by a nonlinear electrodynamic field. While previous studies have focused on the computation of quasi-normal frequencies for the wave or higher spin equation on a fixed background geometry described by such black holes, here we compute for the first time the quasi-normal frequencies for the coupled electromagnetic-gravitational linear perturbations. To this purpose, we consider a parametrized family of Lagrangians for the electromagnetic field which contains the Maxwell Lagrangian as a special case. In the Maxwell case, t…
We investigate a method to evaluate quasi-normal modes of D3-brane black holes by wave interpretation of fields on D3-brane based on the Feynman's space-time approach. We perturbatively solve the wave equation which describes propagation of a dilaton wave in a bulk space and its interaction with the D3-brane. The obtained condition for the quasi-normal modes are qualitatively equivalent to that evaluated in the usual scattering of the dilaton in the black 3-brane spacetime in the corresponding supergravity description.
We study the perturbative behaviour of topological black holes with scalar hair. We calculate both analytically and numerically the quasi-normal modes of the electromagnetic perturbations. In the case of small black holes we find evidence of a second-order phase transition of a topological black hole to a hairy configuration.
We calculate the quasi-normal modes of black holes localized on a 3-brane of finite tension embedded in six-dimensional flat spacetime. Using the WKB approximation, we obtain the frequencies of various types of perturbation both analytically and numerically for arbitrary brane tension. We thus extend earlier analytical results which were derived for a brane of small tension.
In this paper we study the stability and quasi-normal modes of scalar perturbations of black holes. The static charged black hole considered here is a solution to Born-Infeld electrodynamics coupled to gravity. We conclude that the black hole is stable. We also compare the stability of it with its linear counter-part Reissner-Nordstrom black hole. The quasi-normal modes are computed using the WKB method. The behavior of these modes with the non-linear parameter, temperature, mass of the scalar field and the spherical index are analyzed in detail.
We provide an algebraic way to calculate the quasi-normal modes of a black hole, which possesses a hidden conformal symmetry. We construct an infinite tower of quasi-normal modes from the highest-weight mode, in a simple and elegant way. For the scalar, the hidden conformal symmetry manifest itself in the fact that the scalar Laplacian could be rewritten in terms of the $SL(2,R)$ quadratic Casimir. For the vector and the tensor, the hidden conformal symmetry acts on them through Lie derivatives. We show that for three-dimensional black holes, with appropriate combination of the components the …
In this paper we use the conformal properties of the spinor field to show how we can obtain the fermion quasi-normal modes for a higher dimensional Schwarzschild black hole. These modes are of interest in so called split fermion models, where quarks and leptons are required to exist on different branes in order to keep the proton stable. As has been previously shown, for brane localized fields, the larger the number of dimensions the faster the black hole damping rate. Moreover, we also present the analytic forms of the quasi-normal frequencies in both the large angular momentum and the large …
We calculate the quasinormal modes and associated frequencies of the Banados, Zanelli and Teitelboim (BTZ) non-rotating black hole. This black hole lives in 2+1-dimensions in an asymptotically anti-de Sitter spacetime. We obtain exact results for the wavefunction and quasi normal frequencies of scalar, electromagnetic and Weyl (neutrino) perturbations.
Highly-damped quasi-normal frequencies are very often of the form omega_n = (offset) + i n (gap). We investigate the genericity of this phenomenon by considering a model potential that is piecewise Eckart (piecewise Poeschl-Teller), and developing an analytic "quantization condition" for the highly-damped quasi-normal frequencies. We find that this omega_n = (offset) + i n (gap) behaviour is generic but not universal, with the controlling feature being whether or not the ratio of the rates of exponential falloff in the two asymptotic directions is a rational number. These observations are of d…
In this work, we calculate the quasi-normal frequencies from a bumblebee traversable wormhole. The bumblebee wormhole model is based on the bumblebee gravity, which exhibits a spontaneous Lorentz symmetry breaking. Supporting by the Lorentz violation parameter $λ$, this model allows the fulfillment of the flare-out and energy conditions, granted non-exotic matter to the wormhole. We analyze the parameters of bumblebee wormhole in order to obtain a Reege-Wheeler's equation with a bell-shaped potential. We obtain the quasi-normal modes (QNMs) via the WKB approximation method for both scalar and …
We discuss a systematic method of analytically calculating the asymptotic form of quasi-normal frequencies of a four-dimensional Schwarzschild black hole by expanding around the zeroth-order approximation to the wave equation proposed by Motl and Neitzke. We obtain an explicit expression for the first-order correction and arbitrary spin. Our results are in agreement with the results from WKB and numerical analyses in the case of gravitational waves.
We obtain exact expressions for the quasi-normal modes of various spin for the BTZ black hole. These modes determine the relaxation time of black hole perturbations. Exact agreement is found between the quasi-normal frequencies and the location of the poles of the retarded correlation function of the corresponding perturbations in the dual conformal field theory. This then provides a new quantitative test of the AdS/CFT correspondence.
Born-Infeld electrodynamics has attracted considerable interest due to its relation to strings and D-branes. In this paper the gravitational perturbations of electrically charged black holes in Einstein-Born-Infeld gravity are studied. Effective potentials for axial perturbations are derived and discussed. The quasi normal modes for the gravitational perturbations are computed using the WKB method. The modes are compared with those of the Reissner-Nordstrom black hole. The relation of the quasi normal modes with the non-linear parameter and the spherical index are also investigated. Comments o…
We study the quasi-normal modes (QNM) of scalar, electromagnetic and gravitational perturbations of black holes in general relativity whose horizons have toroidal, cylindrical or planar topology in an asymptotically anti-de Sitter (AdS) spacetime. The associated quasinormal frequencies describe the decay in time of the corresponding test field in the vicinities of the black hole. In terms of the AdS/CFT conjecture, the inverse of the frequency is a measure of the dynamical timescale of approach to thermal equilibrium of the corresponding conformal field theory.
We reexamined the argument that the quasinormal modes could be a probe of the phase transition of a topological black hole to a hairy configuration by investigating general scalar perturbations. We found further evidence in the quasinormal modes for this phase transition. For the general black hole configurations, we observed that although the quasinormal modes can present us different phases of different configurations, there is no dramatic change in the slope of quasinormal frequencies at the critical point of the phase transition. More detailed studies of quasinormal modes are needed to rev…
It is well-known that there is a geometric correspondence between high-frequency quasi-normal modes (QNMs) and null geodesics (spherical photon orbits). In this paper, we generalize such correspondence to charged scalar field in Kerr-Newman space-time. In our case, the particle and black hole are all charged, so one should consider non-geodesic orbits. Using the WKB approximation, we find that the real part of quasi-normal frequency corresponds to the orbits frequency, the imaginary part of the frequency corresponds to the Lyapunov exponent of these orbits and the eigenvalue of angular equatio…
We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr-Newman-de Sitter black holes. The resonances split in a way similar to the Zeeman effect. The method is based on the extension to Dirac operators of techniques applied by Dyatlov 2011, 2012 to the (uncharged) Kerr-de Sitter black holes. We show that the mass of the Dirac field does not have effect on the two leading terms in the expansions of resonances.
With the help of two new intrinsic tensor fields associated with the $SL(2,R)$ quadratic Casimir of Killing fields, we uncover the $SL(2,R)$ symmetry satisfied by the solutions to the equations of motion for various fields in the BTZ black hole in a uniform way by performing tensor and spinor analysis without resorting to any specific coordinate system. Then with the standard algebraic method developed recently, we determine the quasi-normal modes for various fields in the BTZ black hole. As a result, the quasi-normal modes are given by the infinite tower of descendants of the chiral highest w…
Here we present a fast review of some of developments and new results concerning applications of gravity in the context of the AdS/CFT correspondence, in brane world perturbations as well as in holographic superconductors. We also discuss the structure of the phase transitions in a more general set up defined by the quasi-normal oscillations in the bulk, which signalize a very complex structure of the phase transition at the border.
We calculate QNMs of the scalar hairy black hole in the AdS background using Horowitz-Hubeny method for the potential that is not known in analytical form. For some black hole parameters we found pure imaginary frequencies. Increasing of the scalar field mass does not cause the imaginary part to vanish, it reaches some minimum and then increases, thus in the case under consideration the infinitely long living modes (quasi-resonances) do not appear.
The (in)stability on total transmission modes with small bumps
Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；University of Science and Technology of China（中国科学技术大学）
Normal modes, quasi-normal modes and super-radiant modes are studied to clarify the total dynamics for complex scalar fields in Kerr anti-de Sitter black hole spacetime. Orthonormal relations and quasi-orthonormal relations are obtained for normal modes and quasi-normal modes. Mode expansions are done and the conserved quantities are studied. Any modes are shown to be separated into two groups, physical modes and unphysical modes, by the zero mode line. Zero modes themselves do not exist as normalizable modes with the correct boundary condition. The allowed physical modes exclude the super-rad…
We study the quasi-normal modes of asymptotically anti-de Sitter black holes in a class of shift-symmetric Horndeski theories where a gravitational scalar is derivatively coupled to the Einstein tensor. The space-time differs from exact Schwarzschild-anti-de Sitter, resulting in a different effective potential for the quasi-normal modes and a different spectrum. We numerically compute this spectrum for a massless test scalar coupled both minimally to the metric, and non-minimally to the gravitational scalar. We find interesting differences from the Schwarzschild-anti-de Sitter black hole found…
In this work we derive the asymptotic quasi-normal modes of a BTZ black hole using a quantum field theoretic Lagrangian. The BTZ black hole is a very popular system in the context of $2+1$-dimensional quantum gravity. However to our knowledge the quasi-normal modes of the BTZ black hole have been studied only in the classical domain. Here we show a way to quantize the quasi-normal modes of the BTZ black hole by mapping it to the Bateman-Feschbach-Tikochinsky oscillator and the Caldirola-Kanai oscillator. We have also discussed a couple of other black hole potentials to which this method can be…
It is known that the imaginary parts of the frequencies of the quasi normal modes of the Schwarzschild black hole are equally spaced, with the level spacing dependent only on the surface gravity. We generalize this result to a wider class of spacetimes and provide a simple derivation of the imaginary parts of the frequencies. The analysis shows that the result is closely linked to the thermal nature of horizons and arises from the exponential redshift of the wave modes close to the horizon.
A perturbed black hole has characteristic frequencies (quasi-normal modes). Here I apply a quantum measurement analysis of the quasi-normal mode frequency in the limit of high damping. It turns out that a measurement of this mode necessarily adds noise to it. For a Schwarzschild black hole, this corresponds exactly to the Hawking temperature. The situation for other black holes is briefly discussed.
