The geometric optics approximation provides an interpretation for eikonal correspondence that, in black-hole-containing spacetimes, connects high-frequency black hole quasinormal modes with closed photon orbits around said black hole. This correspondence has been identified explicitly for Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman black holes, the violation of which can be a potential hint toward physics beyond General Relativity. Notably, the aforementioned black hole spacetimes have sufficient symmetries such that both the geodesic equations and the master wave equations are separable. The identification of the correspondence seems to largely rely on these symmetries. One naturally asks how the eikonal correspondence would appear if the spacetime were less symmetric. For a pioneering work in this direction, we consider in this paper a deformed Schwarzschild spacetime retaining only axisymmetry and stationarity. We show that up to the first order of spacetime deformations the eikonal correspondence manifests through the definition of the \textit{averaged} radius of trapped photon orbits along their one period. This averaged radius overlaps the potential peak in the master wave equation, which can be defined up to the first order of spacetime deformations, allowing the explicit identification of the eikonal correspondence.
In Phys. Rev. D 79, 064016 (2009) it was claimed that quasinormal modes which any stationary, spherically symmetric and asymptotically flat black hole emits in the eikonal regime are determined by the parameters of the circular null geodesic: the real and imaginary parts of the quasinormal mode are multiples of the frequency and instability timescale of the circular null geodesics respectively. We shall consider asymptotically flat black hole in the Einstein-Lovelock theory, find analytical expressions for gravitational quasinormal modes in the eikonal regime and analyze the null geodesics. Co…
Much of our physical intuition about black hole quasinormal modes in general relativity comes from the eikonal/geometric optics approximation. According to the well-established eikonal model, the fundamental quasinormal mode represents wavepackets orbiting in the vicinity of the black hole's geodesic photon ring, slowly peeling off towards the event horizon and infinity. Besides its strength as a "visualisation" tool, the eikonal approximation also provides a simple quantitative method for calculating the mode frequency, in close agreement with rigorous numerical results. In this paper we move…
Quasinormal modes in the high frequency (eikonal) regime can be obtained analytically as the Mashhoon-Will-Schiutz WKB formula is exact in this case. This regime is interesting because of the correspondence between eikonal quasinormal modes and null geodesics, as well as due to existence of potential eikonal instabilities in some theories of gravity. At the same time in a number of studies devoted to quasinormal modes of spherically symmetric black holes this opportunity was omitted. Here we find analytical quasinormal modes of black holes in various alternative and extended theories of gravit…
We compute the quasinormal frequencies of $d$-dimensional spherically symmetric black holes with leading string $α'$ corrections in the eikonal limit for tensorial gravitational perturbations and scalar test fields. We find that, differently than in Einstein gravity, the real parts of the frequency are no longer equal for these two cases. The corresponding imaginary parts remain equal to the principal Lyapunov exponent corresponding to circular null geodesics, to first order in $α'$. We also compute the radius of the shadow cast by these black holes.
Black hole `spectroscopy', i.e. the identification of quasinormal mode frequencies via gravitational wave observations, is a powerful technique for testing the general relativistic nature of black holes. In theories of gravity beyond general relativity perturbed black holes are typically described by a set of coupled wave equations for the tensorial field and the extra scalar/vector degrees of freedom, thus leading to a theory-specific quasinormal mode spectrum. In this paper we use the eikonal/geometric optics approximation to obtain analytic formulae for the frequency and damping rate of the…
We show that the quasinormal spectrum of gravitational perturbations of Schwarzschild - de Sitter black holes contains a new branch of purely imaginary modes. These modes are not algebraically special and we showed that the sum of them form the well-known in the literature exponential asymptotic tail. When the ratio of the event horizon radius to the cosmological horizon vanishes, these quasinormal modes approach modes of empty de Sitter spacetime. Thus, the spectrum consists of the two branches: Schwarzschild branch deformed by the cosmological constant and de Sitter branch deformed by the bl…
We study the scattering of gravitational waves by a Schwarzschild black hole and its perturbed siblings to investigate influences of proposed spectral instability of quasinormal modes on the ringdown signal. Our results indicate that information of dominant ringdown signals, which are ascribed to the fundamental (i.e., least damping) quasinormal mode of unperturbed Schwarzschild black holes, is imprinted in the phase shift defined from the transmission amplitude (1/A_{in} in our notation). This approximately parallels the fact that the resonance of quantum systems is imprinted in the phase shi…
In the scope of black hole spectroscopy, several attempts have been made in the past decades in order to test black holes or gravitational theories via black hole quasinormal modes. In the eikonal approximation, the quasinormal modes are generically associated with the photon ring of the black hole. This correspondence is valid for most cases in general relativity, but may not be true in other theories of gravity. In this paper, we consider the generalized energy-momentum squared gravity in which matter fields are non-minimally coupled to geometry. We investigate the axial perturbations of the…
Classical (quasinormal) and quantum (Hawking) radiations are investigated for test fields in the background of a four dimensional, spherically symmetric and asymptotically flat black hole in the Einstein-dilaton-Gauss-Bonnet (EdGB) theory. The geometry of the EdGB black hole deviates from the Schwarzschild geometry only slightly. Therefore, here we observe that the quasinormal spectrum also deviates from its Schwarzschild limit at most moderately, allowing for a $9\%$ decrease in the damping rate and up to a $6\%$ decrease in the real oscillation frequency. However, the intensity of Hawking ra…
In a recent series of papers we have shown how the eikonal/geometrical optics approximation can be used to calculate analytically the fundamental quasinormal mode frequencies associated with coupled systems of wave equations, which arise, for instance, in the study of perturbations of black holes in gravity theories beyond General Relativity. As a continuation to this series, we here focus on the quasinormal modes of nonrotating black holes in scalar Gauss-Bonnet gravity assuming a small-coupling expansion. We show that the axial perturbations are purely tensorial and are described by a modifi…
Confronting eikonal and post-Kerr methods with numerical evolution of scalar field perturbations in spacetimes beyond Kerr
Università di Napoli “Federico II”（那不勒斯费德里克二世大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；University of Tübingen（蒂宾根大学）；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））；Ministero dell’Istruzione e del Merito (M.I.M., ex M.I.U.R.)（意大利教育与功绩部）；Scuola Superiore Meridionale（南方高等学院）
In this work, we study the relation of the eikonal quasinormal modes (EQNMs) and the unstable fundamental photon orbits (UFPOs) in the Kerr-Newman spacetime. We find that in the eikonal limit the gravitational and electromagnetic perturbations of the Kerr-Newman black hole are naturally decoupled, and a single one-dimensional Schrödinger-like equation encoding the QNM spectrum can be derived. We then show that the decoupled Teukolsky master equation and the Klein-Gordon equation for the massless scalar field in the Kerr-Newman spacetime are of the same form in the eikonal limit. As a direct co…
We analytically calculate to second order the correction to the asymptotic form of quasinormal frequencies of four dimensional Schwarzschild black holes based on the monodromy analysis proposed by Motl and Neitzke. Our results are in good agreement with those obtained from numerical calculation.
We investigate the interior structure, perturbations, and the associated quasinormal modes of a quantum black hole model recently proposed by Bodendorfer, Mele, and Münch (BMM). Within the framework of loop quantum gravity, the quantum parameters in the BMM model are introduced through polymerization, consequently replacing the Schwarzschild singularity with a spacelike transition surface. By treating the quantum geometry corrections as an `effective' matter contribution, we first prove the violation of energy conditions (in particular the null energy condition) near the transition surface and…
The deep reason why the equations describing axial and polar perturbations of Schwarzschild black holes have the same spectrum is far from trivial. In this article, we revisit the original proof and try to make it clearer. Still focusing on uncharged and non-rotating black holes, we extend the results to spacetimes including a cosmological constant, which have so far mostly been investigated numerically from this perspective.
We show that the hidden conformal symmetry of the Schwarzschild black hole is realized from the AdS$_2$ sector of the AdS$_2\times S^2$, but not from the Rindler spacetime which is the genuine near-horizon geometry of the Schwarzschild black hole. This implies that purely imaginary quasinormal frequencies obtained using the hidden conformal symmetry is not suitable for describing the largely damped modes around the Schwarzschild black hole.
Recently an anomalous decay rate of the quasinormal modes of a massive scalar field in Schwarzschild black holes backgrounds was reported in which the longest-lived modes are the ones with higher angular number, for a scalar field mass smaller than a critical value, while that beyond this value the behaviour is inverted. In this work, we extend the study to other asymptotic geometries, such as, Schwarzschild-de Sitter and Schwarzschild-AdS black holes. Mainly, we found that such behaviour and the critical mass are present in the Schwarzschild-de Sitter background. Also, we found that the value…
We consider quantum corrections for the Schwarzschild black hole metric by using the generalized uncertainty principle (GUP) to investigate quasinormal modes, shadow and their relationship in the eikonal limit. We calculate the quasinormal frequencies of the quantum-corrected Schwarzschild black hole by using the sixth-order Wentzel-Kramers-Brillouin (WKB) approximation, and also perform a numerical analysis that confirms the results obtained from this approach. We also find that the shadow radius is nonzero even at very small mass limit for finite GUP parameter.
It is well established that the response of a black hole to a generic perturbation is characterized by a spectrum of damped resonances, called quasinormal modes; and that, in the limit of large angular momentum ($l \gg 1$), the quasinormal mode frequency spectrum is related to the properties of unstable null orbits. In this paper we develop an expansion method to explore the link. We obtain new closed-form approximations for the lightly-damped part of the spectrum in the large-$l$ regime. We confirm that, at leading order in $l$, the resonance frequency is linked to the orbital frequency, and …
We obtain an analytic expression for the highly damped asymptotic quasinormal mode frequencies of the $d\geq 5$-dimensional Schwarzschild black hole modified by the Gauss-Bonnet term, which appears in string derived models of gravity. The analytic expression is obtained under the string inspired assumption that there exists a minimum length scale in the system and in the limit when the coupling in front of the Gauss-Bonnet term in the action is small. Although there are several similarities of this geometry with that of the Schwarzschild black hole, the asymptotic quasinormal mode frequencies …
We compute the quasinormal spectra for scalar, Dirac and electromagnetic perturbations of the Schwarzschild-de Sitter geometry in the framework of scale-dependent gravity, which is one of the current approaches to quantum gravity. Adopting the widely used WKB semi-classical approximation, we investigate the impact on the spectrum of the angular degree, the overtone number as well as the scale-dependent parameter for fixed black hole mass and cosmological constant. We summarize our numerical results in tables, and for better visualization, we show them graphically as well. All modes are found t…
We determine the quasinormal frequencies for all gravitational perturbations of the d-dimensional Schwarzschild black hole, in the infinite damping limit. Using the potentials for gravitational perturbations derived recently by Ishibashi and Kodama, we show that in all cases the asymptotic real part of the frequency is proportional to the Hawking temperature with a coefficient of log 3. Via the correspondence principle, this leads directly to an equally spaced entropy spectrum. We comment on the possible implications for the spacing of eigenvalues of the Virasoro generator in the associated ne…
We study Dirac quasinormal modes of Schwarzschild-Anti-de Sitter (Schwarzschild-AdS) black holes, following the generic principle for allowed boundary conditions proposed in \cite{PhysRevD.92.124006}. After deriving the equations of motion for Dirac fields on the aforementioned background, we impose vanishing energy flux boundary conditions to solve these equations. We find a set of two Robin boundary conditions are allowed. These two boundary conditions are used to calculate Dirac normal modes on empty AdS and quasinormal modes on Schwarzschild-AdS black holes. In the former case, we recover …
We reformulate the Chern-Simons modified gravity in the metric-affine formalism, by enlarging the Pontryagin density with homothetic curvature terms which restore projective invariance without spoiling topologicity. The latter is then violated by promoting the coupling of the Chern-Simons term to a (pseudo)-scalar field. We derive the perturbative solutions for torsion and nonmetricity from the background fields, and we describe the dynamics for the resulting linearized metric and the scalar fields in a Schwarzschild black hole background. Then, by adopting numerical techniques we compute the …
A First-Order Eikonal Framework for Quasinormal Modes, Shadows, Strong Lensing, and Grey-Body Factors in a Scalarized Black-Hole Metric
University of Hradec Králové（赫拉德茨克拉洛韦大学）；National University of Uzbekistan（乌兹别克斯坦国立大学）；University of Tashkent for Applied Sciences（塔什干应用科学大学）；Tashkent State Technical University（塔什干国立技术大学）；New Uzbekistan University（新乌兹别克斯坦大学）；Tashkent International University of Education（塔什干国际教育大学）；Kimyo International University in Tashkent（塔什干基莫国际大学）；Mamun University（马蒙大学）
The leading term of the asymptotic of quasinormal modes in the Schwarzschild background, omega_n = - i n/2, is obtained in two straightforward analytical ways for arbitrary spins. One of these approaches requires almost no calculations. As simply we demonstrate that for any odd integer spin, described by the Teukolsky equation, the first correction to the leading term vanishes. Then, this correction for half-integer spins is obtained in a slightly more intricate way. At last, we derive analytically the general expression for the first correction for all spins, described by the Teukolsky equati…
We find the asymptotic formula for quasinormal mode frequencies omega_n of the Dirac equation in a Schwarzschild-AdS_D background in space-time dimension D > 3, in the large black-hole limit appropriate to many applications of the AdS/CFT correspondence. By asymptotic, we mean large overtone number n with everything else held fixed, and we find the O(n^0) correction to the known leading O(n) behavior of omega_n. The result has the schematic form omega_n =~ n Delta(omega) + A ln(n) + B, where Delta(omega) and A are constants and B depends logarithmically on the (D-2)-dimensional spatial momentu…
We obtain an analytical expression for the electromagnetic quasinormal spectrum of the higher-dimensional nearly-extremal Schwarzschild-de Sitter black hole. The WKB method is used to verify the results, and a comparison with known results from previous works is briefly made as well.
In this article we show that the asymptotic iteration method (AIM) allows one to numerically find the quasinormal modes of Schwarzschild and Schwarzschild de Sitter (SdS) black holes. An added benefit of the method is that it can also be used to calculate the Schwarzschild anti-de Sitter (SAdS) quasinormal modes for the case of spin zero perturbations. We also discuss an improved version of the AIM, more suitable for numerical implementation.
The quasinormal modes (QNMs) associated with the decay of Dirac field perturbation around a Schwarzschild black hole is investigated by using continued fraction and Hill-determinant approaches. It is shown that the fundamental quasinormal frequencies become evenly spaced for large angular quantum number and the spacing is given by $ω_{λ+1}- ω_λ=0.38490-0.00000i$. The angular quantum number has the surprising effect of increasing real part of the quasinormal frequencies, but it almost does not affect imaginary part, especially for low overtones. In addition, the quasinormal frequencies also bec…
The large dimension ($D$) limit of general relativity has been used in problems involving black holes as an analytical approximation tool. Further it has been proposed that both linear and nonlinear problems involving black holes can be systematically studied in a $1/D$ expansion. Certain quasinormal modes of higher-dimensional Schwarzschild black holes with $ω\sim \mathcal{O}(1)$ were studied in the large $D$ limit using a $1/D$ expansion for the mode function. In this paper, we revisit this linear perturbation problem and obtain an analytical equation for the vector quasinormal modes $ω\sim …
We analyze analytically the asymptotic regions of the quasinormal mode frequency spectra with infinitely large overtone numbers for $D$-dimensional Schwarzschild black holes in anti de Sitter spacetimes. In this limit, we confirm the analytic results obtained previously in the literature using different methods. In addition, we show that in certain spacetime dimensions these techniques imply the existence of other regions of the asymptotic quasinormal mode frequency spectrum which have not previously appeared in the literature. For large black holes, some of these modes have a damping rate of …
We generalize our previous studies on the Maxwell quasinormal modes around Schwarzschild-anti-de-Sitter black holes with Robin type vanishing energy flux boundary conditions, by adding a global monopole on the background. We first formulate the Maxwell equations both in the Regge-Wheeler-Zerilli and in the Teukolsky formalisms and derive, based on the vanishing energy flux principle, two boundary conditions in each formalism. The Maxwell equations are then solved analytically in pure anti-de Sitter spacetimes with a global monopole, and two different normal modes are obtained due to the existe…
We present an exact expression for the quasinormal modes of scalar, electromagnetic and gravitational perturbations of a near extremal Scwarzschild-de Sitter black hole and we show why a previous approximation holds exactly in this near extremal regime. In particular, our results give the asymptotic behavior of the quasinormal frequencies for highly damped modes, which as recently attracted much attention due to the proposed identification of its real part with the Barbero-Immirzi parameter.
The quasinormal modes of the electromagnetic and gravitational perturbation on Schwarzschild-AdS black hole calculated by Cardoso and Lemos has been revisited. Although the equations of motion are correct some frequencies calculated previously by the authors are not. We present the new values of quasinormal modes and discuss the possible sources of problems and implications on the conclusions presented.
We investigate quasinormal mode frequencies $ω_n$ of gravitinos and generic massive spin-3/2 fields in a Schwarzschild-AdS$_D$ background in spacetime dimension $D>3$, in the black brane (large black hole) limit appropriate to many applications of the AdS/CFT correspondence. First, we find asymptotic formulas for $ω_n$ in the limit of large overtone number $n$. Asymptotically, $ω_n \simeq n Δω+ O(\ln n) + O(n^0)$, where $Δω$ is a known constant, and here we compute the $O(\ln n)$ and $O(n^0)$ corrections to the leading $O(n)$ behavior. Then we compare to numerical calculations of exact quasino…
The spherically symmetric deformation of the Schwarzschild solution owing to the quantum corrections to gravity is known as Kazakov-Solodukhin black-hole metric. Neglecting non-spherical deformations of the background the problem was solved non-perturbatively. Here we analyze the basic characteristics of this geometry, such as: quasinormal modes and grey-body factors of fields of various spin and shadow cast by this black hole. The WKB approach and time-domain integration method, which we used for calculation of quasinormal modes, are in a good concordance. The analytical formula for quasinorm…
The fundamental resonances of rapidly rotating Kerr black holes in the eikonal limit are derived analytically. We show that there exists a critical value, $μ_c=\sqrt{{15-\sqrt{193}\over{2}}}$, for the dimensionless ratio $μ\equiv m/l$ between the azimuthal harmonic index $m$ and the spheroidal harmonic index $l$ of the perturbation mode, above which the perturbations become long lived. In particular, it is proved that above $μ_c$ the imaginary parts of the quasinormal frequencies scale like the black-hole temperature: $ω_I(n;μ>μ_c)=2πT_{BH}(n+{1\over 2})$. This implies that for perturbations m…
The quasinormal modes (QNMs) of a regular black hole with charge are calculated in the eikonal approximation. In the eikonal limit the QNMs of black hole are determined by the parameters of the unstable circular null geodesics. The behaviors of QNMs are compared with QNMs of Reisner-Nordström black hole, it is done by fixing some of the parameters that characterize the black holes and varying another. We observed that the parameter that is related one effective cosmological constant at small distances , determines the behaviors of the QNMs of regular black hole with charge.
In this paper, the quasinormal modes of gravitational perturbation around a Schwarzschild black hole surrounded by quintessence were evaluated by using the third-order WKB approximation. Due to the presence of quintessence, the gravitational wave damps more slowly.
A recent investigation has led to the possibility of the existence of an interesting region of the asymptotic quasinormal mode spectrum of Schwarzschild-anti de Sitter black holes. In this asymptotic region, the real part of quasinormal mode frequencies approaches infinity while the damping rate approaches a finite value. These quasinormal mode frequencies were calculated using an analytic technique based on the complex coordinate WKB method. In this paper, we use a different analytic technique to calculate such quasinormal mode frequencies. The results of this paper provide further support of…
Asymptotically safe gravity is based on the idea that the main contribution to the Schwarzschild-like black hole spacetime is due to the value of the gravitational coupling which depends on the distance from the origin and approaches its classical value in the far zone. However, at some stage this approach has an arbitrariness of choice of some identification parameter. The two cases of identification are considered here: first, by the modified proper length (the Bonanno-Reuter metric), and second, by the Kretschmann scalar (the metric for this case coincides, up to the redefinition of constan…
The quasinormal modes (QNMs) associated with the decay of Dirac field perturbation around a Schwarzschild-anti-de Sitter (SAdS) black hole is investigated by using Horowitz-Hubeny approach. We find that both the real and the imaginary parts of the fundamental quasinormal frequencies for large black holes are linear functions of the Hawking temperature, and the real part of the fundamental quasinormal frequencies for intermediate and small black holes approximates a temperature curve and the corresponding imaginary part is almost a linear function of the black hole radius. The angular quantum n…
We consider the equivalence of quasinormal modes and geodesic quantities recently brought back due to the black hole shadow observation by Event Horizon Telescope. Using WKB method we found an analytical relation between the real part of quasinormal frequencies at the eikonal limit and black hole shadow radius. We verify this correspondence with two black hole families in $4$ and $D$ dimensions, respectively.
We address the concerns raised by the author of "Comment: Some exact quasinormal frequencies of a massless scalar field in Schwarzschild spacetime" appeared in arXiv:1807.05940. In particular, we explain why the analysis in our paper [Phys. Rev. D{\bf{98}}, 024017 (2018)] is relevant and correct.
A new branch of quasinormal modes for a massless scalar field propagating on the Schwarzschild spacetime was recently announced [1807.04513]. We review the quasinormal modes characterisation and arguments and identify the flaws in their proof. Then, we preset explicit counterexamples to such arguments. Finally, we study the modes via alternative methods and do not find the new branch. We conclude against their interpretation.
Recently, the anomalous decay rate of quasinormal modes has been studied for some geometries under scalar field perturbations, which occurs when the longest-lived modes are the ones with higher angular number, as well as, the existence of a critical scalar field mass, i.e the value of scalar field mass such that the decay rate does not depend appreciably on the angular number, and beyond which the behaviour of the decay rate is inverted. Here, we consider the propagation of fermionic fields in the background of Schwarzschild de Sitter black holes, and we show that the anomalous decay rate beha…
