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 to be stable. Our findings show that both the real part and the absolute value of the imaginary part of the frequencies increase with the parameter $ε$ that measures the deviation from the classical geometry. Therefore, in the framework of scale-dependent gravity the modes oscillate and decay faster in comparison with their classical counterparts.
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…
We present the improved Mathematica code which computes quasinormal frequencies with the help of the Bernstein spectral method for a general class of black holes, allowing for asymptotically flat, de Sitter or anti-de Sitter asymptotic. The method is especially efficient when searching for purely imaginary and unstable modes and here it is used for detecting the instability region of a charged scalar field in the background of the charged asymptotically de Sitter dilatonic black hole. We show that the instability has superradiant nature and the dilaton field essentially influences the region o…
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.
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 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…
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.
Quasinormal modes of Kerr-Newman-de Sitter black holes from Heun connection formulae
Toyota Technological Institute（丰田技术研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所交叉理论数学科学中心（iTHEMS））；Tokyo Metropolitan University（东京都立大学）；Nagoya University（名古屋大学）
We construct higher spin quasinormal modes algebraically in $D$-dimensional de Sitter spacetime using the ambient space formalism. The quasinormal modes fall into two nonunitary lowest-weight representations of $\mathfrak{so}(1, D)$. From a local QFT point of view, the lowest-weight quasinormal modes of massless higher spin fields are produced by gauge-invariant boundary conserved currents and boundary higher-spin Weyl tensors inserted at the southern pole of the past boundary. We also show that the quasinormal spectrum of a massless/massive spin-$s$ field is precisely encoded in the Harish-Ch…
The quasinormal resonant modes of massless neutral fields in near-extremal Kerr-Newman-de Sitter black-hole spacetimes are calculated in the eikonal regime. It is explicitly proved that, in the angular momentum regime ${\bar a}> \sqrt{{1-2{\barΛ}\over{4+{\barΛ}/3}}}$, the black-hole spacetimes are characterized by slowly decaying resonant modes which are described by the compact formula $\Imω(n)=κ_+\cdot(n+{1\over2})$ [here the physical parameters $\{{\bar a},κ_+,{\barΛ},n\}$ are respectively the dimensionless angular momentum of the black hole, its characteristic surface gravity, the dimensio…
We have studied scalar perturbations as well as fermion perturbations in pure de Sitter space-times. For scalar perturbations we have showed that well-defined quasinormal modes can exist provided that the mass of scalar field $m>\frac{d-1}{2l}$. The quasinormal frequencies of fermion perturbations in three and four dimensional cases have also been presented. We found that different from other dimensional cases, in three dimensional pure de Sitter spacetime there is no quasinormal mode for the s-wave. This interesting difference caused by the spacial dimensions is true for both scalar and fermi…
In this work, we present a numerical scheme to study the quasinormal modes of the time-dependent Vaidya black hole metric in asymptotically anti-de Sitter spacetime. The proposed algorithm is primarily based on a generalized matrix method for quasinormal modes. The main feature of the present approach is that the quasinormal frequency, as a function of time, is obtained by a generalized secular equation and therefore a satisfactory degree of precision is achieved. The implications of the results are discussed.
Quasinormal modes of scalar, electromagnetic, and gravitational fields in the extreme Schwarzschild-de Sitter background are known to be expressed in analytic form as eigenvalues of the Pöschl-Teller wavelike equation. We show that perturbations of fermionic fields (given by Dirac and Rarita-Schwinger equations) do not lead to the Pöschl-Teller effective potential. Nevertheless, using the Frobenius method we find quasinormal modes analytically in this case as well. We write down the analytical formula for quasinormal frequencies of the near-extreme Schwarzschild-de Sitter black holes, which is…
Modifying a method by Horowitz and Hubeny for asymptotically anti-de Sitter black holes, we establish the classical stability of the quasinormal modes of the de Sitter spacetime. Furthermore using a straightforward method we calculate the de Sitter quasinormal frequencies of the gravitational perturbations and discuss some properties of the radial functions of these quasinormal modes.
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 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 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.
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 …
We study the Maxwell quasinormal spectrum on asymptotically anti-de Sitter black holes with a set of two Robin type boundary conditions, by requiring the energy flux to vanish at asymptotic infinity. Focusing, for illustrative purposes, on Schwarzschild-anti-de Sitter black holes both without and with a global monopole, we unveil that, on the one hand, the Maxwell quasinormal spectrum bifurcates as the black hole radius increases for both boundary conditions, which is termed the mode split effect; while on the other hand, with an appropriate fixed black hole radius but increasing the monopole …
We analytically investigate the quasinormal modes of the massive scalar field with a nonminimal coupling in the higher-dimensional de Sitter black hole with a single rotation. According to the separated scalar field equation, the boundary conditions of quasinormal modes are well constructed at the outer and cosmological horizons. Then, under near-extremal conditions, where the outer horizon closes to the cosmological horizon, the quasinormal frequencies are obtained and generalized to universal form in the higher-dimensional spacetime. Here, the real part of the frequency includes the scalar f…
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…
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 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 se…
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.
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.
In this note we compute the 1-loop partition function of spin-$s$ fields on Euclidean de Sitter space $S^{2n+1}$ using the quasinormal mode method. Instead of computing the quasinormal mode frequencies from scratch, we use the analytic continuation prescription $L_{\text{AdS}}\to iL_{\text{dS}}$, appearing in the dS/CFT correspondence, and Wick rotate the normal mode frequencies of fields on thermal $\text{AdS}_{2n+1}$ into the quasinormal mode frequencies of fields on de Sitter space. We compare the quasinormal mode and heat kernel methods of calculating 1-loop determinants, finding exact agr…
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 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.
Quasinormal mode spectra for gravitational perturbations of black holes in four dimensional de Sitter and anti-de Sitter space are investigated. The anti-de Sitter case is relevant to the ADS-CFT correspondence in superstring theory. The ADS-CFT correspondence suggests a prefered set of boundary conditions.
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 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 …
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.
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…
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…
Using the Pöshl-Teller approximation, we evaluate the neutrino quasinormal modes (QNMs) of a Kerr-Newman-de Sitter black hole. The result shows that for a Kerr-Newman-de Sitter black hole, massless neutrino perturbation of large $Λ$, positive $m$ and small value of $n$ will decay slowly.
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.
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 calculate the exact values of the quasinormal frequencies for an electromagnetic field and a gravitational perturbation moving in $D$-dimensional de Sitter spacetime ($D \geq 4$). We also study the quasinormal modes of a real massive scalar field and we compare our results with those of other references.
The quasinormal modes of the Reissner-Nordström de Sitter black hole for the massless Dirac fields are studied using the Pöshl-Teller potential approximation. We find that the magnitude of the imaginary part of the quasinormal frequencies decreases as the cosmological constant or the orbital angular momentum increases, but it increases as the charge or the overtone number increases. An interesting feature is that the imaginary part is almost linearly related to the real part as the cosmological constant changes for fixed charge, and the linearity becomes better as the orbital angular momentum …
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…
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…
We study two fermionic systems that have an underlying supersymmetric structure, namely a color superconductor and Dirac fermion in a Reissner-Nordström-anti-de Sitter gravitational background. In the chiral limit of the color superconductor, the localized fermionic zero modes around the vortex form an N=2 with zero central charge $d=1$ quantum algebra, with all the operators being Fredholm. We compute the Witten index of this algebra and we find an unbroken supersymmetry. The fermionic gravitational system in the chiral limit too, has two underlying unbroken N=2, $d=1$ supersymmetric algebras…
Analysis of time-domain profiles for gravitational perturbations shows that Gauss-Bonnet black holes in a de Sitter world possess a new kind of dynamical instability which does not take place for asymptotically flat Einstein-Gauss-Bonnet black holes. The new instability is in the gravitational perturbations of the scalar type and is due to the nonvanishing cosmological constant. Analysis of the quasinormal spectrum in the stability sector shows that although the scalar type of gravitational perturbations alone does not obey Hod's conjectural bound, connecting the damping rate and the Hawking t…
We investigate the quasinormal mode frequencies for the massless Dirac field in static four dimensional $AdS$ space-time. The separation of the Dirac equation is achieved for the first time in $AdS$ space. Besides the relevance that this calculation can have in the framework of the $AdS/CFT$ correspondence between M-theory on $AdS_4\times S^7$ and SU(N) super Yang-Mills theory on $M_3$, it also serves to fill in a gap in the literature, which has only been concerned with particles of integral spin $0,1,2$.
We generalize the Chern-Simons modified gravity to the metric-affine case and impose projective invariance by supplementing the Pontryagin density with homothetic curvature terms which do not spoil topologicity. The latter is then broken by promoting the coupling of the Chern-Simons term to a (pseudo)-scalar field. The solutions for torsion and nonmetricity are derived perturbatively, showing that they can be iteratively obtained from the background fields. This allows us to describe the dynamics for the metric and the scalar field perturbations in a self-consistent way, and we apply the forma…
We have studied the quasinormal modes and the late-time tail behaviors of scalar, electromagnetic and gravitational perturbations in the Schwarzschild black hole pierced by a cosmic string. Although the metric is locally identical to that of the Schwarzschild black hole so that the presence of the string will not imprint in the motion of test particles, we found that quasinormal modes and the late-time tails can reflect physical signatures of the cosmic string. Compared with the scalar and electromagnetic fields, the gravitational perturbation decays slower, which could be more interesting to …
We extensively study the exact solutions of the massless Dirac equation in 3D de Sitter spacetime that we published recently. Using the Newman-Penrose formalism, we find exact solutions of the equations of motion for the massless classical fields of spin s=1/2,1,2 and to the massive Dirac equation in 4D de Sitter metric. Employing these solutions, we analyze the absorption by the cosmological horizon and de Sitter quasinormal modes. We also comment on the results given by other authors.
We study quasinormal modes for massive scalar fields in Schwarzschild-anti-de Sitter black holes. When the mass-squared is above the Breitenlohner-Freedman bound we show that for large angular momenta, $\ell$, there exist quasinormal modes with imaginary parts of size $\exp(-\ell/C)$. We provide an asymptotic expansion for the real parts of the modes closest to the real axis and identify the vanishing of certain coefficients depending on the dimension.
