We analytically investigate the spin-1 quasinormal mode frequencies of Schwarzschild black hole space-time. We formally determine these frequencies to arbitrary order as an expansion for large imaginary part (i.e., large-n, where n is the overtone number). As an example of the practicality of this formal procedure, we explicitly calculate the asymptotic behaviour of the frequencies up to order $n^{-5/2}$.
Using the Frobenius method, we find high overtones of the Dirac quasinormal spectrum for the Schwarzschild black hole. At high overtones, the spacing for imaginary part of $ω_{n}$ is equidistant and equals to $\Im{ω_{n+1}}-\Im{ω_{n}} =i/8M$, ($M$ is the black hole mass), which is twice less than that for fields of integer spin. At high overtones, the real part of $ω_{n}$ goes to zero. This supports the suggestion that the expected correspondence between quasinormal modes and Barbero-Immirzi parameter in Loop Quantum Gravity is just a numerical coincidence.
In this paper we show that with standard methods it is possible to obtain highly precise results for QNMs. In particular, secondary modes are obtained by numerical integration. We compare several results making a detailed analysis.
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…
We present a thorough analysis for the quasinormal (QN) behavior, associated with the decay of scalar, electromagnetic and gravitational perturbations, of Schwarzschild-anti-de Sitter black holes. As it is known the anti-de Sitter (AdS) QN spectrum crucially depends on the relative size of the black hole to the AdS radius. There are three different types of behavior depending on whether the black hole is large, intermediate, or small. The results of previous works, concerning lower overtones for large black holes, are completed here by obtaining higher overtones for all the three black hole re…
Channel-Resolved High-Overtone Quasinormal Modes Decode Kasner Scaling inside Black Holes
International Centre for Theoretical Physics Asia-Pacific (ICTP-AP), University of Chinese Academy of Sciences（国际理论物理中心亚太分部，中国科学院大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
We investigate the scalar, gravitational, and electromagnetic quasinormal mode spectra of Schwarzschild anti-de Sitter black holes using the numerical continued fraction method. The spectra have similar, almost linear structures. With a few exceptions, the low overtone quasinormal modes are consistent with previously obtained results in the literature that use other numerical techniques. The intermediate and high overtone quasinormal modes, in comparison to the Schwarzschild case, converge very quickly to the asymptotic formulas previously obtained by analytic monodromy techniques. In addition…
Vanishing of all redshift modes in Schwarzschild ringdown
CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL（里斯本大学高等理工学院物理系CENTRA）；Dipartimento di Fisica, Sapienza Università di Roma（罗马智慧大学物理系）；INFN, Sezione di Roma（罗马意大利国家核物理研究所）
We consider the perturbations of the massive vector field around Schwarzschild black hole, (generally, with non-vanishing $Λ$ - term). The monopole massive vector perturbation equations can be reduced to a single wave-like equation. We have proved the stability against these perturbations and investigated the quasinormal spectrum. The quasinormal behaviour for Schwarzschild black hole is quite unexpected: the fundamental mode and all higher overtones shows totally different dependence on the mass of the field $m$: as $m$ is increasing, the damping rate of the fundamental mode is decreasing, wh…
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…
Probing higher curvature gravity via ringdown with overtones
Kaichi Tokorozawa Secondary School（东京都立小平高等学校）；Department of Physics, Rikkyo University（立教大学物理系）；Department of Information, Artificial Intelligence and Data Science, Daiichi Institute of Technology（第一工业大学信息、人工智能与数据科学系）；Department of Physics, Tokyo Metropolitan University（东京都立大学物理系）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Department of Physics, College of Humanities and Sciences, Nihon University（日本大学人文学部物理系）
研究高曲率引力中黑洞的度规扰动，发现高曲率修正使有效势近邻区域变形，泛音模式的 quasinormal mode 频率偏离 GR 值更显著，分析高曲率引力模型中的回波波形。
The imaginary part of the quasinormal frequencies spectrum for a static and spherically symmetric black hole is analytically known to be equally spaced, both for the highly damped and the weakly damped families of quasinormal modes. Some interesting attempts have been made in the last twenty years to understand in simple ways this level spacing for the only case of highly damped quasinormal frequencies. Here, we show that the overtone level spacing, for both the highly damped and weakly damped families of quasinormal modes, can simply be understood as a fingerprint of a hidden local $SL(2,\mat…
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 …
Using the third-order WKB approximation and monodromy methods, we investigate the influence of Lorentz violating coefficient $b$ (associated with a special axial-vector $b_μ$ field) on Dirac quasinormal modes in the Schwarzschild black hole spacetime. At fundamental overtone, the real part decreases linearly as the parameter $b$ increases. But the variation of the imaginary part with $b$ becomes more complex. For the larger multiple moment $k$, the magnitude of imaginary part increases with the increase of $b$, which means that presence of Lorentz violation makes Dirac field damps more rapidly…
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…
Spectral instability of parametrized black hole quasinormal modes in the high-overtone limit via the exact WKB analysis
Osaka Metropolitan University（大阪公立大学）；Osaka Central Advanced Mathematical Institute (OCAMI)（大阪中央先进数学研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学中心交叉理论与数学科学中心）；Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo（东京大学理学研究科早期宇宙研究中心）；Department of Physics , Kyoto University（京都大学物理系）；Center for Gravitational Physics and Quantum Information,Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理中心引力物理与量子信息）；The Hakubi Center for Advanced Research, Kyoto University（京都大学白兔高级研究中心）
Validating the black-hole no-hair theorem with gravitational-wave observations of compact binary coalescences provides a compelling argument that the remnant object is indeed a black hole as described by the general theory of relativity. This requires performing a spectroscopic analysis of the post-merger signal and resolving the frequencies of either different angular modes or overtones (of the same angular mode). For a nearly-equal mass binary black-hole system, only the dominant angular mode ($l=m=2$) is sufficiently excited and the overtones are instrumental to perform this test. Here we i…
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 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…
Motivated by recent physical interpretation on quasinormal modes presented by Maggiore, the adiabatic quantity method given by Kunstatter is used to calculate the spectrums of a non-extremal Schwarzschild de Sitter black hole in this paper, as well as electrically charged case. According to highly damped Konoplya and Zhidenko's numerical observational results for high overtone modes\cite{Konoplya}, we found that the asymptotic non-flat spacetime structure leads two interesting facts as followings: (i) near inner event horizon, the area and entropy spectrums, which are given by $A_{en} = 8 n_1 …
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…
We study the propagation of scalar fields in the background of an asymptotically de-Sitter black hole solution in $f(R)$ gravity. The aim of this work is to analyze in modified theories of gravity the existence of an anomalous decay rate of the quasinormal modes (QNMs) of a massive scalar field which was recently reported in Schwarzschild black holes backgrounds, 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 behavior is inverted. We study the QNMs for various overtone numbers and…
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 evaluate both the massless and the massive Dirac quasi-normal mode frequencies in the Schwarzschild black hole spacetime using the WKB approximation. For the massless case, we find that, similar to those for the integral spin fields, the real parts of the frequencies increase with the angular momentum number $κ$, while the imaginary parts or the dampings increase with the mode number $n$ for fixed $κ$. For the massive case, the oscillation frequencies increase with the mass $m$ of the field, while the dampings decrease. Fields with higher masses will therefore decay more slowly.
The Dirac quasinormal modes (QNMs) of the Kerr-Newman black hole are investigated using continued fraction approach. It is shown that the quasinormal frequencies in the complex $ω$ plane move counterclockwise as the charge or angular momentum per unit mass of the black hole increases. They get a spiral-like shape, moving out of their Schwarzschild or Reissner-Nordström values and "looping in" towards some limiting frequencies as the charge and angular momentum per unit mass tend to their extremal values. The number of the spirals increases as the overtone number increases but decreases as the …
We revisit the problem of calculating the quasinormal modes of spin $0$, $1/2$, $1$, $3/2$, $2$, and spin $5/2$ fields in the asymptotically flat Schwarzschild black hole spacetime. Our aim is to investigate the problem from the numerical point of view, by comparing some numerical methods available in the literature and still not applied for solving the eigenvalue problems arising from the perturbation equations in the Schwarzschild black hole spacetime. We focus on the pseudo-spectral and the asymptotic iteration methods. These numerical methods are tested against the available results in the…
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 …
Linear field perturbations of a black hole are described by the Green function of the wave equation that they obey. After Fourier decomposing the Green function, its two natural contributions are given by poles (quasinormal modes) and a largely unexplored branch cut in the complex-frequency plane. We present new analytic methods for calculating the branch cut on a Schwarzschild black hole for arbitrary values of the frequency. The branch cut yields a power-law tail decay for late times in the response of a black hole to an initial perturbation. We determine explicitly the first three orders in…
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.
In general relativity, the remnant object originating from an uncharged black hole merger is a Kerr black hole. The approach to this final state is reached through the emission of a late train of radiation known as the black hole ringdown. The ringdown morphology is described by a countably infinite set of damped sinusoids, whose complex frequencies are solely determined by the final black hole's mass and spin. Recent results advocate that ringdown waveforms from numerical relativity can be fully described from the peak of the strain onwards if quasi-normal mode models with $N_{max}=7$ overton…
The quasinormal modes (QNMs) and the late-time behavior of arbitrary spin fields are studied in the background of a Schwarzschild black hole with a global monopole (SBHGM). It has been shown that the real part of the QNMs for a SBHGM decreases as the symmetry breaking scale parameter $H$ increases but imaginary part increases instead. For large overtone number $n$, these QNMs become evenly spaced and the spacing for the imaginary part equals to $-i(1-H)^{3/2}/(4M)$ which is dependent of $H$ but independent of the quantum number $l$. It is surprisingly found that the late-time behavior is domin…
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 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 show that the scalar wave equation at low frequencies in the Schwarzschild geometry enjoys a hidden SL(2,R) invariance, which is not inherited from an underlying symmetry of the spacetime itself. Contrary to what happens for Kerr black holes, the vector fields generating the SL(2,R) are globally defined. Furthermore, it turns out that under an SU(2,1) Kinnersley transformation, which maps the Schwarzschild solution into the near horizon limit AdS_2 x S^2 of the extremal Reissner-Nordstr"om black hole (with the same entropy), the Schwarzschild hidden symmetry generators become exactly the is…
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…
It is known that the imaginary parts of the quasi normal mode (QNM) frequencies for the Schwarzschild black hole are evenly spaced with a spacing that depends only on the surface gravity. On the other hand, for massless minimally coupled scalar fields, there exist no QNMs in the pure DeSitter spacetime. It is not clear what the structure of the QNMs would be for the Schwarzschild-DeSitter (SDS) spacetime, which is characterized by two different surface gravities. We provide a simple derivation of the imaginary parts of the QNM frequencies for the SDS spacetime by calculating the scattering amp…
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.
It is shown here that the extraction of quasinormal modes (QNMs) within the first Born approximation of the scattering amplitude is mathematically not well founded. Indeed, the constraints on the existence of the scattering amplitude integral lead to inequalities for the imaginary parts of the QNM frequencies. For instance, in the Schwarzschild case, $0 \leq ω_I < κ$ (where $κ$ is the surface gravity at the horizon) invalidates the poles deduced from the first Born approximation method, namely, $ω_n = i n κ$.
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.
