Quasinormal modes are eigenmodes of dissipative systems. For instance, if a spacetime with an event or cosmological horizon is perturbed from its equilibrium state, quasinormal modes arise as damped oscillations with a spectrum of complex frequencies, called quasinormal frequencies, that depends just on the charges which define the geometry of the spacetime in which the perturbation is propagating, such as the mass, electric charge, and angular momentum. Mathematically, this discrete spectrum of quasinormal modes stems from the fact that certain boundary conditions must be imposed to the physical fields propagating in such a spacetime. In this thesis, we shall consider a higher-dimensional generalization of the charged Nariai spacetime that is comprised of the direct product of the two-dimensional de Sitter space, $dS_2$, with an arbitrary number of two-spheres, $S^2$, and investigate the dynamics of spin-$s$ field perturbations for $s = 0, 1/2, 1$ and $2$. The boundary conditions leading to quasinormal modes are analyzed and the quasinormal frequencies are analytically obtained.
In this work we analytically obtain the quasinormal spectrum for the gravitational perturbation on a higher-dimensional generalization of the Nariai spacetime that is comprised of the direct product of the two-dimensional de Sitter space with several two-spheres. A key step in order to attain this result is to use a suitable basis for the angular functions depending on the rank of the tensorial degree of freedom that one needs to describe. Here we define such a basis, which is a generalization of the tensor spherical harmonics that is suited for spaces that are the product of several spaces of…
The perturbations of fields with spin 0, 1/2, and 1 propagating in a higher-dimensional generalization of the charged Nariai spacetime are investigated. The boundary conditions leading to quasinormal modes are analyzed and the quasinormal frequencies are analytically obtained.
We show the existence of static, spherically symmetric spacetimes containing two stars of incompressible matter, possibly oppositely charged. The stars are held apart by the negative pressure of a positive cosmological constant but there is no cosmological horizon separating them. The spacetime between the stars is given by the Nariai solution, or a slight generalisation thereof in the charged case.
Exceptional Lines and Excitation of (Nearly) Double-Pole Quasinormal Modes: A Semi-Analytic Study in the Nariai Black Hole
Kyoto University（京都大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；The Hakubi Center for Advanced Research, Kyoto University（京都大学博雅中心）；RIKEN iTHEMS（理化学研究所iTHEMS）；Kindai University（近畿大学）
The product spacetimes of constant curvature describe in Einstein gravity, which is linear in Riemann curvature, Nariai metric which is a solution of $Λ$-vacuum when curvatures are equal, $k_1=k_2$, while it is Bertotti-Robinson metric describing uniform electric field when curvatures are equal and opposite, $k_1=-k_2$. We probe pure Lovelock gravity by these simple product spacetimes and prove that the same characterization of these solutions is indeed true in general for pure Lovelock gravitational equation of order $N$ in $d=2N+2$ dimension. We also consider these solutions for the conventi…
We will show that the Nariai metric, i.e. the static spherically symmetric vacuum spacetime with a cosmological constant, admits a conformally Kerr-Schild spacetime representation. We find the vacuum solutions of the Einstein-Maxwell equations for the Nariai metric using the Horsky-Mitskievich generating conjecture.
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 discuss the problem of wormholes from the viewpoint of gluing together two Reissner--Nordström-type universes while putting between them a segment of the Nariai-type world (in both cases there are also present electromagnetic fields as well as the cosmological constant). Such a toy wormhole represents an example of one-way topological communication free from causal paradoxes, though involving a travel to next spacetime sheet since one has to cross at least a pair of horizons through which the spacetimes' junction occurs. We also consider the use of thin shells in these constructions. Such a…
Higher-dimensional solutions for Einstein-Maxwell equations that generalize the charged Nariai spacetime are obtained. The solutions presented here are made from the direct product of several 2-spaces of constant curvature. These solutions turn out to have many magnetic charges, contrary to the usual higher-dimensional generalization of the Nariai spacetime, which has no magnetic charge at all. These solutions are then used to generate black hole metrics. Finally, it is analyzed how these generalized Nariai solutions are modified in more general theories of gravity.
In this paper, we construct and study solutions of Einstein's equations in vacuum with a positive cosmological constant which can be considered as inhomogeneous generalizations of the Nariai cosmological model. Similar to this Nariai spacetime, our solutions are at the borderline between gravitational collapse and de-Sitter-like exponential expansion. Our studies focus in particular on the intriguing oscillatory dynamics which we discover. Our investigations are carried out both analytically (using heuristic mode analysis arguments) and numerically (using the numerical infrastructure recently …
We consider the five-dimensional bulk spacetime with negative Lambda described by the Nariai metric (which is not conformally flat) and match it with a vacuum brane satisfying the proper boundary conditions. It is shown that the brane metric corresponds to a cloud of string dust of constant energy density.
We study charged massless fermionic perturbations in the background of $4$-dimensional linear dilaton black holes in Einstein-Maxwell-dilaton theory with double Liouville-type potentials. We present the analytical fermionic quasinormal modes, whose Dirac equations are solved in terms of hypergeometric functions. We also discuss the stability of these black holes under the charged fermionic perturbations.
Recently Hod proposes a lower bound on the relaxation time of a perturbed thermodynamic system. For gravitational systems this bound transforms into a condition on the fundamental quasinormal frequency. We test the bound in some spacetimes whose quasinormal frequencies are calculated exactly, as the three-dimensional BTZ black hole, the D-dimensional de Sitter spacetime, and the D-dimensional Nariai spacetime. We find that for some of these spacetimes their fundamental quasinormal frequencies do not satisfy the bound proposed by Hod.
By using the sixth order WKB approximation we calculate for an electromagnetic field propagating in D-dimensional Schwarzschild and Schwarzschild de Sitter black holes its quasinormal frequencies for the fundamental mode and first overtones. We study the dependence of these QN frequencies on the value of the cosmological constant and the spacetime dimension. We also compare with the known results for the gravitational perturbations propagating in the same background. Moreover we exactly compute the QN frequencies of the electromagnetic field propagating in D-dimensional massless topological bl…
The exact computation of asymptotic quasinormal frequencies is a technical problem which involves the analytic continuation of a Schrodinger-like equation to the complex plane and then performing a method of monodromy matching at the several poles in the plane. While this method was successfully used in asymptotically flat spacetime, as applied to both the Schwarzschild and Reissner-Nordstrom solutions, its extension to non-asymptotically flat spacetimes has not been achieved yet. In this work it is shown how to extend the method to this case, with the explicit analysis of Schwarzschild de Sit…
A detailed study of the geometries that emerge by a gravitating generalized Yang monopole in even dimensions is carried out. In particular, those which present black hole and cosmological horizons. This two-horizon system is thermally unstable. The process of thermalization will drive both horizons to coalesce. This limit is what is profusely studied in this paper. It is shown that eventhough coordinate distance shrinks to zero, physical distance does not. So, there is some remaining space which geometry has been computed and identified as a generalized Nariai solution. The thermal properties …
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.
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…
We find an exact black hole solution with a minimally coupled scalar field. The corresponding spacetime has two horizons and one of the them is the black hole event horizon and the other is the cosmic horizon. In this sense, the solution is analogous to the Schwarzschild-de Sitter (or anti-de Sitter) spacetime. We investigate the thermodynamics and construct the first law of thermodynamics. At the same time, we make a study on the shadow and quasinormal modes of this black hole solution.
To seek a deeper understanding of wave propagation on the Schwarzschild spacetime, we investigate the relationship between (i) the lightcone of an event and its caustics (self-intersections), (ii) the large-$l$ asymptotics of quasinormal (QN) modes, and (iii) the singular structure of the retarded Green function (GF) for the scalar field. First, we recall that the GF has a (partial) representation as a sum over QN modes. Next, we extend a recently-developed expansion method to obtain asymptotic expressions for QN wavefunctions and their residues. We employ these asymptotics to show (approximat…
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.
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 …
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…
We study the stability under scalar perturbations, and we compute the quasinormal modes of the Einstein-Born-Infeld dilaton spacetime in 1+3 dimensions. Solving the full radial equation in terms of hypergeometric functions, we provide an exact analytical expression for the spectrum. We find that the frequencies are purely imaginary, and we confirm our results by computing them numerically. Although the scalar field that perturbs the black hole is electrically neutral, an instability similar to that seen in charged scalar perturbations of the Reissner-Nordström black hole is observed.
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.
The numerical study indicates that there exists a relation between the quasinormal modes and the Davies point for a black hole. In this paper, we analytically study this relation for the charged Reissner-Nordström black holes in asymptotically flat and dS spacetimes. In the eikonal limit, the angular velocity $Ω$ and the Lyapunov exponent $λ$ of the photon sphere, respectively, corresponding to the real and imaginary parts of the quasinormal modes are obtained from the null geodesics. Both in asymptotically flat and dS spacetimes, we observe the spiral-like shapes in the complex quasinormal mo…
We present and analyze exact gyraton and nonexpanding gravitational wave solutions of algebraic type II on backgrounds which are a direct-product of two 2-spaces of constant curvature, or more general type D spacetimes. This family of electro-vacuum background spacetimes contains the Nariai, anti-Nariai and Plebanski-Hacyan universes, conformally flat Bertotti-Robinson and Minkowski spaces. The gyraton solutions are given in a simple Kundt metric form. They belong to the recently discussed class of spacetimes with constant scalar invariants (CSI) of the curvature tensor. We show that the Einst…
Null geodesics, quasinormal modes of a massless scalar field perturbation and the correspondence with shadow radii are investigated in the background spacetime of high-dimensional Einstein-Yang-Mills black holes. Based on the properties of null geodesics, we obtain the connection between the radius of a photon sphere and the radius of a horizon in the five- and six-dimensional Einstein-Yang-Mills spacetimes. Especially in the five-dimensional case, there exist two branches for the radius of a photon sphere, but only the branch outside the event horizon satisfies the condition of circular null …
We have studied the quasinormal modes of a massive scalar field coupling to Einstein's tensor in the spacelike stretched $AdS_3$ black hole spacetime. We find that both the right-moving and left-moving quasinormal frequencies depend not only on the warped parameter $v$ of black hole, but also on the coupling between the scalar field and Einstein's tensor. Moreover, we also discuss the warped $AdS/CFT$ correspondence from the quasinormal modes and probe the effects of the coupling on the left and right conformal weights $h_L$ and $h_R$ of the operators dual to the scalar field in the boundary.
We study the spectrum of quasinormal mode frequencies for a Proca field on a rotating black hole spacetime. First, we review how the introduction of field mass modifies the spectrum in the scalar-field case, leading to evanescent modes and to quasiresonance. Next, we examine the three physical polarizations of the Proca field and their relation to the electromagnetic field modes in the massless limit. Exploiting a separation of variables, we obtain a five-term recurrence relation from an appropriate ansatz for the radial function. Gaussian elimination and the modified Lentz algorithm are appli…
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.
We demonstrate the separability of the massive vector (Proca) field equation in general Kerr-NUT-(A)dS black hole spacetimes in any number of dimensions, filling a long-standing gap in the literature. The obtained separated equations are studied in more detail for the four-dimensional Kerr geometry and the corresponding quasinormal modes are calculated. Two of the three independent polarizations of the Proca field are shown to emerge from the separation ansatz and the results are found in a perfect agreement with those of the recent numerical study where the full coupled partial differential e…
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 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…
A scalar field in four-dimensional deSitter spacetime (dS_4) has quasinormal modes which are singular on the past horizon of the south pole and decay exponentially towards the future. These are found to lie in two complex highest-weight representations of the dS_4 isometry group SO(4,1). The Klein-Gordon norm cannot be used for quantization of these modes because it diverges. However a modified `R-norm', which involves reflection across the equator of a spatial S^3 slice, is nonsingular. The quasinormal modes are shown to provide a complete orthogonal basis with respect to the R-norm. Adopting…
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 analytic forms of the asymptotic quasinormal frequencies of a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime is investigated by using the monodromy technique proposed by Motl and Neitzke. It is found that the asymptotic quasinormal frequencies depend not only on the structure parameters of the background spacetime, but also on the coupling between the scalar fields and gravitational field. Moreover, our results show that only in the minimal couple case, i.e., $ξ$ tends zero, the real parts of the asymptotic quasinormal frequencies agrees with the Hod's conjectu…
One parameter quantization ambiguity is existed in Loop quantum gravity which is called the Immirzi parameter. In this paper, we fix this free paremater by considering the quasinormal mode spectrum of black holes in four and higher spacetime dimensions. As a consequence, our result consistents with Bekenstein-Hawking entropy of a black hole. Moreover, we also give a possible quantum gravity explanation of the universal $\ln(3)$ behavior of the quasinormal mode spectrum.
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.
Perturbations of asymptotically Anti-de-Sitter (AdS) spacetimes are often considered by imposing field vanishing boundary conditions (BCs) at the AdS boundary. Such BCs, of Dirichlet-type, imply a vanishing energy flux at the boundary, but the converse is, generically, not true. Regarding AdS as a gravitational box, we consider vanishing energy flux (VEF) BCs as a more fundamental physical requirement and we show that these BCs can lead to a new branch of modes. As a concrete example, we consider Maxwell perturbations on Kerr-AdS black holes in the Teukolsky formalism, but our formulation appl…
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.
Bardeen's 1968 idea of a regular black hole spacetime was revived by Hayward in 2006 through the construction of a new example of such a geometry. Later it was realised by Neves and Saa, that a wider, two-parameter class exists, with Bardeen and Hayward spacetimes as special cases. In this article, we revisit and generalise the Hayward spacetime by applying the Damour-Solodukhin (DS) prescription. Recalling the DS suggestion of a deformed Schwarzschild spacetime where $g_{tt} = -\left (1-\frac{2M_1}{r}\right )$, $g_{rr} = \left (1-\frac{2M_2}{r}\right )^{-1}$ and $M_1\neq M_2$, we propose a si…
The fundamental quasinormal resonances of charged Reissner-Nordström black holes due to charged scalar perturbations are derived {\it analytically}. In the WKB regime, $qQ\gg\hbar$, we obtain a simple expression for the fundamental quasinormal resonances: $ω=qQ/r_+-i2πT_{\text{BH}}(n+{1 \over 2})$, where $T_{\text{BH}}$ and $Q$ are the temperature and electric charge of the black hole and $q$ is the electric charge of the field. Remarkably, our results show that the relaxation dynamics of a perturbed Reissner-Nordström black hole may saturate the recently proposed universal relaxation bound.
It has been observed that many spacetimes which feature a near-extremal horizon exhibit the phenomenon of zero-damped modes. This is characterised by the existence of a sequence of quasinormal frequencies which all converge to some purely imaginary number $iα$ in the extremal limit and cluster in a neighbourhood of the line $\Im s=α$. In this paper, we establish that this property is present for the conformal Klein-Gordon equation on a Reissner-Nordström-de Sitter background. This follows from a similar result that we prove for a class of spherically symmetric black hole spacetimes with a cosm…
The resonant mode spectrum of the Kerr-Newman spacetime is presently unknown. These modes, called the quasinormal modes, play a central role in determining the stability of Kerr-Newman black holes and their response to perturbations. We present a new formalism, generalized from time-independent perturbation theory in quantum mechanics, for calculating the quasinormal mode frequencies of weakly charged Kerr-Newman spacetimes of arbitrary spin. Our method makes use of an original technique for applying perturbation theory to zeroth-order solutions that are not square- integrable, and it can be a…
We study quasinormal modes (QNMs) of massive Klein-Gordon fields in static Bañados-Teitelboim-Zanelli (BTZ) black holes in terms of ladder operators constructed from spacetime conformal symmetries. Because the BTZ spacetime is locally isometric to the three-dimensional anti-de Sitter spacetime, ladder operators, which map a solution of the massive Klein-Gordon equation into that with different mass squared, can be constructed from spacetime conformal symmetries. In this paper, we apply the ladder operators to the QNMs of the Klein-Gordon equations in the BTZ spacetime. We demonstrate that the …
The quasinormal modes (QNMs) of Dirac field perturbations of a Reissner-Nordström black hole in an asymptotically Anti-de Sitter spacetime are investigated. We find that both the real and imaginary parts of the fundamental quasinormal frequencies for large black holes are the linear functions of the Hawking temperature, and the slope of the lines for the real parts decreases while that for the magnitude of the imaginary parts increases as the black hole charge increases. According to the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, the fact shows that different charge presen…
We consider quasinormal modes of the massive Dirac field in the background of a Schwarzschild- Tangherlini black hole. Different dimensions of the spacetime are considered, from d = 4 to d = 9. The quasinormal modes are calculated using two independent methods: WKB and continued fraction. We obtain the spectrum of quasinormal modes for different values of the overtone number and angular quantum number. An analytical approximation of the spectrum valid in the case of large values of the angular quantum number and mass is calculated. Although we don't find unstable modes in the spectrum, we show…
In this work, we investigate the quasinormal modes for a massive scalar field with a nonminimal coupling with gravity in the spacetime of a loop quantum black hole, known as the Self-Dual Black Hole. In this way, we have calculated the characteristic frequencies using the 3rd order WKB approach, where we can verify a strong dependence with the mass of scalar field, the parameter of nonminimal coupling with gravity, and parameters of the Loop Quantum Gravity. From our results, we can check that the Self-Dual Black Hole is stable under the scalar perturbations when assuming small values for the …
