The traditional approach to perturbations of nonrotating black holes in General Relativity uses the reformulation of the equations of motion into a radial second-order Schrödinger-like equation, whose asymptotic solutions are elementary. Imposing specific boundary conditions at spatial infinity and near the horizon defines, in particular, the quasi-normal modes of black holes. For more complicated equations of motion, as encountered for instance in modified gravity models with different background solutions and/or additional degrees of freedom, we present a new approach that analyses directly the first-order differential system in its original form and extracts the asymptotic behaviour of perturbations, without resorting to a second-order reformulation. As a pedagogical illustration, we apply this treatment to the perturbations of Schwarzschild black holes and then show that the standard quasi-normal modes can be obtained numerically by solving this first-order system with a spectral method. This new approach paves the way for a generic treatment of the asymptotic behaviour of black hole perturbations and the identification of quasi-normal modes in theories of modified gravity.
We study the quasinormal frequencies of three effective geometries of nonrotating regular black holes derived from loop quantum gravity. Concretely, we consider the Ashtekar-Olmedo-Singh and two Gambini-Olmedo-Pullin prescriptions. We compute the quasinormal frequencies of axial and polar perturbations adopting a WKB method. We show that they differ from those of classical general relativity and, more importantly, that isospectrality is broken. Nevertheless, these deviations are tiny, even for microscopic black holes, and they decay following an inverse power law of the size of the mass of the…
We study scalar and electromagnetic perturbations of a family of nonsingular nonrotating black hole spacetimes that are solutions in a large class of conformally invariant theories of gravity. The effective potential for scalar perturbations depends on the exact form of the scaling factor. Electromagnetic perturbations do not feel the scaling factor, and the corresponding quasinormal mode spectrum is the same as in the Schwarzschild metric. We find that these black hole metrics are stable under scalar and electromagnetic perturbations. Assuming that the quasinormal mode spectrum for scalar per…
Linear perturbations of spherically symmetric spacetimes in general relativity are described by radial wave equations, with potentials that depend on the spin of the perturbing field. In previous work we studied the quasinormal mode spectrum of spacetimes for which the radial potentials are slightly modified from their general relativistic form, writing generic small modifications as a power-series expansion in the radial coordinate. We assumed that the perturbations in the quasinormal frequencies are linear in some perturbative parameter, and that there is no coupling between the perturbation…
Quasinormal modes of rotating black holes beyond general relativity in the WKB approximation
Observatoire de Paris, Universite PSL（巴黎天文台，巴黎文理研究大学）；CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL（里斯本大学，技术研究所物理系与理论研究中心）；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））；Johns Hopkins University（约翰斯·霍普金斯大学）
We study the linear perturbations about a nonrotating black hole solution of Horndeski's theory, using a systematic approach that extracts the asymptotic behaviour of perturbations (at spatial infinity and near the horizon) directly from the first-order radial differential system governing these perturbations instead of finding Schrödinger-like equations for their dynamics. We illustrate this method in the case of a specific black hole solution. The knowledge of the asymptotic behaviours of the perturbations paves the way for a numerical computation of the quasinormal modes. Finally, the asymp…
Quasinormal modes and their excitation beyond general relativity. II: isospectrality loss in gravitational waveforms
The Grainger College of Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格兰杰工程学院）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；ETH Zürich（苏黎世联邦理工学院）；Universidad de Murcia（穆尔西亚大学）；University of Tübingen（蒂宾根大学）；University of Virginia（弗吉尼亚大学）
We study the quasi-normal modes of asymptotically anti-de Sitter black holes in a class of shift-symmetric Horndeski theories where a gravitational scalar is derivatively coupled to the Einstein tensor. The space-time differs from exact Schwarzschild-anti-de Sitter, resulting in a different effective potential for the quasi-normal modes and a different spectrum. We numerically compute this spectrum for a massless test scalar coupled both minimally to the metric, and non-minimally to the gravitational scalar. We find interesting differences from the Schwarzschild-anti-de Sitter black hole found…
Despite recent progress, the complete understanding of the perturbations of charged, rotating black holes as described by the Kerr-Newman metric remains an open and fundamental problem in relativity. In this study, we explore the existence of families of quasinormal modes of Kerr-Newman black holes whose decay rates limit to zero at extremality, called zero-damped modes in past studies. We review the nearly extremal and WKB approximation methods for spin-weighted scalar fields (governed by the Dudley-Finley equation) and give an accounting of the regimes where scalar zero-damped and damped mod…
We discuss the quasi-normal oscillations of black holes which are sourced by a nonlinear electrodynamic field. While previous studies have focused on the computation of quasi-normal frequencies for the wave or higher spin equation on a fixed background geometry described by such black holes, here we compute for the first time the quasi-normal frequencies for the coupled electromagnetic-gravitational linear perturbations. To this purpose, we consider a parametrized family of Lagrangians for the electromagnetic field which contains the Maxwell Lagrangian as a special case. In the Maxwell case, t…
The odd parity gravitational Quasi-Normal Mode spectrum of black holes with non-trivial scalar hair in Horndeski gravity is investigated. We study `almost' Schwarzschild black holes such that any modifications to the spacetime geometry (including the scalar field profile) are treated as small quantities. A modified Regge-Wheeler style equation for the odd parity gravitational degree of freedom is presented to quadratic order in the scalar hair and spacetime modifications, and a parameterisation of the modified Quasi-Normal Mode spectrum is calculated. In addition, statistical error estimates f…
In this paper we study the stability and quasi-normal modes of scalar perturbations of black holes. The static charged black hole considered here is a solution to Born-Infeld electrodynamics coupled to gravity. We conclude that the black hole is stable. We also compare the stability of it with its linear counter-part Reissner-Nordstrom black hole. The quasi-normal modes are computed using the WKB method. The behavior of these modes with the non-linear parameter, temperature, mass of the scalar field and the spherical index are analyzed in detail.
Highly-damped quasi-normal frequencies for piecewise Eckart potentials
高阻尼的准正常频率通常为ω_n =（偏移）+ i n（间隙）的形式。我们调查这种现象的通用性，通过考虑模型的潜力，是分段Eckart（分段Poeschl-Teller），并开发一个分析的“量化条件”的高阻尼准正常频率。我们发现，这种Ω_n =（offset）+ i n（gap）的行为是一般的，但不是普遍的，其控制特征是两个渐近方向上的指数衰减率之比是否是有理数。这些意见是直接相关的任何物理情况下，高阻尼准正常模式（阻尼模式）是重要的-特别是（但不限于）黑洞物理，理论和观测。
In this work, we calculate the quasi-normal frequencies from a bumblebee traversable wormhole. The bumblebee wormhole model is based on the bumblebee gravity, which exhibits a spontaneous Lorentz symmetry breaking. Supporting by the Lorentz violation parameter $λ$, this model allows the fulfillment of the flare-out and energy conditions, granted non-exotic matter to the wormhole. We analyze the parameters of bumblebee wormhole in order to obtain a Reege-Wheeler's equation with a bell-shaped potential. We obtain the quasi-normal modes (QNMs) via the WKB approximation method for both scalar and …
Born-Infeld electrodynamics has attracted considerable interest due to its relation to strings and D-branes. In this paper the gravitational perturbations of electrically charged black holes in Einstein-Born-Infeld gravity are studied. Effective potentials for axial perturbations are derived and discussed. The quasi normal modes for the gravitational perturbations are computed using the WKB method. The modes are compared with those of the Reissner-Nordstrom black hole. The relation of the quasi normal modes with the non-linear parameter and the spherical index are also investigated. Comments o…
Quasi-normal modes of toroidal, cylindrical and planar black holes in anti-de Sitter spacetimes: scalar, electromagnetic and gravitational perturbations
We study the perturbations to General Relativistic black holes (i.e. those without scalar hair) in Horndeski scalar-tensor gravity. First, we derive the equations of odd and even parity perturbations of both the metric and scalar field in the case of a Schwarzschild black hole, and show that the gravitational waves emitted from such a system contain a mixture of quasi-normal mode frequencies from the usual General Relativistic spectrum and those from the new scalar field spectrum, with the new scalar spectrum characterised by just two free parameters. We then specialise to the sub-family of Ho…
$SL(2,R)$ symmetry and quasi-normal modes in the BTZ black hole
借助于$SL(2,R)$二次Casimir of Killing场的两个新的内禀张量场，我们在不借助任何特定坐标系的情况下，通过张量和旋量分析，统一地揭示了BTZ黑洞中各种场运动方程的解所满足的$SL(2,R)$对称性.然后利用最近发展起来的标准代数方法，我们确定了BTZ黑洞中各种场的准正规模。结果表明，准简正模是由手征最高权模的无穷多个子模组成的，这与以往精确求解运动方程得到的解析结果是一致的.
We analyze the instability of the non-rotating fermion vacuum in Kerr spacetimes. We describe how the co-rotating Fermi sea is formed as a result of a spontaneous vacuum decay. Most significantly, and drawing upon intuition gained from analogous electrodynamic processes in supercritical fields, we show that this decay process is encoded entirely in a subset of quasi-normal fermion modes.
The low-laying frequencies of characteristic quasi-normal modes (QNM) of Schwarzschild-de Sitter (SdS) black holes have been calculated for fields of different spin using the 6th-order WKB approximation and the approximation by the Pöshl-Teller potential. The well-known asymptotic formula for large $l$ is generalized here on a case of the Schwarzchild-de Sitter black hole. In the limit of the near extreme $Λ$ term the results given by both methods are in a very good agreement, and in this limit fields of different spin decay with the same rate.
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…
