We study gravitational perturbations on the near-horizon region of extremal and near-extremal rotating black holes in a general higher-derivative extension of Einstein gravity. We find a decoupled modified Teukolsky equation that rules the gravitational perturbations and that separates into an angular and a radial equation. The angular equation leads to a deformation of the spin-weighted spheroidal harmonics, while the radial equation takes the same form as in Kerr except for a modification of the angular separation constants. We provide a detailed analysis of the corrections to these angular separation constants and find analytic results for axisymmetric modes as well as in the eikonal limit. As an application, we reproduce recent results that show that extremal Kerr black holes in higher-derivative gravity become singular under certain deformations and extend them by including parity-breaking corrections, which we show lead to the same effect. Finally, we obtain constraints on the form of the full modified Teukolsky radial equation by demanding that it has the right near-horizon limit. These results serve as an stepping stone towards the study of quasinormal modes of near-extremal black holes in higher-derivative extensions of GR.
We conduct an analysis of the quasi-normal modes for generic spin perturbations of the Kerr black hole using the isomonodromic method. The strategy consists of solving the Riemann-Hilbert map relating the accessory parameters of the differential equations involved to monodromy properties of the solutions, using the $τ$-function for the Painlevé V transcendent. We show good accordance of the method with the literature for generic rotation parameter $a<M$. In the extremal limit, we determined the dependence of the modes with the black hole temperature and establish that the extremal values of th…
We prove that there are no exponentially growing modes nor modes on the real axis for the Teukolsky equation on Kerr black hole spacetimes, both in the extremal and subextremal case. We also give a quantitative refinement of mode stability. As an immediate application, we show that the transmission and reflection coefficients of the scattering problem are bounded, independently of the specific angular momentum of the black hole, in any compact set of real frequencies excluding zero frequency and the superradiant threshold. While in the subextremal setting these results were known previously, t…
We study the quasinormal mode eigenvalues and eigenfunctions for the Teukolsky equation in a horizon penetrating, hyperboloidally compactified (HPHC) coordinate system. Following earlier work by Zenginoğlu (arXiv:1102.2451), we show that the quasinormal eigenfunctions for the Teukolsky equation are regular from the black hole horizon to future null infinity in these coordinates. We then present several example quasinormal eigenfunction solutions, and study some of their properties in the near-extremal Kerr limit.
We develop a formalism to treat higher order (nonlinear) metric perturbations of the Kerr spacetime in a Teukolsky framework. We first show that solutions to the linearized Einstein equation with nonvanishing stress tensor can be decomposed into a pure gauge part plus a zero mode (infinitesimal perturbation of the mass and spin) plus a perturbation arising from a certain scalar ("Debye-Hertz") potential, plus a so-called "corrector tensor." The scalar potential is a solution to the spin $-2$ Teukolsky equation with a source. This source, as well as the tetrad components of the corrector tensor…
We study here the quasi-normal mode stability of a near-extremal Kerr superspinar, an exotic spinning compact object that exceeds the Kerr bound, under gravitational perturbations. Despite previous beliefs that these objects would be mode unstable, we show by analytically treating the Teukolsky equations that these objects are in fact mode stable under almost all (barring a zero-measure set) boundary conditions.
This work outlines a fast, high-precision time-domain solver for scalar, electromagnetic and gravitational perturbations on hyperboloidal foliations of Kerr space-times. Time-domain Teukolsky equation solvers have typically used explicit methods, which numerically violate Noether symmetries and are Courant-limited. These restrictions can limit the performance of explicit schemes when simulating long-time extreme mass ratio inspirals, expected to appear in LISA band for 2-5 years. We thus explore symmetric (exponential, Padé or Hermite) integrators, which are unconditionally stable and known to…
We prove boundedness and polynomial decay statements for solutions of the spin $\pm2$ Teukolsky equation on a Kerr exterior background with parameters satisfying $|a|\ll M$. The bounds are obtained by introducing generalisations of the higher order quantities $P$ and $\underline{P}$ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters $|a|<M$. As in the Schwarzschild …
We study the existence and nature of Aretakis "hair" and its potentially observable imprint at a finite distance from the horizon (Ori-coefficient) in near-extremal black hole backgrounds. Specifically, we consider the time evolution of horizon penetrating scalar and gravitational perturbations with compact support on near-extremal Reissner-Nordstrom (NERN) and Kerr (NEK). We do this by numerically solving the Teukolsky equation and determining the Aretakis charge values on the horizon and at a finite distance from the black hole. We demonstrate that these values are no longer strictly conserv…
Constraining deviations from the Teukolsky equation with GW250114
University of Tübingen（蒂宾根大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Universidade de Lisboa（里斯本大学）
Parametrized beyond-Teukolsky framework in the time domain
Universitá di Napoli “Federico II”（那不勒斯费德里科二世大学）；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）；Scuola Superiore Meridionale（南方高等学院）
Existing constraints on the graviton mass from gravitational-wave detections rely on the phase difference developed between different frequencies during the propagation. Effects on the quasinormal-mode frequencies of the black-hole ringdown due to the graviton mass are often ignored. While perturbation theories of black holes have been well developed in the context of general relativity, this is not the case for modified gravity theories. We propose a phenomenological modification to the Teukolsky equation of perturbed black holes to include the dispersion relation due to a gravitational field…
This paper establishes a mathematical proof of the blue-shift instability at the sub-extremal Kerr Cauchy horizon for the linearised vacuum Einstein equations. More precisely, we exhibit conditions on the $s=+2$ Teukolsky field, consisting of suitable integrated upper and lower bounds on the decay along the event horizon, that ensure that the Teukolsky field, with respect to a frame that is regular at the Cauchy horizon, becomes singular. The conditions are in particular satisfied by solutions of the Teukolsky equation arising from generic and compactly supported initial data by the recent wor…
Motivated by the need of a robust geometrical framework for the calculation of long, and highly accurate waveforms for extreme-mass-ratio inspirals, this work presents an extensive study of the hyperboloidal formalism for the Kerr spacetime and the Teukolsky equation. In a first step, we introduce a generic coordinate system foliating the Kerr spacetime into hypersurfaces of constant time extending between the black-hole horizon and future null infinity, while keeping track of the underlying degrees of freedom. Then, we express the Teukolsky equation in terms of these generic coordinates with …
Long-Lived Ringing of Near-Extremal Kerr Black Holes Resonantly Driven by Extreme-Mass-Ratio Inspirals
State Key Laboratory of Radio Astronomy and Technology, Shanghai Astronomical Observatory, CAS（中国科学院上海天文台射电天文技术国家重点实验室）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（中国科学院大学基础科学与数学学院杭州高等研究院）；School of Astronomy and Space Science, University of Chinese Academy of Sciences（中国科学院大学天文与空间科学学院）
We examine the problem of overcharging extremal and near-extremal black hole solutions of Einstein-Gauss-Bonnet gravity in any dimension, generalizing the result in general relativity. We show that as in the case of general relativity, it is not possible to create a naked singularity by overcharging an extremal black hole in Einstein-Gauss-Bonnet gravity using a charged test particle. Our result suggests that the validity of the cosmic censorship hypothesis transcends beyond general relativity to well motivated higher curvature gravity.
Kerr Quasinormal Modes without Variable Separation: A Two-Dimensional Hyperboloidal Teukolsky Solver with Physics-Informed Neural Networks
University of Valencia（瓦伦西亚大学）；Eberhard Karls University of Tübingen（蒂宾根大学）；Sofia University（索非亚大学）；Bulgarian Academy of Sciences（保加利亚科学院）；Instituto de Física Corpuscular CSIC-UV（CSIC-UV 粒子物理研究所）
We present a general framework for calculating post-Minskowskian, classical, conservative Hamiltonians for $N$ non-spinning bodies in general relativity from relativistic scattering amplitudes. Novel features for $N>2$ are described including the subtraction of tree-like iteration contributions and the calculation of non-trivial many-body Fourier transform integrals needed to construct position space potentials. A new approach to calculating these integrals as an expansion in the hierarchical limit is described based on the method of regions. As an explicit example, we present the $\mathcal{O}…
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）
The Weak Cosmic Censorship (WCC) conjecture can be used as a consistency criterion for any viable modification over general relativity (GR). We employ this idea to show that in contrast to the black holes in GR, it is indeed possible to overcharge modified extremal black hole solutions. Demanding the validity of WCC, we put some stringent constraints on various parameters of the theory. In particular, for charged Einstein-aether black hole, our method is strong enough to reproduce the identical bound on the parameter obtained previously from entirely different considerations.
We study linear perturbations of a rotating black hole solution that has been recently discovered in degenerate higher-order scalar-tensor (DHOST) theories. We find a parametrization which permits the explicit resolution of the scalar perturbation while the tensor perturbation is obtained as a Teukolsky equation supplemented by an effective source term. The effective source term is related to the black hole hair and can be computed exactly for any value of the black hole spin. We discuss how the perturbations of the geometry and thus the emitted gravitational waves could be modified in compari…
Conserved currents associated with the time translation and axial symmetries of the Kerr spacetime and with scaling symmetry are constructed for the Teukolsky Master Equation (TME). Three partly different approaches are taken, of which the third one applies only to the spacetime symmetries. The results yielded by the three approaches, which correspond to three variants of Noether's theorem, are essentially the same, nevertheless. The construction includes the embedding of the TME into a larger system of equations, which admits a Lagrangian and turns out to consist of two TMEs with opposite spi…
Numerical relativity (NR) simulations of binary black hole (BBH) systems provide the most accurate gravitational wave predictions, but at a high computational cost -- especially when the black holes have nearly extremal spins (i.e. spins near the theoretical upper limit) or very unequal masses. Recently, the technique of Reduced Order Modeling (ROM) has enabled the construction of surrogate models trained on an existing set of NR waveforms. Surrogate models enable the rapid computation of the gravitational waves emitted by BBHs. Typically these models are used for interpolation to compute grav…
Perturbative corrections to General Relativity alter the expressions for both the entropy of black holes and their extremality bounds. We prove a universal relation between the leading corrections to these quantities. The derivation is purely thermodynamic and the result also applies beyond the realm of gravitational systems. In scenarios where the correction to the entropy is positive, our result proves that the perturbations decrease the mass of extremal black holes, when holding all other extensive variables fixed in the comparison. This implies that the extremality relations of a wide clas…
We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, $ |\mathcal{Q}|=M, $ Reissner-Nordström spacetime, as a solution to the Einstein-Maxwell equations. Our work uses and extends the framework \cite{giorgi2019boundedness,giorgie2020boundedness} of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon $ \mathcal{H}^+ $. In particular, we prove decay, non-decay, and polynomial blow-up estimates asymptotically along $ \mathcal{H}^+ $, the exact behavio…
A Buchdahl star is a highly compact star for which the boundary radius $R$ obeys $R=\frac98 r_+$, where $r_+$ is the gravitational radius of the star itself. A quasiblack hole is a maximum compact star, or more generically a maximum compact object, for which the boundary radius $R$ obeys $R=r_+$. Quasiblack holes are objects on the verge of becoming black holes. Continued gravitational collapse ends in black holes and has to be handled with the Oppenheimer-Snyder formalism. Quasistatic contraction ends in a quasiblack hole and should be treated with appropriate techniques. Quasiblack holes, no…
