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, are obtained by solving certain decoupled ordinary differential equations involving the stress tensor. As we show, solving these ordinary differential equations reduces simply to integrations in the coordinate $r$ in outgoing Kerr-Newman coordinates, so in this sense, the problem is reduced to the Teukolsky equation with source, which can be treated by a separation of variables ansatz. Since higher order perturbations are subject to a linearized Einstein equation with a stress tensor obtained from the lower order perturbations, our method also applies iteratively to the higher order metric perturbations, and could thus be used to analyze the nonlinear coupling of perturbations in the near-extremal Kerr spacetime, where weakly turbulent behavior has been conjectured to occur. Our method could also be applied to the study of perturbations generated by a pointlike body traveling on a timelike geodesic in Kerr, which is relevant to the extreme mass ratio inspiral problem.
We use Heun type solutions given in \cite{Suzuki} for the radial Teukolsky equation, written in the background metric of the Kerr-Newman-de Sitter geometry, to calculate the quasinormal frequencies for polynomial solutions and the reflection coefficient for waves coming from the de Sitter horizon and reflected at the outer horizon of the black hole.
This paper concludes the study, initiated by the authors in arXiv:2007.07211, of the Teukolsky equation of spin $\pm 1$ and spin $\pm 2$ on Kerr backgrounds in the full subextremal range of parameters $|a| < M$. In our previous arXiv:2007.07211, we obtained uniform-in-frequency estimates for the ODEs governing separable solutions to the Teukolsky equation. In this paper, by adapting the techniques developed by the first author with Dafermos and Rodnianski for scalar waves, we show that our ODE estimates can be upgraded to estimates for the Teukolsky PDE. In particular, we conclude the proof 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…
This paper is the first of a series regarding the Teukolsky equation of spin $\pm 1$ and spin $\pm 2$ on Kerr backgrounds in the full subextremal range of parameters $|a|<M$. In the present paper, we study fixed frequency solutions of the transformed system of equations introduced by Dafermos, Holzegel and Rodnianski, obtaining estimates which are uniform in the separation parameters. A corollary of our result, to be laid out in the second paper of the series, is that solutions of the Teukolsky equation on subextremal Kerr arising from regular initial data remain bounded and decay in time. Thi…
We derive the equations governing the linear stability of Kerr-Newman spacetime to coupled electromagnetic-gravitational perturbations. The equations generalize the celebrated Teukolsky equation for curvature perturbations of Kerr, and the Regge-Wheeler equation for metric perturbations of Reissner-Nordström. Because of the "apparent indissolubility of the coupling between the spin-1 and spin-2 fields", as put by Chandrasekhar, the stability of Kerr-Newman spacetime can not be obtained through standard decomposition in modes. Due to the impossibility to decouple the modes of the gravitational …
We derive the large time asymptotics of initially regular and localized solutions of the Teukolsky equation on the exterior of a subextremal Kerr black hole for any half integer spin. More precisely, we obtain the leading order term (predicted by Price's law) in the large time regime assuming that the initial data have compact support and have enough (but finite) Sobolev regularity. For initial data with less spatial decay (typically decaying like r^{--1--$α$} with $α$ $\in$ (0, 1)), we prove that the solution has a pointwise decay of order t^{--1--$α$--s--|s|+} on spatially compact regions. I…
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…
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 …
We use the frequency and time domain Teukolsky formalism to calculate gravitational-wave fluxes from a spinning body on a bound eccentric equatorial orbit around a Kerr black hole. The spinning body is represented as a point particle following the pole-dipole approximation of the Mathisson-Papapetrou-Dixon equations. Reformulating these equations we are not only able to find the trajectory of a spinning particle in terms of its constants of motion, but also to provide a method to calculate the azimuthal and the radial frequency of this trajectory. Using these orbital quantities, we introduce t…
We present numerical waveforms of gravitational-wave echoes from spinning exotic compact objects (ECOs) that result from binary black hole coalescence. We obtain these echoes by solving the Teukolsky equation for the $ψ_4$ associated with gravitational waves that propagate toward the horizon of a Kerr spacetime, and process the subsequent reflections of the horizon-going wave by the surface of the ECO, which lies right above the Kerr horizon. The trajectories of the infalling objects are modified from Kerr geodesics, such that the gravitational waves propagating toward future null infinity mat…
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.
The Teukolsky Master Equation describes the dynamics of massless fields with spin on a Kerr black hole. Under separation of variables, spin-reversal for this equation is accomplished through the so-called Teukolsky--Starobinsky identities. These identities are associated to the so-called Teukolsky--Starobinsky constants, which are spin-dependent. We collect some properties of the Teukolsky--Starobinsky constants and dispel some myths present in the literature. We show that, contrary to popular belief, these constants can be negative for spin larger than 2. Such fields thus exhibit a novel form…
A self-consistent EOB--Teukolsky framework for generic extreme mass-ratio inspirals
Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院，中国科学院大学）；Shanghai Astronomical Observatory（上海天文台）；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）；Shanghai University of Engineering Science（上海工程技术大学）
Quasinormal modes of usual, four dimensional, Kerr black holes are described by certain solutions of a confluent Heun differential equation. In this work, we express these solutions in terms of the connection matrices for a Riemann-Hilbert problem, which was recently solved in terms of the Painlevé V transcendent. We use this formulation to generate the small-frequency expansion for the angular spheroidal harmonic eigenvalue, and derive conditions on the monodromy properties for the radial modes. Using exponentiation, we relate the accessory parameter to a semi-classical conformal description …
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…
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）
