Mode stability on the real axis

Andersson, Lars; Ma, Siyuan; Paganini, Claudio F.; Whiting, B. F.

doi:10.1063/1.4991656

Cited by 43 publications

(112 citation statements)

References 33 publications

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“…This equation can be solved by noting that it is like a fourth order heat equation with source on S 2 . The operator replacing the Laplacian in the ordinary heat equation is 1 4 (ððð ð +ð ð ðð ), where in Schwarzschild,ð,ð are given by (48) with a = 0.…”

Section: Theorem 2 Suppose H Ab Satisfies the Inhomogeneous Linearizmentioning

confidence: 99%

Teukolsky formalism for nonlinear Kerr perturbations

Green

Hollands

Zimmerman

2020

Class. Quantum Grav.

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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.

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Section: Theorem 2 Suppose H Ab Satisfies the Inhomogeneous Linearizmentioning

confidence: 99%

Teukolsky formalism for nonlinear Kerr perturbations

Green

Hollands

Zimmerman

2020

Class. Quantum Grav.

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“…This can be seen from a basic conservation-of-energy argument which, in fact, also applies to all real-frequency non-superradiant modes in either extremal or sub-extremal Kerr [36,41,70]. In sub-extremal Kerr, this result has been extended to the superradiant regime: the only mode with ω ∈ R and no incoming radiation is the trivial mode [79]. This then implies the non-existence of exponentially-growing modes in sub-extremal Kerr [64,79], a result which had been previously proven in [31] in a different way.…”

Section: Properties Of Qnms In Nekmentioning

confidence: 99%

“…Finally, we note that the result in [79] that no poles can lie on the real axis for a < M together with the fact that there cannot be a pole at the branch point ω SR [36,83] provides an argument -although not a rigorous proofthat no poles cross the real axis as a increases from 0 to M and so for the absence of poles in the upper plane in extremal Kerr.…”

Section: B Search For Unstable Modesmentioning

confidence: 99%

Spectroscopy of extremal and near-extremal Kerr black holes

Casals

Micchi

2019

Phys. Rev. D

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We investigate linear, spin-field perturbations of Kerr black holes in the extremal limit throughout the complex-frequency domain. We calculate quasi-normal modes of extremal Kerr, as well as of near-extremal Kerr, via a novel approach: using the method of Mano, Suzuki and Takasugi (MST). We also show how, in the extremal limit, a branch cut is formed at the superradiant-bound frequency, ωSR, via a simultaneous accumulation of quasi-normal modes and totally-reflected modes. For real frequencies, we calculate the superradiant amplification factor, which yields the amount of rotational energy that can be extracted from a black hole. In the extremal limit, this factor is the largest and it displays a discontinuity at ωSR for some modes. Finally, we find no exponentially-growing modes nor branch points on the upper-frequency plane in extremal Kerr after a numerical investigation, thus providing evidence of the mode-stability of this space-time away from the horizon.PACS numbers: * Electronic address: mcasals@cbpf.br,marc.casals@ucd.ie † Electronic address: luisflm@cbpf.br Apart from the superradiant BC stemming from ω SR , which is present in extremal Kerr only, there is another BC stemming from the frequency at the origin (ω = 0), which exists both in sub-extremal and extremal Kerr. The contribution from the BC near the origin to the full field perturbation is a power-law decay at late-times, in sub-extremal Kerr [19] (see [20][21][22][23][24] for higher-order contributions which include logarithmic terms) as well as in extremal Kerr [16]. This power-law decay, originally observed by Price [25], typically kicks in after the QNM exponential decay.Both superradiance and BCs have important conse-1 We consider the behaviour in time t of a field mode of frequency ω to be e −iωt .arXiv:1901.04586v2 [gr-qc]

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“…It has the useful property that it is of variational form, meaning that it can be derived from an action principle. Indeed, choosing the Dirichlet action 4 x is the volume measure induced by the Lorentzian metric), demanding criticality for first variations gives the scalar wave equation…”

Section: The Scalar Wave Equation In the Kerr Geometrymentioning

confidence: 99%

“…The remaining issue is that the integrands in this representation might have poles on the real axis. These so-called radiant modes are ruled out by a causality argument (see [30,Section 11]; for an alternative proof see [4]). We thus obtain the following result (see [30,Theorem 12.1]).…”

Section: Separation Of the Resolvent And Contour Deformationsmentioning

confidence: 99%

Lectures on Linear Stability of Rotating Black Holes

Finster

2019

Einstein Equations: Physical and Mathematical Aspects of General Relativity

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These lecture notes are concerned with linear stability of the nonextreme Kerr geometry under perturbations of general spin. After a brief review of the Kerr black hole and its symmetries, we describe these symmetries by Killing fields and work out the connection to conservation laws. The Penrose process and superradiance effects are discussed. Decay results on the long-time behavior of Dirac waves are outlined. It is explained schematically how the Maxwell equations and the equations for linearized gravitational waves can be decoupled to obtain the Teukolsky equation. It is shown how the Teukolsky equation can be fully separated to a system of coupled ordinary differential equations. Linear stability of the non-extreme Kerr black hole is stated as a pointwise decay result for solutions of the Cauchy problem for the Teukolsky equation. The stability proof is outlined, with an emphasis on the underlying ideas and methods.

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Mode stability on the real axis

Cited by 43 publications

References 33 publications

Teukolsky formalism for nonlinear Kerr perturbations

Teukolsky formalism for nonlinear Kerr perturbations

Spectroscopy of extremal and near-extremal Kerr black holes

Lectures on Linear Stability of Rotating Black Holes

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