Instabilities of Extremal Rotating Black Holes in Higher Dimensions

Hollands, Stefan; Ishibashi, Akihiro

doi:10.1007/s00220-015-2410-0

Cited by 22 publications

(51 citation statements)

References 99 publications

(320 reference statements)

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“…Our analysis also applies to axisymmetric electromagnetic perturbations with compact support initial data on stationary-axisymmetric (asymptotically flat or deSitter) extremal black hole backgrounds. In particular, our results imply that the criterion for instability, in terms of violating the near-horizon "effective Breitenlöhner-Freedman bound", given in Sec.5.2 and 6.4 [25] is never satisfied for axisymmetric electromagnetic perturbations, in agreement with the numerical evidence in Sec.III.E [26].…”

Section: Positivity Of E For Axisymmetric Perturbationssupporting

confidence: 88%

Stability of stationary-axisymmetric black holes in vacuum general relativity to axisymmetric electromagnetic perturbations

Prabhu

Wald

2017

Class. Quantum Grav.

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We consider arbitrary stationary and axisymmetric black holes in general relativity in (d + 1) dimensions (with d ≥ 3) that satisfy the vacuum Einstein equation and have a non-degenerate horizon. We prove that the canonical energy of axisymmetric electromagnetic perturbations is positive definite. This establishes that all vacuum black holes are stable to axisymmetric electromagnetic perturbations. Our results also hold for asymptotically deSitter black holes that satisfy the vacuum Einstein equation with a positive cosmological constant. Our results also apply to extremal black holes provided that the initial perturbation vanishes in a neighborhood of the horizon.

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Section: Positivity Of E For Axisymmetric Perturbationssupporting

confidence: 88%

Stability of stationary-axisymmetric black holes in vacuum general relativity to axisymmetric electromagnetic perturbations

Prabhu

Wald

2017

Class. Quantum Grav.

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“…We treat electromagnetic and gravitational perturbations using the Hertz potential formalism [36,[40][41][42][43][44]. We choose any stationary, axisymmetric null basis for spacetime that reduces to Eqs.…”

Section: E Electromagnetic and Gravitational Perturbationsmentioning

confidence: 99%

Scaling and universality in extremal black hole perturbations

Gralla

Zimmerman

2018

J. High Energ. Phys.

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We show that the emergent near-horizon conformal symmetry of extremal black holes gives rise to universal behavior in perturbing fields, both near and far from the black hole horizon. The scale-invariance of the near-horizon region entails power law timedependence with three universal features: (1) the decay off the horizon is always precisely twice as fast as the decay on the horizon; (2) the special rates of 1/t off the horizon and 1/ √ v on the horizon commonly occur; and (3) sufficiently high-order transverse derivatives grow on the horizon (Aretakis instability). The results are simply understood in terms of nearhorizon (AdS 2 ) holography. We first show how the general features follow from symmetry alone and then go on to present the detailed universal behavior of scalar, electromagnetic, and gravitational perturbations of d-dimensional electrovacuum black holes.

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“…However, we can effectively expand 15 Note that since Y ⊂ X , this and the previous set inclusion can hold only if equality holds. 16 Prop. 3.1 was proven for smooth γ, whereas, a priori, the γ appearing in Eq.…”

Section: Generation Of Metric Perturbations By Hertz Potentialsmentioning

confidence: 99%

Canonical energy and Hertz potentials for perturbations of Schwarzschild spacetime

Prabhu

Wald

2018

Class. Quantum Grav.

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Canonical energy is a valuable tool for analyzing the linear stability of black hole spacetimes; positivity of canonical energy for all perturbations implies mode stability, whereas the failure of positivity for any perturbation implies instability. Nevertheless, even in the case of 4-dimensional Schwarzschild spacetime -which is known to be stable -manifest positivity of the canonical energy is difficult to establish, due to the presence of constraints on the initial data as well as the gauge dependence of the canonical energy integrand. Consideration of perturbations generated by a Hertz potential would appear to be a promising way to improve this situation, since the constraints and gauge dependence are eliminated when the canonical energy is expressed in terms of the Hertz potential. We prove that the canonical energy of a metric perturbation of Schwarzschild that is generated by a Hertz potential is positive. We relate the energy quantity arising in the linear stability proof of Dafermos, Holzegel and Rodnianski (DHR) to the canonical energy of an associated metric perturbation generated by a Hertz potential. We also relate the Regge-Wheeler variable of DHR to the ordinary Regge-Wheeler and twist potential variables of the associated perturbation. Since the Hertz potential formalism can be generalized to a Kerr black hole, our results may be useful for the analysis of the linear stability of Kerr. * kartikprabhu@cornell.edu † rmwa@uchicago.edu not been able to directly establish the positivity of the potential energy U , Eq. 1.3, even for the Schwarzschild background. 3 The two main obstacles to showing positivity of U appear to be the following: (1) The variable q ab is not free data but is subject to the linearized Hamiltonian constraint Eq. 1.4b.(2) Even though U is gauge-invariant, the integrand in Eq. 1.3 is not. Thus, it would seem that to show positivity of U one would need to utilize the constraint equations effectively and make a suitable choice of gauge (as was done for the proof of positivity of K in [8]). However, we have not, as yet, found a way to do this. A promising strategy to prove positivity would be rewrite the canonical energy in terms of unconstrained variables. In 4-dimensions, for the case of algebraically special spacetimes such as Schwarzschild and Kerr, a possible choice of such variables are the Hertz-Bromowich-Debye-Whittaker-Penrose potentials (henceforth Hertz potentials) [11][12][13][14]. The Hertz potentials, which solve the Teukolsky equation [15], can be used to generate (complex) metric perturbations that solve the Einstein equation. The initial data corresponding to the Hertz potentials is unconstrained, and one could attempt to prove positivity of canonical energy for perturbations generated by a Hertz potential. 4 The main purpose of this paper is to show that for perturbations of Schwarzschild generated by a Hertz potential, the canonical energy is indeed positive. Since the Hertz potential formalism can be straightforwardly extended to Kerr spacetime, it is possible t...

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Instabilities of Extremal Rotating Black Holes in Higher Dimensions

Cited by 22 publications

References 99 publications

Stability of stationary-axisymmetric black holes in vacuum general relativity to axisymmetric electromagnetic perturbations

Stability of stationary-axisymmetric black holes in vacuum general relativity to axisymmetric electromagnetic perturbations

Scaling and universality in extremal black hole perturbations

Canonical energy and Hertz potentials for perturbations of Schwarzschild spacetime

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