Linear stability of Schwarzschild spacetime: Decay of metric coefficients

Hung, Pei Ken; Keller, Jordan; Wang, Mu-Tao

doi:10.4310/jdg/1606964416

Cited by 28 publications

(34 citation statements)

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“…Recently, there have been significant advances towards a mathematical proof of the Kerr Stability Conjecture in linearized settings, see [28,29,33,53,61] and also [2,45,51]. 2 This conjecture asserts that initially small and localized metric perturbations of sub-extremal Kerr black hole solutions to (1.1) should decay in time and the metric should asymptotically approach a nearby Kerr solution.…”

Section: Introductionmentioning

confidence: 99%

Quasinormal Modes in Extremal Reissner–Nordström Spacetimes

Gajic

Warnick

2021

Commun. Math. Phys.

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We present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner–Nordström black holes. We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing. The main difficulty that is present in the asymptotically flat setting, but is absent in the previously studied asymptotically de Sitter or anti de Sitter sub-extremal black hole spacetimes, is that $$L^2$$ L 2 -based Sobolev spaces are not suitable Hilbert space choices. Instead, we consider Hilbert spaces of functions that are additionally Gevrey regular at infinity and at the event horizon. We introduce $$L^2$$ L 2 -based Gevrey estimates for the wave equation that are intimately connected to the existence of conserved quantities along null infinity and the event horizon. We relate this new framework to the traditional interpretation of quasinormal frequencies as poles of the meromorphic continuation of a resolvent operator and obtain new quantitative results in this setting.

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Section: Introductionmentioning

confidence: 99%

Quasinormal Modes in Extremal Reissner–Nordström Spacetimes

Gajic

Warnick

2021

Commun. Math. Phys.

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“…In the end, we draw attention to the progress on black hole stability problem in recent years. Linear stability of a Schwarzschild or a subextremal Reissner-Nordström spacetime has been shown by [28,52,7,53,50,51,38], and linear stability of a slowly rotating Kerr spacetime is proven in [3,44,4]. For nonlinear stability results, we refer to [61,29] for Schwarzschild, [48] for slowly rotating Kerr-de Sitter, and [60,39,62] for slowly rotating Kerr.…”

Section: Introductionmentioning

confidence: 99%

Sharp decay for Teukolsky equation in Kerr spacetimes

Ma,

Zhang

2021

Preprint

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In this work, we derive the global sharp decay, as both a lower and an upper bounds, for the spin ±s components, which are solutions to the Teukolsky equation, in the black hole exterior and on the event horizon of a slowly rotating Kerr spacetime. These estimates are generalized to any subextreme Kerr background under an integrated local energy decay estimate. Our results apply to the scalar field (s = 0), the Maxwell field (s = 1) and the linearized gravity (s = 2) and confirm the Price's law decay that is conjectured to be sharp. Our analyses rely on a novel global conservation law for the Teukolsky equation, and this new approach can be applied to derive the precise asymptotics for solutions to semilinear wave equations.

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“…In brief, the problem is to show that, for any initial data that generates a solution containing a Kerr exterior, any sufficiently small perturbation of such initial data will generate a solution which contains a region that, in the future, converges to some Kerr exterior. So far, most work has focused on the linearized Einstein equation and models for it, such as the wave and Maxwell equations [6,8,13,7,5,31,2,3,14,27,22] and the linearized gravity [11,19,23,10,1,17]. Quite recently, a few works [20,12,21] have made important progress on the full nonlinear stability of Kerr spacetimes.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear radiation gauge for near Kerr spacetimes

Andersson,

Bäckdahl,

Blue

et al. 2021

Preprint

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In this paper, we introduce and explore the properties of a new gauge choice for the vacuum Einstein equation inspired by the ingoing and outgoing radiation gauges (IRG, ORG) for the linearized vacuum Einstein equation introduced by Chrzanowski in his work on metric reconstruction [9] on the Kerr background. It has been shown by Price, Shankar and Whiting [29] that IRG/ORG are consistent gauges for the linearized vacuum Einstein equation on Petrov type II backgrounds. In [1] ORG was used in proving linearized stability for the Kerr spacetime, and the new non-linear radiation gauge introduced here is a direct generalization of that gauge condition, and is intended to be used to study the stability of Kerr black holes under the evolution generated by the vacuum Einstein equation.

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Linear stability of Schwarzschild spacetime: Decay of metric coefficients

Cited by 28 publications

References 0 publications

Quasinormal Modes in Extremal Reissner–Nordström Spacetimes

Quasinormal Modes in Extremal Reissner–Nordström Spacetimes

Sharp decay for Teukolsky equation in Kerr spacetimes

Nonlinear radiation gauge for near Kerr spacetimes

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