Late-time asymptotics for the wave equation on extremal Reissner–Nordström backgrounds

Angelopoulos, Yannis; Aretakis, Stefanos; Gajic, Dejan

doi:10.1016/j.aim.2020.107363

Cited by 36 publications

(57 citation statements)

References 152 publications

(316 reference statements)

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“…In [5] further results were obtained regarding (1.2) on extremal Reissner-Nordström, exploiting heavily the aforementioned connection between null infinity and the event horizon and the presence of conserved quantities along both. In particular, the precise leading-order behaviour in time was obtained for solutions to (1.2), demonstrating the presence of polynomial tails, first predicted in heuristics and numerics [60,68,72].…”

Section: Uniform Decay Estimatesmentioning

confidence: 99%

“…We refer to [7,31,33,64,71] and references therein for energy decay results along asymptotically hyperboloidal or null hypersurfaces in the asymptotically flat setting. Of particular relevance to the setting of the present paper is the Dafermos-Rodnianski r p -weighted energy method [31] and the extended methods in [5,7], which relate the existence of hierarchies of r -weighted energy estimates along asymptotically hyperboloidal or null hypersurfaces to polynomial energy time-decay rates.…”

Section: Uniform Decay Estimatesmentioning

confidence: 99%

“…In fact, early heuristic and numerical analysis of the linearized problem initiated by Price [44,59,69] suggested that one can do no better than proving inverse polynomial decay estimates, because at sufficiently late times, the leading-order behaviour should be exactly inverse polynomial (this is sometimes referred to as "Price's law"). The presence of so-called polynomial tails in the context of the linear wave equation on asymptotically flat, spherically symmetric black hole backgrounds has recently been proved in a mathematically rigorous setting [4][5][6], where it has moreover been connected to the existence of conservation laws along null hypersurfaces, first discovered by Newman and Penrose at null infinity [65] and discovered in a different guise by Aretakis at the event horizons of extremal black holes 3 [10][11][12][13]. 4 In Fourier space, polynomial tails can alternatively be related to the precise behaviour of resolvent operators corresponding to the wave equation near the zero time frequency, see [48,59,75].…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, in the proof of the nonlinear stability of slowly rotating Kerr-de Sitter black holes [49], it was shown that in the cosmological setting there are no polynomial tails and the asymptotic behaviour of gravitational radiation is instead determined by QNMs. 5 It remains an open problem in the asymptotically flat setting to reconcile the expected polynomial tails in gravitational radiation with the observed exponentially decaying and oscillating QNMs. In order to discern the dominant behaviour in a given time interval and to determine how this behaviour depends on the type of initial data, it is necessary to establish in the first place a suitably robust and quantitative understanding of QNMs in the context of the linear wave equation…”

Section: Introductionmentioning

confidence: 99%

“…See[14] for a comprehensive overview on extremal black holes 4. The connection between the conserved quantities of Newman-Penrose and Aretakis was first observed in[21,60] 5. We refer to Sect.…”

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confidence: 99%

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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: Uniform Decay Estimatesmentioning

confidence: 99%

Section: Uniform Decay Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Quasinormal Modes in Extremal Reissner–Nordström Spacetimes

Gajic

Warnick

2021

Commun. Math. Phys.

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The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

Kehrberger

2021

Ann. Henri Poincaré

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This paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity. We begin with a brief review of an argument due to Christodoulou (in: The Ninth Marcel Grossmann Meeting, World Scientific Publishing Company, Singapore, 2002) stating that Penrose’s proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by N infalling masses coming from past timelike infinity $$i^-$$ i - . Modelling gravitational radiation by scalar radiation, we then take a first step towards a dynamical understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein–Scalar field equations that arise from polynomially decaying boundary data, $$r\phi \sim t^{-1}$$ r ϕ ∼ t - 1 as $$t\rightarrow -\infty $$ t → - ∞ , on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, $$r\partial _v\phi =0$$ r ∂ v ϕ = 0 , on past null infinity. We show that if the initial Hawking mass at $$i^-$$ i - is nonzero, then, in accordance with the non-smoothness of $${\mathcal {I}}^+$$ I + , the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + reads $$\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})$$ ∂ v ( r ϕ ) = C r - 3 log r + O ( r - 3 ) for some non-vanishing constant C. In fact, the same logarithmic terms appear already in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background. As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting compactly supported scattering data for the linear (or coupled) wave equation on $${\mathcal {I}}^-$$ I - and on $${\mathcal {H}}^-$$ H - , we find that the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + generically contains logarithmic terms at second order, i.e. at order $$r^{-4}\log r$$ r - 4 log r .

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Existence of Zero-Damped Quasinormal Frequencies for Nearly Extremal Black Holes

Joykutty

2022

Ann. Henri Poincaré

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It has been observed that many spacetimes which feature a near-extremal horizon exhibit the phenomenon of zero-damped modes. This is characterised by the existence of a sequence of quasinormal frequencies which all converge to some purely imaginary number $$i\alpha $$ i α in the extremal limit and cluster in a neighbourhood of the line $${{\,\mathrm{Im}\,}}s=\alpha $$ Im s = α . In this paper, we establish that this property is present for the conformal Klein–Gordon equation on a Reissner–Nordström–de Sitter background. This follows from a similar result that we prove for a class of spherically symmetric black hole spacetimes with a cosmological horizon. We also show that the phenomenon of zero-damped modes is stable to perturbations that arise through adding a potential.

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Late-time asymptotics for the wave equation on extremal Reissner–Nordström backgrounds

Cited by 36 publications

References 152 publications

Quasinormal Modes in Extremal Reissner–Nordström Spacetimes

Quasinormal Modes in Extremal Reissner–Nordström Spacetimes

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

Existence of Zero-Damped Quasinormal Frequencies for Nearly Extremal Black Holes

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