A Non-degenerate Scattering Theory for the Wave Equation on Extremal Reissner–Nordström

Angelopoulos, Yannis; Aretakis, Stefanos; Gajic, Dejan

doi:10.1007/s00220-020-03857-3

Cited by 13 publications

(23 citation statements)

References 48 publications

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“…When |e| = M, the mode solutions corresponding to any homogeneous solution to the ODE are smooth at the horizon and at infinity with respect to 1/r , which is consistent with the scattering theory viewpoint in[8] where exponentially decaying data along future null infinity and the event horizon that lead to smooth solutions can be chosen independently.…”

supporting

confidence: 78%

“…See also [6] for an illustration of the importance of conserved quantities along future null infinity in the sub-extremal setting. [8], a scattering theory is developed in extremal Reissner-Nordström involving non-degenerate energies. As an application of this theory, it is shown that for any smooth, superpolynomially decaying scattering data along the event horizon and future null infinity, there exists a corresponding unique solution to (1.2) that is smooth along an asymptotically null foliation and is moreover smooth with respect to the coordinate 1 r at r = ∞.…”

Section: Uniform Decay Estimatesmentioning

confidence: 99%

“…7 The exploitation of the aforementioned conservation laws along future null infinity and the event horizon plays a fundamental role in the new type of L 2 -based "Gevrey estimates" that are introduced in this paper and involve functions that are Gevrey regular at infinity and near the horizon, a notion that first appeared in [43]. 8 In future work, we hope to explore applications and extensions of the methods introduced in the present paper to a quantitative study of QNMs in sub-extremal Reissner-Nordström and Kerr spacetimes. 9 A key feature of the methods introduced in the present paper is the demonstration that, in contrast with previous approaches, QNMs may be interpreted as honest eigenfunctions of the infinitesimal generator of time translations acting on a suitable Hilbert space of initial data, thus placing them on a similar footing to normal modes.…”

Section: Introductionmentioning

confidence: 99%

“…The results of the present paper demonstrate that such spaces are natural candidate initial data spaces for investigating the role of quasinormal modes in the time evolution of solutions to (1.2). Note that weighted L 2 -based Sobolev spaces are not well-suited for this purpose, see Theorem 4.5 and Remark 4.4 in [8] and the discussion in Sect. 1.5.2.…”

Section: Introductionmentioning

confidence: 99%

“…2.1 for more details). 8 The spaces of Gevrey functions can be viewed as lying "between" smooth and analytic functions. See Sect.…”

mentioning

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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supporting

confidence: 78%

Section: Uniform Decay Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…2.1 for more details). 8 The spaces of Gevrey functions can be viewed as lying "between" smooth and analytic functions. See Sect.…”

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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A Non-degenerate Scattering Theory for the Wave Equation on Extremal Reissner–Nordström

Angelopoulos

Aretakis

Gajic

2020

Commun. Math. Phys.

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It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction in time. In the extremal case, however, the redshift effect degenerates and hence yields a much milder blueshift effect when viewed in the backwards direction. In this paper, we construct a definitive (bijective) non-degenerate scattering theory for the wave equation on extremal Reissner–Nordström backgrounds. We make use of physical-space energy norms which are non-degenerate both at the event horizon and at null infinity. As an application of our theory we present a construction of a large class of smooth, exponentially decaying modes. We also derive scattering results in the black hole interior region.

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A Non-degenerate Scattering Theory for the Wave Equation on Extremal Reissner–Nordström

Cited by 13 publications

References 48 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

A Non-degenerate Scattering Theory for the Wave Equation on Extremal Reissner–Nordström

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