Abstract
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-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-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.
Highlights
An important problem in the theory of general relativity is to classify the behaviour of gravitational radiation emitted by dynamical solutions to the vacuum Einstein equations Ric[g] = 0, (1.1)where g is a Lorentzian metric and Ric[g] is the corresponding Ricci tensor
It is expected that a significant proportion of the gravitational radiation emitted by dynamical black hole solutions to (1.1) may be dominated by quasinormal modes (QNMs), known as resonant states as they settle down to a stationary Kerr black hole solution [54]; see for example the numerics in [19,23], the first experimental observations of gravitational radiation in [76] and subsequent further analysis in [52]
QNMs are exponentially damped, oscillating solutions to linear wave equations on fixed stationary spacetime backgrounds that are characterized by a discrete set of complex frequencies called quasinormal frequencies (QNFs) or scattering resonances
Summary
Where g is a Lorentzian metric and Ric[g] is the corresponding Ricci tensor. It is expected that a significant proportion of the gravitational radiation emitted by dynamical black hole solutions to (1.1) may be dominated by quasinormal modes (QNMs), known as resonant states as they settle down to a stationary Kerr black hole solution [54]; see for example the numerics in [19,23], the first experimental observations of gravitational radiation in [76] and subsequent further analysis in [52]. In order to make use of a contour deformation for the inverse Laplace transform, as, for determining the role of quasinormal modes in the temporal behaviour of solutions to (1.2) arising from suitably regular, localized and generic initial data, additional information is needed with regards to the distribution of scattering resonances and further properties of the resolvent operator R (s). The corresponding radiation field r ψ m, with ψ m a solution to (1.2), is smooth along suitable hyperboloidal/asymptotically null hypersurfaces intersecting the horizon to the future of the bifurcation sphere.17 This implies that, in contrast with the = 0 settings, a restriction to an arbitrarily regular Sobolev space will not ensure discreteness of the set of eigenvalues of A in the sub-extremal ( = 0) setting.
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