Characterisation of the Energy of Gaussian Beams on Lorentzian Manifolds - with Applications to Black Hole Spacetimes

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Abstract

It is known that using the Gaussian beam approximation one can show that there exist solutions of the wave equation on a general globally hyperbolic Lorentzian manifold whose energy is localised along a given null geodesic for a finite, but arbitrarily long time. In this paper, we show that the energy of such a localised solution is determined by the energy of the underlying null geodesic. This result opens the door to various applications of Gaussian beams on Lorentzian manifolds that do not admit a globally timelike Killing vector field. In particular we show that trapping in the exterior of Kerr or at the horizon of an extremal Reissner-Nordström black hole necessarily leads to a `loss of derivative' in a local energy decay statement. We also demonstrate the obstruction formed by the red-shift effect at the event horizon of a Schwarzschild black hole to scattering constructions from the future (where the red-shift turns into a blue-shift): we construct solutions to the backwards problem whose energies grow exponentially for a finite, but arbitrarily long time. Finally, we give a simple mathematical realisation of the heuristics for the blue-shift effect near the Cauchy horizon of sub-extremal and extremal black holes: we construct a sequence of solutions to the wave equation whose initial energies are uniformly bounded, whereas the energy near the Cauchy horizon goes to infinity.
Original languageEnglish
Article number46
Pages (from-to)1379-1420
JournalAnalysis & PDE
Volume8
Issue number6
DOIs
Publication statusPublished - 5 Sep 2015

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