Vol. 8, No. 6, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes

Jan Sbierski

Vol. 8 (2015), No. 6, 1379–1420
DOI: 10.2140/apde.2015.8.1379
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. 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 subextremal 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.

Keywords
Gaussian beams, characterisation of energy, spacetime
Mathematical Subject Classification 2010
Primary: 35R01
Secondary: 83C57
Milestones
Received: 7 May 2014
Revised: 24 September 2014
Accepted: 30 April 2015
Published: 5 September 2015
Authors
Jan Sbierski
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Wilberforce Road
Cambridge
CB3 0WA
United Kingdom