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Characterisation of the
energy of Gaussian beams on Lorentzian manifolds: with
applications to black hole spacetimes
Jan Sbierski |
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Vol. 8 (2015), No. 6, 1379–1420
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DOI: 10.2140/apde.2015.8.1379
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 35R01
Secondary: 83C57
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Milestones
Received: 7 May 2014
Revised: 24 September 2014
Accepted: 30 April 2015
Published: 5 September 2015
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Authors |
| Jan Sbierski | |
| Department of Applied Mathematics
and Theoretical Physics University of Cambridge Wilberforce Road Cambridge CB3 0WA United Kingdom |
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