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

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.

46 pages, v2 extends the version accepted for publication

References in corpus (3)

• Gravitational instability of an extreme Kerr black hole
• What happens at the horizon(s) of an extreme black hole?
• Caustic echoes from a Schwarzschild black hole

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• The Strauss conjecture on asymptotically flat space-times
• Mass inflation and the $C^2$-inextendibility of spherically symmetric charged scalar field dynamical black holes
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• Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes
• Spin Hall effects and the localization of massless spinning particles
• Gaussian beams and caustic avoidance in gravitational optics