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. 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.