Characterisation of the Energy of Gaussian Beams on Lorentzian Manifolds - with Applications to Black Hole Spacetimes
Jan Sbierski
TL;DR
The paper develops a rigorous Gaussian beam framework on globally hyperbolic Lorentzian manifolds, proving that high-frequency wave packets can have energy localized along null geodesics and that the beam energy tracks the geodesic energy, enabling precise control in spacetimes without a global timelike Killing field. By constructing explicit Gaussian beams and analyzing their energy via the N-energy, it provides a geometric energy characterisation and general limit theorems for the wave equation, forming a powerful tool beyond geometric optics. The authors then apply these results to black hole spacetimes (Schwarzschild, Reissner-Nordström, and Kerr), showing intrinsic obstructions to local energy decay due to trapping, elucidating red-shift and blue-shift effects, and presenting forward/backward constructions that demonstrate exponential or polynomial energy growth near horizons, with implications for strong cosmic censorship and horizon stability.
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.