Logarithmic catastrophes and Stokes’s phenomenon in waves at horizons

Abstract

Waves propagating near an event horizon display interesting features including logarithmic phase singularities and caustics. We consider an acoustic horizon in a flowing Bose-Einstein condensate where the elementary excitations obey the Bogoliubov dispersion relation. In the hamiltonian ray theory the solutions undergo a broken pitchfork bifurcation near the horizon and one might therefore expect the associated wave structure to be given by a Pearcey function, this being the universal wave function that dresses catastrophes with two control parameters. However, the wave function is in fact an Airy-type function supplemented by a logarithmic phase term, a novel type of wave catastrophe. Similar wave functions arise in aeroacoustic flows from jet engines, path integrals in radio astronomy, and also gravitational horizons if dispersion which violates Lorentz symmetry in the UV is included. The approach we take differs from most previous authors in that we analyze the behaviour of the integral representation of the wave function using exponential coordinates. This allows for a different treatment of the branch cuts and gives rise to an analysis based purely on saddlepoint expansions. We are thereby able to resolve the multiple real and complex waves that interact at the horizon and its companion caustic. We find that the horizon is a physical manifestation of a Stokes surface, marking the place where a wave is born, and that the horizon and the caustic do not in general coincide: the finite spatial region between them delineates a broadened horizon.

Duality between the quantum inverted harmonic oscillator and inverse square potentials

“…When doing so it is crucial that for general Ê the solution (19) has a logarithmic branch point at Q = 0, leading to both an amplitude singularity and a logarithmic phase singularity at Q = 0 (as can be seen in figure 5). Such singularities are generic signatures of quantum catastrophes [105,106], and appear in many other interesting physical systems like waves near black-hole event horizons, in accelerated frames, and so on [4,7,8,107,108].…”

Quantum electrodynamics of time-varying gratings

“…The relation of these singular points to wave catastrophes was recently discussed in ref. 27 . Transluminal space-time gratings can therefore be understood as an effective space-time geometry for light, containing an alternating sequence of optical black and white hole event horizons.…”