Logarithmic catastrophes and Stokes's phenomenon in waves at horizons

TL;DR

This paper investigates wave behavior near horizons in flowing Bose-Einstein condensates, revealing a novel wave catastrophe characterized by an Airy-type function with a logarithmic phase, and links to Stokes phenomena and caustics.

Contribution

It introduces a new wave catastrophe description involving an Airy-type function with a logarithmic phase, differing from the expected Pearcey function, and analyzes the integral representation using exponential coordinates.

Findings

Horizon acts as a Stokes surface where waves are generated.

Wave functions near horizons are characterized by an Airy-type function with a logarithmic phase.

The horizon and caustic generally do not coincide, defining a broadened horizon.

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 and also gravitational horizons if dispersion which violates Lorentz symmetry in the UV is included. The approach we take differs from 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 branches that gives rise to an analysis based purely on saddlepoint expansions, which 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.