Logarithmic catastrophes and Stokes’s phenomenon in waves at horizons
Logarithmic catastrophes and Stokes’s phenomenon in waves at horizons
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 behavior 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.
analogue black hole, catastrophe theory, event horizon, Hawking radiation, Stokes’s phenomenon
Farrell, L. M.
e97745dd-7b20-43ec-9e84-02cf1f55e8d2
Howls, C. J.
66d3f0f0-376c-4f7a-a206-093935e6c560
O’Dell, D. H.J.
d9b9f036-49d6-4b63-9e53-d5a6106696eb
8 February 2023
Farrell, L. M.
e97745dd-7b20-43ec-9e84-02cf1f55e8d2
Howls, C. J.
66d3f0f0-376c-4f7a-a206-093935e6c560
O’Dell, D. H.J.
d9b9f036-49d6-4b63-9e53-d5a6106696eb
Farrell, L. M., Howls, C. J. and O’Dell, D. H.J.
(2023)
Logarithmic catastrophes and Stokes’s phenomenon in waves at horizons.
Journal of Physics A: Mathematical and Theoretical, 56 (4), [044001].
(doi:10.1088/1751-8121/acb29e).
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 behavior 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.
Text
Farrell_2023_J._Phys._A__Math._Theor._56_044001
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Accepted/In Press date: 12 January 2023
Published date: 8 February 2023
Additional Information:
Funding Information:
The authors would like to thank two anonymous referees for their valuable feedback and suggestions. They also gratefully acknowledge the Isaac Newton Institute (INI) of Cambridge University for hosting the Applicable Resurgent Asymptotics Programme where part of this work was undertaken, and the Natural Sciences and Engineering Research Council of Canada (NSERC) for funding DO and LF.
Publisher Copyright:
© 2023 The Author(s). Published by IOP Publishing Ltd.
Keywords:
analogue black hole, catastrophe theory, event horizon, Hawking radiation, Stokes’s phenomenon
Identifiers
Local EPrints ID: 476035
URI: http://eprints.soton.ac.uk/id/eprint/476035
ISSN: 1751-8113
PURE UUID: 9cbfc707-c213-4c02-b94c-93419675755a
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Date deposited: 04 Apr 2023 16:55
Last modified: 06 Jun 2024 01:38
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Author:
L. M. Farrell
Author:
D. H.J. O’Dell
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