Trans-Planckian issues & Analogue Gravity: from BEC to acoustic Black Holes

Abstract

To account for the non-local interactions in a Bose-Einstein Condensate (BEC), an addition of a minimal correction term to the standard Gross-Pitaevskii model effectively can make the healing length or characteristic length ( ) decrease more rapidly with the increase of s-wave scattering length (a). From emergent gravity perspectives, this shrinking of via tuning a through Feshbach resonance, in principle, does make the short-wavelength (i.e. high energy) regime more accessible experimentally by pushing the Lorentz-breaking dispersion even more towards the UV side [1]. The effects of the Lorentz-breaking quantum potential term in the BEC-dynamics on independent multiple scales can be captured through a UV-IR coupling of the phonon-excitation-modes with the emergence of the massive minimally coupled free Klein- Gordon field. The analysis was argued on a (3+1)D flat spacetime where the presence of the mass term gives a hint to cure the infrared divergences through a nonzero threshold for the large-wavelength phonon excitations (say, Kz , 0 modes as perceived by an observer sitting on x 􀀀 y plane) on lower dimensional models of emergent spacetimes [2]. The analysis was extended for a canonical acoustic black hole on a (3+1)D curved spacetime through presenting an emergent gravity model up to O( 2) accuracy. In our formalism, the growth rate of the large-wavelength ‘secondary’ ! modes is found to hold the clue to extract the lost information regarding the short-wavelength ‘primary’ !1 modes. And hence this can actually reveal the relative abundance of the originally Hawking radiated quanta in a (3+1)D canonically curved background [3]. Thiswaywe could address the trans-Planckian issues in Physics via introducing the analogue Planck length (the healing length of the underlying cond.mat system, say BEC in our case, is the analogue of the Planck length `p 10􀀀35m. in Gravity) and shrinking this to an arbitrary small value by experimentally (Feshbach resonances) increasing the s-wave scattering length a. We showed that the UV-IR coupling between two different bands of phonon excitations is inevitable as perceived by an observer in the emergent gravity experiments. And finally we tackled the way to retrieve the “information loss” in Hawking radiation within the scope emergent spacetimes.