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The current article explores the problem of a Bose-Einstein condensate (BEC)trapped in an anisotropic three-dimensional harmonic-oscillator potential, withpositive effective interatomic interactions, and the analogue gravity acousticmetric emerging from it. We find that the latter gives rise to a family ofconformally equivalent metrics and the space-time interval corresponds to thatof an accelerated observer subject to a position-dependent acceleration.Furthermore, such acceleration can be deduced differentiating the potential ofthe trap, which leads us to infer that, for a Bose-Einstein condensate in anarbitrary trap, the space-time interval will correspond to the conformalequivalent of that of an accelerated observer, being the acceleration in anydirection proportional to the partial derivative of the trap potential of theBEC with respect to that coordinate. Finally, we compute the space-timeinterval, the Einstein tensor and the geodesic equations, identifying some ofthe Killing vectors, for three cases: the spherically symmetric, the axiallysymmetric and the asymmetric traps.

Accelerated observers emerging from a Bose-Einstein condensate through analogue gravity