Abstract
In this work we consider the Gross-Pitaevskii equation for Bose-Einstein condensates (BECs) in a general Riemannian metric. Given initial conditions dictated by an external potential, we consider the free expansion of the condensate when the external potential is turned off. Focusing on the forces associated with the geometry of the initial configuration, we show how these are related to the Ricci curvature tensor and the Ricci scalar and we find an Einstein field equation governing the steady flow. Some important correlations between the study of defects in BECs and the appearance of cosmological singularities will be addressed, in particular the emergence of an effective Lorentzian spacetime geometry, which is what is needed to obtain Hawking radiation effects.
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