Scalar field as a Bose-Einstein condensate?

We discuss the analogy between a classical scalar field with a self-interacting potential, in a curved spacetime described by a quasi-bounded state, and a trapped Bose-Einstein condensate. In this context, we compare the Klein-Gordon equation with the Gross-Pitaevskii equation. Moreover, the introduction of a curved background spacetime endows, in a natural way, an equivalence to the Gross-Pitaevskii equation with an explicit confinement potential. The curvature also induces a position dependent self-interaction parameter. We exploit this analogy by means of the Thomas-Fermi approximation, commonly used to describe the Bose-Einstein condensate, in order to analyze the quasi bound scalar field distribution surrounding a black hole.