Analogue gravity from Bose-Einstein condensates

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Abstract
We analyse prospects for the use of Bose-Einstein condensates as condensed-matter systems suitable for generating a generic `effective metric', and for mimicking kinematic aspects of general relativity. We extend the analysis due to Garay et al (2000 Phys. Rev. Lett. 85 4643, 2001 Phys. Rev. A 63 023611). Taking a long-term view, we ask what the ultimate limits of such a system might be. To this end, we consider a very general version of the nonlinear Schrödinger equation (with a 3-tensor position-dependent mass and arbitrary nonlinearity). Such equations can be used, for example, in discussing Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We demonstrate that at low momenta linearized excitations of the phase of the condensate wavefunction obey a (3 + 1)-dimensional d'Alembertian equation coupling to a (3 + 1)-dimensional Lorentzian-signature `effective metric' that is generic, and depends algebraically on the background field. Thus at low momenta this system serves as an analogue for the curved spacetime of general relativity. In contrast, at high momenta we demonstrate how one can use the eikonal approximation to extract a well controlled Bogoliubov-like dispersion relation, and (perhaps unexpectedly) recover non-relativistic Newtonian physics at high momenta. Bose-Einstein condensates appear to be an extremely promising analogue system for probing kinematic aspects of general relativity.
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