Analogue Gravity Models From Conformal Rescaling

Abstract: Analogue gravity is based on a mathematical identity between quantum field theory in curved space-time and the propagation of perturbations in certain condensed matter systems. But not every curved space-time can be simulated in such a way, because one does not only need a condensed matter system that generates the desired metric tensor, but that system then also has to obey its own equations of motion. And specifying the metric tensor that one wishes to realize usually overdetermines the underlying condensed matter system, such that its equations of motion are in general not fulfilled, in which case the desired metric does not have an analogue. Here, we show that the class of metrics that have an analogue is bigger than what a first cursory consideration might suggest. This is due to the analogue metric only being defined up to a choice of parametrization of the perturbation in the underlying condensed matter system. In this way, the class of analogue gravity models can be vastly expanded. In particular, we demonstrate how this freedom of choice can be used to insert an intermediary conformal factor. Then, as a corollary, we find that any metric conformal to a Painlev\'e--Gullstrand type line element can, potentially, result as an analogue of a perturbation propagating in a non-viscous, barotropic fluid.