Vorticity in analogue spacetimes

Analogue spacetimes can be used to probe and study physically interesting spacetime geometries by constructing, either theoretically or experimentally, some notion of an effective Lorentzian metric $[g_{\mathrm{eff}}(g,V,\mathrm{\Xi }){]}_{ab}$. These effective metrics generically depend on some physical background metric $g_{ab}$, often flat Minkowski space ${\eta }_{ab}$, some “medium” with 4-velocity $V^a$, and possibly some additional background fields and parameters $\mathrm{\Xi }$. (These might include signal propagation speeds and the like.) Analogue spacetimes based on electromagnetic media date back to Gordon’s work in the 1920s, analogue spacetimes based on acoustics in fluids date back to Unruh’s work in the 1980s, and BEC-based analogue spacetimes date back to various authors in the 1990s. The analogue spacetimes based on acoustic propagation in bulk fluids have perhaps the most rigorous mathematical formulation, and these acoustics-based analogue models really work best in the absence of vorticity, when the medium has an irrotational flow. This physical restriction makes it difficult to mimic the particularly interesting case of rotating astrophysical spacetimes, spacetimes with nonzero angular momentum, and in the current article we explore the extent to which one might hope to be able to develop an analogue model for astrophysical spacetimes with angular momentum (thereby implying vorticity in the 4-velocity of the medium). We shall focus on two particular analogue models: (1) the use of a charged BEC as the background medium, where new results concerning the interplay between healing length and London penetration depth are a key technical improvement, and (2) new results regarding the Gordon metric associated with an isotropic fluid medium.