Analogue model for anti-de Sitter as a description of point sources in fluids

We introduce an analogue model for a nonglobally hyperbolic spacetime in terms of a two-dimensional fluid. This is done by considering the propagation of sound waves in a radial flow with constant velocity. We show that the equation of motion satisfied by sound waves is the wave equation on ${\mathrm{AdS}}_2\times S^1$. Since this spacetime is not globally hyperbolic, the dynamics of the Klein-Gordon field is not well defined until boundary conditions at the spatial boundary of ${\mathrm{AdS}}_2$ are prescribed. On the analogue model end, those extra boundary conditions provide an effective description of the point source at $r=0$. For waves with circular symmetry, we relate the different physical evolutions to the phase difference between ingoing and outgoing scattered waves. We also show that the fluid configuration can be stable or unstable depending on the chosen boundary condition.

Figure 1

The phase difference between the incoming and the outgoing waves as a function of the dimensionless parameter $z=\frac{\beta \omega }{c(1-{\alpha }^2)}$.