Role of the Gullstrand-Painlevè metric in acoustic black holes

Analog models of general relativity have received great attention in the last few years, since it is believed that these models are shedding light on possible experimental verifications of some fundamental problems in black hole physics, such as the evaporation of black holes and semiclassical quantities. The idea of using supersonic acoustic flows as analog systems to mimic some properties of black hole physics was proposed for the first time by Unruh. The basis of the analogy between gravitational black hole and sonic black holes comes from considering the propagation of acoustic disturbances on a barotropic, inviscid, inhomogeneous, and irrotational (at least locally) fluid flow. It is well known, that the equation of motion for this acoustic disturbance (described by its velocity potential) is identical to the Klein-Gordon equation for a massless scalar field minimally coupled to gravity in a curved spacetime. In this letter, we discuss the role of the Gullstrand-Painlevè metric in acoustic geometry and the physical interpretation of these models.