Using the action principle we first review how linear density perturbations
(sound waves) in an Eulerian fluid obey a relativistic equation: the d'Alembert
equation. This analogy between propagation of sound and that of a massless
scalar field in a Lorentzian metric also applies to non-homogeneous flows. In
these cases, sound waves effectively propagate in a curved four-dimensional
''acoustic'' metric whose properties are determined by the flow. Using this
analogy, we consider regular flows which become supersonic, and show that the
acoustic metric behaves like that of a black hole. The analogy is so good that,
when considering quantum mechanics, acoustic black holes should produce a
thermal flux of Hawking phonons.
We then focus on two interesting questions related to Hawking radiation which
are not fully understood in the context of gravitational black holes due to the
lack of a theory of quantum gravity. The first concerns the calculation of the
modifications of Hawking radiation which are induced by dispersive effects at
short distances, i.e., approaching the atomic scale when considering sound. We
generalize existing treatments and calculate the modifications caused by the
propagation near the black hole horizon. The second question concerns
backreaction effects. We return to the Eulerian action, compute second order
effects, and show that the backreaction of sound waves on the fluid's flow can
be expressed in terms of their stress-energy tensor. Using this result in the
context of Hawking radiation, we compute the secular effect on the background
flow.Comment: 60 pages, 6 figures. Review submitted to "La Rivista del Nuovo
Cimento
