We demonstrate here that the metric of a planar black hole in asymptotic anti-de Sitter space can, on a slice of dimension 3. +. 1, be reproduced as a relativistic acoustic metric. This completes an earlier calculation in which the non-relativistic limit was used, and also serves to obtain a concrete form of the Lagrangian.
Hossenfelder, S. (2016). A relativistic acoustic metric for planar black holes. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 752, 13–17. https://doi.org/10.1016/j.physletb.2015.11.026