  Planar black holes in AdS, which are holographically dual to compressible
relativistic fluids, have a long-lived phonon mode that captures the physics of
attenuated sound propagation and transports energy in the plasma. We describe
the open effective field theory of this fluctuating phonon degree of freedom.
The dynamics of the phonon is encoded in a single scalar field whose
gravitational coupling has non-trivial spatial momentum dependence. This
description fits neatly into the paradigm of classifying gravitational modes by
their Markovianity index, depending on whether they are long-lived. The sound
scalar is a non-Markovian field with index (3-d) for a d-dimensional fluid. We
reproduce (and extend) the dispersion relation of the holographic sound mode to
quartic order in derivatives, constructing in the process the effective field
theory governing its attenuated dynamics and associated stochastic
fluctuations. We also remark on the presence of additional spatially
homogeneous zero modes in the gravitational problem, which remain disconnected
from the phonon Goldstone mode.

High Energy Physics - Theory;Statistical Mechanics;General Relativity and Quantum Cosmology