  Motivated by recent progress in developing action formulations of
relativistic hydrodynamics, we use holography to derive the low energy
dissipationless effective action for strongly coupled conformal fluids. Our
analysis is based on the study of novel double Dirichlet problems for the
gravitational field, in which the boundary conditions are set on two
codimension one timelike hypersurfaces (branes). We provide a geometric
interpretation of the Goldstone bosons appearing in such constructions in terms
of a family of spatial geodesics extending between the ultraviolet and the
infrared brane. Furthermore, we discuss supplementing double Dirichlet problems
with information about the near-horizon geometry. We show that upon coupling to
a membrane paradigm boundary condition, our approach reproduces correctly the
complex dispersion relation for both sound and shear waves. We also demonstrate
that upon a Wick rotation, our formulation reproduces the equilibrium partition
function formalism, provided the near-horizon geometry is properly accounted
for. Finally, we define the conserved hydrodynamic entropy current as the
Noether current associated with a particular transformation of the Goldstone
bosons.
