We develop a Schwinger-Keldysh effective field theory describing the
hydrodynamics of a fluid with conserved charge and dipole moments, together
with conserved momentum. The resulting hydrodynamic modes are highly unusual,
including sound waves with quadratic (magnon-like) dispersion relation and
subdiffusive decay rate. Hydrodynamics itself is unstable below four spatial
dimensions. We show that the momentum density is, at leading order, the
Goldstone boson for a dipole symmetry which appears spontaneously broken at
finite charge density. Unlike an ordinary fluid, the presence or absence of
energy conservation qualitatively changes the decay rates of the hydrodynamic
modes. This effective field theory naturally couples to curved spacetime and
background gauge fields; in the flat spacetime limit, we reproduce the "mixed
rank tensor fields" previously coupled to fracton matter.Comment: 20+10 pages. v2, v3: minor edit
