  We write down a Schwinger-Keldysh effective field theory for non-relativistic
(Galilean) hydrodynamics. We use the null background construction to
covariantly couple Galilean field theories to a set of background sources. In
this language, Galilean hydrodynamics gets recast as relativistic hydrodynamics
formulated on a one-dimension higher spacetime admitting a null Killing vector.
This allows us to import the existing field-theoretic techniques for
relativistic hydrodynamics into the Galilean setting, with minor modifications
to include the additional background vector field. We use this formulation to
work out an interacting field theory describing stochastic fluctuations of
energy, momentum, and density modes around thermal equilibrium. We also present
a translation of our results to the more conventional Newton-Cartan language
and discuss how the same can be derived via a non-relativistic limit of the
effective field theory for relativistic hydrodynamics.
