  We show that the relativistic gravity theory can offer a framework to
formulate the non-relativistic effective field theory in a general coordinate
invariant way. We focus on the parity violating case in 2+1 dimensions which is
particularly appropriate for the study on quantum Hall effects and chiral
superfluids. We discuss how the non-relativistic spacetime structure emerges
from relativistic gravity. We present covariant maps and constraints that
relate the field contents in the two theories, which also serve as the
holographic dictionary in context of gauge/gravity duality. A low energy
effective action for fractional quantum Hall states is constructed, which
captures universal geometric properties and generates non-universal corrections
systematically. We give another holographic example with dyonic black brane
background to calculate thermodynamic and transport properties of strongly
coupled non-relativistic fluids in magnetic field. In particular, by
identifying the shift function in the gravity as minus of guiding center
velocity, we obtain the Hall viscosity with its relation to Landau orbital
angular momentum density proportional to Wen-Zee shift. Our formalism has a
good projection to lowest Landau level.
