  We argue that the Fermi-Hubbard Hamiltonian describing the physics of
ultracold atoms on optical lattices in the presence of artificial non-Abelian
gauge fields, is exactly equivalent to the gauge theory Hamiltonian describing
Dirac fermions in the lattice. We show that it is possible to couple the Dirac
fermions to an "artificial" gravitational field, i.e. to consider the Dirac
physics in a curved spacetime. We identify the special class of spacetime
metrics that admit a simple realization in terms of a Fermi-Hubbard model
subjected to an artificial SU(2) field, corresponding to position dependent
hopping matrices. As an example, we discuss in more detail the physics of the
2+1D Rindler metric, its possible experimental realization and detection.
