  We investigate a model of modified gravity recovering the modified Newtonian
dynamics (MOND) in the non-relativistic limit, based on the introduction of a
preferred time foliation violating Lorentz invariance in the weak-field regime.
Lorentz-invariance violation has been studied in the framework of
Einstein-aether theory, the generalization of which, known as non-canonical
Einstein-aether theory, having been proposed as a relativistic formulation of
MOND. Our model can be seen as a minimal specialization to the hypersurface
orthogonal case, which allows a different interpretation in terms of the
preferred time : it can be either treated as a dynamical scalar field in a 4D
formulation, or chosen as the time coordinate in a 3+1 formulation. We discuss
the equivalence of the two points of view and the non-relativistic limit of the
model.
