Abstract: The modified Newtonian dynamics (MOND) has been proposed as an alternative to the standard approach to dark matter (i.e. cold dark matter or CDM), in which there is no dark matter and we have instead a modification of the law of gravity. Here we advocate that MOND may not result from a modification of gravity, but from a mechanism of "gravitational polarization" of some dipolar medium playing the role of dark matter. We propose a relativistic model in general relativity to describe a dipolar dark matter medium. We show that this model of dark matter recovers both the cosmological model (ΛCDM) at early cosmological times, and the phenomenology of MOND in a galaxy at low redshift. Thus the model reconciles the observations at cosmological scales and at galactic scales in a single relativistic formalism. We argue that the cosmological constant Λ should be of the order of a0², where a0 is the MOND constant acceleration, in good agreement with observations.