Unimodular relativity and the dark matter problem

We introduce a modified divergence law for the energy-momentum tensor in the theory of unimodular relativity. Consequently, an additional equation for the measure field follows from the divergence of the field equations. The equations of motion are derived and the weak-field, low-velocity limit is investigated. It is found that the gravitational acceleration acquires a term that is proportional to the gradient of the mass density. We show that this term can provide the additional acceleration observed on astrophysical scales without the need for dark matter.