Test particle motion in modified gravity theories

We derive the equations of motion of an electrically neutral test particle for modified gravity theories in which the covariant divergence of the ordinary matter energy-momentum tensor dose not vanish (i.e. $\nabla_{\mu}T^{\mu\nu}\neq0$). In fact, we generalize the Mathisson-Papapetrou equations by deriving a general form for the equations of motion of a test particle. Furthermore, using the generalized Mathisson-Papapetrou equations, we investigate the equations of motion of a pole-dipole (spinning) particle in the context of Modified Gravity (MOG).