Field equations and particle motion in covariant emergent gravity

We derive the full set of field equations based on Hossenfelder's recent covariant formulation of the emergent gravity model, along with perturbative and exact solutions. The exact solution describes a static, spherically symmetric spacetime with a nontrivial vector field that plays the role of dark matter under the emergent gravity paradigm. Equations of motion of relativistic test masses are derived and are shown to reduce to modified Newtonian dynamics with additional relativistic corrections. It is also shown that the presence of the vector field gives an additional positive contribution to the bending angle in the deflection of light.