CQW holiday project: linear gravity

Can you construct a Lorentz-invariant linear theory of gravity with the energy-momentum-tensor \(T_{\mu\nu}\) as the source of the gravitational field, using a Lagrangian recipe? how would the classical Poisson equation follow in a slow-motion limit? What would be the corresponding equation of motion, and how would velocity dependent terms look like? would there be gauge-invariance and Lorentz-covariance?