  We obtain the renormalized equations of motion for matter and semi-classical
gravity in an inhomogeneous space-time. We use the functional Schrodinger
picture and a simple Gaussian approximation to analyze the time evolution of
the $\lambda\phi^4$ model, and we establish the renormalizability of this
non-perturbative approximation. We also show that the energy-momentum tensor in
this approximation is finite once we consider the usual mass and coupling
constant renormalizations, without the need of further geometrical
counter-terms.
