Canonical variation of a Lorentzian metric

Given a Lorentzian manifold (M,gL)(M,gL) and a timelike unitary vector field E  , we can construct the Riemannian metric gR=gL+2ω⊗ωgR=gL+2ω⊗ω, ω being the metrically equivalent one form to E. We relate the curvature of both metrics, especially in the case of E   being Killing or closed, and we use the relations obtained to give some results about (M,gL)(M,gL).