Canonical variation of a Lorentzian metric

Given a Lorentzian manifold ( M , g L ) and a timelike unitary vector field E, we can construct the Riemannian metric g R = g L + 2 ω ⊗ 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 , g L ) .