On the Einstein condition for Lorentzian 3-manifolds

It is well known that in Lorentzian geometry there are no compact spherical space forms; in dimension 3, this means there are no closed Einstein 3-manifolds with positive Einstein constant. We generalize this fact here, by proving that there are also no closed Lorentzian 3-manifolds (M,g) whose Ricci tensor satisfiesRic=fg+(f−λ)T♭⊗T♭, for any unit timelike vector field T, any positive constant λ, and any nontrivial function f that never takes the value λ. (Observe that this reduces to the positive Einstein case when f=λ.) We show that there is no such obstruction if λ is negative. Finally, the “borderline” case λ=0 is also examined: we show that if λ=0, then (M,g) must be isometric to (S1×N,−dt2⊕h) with (N,h) a Riemannian manifold.