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 satisfies $$\text{Ric} = fg+(f-\lambda)T^{\flat}\otimes T^{\flat},$$ for any unit timelike vector field $T$, any positive constant $\lambda$, and any smooth function $f$ that never takes the values $0,\lambda$. (Observe that this reduces to the positive Einstein case when $f = \lambda$.) We show that there is no such obstruction if $\lambda$ is negative. Finally, the "borderline" case $\lambda = 0$ is also examined: we show that if $\lambda = 0$ and $f > 0$, then $(M,g)$ must be isometric to $(\mathbb{S}^1\!\times \!N,-dt^2\oplus h)$ with $(N,h)$ a Riemannian manifold.