On the Einstein condition for Lorentzian 3-manifolds

Abstract

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 $$\text{Ric}=fg+(f-\lambda )T^{♭}\otimes T^{♭},$$ 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 $({\mathbb{S}}^1\times N,-d{t}^2\oplus h)$ with $(N,h)$ a Riemannian manifold.

v4: this is a reposting of v2; the previous version (v3) contained an error in the proof of Proposition 2