On the Einstein condition for Lorentzian 3-manifolds

TL;DR

This paper proves that certain Lorentzian 3-manifolds with specific Ricci tensor conditions do not exist when the Einstein constant is positive, extending known results about the non-existence of compact Einstein 3-manifolds in Lorentzian geometry.

Contribution

It generalizes the non-existence of compact Einstein 3-manifolds to a broader class of Ricci tensor conditions involving a function and a timelike vector field.

Findings

No closed Lorentzian 3-manifolds satisfy the Ricci condition for positive Einstein constant.

Existence of such manifolds is possible only when the Einstein constant is negative.

In the borderline case with zero Einstein constant and positive function, the manifold is a product of a circle and a Riemannian manifold.

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 $$ \text{Ric} = fg+(f-\lambda)T^{\flat}\otimes T^{\flat}, $$ 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.