Geroch's splitting theorem

Remarks

The idea of Geroch's proof is to fix a finite measure $\mu$ on $M$ with a positive smooth density and to consider the function $t(p) = \ln\left(\mu(I^-(p)) / \mu(I^+(p))\right)$. Global hyperbolicity makes $t$ continuous, and $t$ increases strictly along every future-directed causal curve and tends to $\pm\infty$ along inextendible ones. Each level set of $t$ is then a Cauchy surface, and flowing along a timelike vector field identifies $M$ with $\R \times \Sigma$.

Bernal and Sánchez proved a smooth version: $M$ is diffeomorphic to $\R \times \Sigma$, with metric $-\beta\, dt^2 + h_t$. Here $\beta > 0$ is a smooth function, and each slice $\set{t} \times \Sigma$ is a smooth spacelike Cauchy surface with Riemannian metric $h_t$.

Example: Minkowski space $\R^{1,3}$ is globally hyperbolic with Cauchy surface $\set{t = 0} \cong \R^3$, and indeed $\R^{1,3} = \R \times \R^3$.

The theorem implies strong topological restrictions: a globally hyperbolic spacetime cannot be compact, since $\R \times \Sigma$ is never compact.

Going deeper

For a student

Geroch's theorem is the bridge between the causal definition of global hyperbolicity and the initial value problem. Einstein's equations are posed on a Cauchy surface $\Sigma$, and the solution lives on $\R \times \Sigma$. It also shows that topology change cannot happen in a globally hyperbolic spacetime. The smooth splitting of Bernal and Sánchez is what one needs in practice to write the metric in the form used by the ADM and other $3 + 1$ formalisms.