Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation? I am wondering what conditions a Lorentzian manifold $(M,g)$ must satisfy to ensure the existence of a global proper-time foliation (i.e. a decomposition of $M$ into spacelike Cauchy hypersurfaces and a global time function whose tangent vector field $\vec{t}$ satisfies $g(\vec{t},\vec{t})=-1$ and $g(\vec{t},\vec{x})=0$ for all $\vec{x}$ tangent to one of the spacelike hypersurfaces.  Is global hyperbolicity enough?  If anyone could recommend a good reference, that would be much appreciated!
Thanks!
 A: Global hyperbolicity is certainly not enough.
Consider the causal diamond
$$ D = \{(t,x) \in \mathbb{R}^{1+1} | |t|+|x| < 1\} $$
in 1+1 dimensional Minkowski space, which is certainly globally hyperbolic. I claim that this set does not admit the so-called "global proper-time foliation".

*

*Observe that with a proper-time foliation, the total elapsed proper-time must be at most 2, this being the length of the longest time-like geodesic you can fit in the causal diamond $D$.


*Supposing such a foliation exists with the global time function denoted $\tau$ (so that the level sets of $\tau$ are Cauchy, and $|\mathrm{d}\tau|_g = 1$), and is at least $C^1$ smooth, we can start with an arbitrary point $p\in D$ and follow the flow of $p$ by $\mathrm{d}\tau^\sharp$. By the first step we know that the total length of the flow-line is finite, and by definition this length is non-zero. Call it $\epsilon_p$.


*Claim: there exists a point $q\in D$ such that any time-like curve $\gamma$ through $q$ must have the total length of $\gamma\cap D$ be strictly less than $\epsilon_p$.
Proof: Consider the point $q = (t = 0, x = 1 - \delta)$ for some $\delta$ to be specified later. Since time-like geodesics maximize length among time-like curves, the maximum length that can be attained by a curve $\gamma$ contained in $D$ through $q$ is bounded above by the lengths of the two segments joining $q$ to $(t = 1, x = 0)$ and $(t = -1,x = 0)$. By taking $\delta$ arbitrarily small we see that this maximum length can be made arbitrarily close to zero.


*Steps 2 and 3 above contradict, since by assumption the flow of $q$ by $\mathrm{d}\tau^\sharp$ should have the same length as the flow of $p$.

The key is step 3: the same argument carries through whenever your global hyperbolic spacetime $D$ satisfies the condition that "for every $\epsilon$ there exists a point $q\in D$ such that no timelike curves through $q$ in $D$ has length greater than $\epsilon$."
A: Global hyperbolicity only gives you a Cauchy time function, whose gradient is past directed timelike. See Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes -  Antonio N. Bernal, Miguel Sanchez - Commun. Math. Phys. 257, 43–50 (2005)
