For non-compact spacetimes of dimension at least three, compact causal diamonds alone imply global hyperbolicity, so the causality condition can be dropped from the definition.
On differentiability of volume time functions
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abstract
We show differentiability of a class of Geroch's volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz time functions which are locally anti-Lipschitz can be uniformly approximated by smooth time functions with timelike gradient. Finally, we prove that in stably causal spacetimes Hawking's time function can be uniformly approximated by smooth time functions with timelike gradient.
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Globally hyperbolic spacetimes can be defined without the 'causal' condition
For non-compact spacetimes of dimension at least three, compact causal diamonds alone imply global hyperbolicity, so the causality condition can be dropped from the definition.