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A closed achronal surface is Cauchy if its domain of dependence is entire manifold. From information given on a Cauchy surface, we can predict what happens throughout all of spacetime.

How do we determine whether a spacetime has a Cauchy surface or not? And how do we specify information on the Cauchy surface?

Also, given a Cauchy surface and the information on it, is it possible to predict whether the spacetime has a singularity or not?

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There's a few handy ways to find out more or less easily if a hypersurface is a Cauchy surface.

Rather than checking every possible timelike curve, it is possible to simply check every inextendible null geodesics. By a theorem of Wald (theorem 8.3.7), for a closed, achronal, edgeless set $\Sigma$, $\Sigma$ is a Cauchy surface if those curves intersect $\Sigma$ and enter $I^+(\Sigma)$ and $I^-(\Sigma)$.

A useful tool for this is also the use of temporal functions. A temporal function $t$ is a function $t : \mathcal M \to \mathbb R$ that is smooth and with a gradient that's a past-directed timelike vector. The level sets of this functions (surfaces such that $t(\Sigma) = \text{const}$) are foliations of the manifold into spacelike hypersurfaces. The existence of a temporal function only guarantees that the spacetime is stably causal (Minkowski space minus a closed set would admit one, after all).

A globally hyperbolic spacetime also requires the splitting of the spacetime as a product $\mathbb R \times \Sigma$, with $g = -\beta^2 dt^2 \oplus h_t$, with $h_t$ a metric on $\Sigma$, $\beta$ a positive function and $t$ said temporal function ($dt$ is the $1$-form obtained from its exterior derivative). If you can obtain such a form for the spacetime it should be a good hint that the surfaces $t^{-1}(\text{const}) = \Sigma$ are Cauchy surfaces. As far as I know, the usual theorem goes the other way : the spacetime being globally hyperbolic guarantees the existence of this form. It is possible that a non-globally hyperbolic spacetime admits such a form, in particular I think anti-de Sitter space does. But that can be a useful tool combined with the null geodesics to find Cauchy surfaces.

And of course, use the invariance of causal structure under Weyl transform : if you have a simpler spacetime conformally related to yours, use that one to do your calculations

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