Determining global hyperbolicity Given a spacetime metric, how can one show that the spacetime is globally hyperbolic? I know that a globally hyperbolic metric has a Cauchy surface, but how can we determine the existence of a Cauchy surface?
For example, given the Kerr metric or Schwarzschild metric, can we find a Cauchy surface? What is the procedure to show that it has a Cauchy surface?
P.S. There is a StackExchange question on the determination of global hyperbolicity without any answer. I am asking for a general procedure like in this question but it would also be helpful if someone can elaborate on how can we know that even a simpler Schwarzschild metric has a Cauchy surface.
 A: The definition I've seen is actually that a spacetime is globally hyperbolic if (1) it doesn't have CTCs, and (2) the intersection of a future lightcone with a past lightcone is always compact. It can then be proved that such a spacetime has Cauchy surfaces (Geroch 1970).
Definitions aren't normally directly associated with a single recipe for testing whether the definition holds. 
You asked about the Kerr and Schwarzschild spacetimes.
For many simple examples, the determination that a spacetime is globally hyperbolic is immediate from its Penrose diagram. For example, if you draw the Penrose diagram for the Schwarzschild spacetime, then conditions 1 and 2 are both clearly true.
I don't think the Kerr spacetime is globally hyperbolic, because it has CTCs.

P.S. There is a StackExchange question on the determination of global hyperbolicity without any answer. 

Please link to it.
A: It is not so simple to prove that a spacetime is globally hyperbolic because it requires a good knowledge of the global behavior of (null) geodesics. Part of that information might be encodable in a Penrose diagram (see for instance arXiv:1211.1718 for a rigorous statement) but one needs to derive it one way or another.
In the case of the Schwarzschild exterior, there are criteria that do the job relatively easily, see for instance arXiv:0712.0600. The proof of the global hyperbolicity of the Kerr exterior region (and its extension to some larger regions) can be found in Appendix C.6 of arXiv:2008.10995 and is significantly more difficult.
Note that the Kerr metric extends to even larger regions and at some point closed time-like curves arise (see for instance the book by O'Neill), so these extensions are not globally hyperbolic. But as a physical model it is the exterior region and its extension across the event horizon that are the most relevant.
