Talk:Globally hyperbolic manifold

Problem on Spacetimes with Boundary

The given definition of global hyperbollicity in reference 1 does not apply correctly to spacetimes with boundary. In this case the condition has to be taken for the interior as a manifold in its own right. Otherwise the existence of a Cauchy surface cannot be guaranteed anymore. One can e.g. cut off a spacetime with non-empty spatial infinity containing a Cauchy surface near spatial infinity. Lets call the result M. Then the timelike spatial boundary of M will be non-empty, which prohibits the existence of a Cauchy surface. The first condition in reference 1 will still be fulfilled as $I^{-}(p)\cap I^{+}(q)$ is just replaced by $(I^{-}(p)\cap I^{+}(q))\cap M$ which is compact as the intersection of two compact sets. $(I^{-}(p)\cap I^{+}(q))\cap int(M)$ however will not be compact if $(I^{-}(p)\cap I^{+}(q))\setminus int(M)\neq \emptyset$. —Preceding unsigned comment added by Doenermaster (talk • contribs) 13:40, 23 October 2010 (UTC)