Global hyperbolicity

In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). This is relevant to Einstein's theory of general relativity, and potentially to other metric gravitational theories.

There are several equivalent definitions of global hyperbolicity. Let M be a smooth connected Lorentzian manifold without boundary. We make the following preliminary definitions:

The following conditions are equivalent:

If any of these conditions are satisfied, we say M is globally hyperbolic. If M is a smooth connected Lorentzian manifold with boundary, we say it is globally hyperbolic if its interior is globally hyperbolic.

Other equivalent characterizations of global hyperbolicity make use of the notion of Lorentzian distance ${\displaystyle d(p,q):=\sup _{\gamma }l(\gamma )}$ where the supremum is taken over all the ${\displaystyle C^{1}}$ causal curves connecting the points (by convention d=0 if there is no such curve). They are

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