Globally hyperbolic manifold

In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is (t and r being the usual variables of time and radius) which is one of the usual equations representing an hyperbola. But this expression is only true relative to the ordinary origin; this article then outline bases for generalizing the concept to any pair of points in spacetime. This is relevant to Albert 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: M is non-totally vicious if there is at least one point such that no closed timelike curve passes through it. M is causal if it has no closed causal curves. M is non-total imprisoning if no inextendible causal curve is contained in a compact set. This property implies causality. M is strongly causal if for every point p and any neighborhood U of p there is a causally convex neighborhood V of p contained in U, where causal convexity means that any causal curve with endpoints in V is entirely contained in V. This property implies non-total imprisonment. Given any point p in M, [resp. ] is the collection of points which can be reached by a future-directed [resp. past-directed] continuous causal curve starting from p. Given a subset S of M, the domain of dependence of S is the set of all points p in M such that every inextendible causal curve through p intersects S. A subset S of M is achronal if no timelike curve intersects S more than once. A Cauchy surface for M is a closed achronal set whose domain of dependence is M. The following conditions are equivalent: The spacetime is causal, and for every pair of points p and q in M, the space of continuous future-directed causal curves from p to q is compact in the topology.