Globally hyperbolic

Globally hyperbolic (also global hyperbolicity) is a term describing the causal structure of a spacetime manifold in Einstein's theory of general relativity, or potentially in other metric gravitational theories.

Open sets are used in discussing the global hyperbolicity of spacetimes. An open set $U$ is said to be globally hyperbolic if the following two conditions holdStephen Hawking and Roger Penrose, "The Nature of Space and Time," Princeton University Press, 1996.]

# For every pair of points $p,q\; in\; U$, $I^-(p)cap\; I^+(q)$ is compact. Here $I^pm(S)$ is the future (past) of a set $S$ in spacetime.
# "Causality" holds on $U$ (no closed timelike curves exist). Classically, a more restrictive and technical assumption is required, named strong causality (no "almost closed" timelike curves exist); but a recent result [http://arxiv.org/abs/gr-qc/0611138] shows that causality suffices.

Global hyperbolicity implies that there is a family of Cauchy surfaces for $U$. Essentially, it means that everything that happens on $U$ is determined by the equations of motion, together with initial data specified on a surface.

ee also

*Causality conditions

References

*cite book | author=Hawking, Stephen; and Ellis, G. F. R. | title = The Large Scale Structure of Space-Time | location= Cambridge | publisher=Cambridge University Press | year=1973 |id = ISBN 0-521-09906-4