Globally hyperbolic manifold

In mathematical physics, a spacetime manifold is globally hyperbolic if it satisfies a condition related to its causal structure. This is relevant to Einstein's theory of general relativity, and potentially to other metric gravitational theories.

To be precise, a spacetime manifold M without boundary is said to be globally hyperbolic if the following two conditions hold

1. For every pair of points p and q in M, the space of all points that can be both reached from p along a past-oriented causal curve and reached from q along a future-oriented causal curve is compact. Note: We denote this compact space

   $J^-(p)\cap J^+(q)$

2. "Causality" holds on M (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 shows that causality suffices.

A spacetime manifold with non-empty boundary is said to be globally hyperbolic if its interior, as a manifold in its own right, is globally hyperbolic.

Global hyperbolicity is completely equivalent to the existence of Cauchy surface. In fact, this implies that a globally hyperbolic spacetime M is foliated by a family of Cauchy surfaces, i.e. the interior of M is topologically isomorphic (diffeomorphic) to the product of the interior of some Cauchy surface Σ and some interval I; the metric structure need not respect this decomposition, however. If the spatial boundary of M is non-empty and the metric non-singular there, it will be the boundary of all Cauchy surfaces of the family above. This essentially looks like a ring used to make soap bubbles with a lot of soap layers.

Summarized, a globally hyperbolic spacetime is a spacetime on which everything is determined by the equations of motion for one hypersurface, together with initial data specified on it. (At least if only fields are present which have a well-defined initial value formulation.) This is also the origin for the name of this property.

See also

• Causality conditions
• Causal structure
• Light cone

References

• Hawking, Stephen; and Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. ISBN 0-521-09906-4. 
• Wald, Robert M. (1984). General Relativity. Chicago: The University of Chicago Press. ISBN 0-226-87033-2.