Globally hyperbolic manifold

Definitions

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:

1. 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 C 0 topology.
2. The spacetime has a Cauchy surface.
3. The spacetime is causal, and for every pair of points p and q in M, the subset J − ( p ) ∩ J + ( q ) is compact.
4. The spacetime is non-total imprisoning, and for every pair of points p and q in M, the subset J − ( p ) ∩ J + ( q ) is contained in a compact set (that is, its closure is compact).

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 d ( p , q ) := sup γ l ( γ ) where the supremum is taken over all the C 1 causal curves connecting the points (by convention d=0 if there is no such curve). They are

Remarks

Global hyperbolicity, in the first form given above, was introduced by Leray in order to consider well-posedness of the Cauchy problem for the wave equation on the manifold. In 1970 Geroch proved the equivalence of definitions 1 and 2. Definition 3 under the assumption of strong causality and its equivalence to the first two was given by Hawking and Ellis.

As mentioned, in older literature, the condition of causality in the first and third definitions of global hyperbolicity given above is replaced by the stronger condition of strong causality. In 2007, Bernal and Sánchez showed that the condition of strong causality can be replaced by causality. In particular, any globally hyperbolic manifold as defined in 3 is strongly causal. However, in definition 3 the closure of J − ( p ) ∩ J + ( q ) seems strong (in fact, the closures of the sets J ± ( p ) imply causal simplicity, the level of the causal hierarchy of spacetimes which stays just below global hyperbolicity) . It is possible to remedy this problem strengthening the causality condition as in definition 4 proposed by Minguzzi in 2009. This version clarifies that global hyperbolicity sets a compatibility condition between the causal relation and the notion of compactness: every causal diamond is contained in a compact set and every inextendible causal curve escapes compact sets. Observe that the larger the family of compact sets the easier for causal diamonds to be contained on some compact set but the harder for causal curves to escape compact sets. Thus global hyperbolicity sets a balance on the abundance of compact sets in relation to the causal structure. Since finer topologies have less compact sets we can also say that the balance is on the number of open sets given the causal relation. Definition 4 is also robust under perturbations of the metric (which in principle could introduce closed causal curves). In fact using this version it has been shown that global hyperbolicity is stable under metric perturbations.

In 2003, Bernal and Sánchez showed that any globally hyperbolic manifold M has a smooth embedded three-dimensional Cauchy surface, and furthermore that any two Cauchy surfaces for M are diffeomorphic. In particular, M is diffeomorphic to the product of a Cauchy surface with R . It was previously well known that any Cauchy surface of a globally hyperbolic manifold is an embedded three-dimensional C 0 submanifold, any two of which are homeomorphic, and such that the manifold splits topologically as the product of the Cauchy surface and R . In particular, a globally hyperbolic manifold is foliated by Cauchy surfaces.

In view of the initial value formulation for Einstein's equations, global hyperbolicity is seen to be a very natural condition in the context of general relativity, in the sense that given arbitrary initial data, there is a unique maximal globally hyperbolic solution of Einstein's equations.