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). This is relevant to Einstein's theory of general relativity, and potentially to other metric gravitational theories.

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:

• M is causal if it has no closed causal curves.
• Given any point p in M, $J^+(p)$ [resp. $J^-(p)$] 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 $J^-(p)\cap J^+(q)$ is compact.
• 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.
• The spacetime has a Cauchy surface.

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.

Remarks

In older literature, the condition of causality in the first two definitions of global hyperbolicity given above is replaced by the stronger condition of strong causality. To be precise, a spacetime M is strongly causal if for any point p in M and any neighborhood U of p, there is a neighborhood V of p contained in U such that any causal curve with endpoints in V is contained in U. In 2007, Bernal and Sánchez^[1] showed that the condition of strong causality can be replaced by causality. In particular, any globally hyperbolic manifold as defined in the previous section is strongly causal.

In 2003, Bernal and Sánchez^[2] 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 $\mathbb{R}$. 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 $\mathbb{R}$. In particular, a globally hyperbolic manifold is foliated by Cauchy surfaces.

Global hyperbolicity, in the second form given above, was introduced by Leray^[3] in order to consider well-posedness of the Cauchy problem for the wave equation on the manifold. In 1970 Geroch^[4] proved the equivalence of the second and third definitions above. The first definition and its equivalence to the other two was given by Hawking and Ellis.^[5]

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.

See also

• Causality conditions
• Causal structure
• Light cone

References

1. Antonio N. Bernal and Miguel Sánchez, "Globally hyperbolic spacetimes can be defined as 'causal' instead of 'strongly causal'", Classical and Quantum Gravity 24 (2007), no. 3, 745–749
2. Antonio N. Bernal and Miguel Sánchez, " On smooth Cauchy hypersurfaces and Geroch's splitting theorem", Communications in Mathematical Physics 243 (2003), no. 3, 461–470
3. Jean Leray, "Hyperbolic Differential Equations." Mimeographed notes, Princeton, 1952.
4. Robert P. Geroch, "Domain of dependence", Journal of Mathematical Physics 11, (1970) 437, 13pp
5. Stephen Hawking and George Ellis, "The Large Scale Structure of Space-Time". Cambridge: Cambridge University Press (1973).

• 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.