nLab globally hyperbolic Lorentzian manifold

Idea

A Lorentzian manifold is called globally hyperbolic if it admits a well-defined time evolution from initial data of physical fields on it.

Definition

There are several equivalent definitions of global hyperbolicity. A simple one is:

Definition

A Lorentzian manifold (without boundary) is called globally hyperbolic if it contains a Cauchy surface.

In this form the characterization of global hyperbolicity appears for instance in the paragraph at the bottom of page 211 in (HE). The equivalence of this to more traditional definitions is (HE, prop. 6.6.3) together with (HE, prop. 6.6.8), due to (Geroch1970). The latter in fact implies the following stronger statement:

Proposition

A Lorentzian manifold (without boundary) is globally hyperbolic if it admits a foliation by Cauchy surfaces.

See also (Baer-Ginoux-Pfaeffle 07, theorem 1.3.10).

Remark

So in particular for a globally hyperbolic spacetime $X$ there is a homeomorphism

$$\phi\colon \mathbb{R} \times \Sigma \to X$$

from the product of the real line with a $(dim X - 1)$-dimensional smooth manifold $\Sigma$ and for each $t \in \mathbb{R}$ the image $\phi(t, \Sigma) \subset X$ is a Cauchy surface of $X$.

A time orientation of a globally hyperbolic Lorentzian spacetime is a choice of orientation of the factor $\mathbb{R}$ under the above homeomorphism.

Related concepts

• AQFT on curved spacetimes, locally covariant perturbative quantum field theory

• Cauchy problem

• Green hyperbolic partial differential equation

References

Concise collection of definitions:

• E. Minguzzi, Miguel Sánchez, §3.11 in: The causal hierarchy of spacetimes, in: Recent Developments in Pseudo-Riemannian Geometry, EMS ESI Lectures in Mathematics and Physics 4 (2008) 299-358 [arXiv:gr-qc/0609119, ISBN:978-3-03719-051-7]

Survey:

• Miguel Sánchez, Globally hyperbolic spacetimes: slicings, boundaries and counterexamples, Gen Relativ Gravit 54 124 (2022) [arXiv:2110.13672, doi:10.1007/s10714-022-03002-6]

See also:

• Hawking, Ellis, section 6.6 of The large-scale structure of Space-Time Cambridge (1973)

• Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (arXiv:0806.1036)

The fact that a single Cauchy surface implies a foliation by Cauchy surfaces is due to:

• Robert Geroch, §5, Thm. 11 in: Domain of Dependence, J. Math. Phys. 11 (1970) 437–449 [doi:10.1063/1.1665157]

The refinement of this statement to a smooth splitting:

• Antonio N. Bernal, Miguel Sánchez, On smooth Cauchy hypersurfaces and Geroch’s splitting theorem, Commun. Math. Phys. 243 (2003) 461-470 [arXiv:gr-qc/0306108v2, doi:10.1007/s00220-003-0982-6]

• Antonio N. Bernal, Miguel Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Commun. Math. Phys. 257 (2005) 43-50 [arXiv:gr-qc/0401112, doi:10.1007/s00220-005-1346-1]

• Antonio N. Bernal, Miguel Sánchez, Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions, Lett. Math. Phys. 77 (2006) 183-197 [arXiv:gr-qc/0512095, doi:10.1007/s11005-006-0091-5]