nLab globally hyperbolic Lorentzian manifold

Context

Riemannian geometry

Riemannian geometry

Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• Theorems

• Poincaré conjecture-theorem
• Applications

• gravity

• Contents

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.

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.

References

Textbook accounts include

• 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

The refinement of this statement to a smooth splitting is in

• Antonio N. Bernal, Miguel Sánchez, On smooth Cauchy hypersurfaces and Geroch’s splitting theorem (arXiv:gr-qc/0306108v2)

Last revised on November 9, 2017 at 15:41:06. See the history of this page for a list of all contributions to it.