globally hyperbolic Lorentzian manifold

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

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

A Lorentzian manifold (without boundary) is called **globally hyperbolic** if it contain 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:

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

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 standard textbook exposition is section 6.6 of

- Hawking, Ellis,
*The large-scale structure of Space-Time*Cambridge (1973)

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

- Robert Geroch, (1970)

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)

Revised on September 9, 2013 21:18:59
by Urs Schreiber
(89.204.155.147)