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 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:

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

$$\varphi :ℝ\times \Sigma \to X$$\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 $t\in ℝ$ the image $\varphi (t,\Sigma )\subset X$ is a Cauchy surface of $X$.

References

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)