nLab Cauchy surface

Idea

A Cauchy surface is a hypersurface in spacetime (so actually a $3$-dimensional region in our $4$-dimensional spacetime) that can profitably be seen as constituting ‘all of space at a given time’.

Definition

For $(X,g)$ a Lorentzian manifold, a Cauchy surface is an embedded submanifold $\Sigma \hookrightarrow X$ such that every timelike curve in $X$ may be extended to a timelike curve that intersects $\Sigma$ precisely in one point.

A Lorentzian manifold that does admit a Cauchy surface is called globally hyperbolic.

Properties

Proposition

For $X^D$ a smooth globally hyperbolic spacetime, it admits a smooth foliation by smooth spacelike Cauchy surfaces $X^d$, exhibited by a diffeomorphism

$$X^D \,\simeq\, \mathbb{R}^{0,1} \times X^d \,,$$

which is isometric with respect to a possibly non-product pseudo-Riemannian metric on the right

(Bernal & Sánchez 2005, Thm. 1.1, following Geroch 1970, reviewed in Sánchez 2022 (3)).

Applications

One way to formulate causality in physics is that the values of all observables at all points on a single Cauchy surface in spacetime is enough information (in the sense of a boundary condition to apply to a differential equation constituting a relevant physical theory) to determine the values of all observables at all points of spacetime. (This is not always an actual theorem of differential equations.) Stated more intuitively, the state of the universe at any given time is enough information to determine the state of the universe at all times.

If spacetime can be equipped with a foliation of Cauchy surfaces, then we may assign a real number $t$ to each surface $\Sigma$, so that we think of $\Sigma$ as ‘space at time $t$’. Of course, there are typically many ways to do this (if any), in accordance with the principle of relativity of simultaneity?. On the other hand, for some spacetimes, this may not be possible at all (because they are not globally hyperbolic).

Related concepts

• Cauchy horizon

References

The existence of a splitting of globally hyperbolic spacetimes into Cauchy surfaces is

in the topological category due to

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

and in the smooth category (needed in practice) due to

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

• 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]