Cauchy surface

Remarks

A Cauchy surface is achronal, since no timelike curve meets it twice, and it is a closed topological hypersurface of $M$. Its domain of dependence, the set of points $p$ such that every inextendible causal curve through $p$ meets $\Sigma$, is all of $M$. Some authors require in addition that $\Sigma$ be a smooth spacelike hypersurface; such Cauchy surfaces exist in every globally hyperbolic spacetime.

Example: in Minkowski space with coordinates $(t, x)$, the hyperplane $\Sigma = \set{t = 0}$ is a Cauchy surface. Indeed, along an inextendible timelike curve the coordinate $t$ is strictly increasing. Since the speed is less than $1$, it takes every real value, so it equals $0$ exactly once.

Non-example: the hyperboloid $H = \set{t = \sqrt{1 + \abs{x}^2}}$ in Minkowski space is spacelike and achronal, but it is not a Cauchy surface. The hyperbolic motion curve $s \mapsto (\sinh s, \cosh s, 0, 0)$ is an inextendible timelike curve with $x_1^2 - t^2 = 1$, while points of $H$ have $t^2 - \abs{x}^2 = 1$, so it never meets $H$.