notation. A time-like or causal curve $\gamma : I\to M$ is called [*future directed*]{} if $g(\mathfrak{t},\dot{\gamma})<0$ and [*past directed*]{} if $g(\mathfrak{t},\dot{\gamma})>0$. The [*chronological future/past*]{} of a point $x\in M$ is the subset $I^\pm_M(x)\subseteq M$ of all points that can be reached from $x$ by future/past directed time-like curves. The [*causal future/past*]{} of a point $x\in M$ is the subset $J^\pm_M(x)\subseteq M$ of all points that can be reached from $x$ by future/past directed causal curves and $x$ itself. Given any subset $S\subseteq M$, we define $I^\pm_M(S) := \bigcup_{x\in S}I^\pm_M(x)$ and $J^\pm_M(S) := \bigcup_{x\in S} J^\pm_M(x)$.

Let $M$ be a time-oriented Lorentzian manifold. A subset $S\subseteq M$ is called [*causally convex*]{} if $J^+_M(S)\cap J^-_M(S)\subseteq S$. Two subsets $S,S^\prime\subseteq M$ are called [*causally disjoint*]{} if $\big(J^+_M(S)\cup J^-_M(S)\big)\cap S^\prime = \emptyset$.

In words, a subset $S\subseteq M$ is causally convex if every causal curve that starts and ends in $S$ is contained entirely in $S$. Two subsets $S,S^\prime\subseteq M$ are causally disjoint if there exists no causal curve in $M$ connecting $S$ and $S^\prime$.

A time-oriented Lorentzian manifold $M$ is called [*globally hyperbolic*]{} if it admits a [*Cauchy surface*]{}, i.e. a subset $\Sigma\subset M$ that is met exactly once by each inextensible time-like 