# nLab causal order

Contents

## In higher category theory

#### Riemannian geometry

Riemannian geometry

# Contents

## Definition

###### Definition

(causal order)

Let $\Sigma$ be a spacetime (a Lorentzian manifold equipped with time orientation).

Consider the relation on the set $P(\Sigma)$ of subsets of spacetime which says a subset $S_1 \subset \Sigma$ is not prior to a subset $S_2 \subset \Sigma$, denoted $S_1 {\vee\!\!\!\wedge} S_2$, if $S_1$ does not intersect the causal past of $S_2$, or equivalently that $S_2$ does not intersect the causal future of $S_1$:

\begin{aligned} S_1 {\vee\!\!\!\wedge} S_2 & \;\coloneqq\; S_1 \cap \overline{V}^-(S_2) = \emptyset \\ & \Leftrightarrow S_2 \cap \overline{V}^+(S_1) = \emptyset \end{aligned} \,.

If $S_1 {\vee\!\!\!\wedge} S_2$ and $S_2 {\vee\!\!\!\wedge} S_1$ we say that the two subsets are spacelike separated and write

$S_1 {\gt\!\!\!\!\lt} S_2 \;\;\;\coloneqq\;\;\; S_1 {\vee\!\!\!\wedge} S_2 \;\text{and}\; S_2 {\vee\!\!\!\wedge} S_1 \,.$