nLab causal order

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

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

(e.g. Epstein-Glaser 73, p. 5 (273))

Related concepts

• causal structure

• causality, microcausality

• causal locality, causal additivity

• causal complement, causal closure

• causal perturbation theory

• time-ordered product

• S-matrix

References

• Henri Epstein, Vladimir Glaser, The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211 (Numdam)