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causal order

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Definition

Definition

(causal order)

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

Consider the relation on the set P(Σ)P(\Sigma) of subsets of spacetime which says a subset S 1ΣS_1 \subset \Sigma is not prior to a subset S 2ΣS_2 \subset \Sigma, denoted S 1S 2S_1 {\vee\!\!\!\wedge} S_2, if S 1S_1 does not intersect the causal past of S 2S_2, or equivalently that S 2S_2 does not intersect the causal future of S 1S_1:

S 1S 2 S 1V¯ (S 2)= S 2V¯ +(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 1S 2S_1 {\vee\!\!\!\wedge} S_2 and S 2S 1S_2 {\vee\!\!\!\wedge} S_1 we say that the two subsets are spacelike separated and write

S 1><S 2S 1S 2andS 2S 1. 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))

References

Last revised on August 2, 2018 at 03:04:49. See the history of this page for a list of all contributions to it.