causal complement



The concept of a causal complement is suitable to establish a causal disjointness relation on index sets where the indices are subsets of a given set. This generalizes the concept of causal complement of subsets of the Minkowski spacetime, see for example Haag-Kastler vacuum representation.


The concept of causal complement, as the name indicates, a concept related to causality on spacetimes, hence a concept associated with Lorentzian manifolds:

One may abstract away some of the key properties of causal complements on Lorentzian manifolds to obtain a more general concept:

On Lorentzian manifolds

Let XX be Lorentzian manifold (a spacetime), hence a pseudo-Riemannian manifold with one time-like dimension in each tangent space.

Then for SXS \subset X any subset, its causal cone J(S)J(S) is the subset of all points in XX which may be joined to a point of SS by a smooth curve γ:[0,1]X\gamma \colon [0,1] \to X which is everywere timelike or lightlike.


(causal complement of subset of Lorentzian manifold)

For SXS \subset X a subset of a Lorentzian manifold, its causal complement S S^\perp is the complement of the causal cone:

S SJ(S). S^\perp \;\coloneqq\; S \setminus J(S) \,.

The causal complement S S^{\perp \perp} of the causal complement S S^\perp is called the causal closure. If

S=S S = S^{\perp \perp}

then the subset SS is called a causally closed subset.

Abstract definition

Let XX be an arbitrary set and MXM \subset X. An assignment

MM X M \mapsto M^{\perp} \subset X

is called a causal complement if the following conditions hold:

(i) MM M \subseteq M^{\perp\perp}

(ii) ( jM j) = j(M j) (\bigcup_j M_j)^{\perp} = \bigcap_j (M_j)^{\perp}

(iii) MM =M \bigcap M^{\perp} = \emptyset

A set MM is causally closed iff M=M M = M^{\perp\perp}.

The set M M^{\perp\perp} is the causal closure of MM.


Causal complements are always causally closed. The intersection of two causally closed sets is again a causally closed set. The causal complement of a set may be empty.

A causal disjointness relation on an index set of subsets of a given set XX can be defined via

M 1M 2iffM 1(M 2) M_1 \perp M_2 \; \text{iff} \; M_1 \subseteq (M_2)^{\perp}

if all sets MM have a causal complement and if

(iv) there is a sequence (Y n) n=1 (Y_n)_{n=1}^{\infty} of mutually different subsets with Y n Y_n^{\perp} \neq \emptyset and Y n=X\bigcup Y_n = X.

The latter condition is needed to get a σ\sigma-bounded poset; the σ\sigma-boundedness is part of the definition of a causal index set.

Revised on September 18, 2017 07:40:09 by Urs Schreiber (