Tim Porter causal site


  • I will give the definition from the Christensen-Crane paper as is in their paper and perhaps expand on it a bit.

  • Their idea is that the regions one should consider should have properties that mirror properties of the regions generated by diamonds in a Lorentzian manifold.


A causal site is a set of ‘’regions’‘ with two binary relations, denoted \subseteq and <\lt satisfying the axioms below. (If ABA\subseteq B, we say that AA is a subset of BB and if A<BA\lt B that AA_ precedes_ BB.)

  1. \subseteq is a partial order on the set of regions;

  2. The partial order \subseteq has a minimum element \emptyset, called the empty region;

  3. The partial order has unions (with usual universal properties);

  4. <\lt is a strict partial order on the non-empty regions.

  5. For all regions A,B,A, B, and CC ABA\subseteq B and B<CB\lt C implies A<CA\lt C;

  6. For all regions A,B,A, B, and CC ABA\subseteq B and C<BC\lt B implies C<AC\lt A;

  7. For all regions A,B,A, B, and CC A<BA\lt B and B<CB\lt C implies AB<CA\cup B\lt C;

  8. For all regions AA and BB, there is a region B AB_A such that

    • B A<AB_A \lt A and B ABB_A \subseteq B;

    • if D<AD\lt A and DBD\subseteq B, then DB AD\subseteq B_A;

  9. If AA and CC are non-empty regions such that A<CA\lt C and there exists a DD with A<D<CA\lt D\lt C, then there is a BB complete with respect to A<CA\lt C. (For complete see later.)

The motivating example for Christensen and Crane comes from a Lorentzian manifold, MM, with no closed timelike curves and a global time orientation. For points pp and rr in MM, write prp\ll r if there is a future directed timelike curve from pp to rr, and let D(p,r)D(p,r) be the set of all points qq with pqrp\ll q\ll r. We say D(p,r)D(p,r) is a diamond. A subset AA of MM is said to be bounded if it is a finite union of diamonds. For AA and BB bounded regions write ABA\subset B when AA is a subset of BB and A<BA\lt B when for every point, aAa\in A and bBb\in B, aba\ll b.

(to be continued)

Last revised on October 19, 2010 at 11:01:58. See the history of this page for a list of all contributions to it.