Tim Porter causal site

Idea

• 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.

Definition

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

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 $A, B,$ and $C$ $A\subseteq B$ and $B\lt C$ implies $A\lt C$;

6. For all regions $A, B,$ and $C$ $A\subseteq B$ and $C\lt B$ implies $C\lt A$;

7. For all regions $A, B,$ and $C$ $A\lt B$ and $B\lt C$ implies $A\cup B\lt C$;

8. For all regions $A$ and $B$, there is a region $B_A$ such that

   • $B_A \lt A$ and $B_A \subseteq B$;

   • if $D\lt A$ and $D\subseteq B$, then $D\subseteq B_A$;

9. If $A$ and $C$ are non-empty regions such that $A\lt C$ and there exists a $D$ with $A\lt D\lt C$, then there is a $B$ complete with respect to $A\lt C$. (For complete see later.)

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

(to be continued)