An observation of Martin and Panangaden links the study of Lorentzian manifolds and the semantics of programming languages via the theory of Scott domains.
Background:
Recall that if $M$ is a time-oriented Lorentzian manifold with no closed causal curves, then there are two important partial orders on $M$, and one additional important relation:
$(M,<)$ is a poset, where $p < q$ if there is a causal path [1] from $p$ to $q$.
$(M, \ll)$ is a poset, where $p \ll q$ if there is a chronological path [1] from $p$ to $q$.
We say that $p$ is horismotic to $q$, and write $p \to q$, if $p < q$ but $p \not \ll q$ [1].
These relations, including various ways of defining one in terms of the other, were first studied in the abstract by Kronheimer and Penrose.
The observation of Martin and Panangaden is that, at least if $M$ is globally hyperbolic, then one way of defining the $\ll$ relation from the $<$ relation (not studied by Kronheimer and Penrose) is precisely the one suggested by the notation from the perspective of domain theory. That is, we have that
$p \ll q$ if and only if $p$ is way below $q$ with respect to the $<$ poset structure [2]. Moreover, $(M,<)$ is a continuous poset [2]. In fact, $(M,<)$ is a local domain [2].
Everything here has a dual statement given by reversing the time orientation, and in fact $M$ is a strongly bicontinuous poset [3].
What I'd like to understand:
What I'd like to understand is just how strong this connection between relativity and domain theory really is. The first thing that stands out to me is that the domains coming from relativity theory don't seem to be "generic" from the point of view of domain theory -- they have various special properties. I'd like to understand what to make of these special properties from the point of view of domain theory. To start with:
Question 1: Is there an intrinsic domain-theoretic reason to study bicontinuous posets [3] rather than just continuous posets? From a domain-theoretic perspective, are there important examples of bicontinuous posets? Are there things one can do and questions one can ask about bicontinuous posets which can't be done with more general continuous posets?
Next, there are various special properties of the horismotic relation in the domains $M$ coming from relativity theory. Most obviously,
- (A) there can be no cycles $x_0 \to x_1 \to \dots \to x_n = x_0$.
There are additional special properties coming from the uniqueness of (lightlike) geodesics. The most succinct way to put it is that
- (B) if $w,x,y,z$ are 4 distinct points of $M$, and if 5 of the 6 unordered pairs of elements of $\{w,x,y,z\}$ are related (in either direction) by the $\to$ relation, then so is the 6th pair [4].
Kronheimer and Penrose called any spacetime satisfying a certain fragment of (B) regular [4]. Note that the analogous condition for 2 distinct points of $M$ rather than 4 would say that $M$ is linearly ordered by $\to$, so (B) is in some sense a weakening of linearity under $\to$. If $(P,<)$ is an arbitrary poset, then we can still define $x \to y$ if and only if $x < y$ but $x \not \ll y$ (where $\ll$ is the way below relation [2]), and then (A) will hold automatically. But (B) seems to be a curious property from the perspective of order theory or domain theory.
Question 2: Does condition (B) "mean anything" domain-theoretically? Is it in any way a natural domain-theoretic condition to consider? Are there perhaps "more-natural" conditions which imply it or are implied by it? What can one "do" with such conditions domain-theoretically?
I could go on, but maybe I'll close with one final question. In a domain $M$ which is a Lorentzian manifold, it's the case, essentially by definition, that if $p \ll q$, then there is a chronological path [1] from $p$ to $q$. From a purely order-theoretic perspective, this is pretty well encapsulated by saying that there exists a map $\gamma: [0,1] \to M$ such that $\gamma(0) = p$, $\gamma(1) = q$, $s < t \Rightarrow \gamma(s) \ll \gamma(t)$, such that $\gamma$ preserves infs and sups.
Question 3: Do "chronological paths" according to the above definition ever come up in domain theory?
Footnotes:
[1] A smooth path $\gamma: [0,1] \to M$ with $\gamma(0) = p$ and $\gamma(1) = q$ is causal if $\gamma'(t)$ lies in the closed future lightcone for all $t \in [0,1]$ and chronological if $\gamma'(t)$ lies in the open future lightcone for all $t \in [0,1]$. I believe this makes it the case that $p \to q$ if and only if there is a future-directed lightlike geodesic from $p$ to $q$.
[2] If $P$ is a poset and $p,q \in P$, then we say that $p$ is way below $q$ and write $p \ll q$ if, for every directed set $D \subseteq P$ with sup $\bigvee D$, if $q \leq \bigvee D$, then there exists $d \in D$ such that $p \leq d$. A poset is $P$ said to be continuous if, for every $p \in P$, the set $\{q \in P \mid q \ll p\}$ is directed and has sup given by $p$. A poset $P$ is said to be a domain if it is continuous and has sups of directed sets, and a local domain if it is continuous and has sups of bounded directed sets.
[3] A strongly bicontinuous poset $(P,<)$ is a poset such that $P$ and $P^{op} = (P,>)$ are both bicontinuous posets and moreover we have that $p$ is way below $q$ with respect to $<$ if and only if $q$ is way below $p$ with respect to $>$.
[4] More explicitly, in light of (A), (B) reduces to the following conditions for any 4 distinct points $w,x,y,z \in M$:
If $w \to x \to y \to z$ and $w \to y$ and $x \to z$, then $w \to z$. This is a sort of "locality" property for the $\to$ relation, allowing us to patch together lightlike geodesics from smaller pieces.
(regularity in the sense of Kronheimer and Penrose) If $w \to x \to y$ and $w \to y$ and $x \to z$ and $w \to z$, then $y \to z$ or $z \to y$. Dually, if $x \to y \to z$ and $x \to z$ and $w \to y$ and $w \to z$, then $w \to x$ or $x \to w$. This says that a lightlike geodesic may be extended (at either end) in at most one way.
If $w \to x$, $y \to z$, $w \to y$, $x \to z$, and $w \to z$, then $x \to y$ or $y \to x$. This says that if $w \to z$, then the interval between them $\{p \in P \mid w \leq p \leq z\}$ is linearly ordered by $\leq$ (or equivalently by $\to$).
