I'm looking for a list of explicit general properties common to all (unrestricted) instances of Synge's world function (SWF) $\sigma : \mathcal U^2 \rightarrow \mathbb R$.
Ideally, the desired list as a whole should constitute sufficient conditions for any specific instance of some generic function $f : \text{(Set }\mathcal S\text{)}^2 \rightarrow \mathbb R$ which has each and all of the listed properties to be therefore equivalent to (at least) one specific (unrestricted) instance of SWF.
The applicable properties should each be explicit by being expressible exclusively in terms of
the values of generic function $f$ themselves, i.e. real numbers $f[ \, x, y \, ]$ for (variable arbitrary, or otherwise specified) arguments $x, y \in \mathcal S$, along with operations (such as comparisons, or arithmetic operations) applicable real numbers, as well as some necessary specific individual real number values (such as real number $0$ etc.), and
the generic domain set $S$ itself, along with set operations (such as selecting one arbitrary element, or selecting two distinct but otherwise arbitrary elements).
(In other words: variants of the implicit description "All properties that follow from the definition of SWFs, along with the properties of each item used in the definition." do not present the desired answer. Nevertheless, the properties I'm asking for should of course each be provable from "the definition of SWFs", along with all properties of symbols and terms appearing therein.)
Further, all applicable properties should be common to all unrestricted SFW instances in the sense that they are satisfied by each and any SWF instance $\sigma : \mathcal U^2 \rightarrow \mathbb R$ for which
- $\exists \, j, k, p, q \in \mathcal U \, \mid \, \sigma[ \, j, k \, ] \, \sigma[ \, p, q \, ] < 0$.
(In other words: the properties should be satisfied by any SWF whose domain set $\mathcal U$ "has, or is assigned, temporal as well as spatial extent"; as opposed to e.g. "being restricted to have only spatial extent", or "being restricted to have only temporal extent", as far as such restrictions might be consistent with the definition of SWFs at all.)
(Note: Related to the distinction between "temporal extent" and "spatial extent" I hope that the properties I'm asking for can be expressed without needing to prescribe or adopt an associated sign convention for the corresponding values of $\sigma$.)
As an example now follows a presumably still insufficient list of five properties (the five "simplest, most obvious" properties) common to all (unrestricted) instances of SWF (including some ostensibly repetitive verbiage for the purpose of explaining and re-inforcing which kind of properties I'm asking for):
- "Categorization":
In order for each specific instance of generic function $f$ to be equivalent to (at least) one specific (unrestricted) instance of SWF, the generic function $f$ (and therefore each of its instances) must be a function
$$f : \mathcal S^2 \rightarrow \mathbb R, \text{where } \mathcal S \text{ denotes a set}. \tag{0a}$$
- "Unrestricted":
In order for each specific instance of generic function $f : \mathcal S^2 \rightarrow \mathbb R$ to be equivalent to (at least) one specific (unrestricted) instance of SWF, the generic domain set $\mathcal S$ must contain at least four (not necessarily all distinct) elements $j, k, p, q \in \mathcal S$ such that the corresponding values of generic function $f$ satisfy:
$$f[ \, j, k \, ] \, f[ \, p, q \, ] < 0. \tag{0b}$$
- "Indiscernability of the Identical":
In order for each specific instance of generic function $f : \mathcal S^2 \rightarrow \mathbb R$ to be equivalent to (at least) one specific (unrestricted) instance of SWF, $f$'s values must satisfy:
$$\forall \, x \in S : f[ \, x, x \, ] = 0. \tag{1}$$
- "Symmetric":
In order for each specific instance of generic function $f : \mathcal S^2 \rightarrow \mathbb R$ to be equivalent to (at least) one specific (unrestricted) instance of SWF, $f$'s values must satisfy:
$$\forall \, x, y \in S : f[ \, x, y \, ] = f[ \, y, x \, ].\tag{2}$$
- "Anti-Heronian":
In order for each specific instance of generic function $f : \mathcal S^2 \rightarrow \mathbb R$ to be equivalent to (at least) one specific (unrestricted) instance of SWF, the generic domain set $\mathcal S$ must contain at least two distinct elements $p, q \in \mathcal S$ such that $f$'s values (for certain arguments $a, b, c \in S$, provided they exist as specified) satisfy:
$$\small (\exists \, a, b, c \in S \, | \, ((f[ \, a, b \, ] \, f[ \, p, q \, ] \gt 0) \text{ and } (f[ \, a, c \, ] \, f[ \, p, q \, ] \gt 0) \text{ and } (f[ \, b, c \, ] \, f[ \, p, q \, ] \gt 0))) \implies \\ \small \left( \frac{f[ \, a, b \, ]}{f[ \, b, c \, ]} \right) + \left( \frac{f[ \, b, c \, ]}{f[ \, a, b \, ]} \right) + \left( \frac{f[ \, a, c \, ]}{f[ \, a, b \, ]} \right) \, \left( \frac{f[ \, a, c \, ]}{f[ \, b, c \, ]} \right) \ge 2 + 2 \, \left( \frac{f[ \, a, c \, ]}{f[ \, a, b \, ]} \right) + 2 \, \left( \frac{f[ \, a, c \, ]}{f[ \, b, c \, ]} \right). \tag{3}$$
My question:
Are there explicit properties in addition to (independent of) the five properties listed above which are common to all (unrestricted) instances of Synge's world function ?
(If so: Please express some or all of them, as requirements on a generic function $f$, following the above example!)