Since a geodesic is understood to be a map $\gamma : \text{real number interval} \rightarrow \text{spacetime } \mathcal S$, with certain additional properties, the image of a particular geodesic is some particular subset of spacetime, i.e. some particular set of events $\mathcal C_{\gamma} \subset \mathcal S$.
The geodesic $\gamma$ provides a certain one-to-one parametrization of its image, $\text{real number interval} \longleftrightarrow \mathcal C_{\gamma}$, while the set of events $\mathcal C_{\gamma}$ itself stands without any particular parametrization or labelling of its elements; it is (overtly) parametrization-independent.
An image of a spacetime geodesic, such as set $\mathcal C_{\gamma}$ as image of geodesic $\gamma$, surely has certain additional properties itself -- besides a suitable cardinality it surely has certain geometric properties. (Therefore any arbitrary event set, even of suitable cardinality, could not be the image of a spacetime geodesic.)
Now, a particular way to express geometric properties of a set of events (and notably invariant under reparametrization) is through values of Synge's world function $\sigma : \mathcal S \times \mathcal S \rightarrow \mathbb R$, which are assigned to certain pairs (i.e. to some, but not necessarily to all pairs) of events of a given spacetime $\mathcal S$.
My question:
Can the geometric properties of the image of a spacetime geodesic be correctly and fully expressed in terms of values $\sigma$ of Synge's world function?
(And if so, how?)
Specificly:
Is a set of events $\mathcal E$ the image of a spacetime geodesic (or in other words: is $\mathcal E \equiv \mathcal C_{\gamma}$ for some geodesic $\gamma$) if:
set $\mathcal E$ is identified by three distinct (but not unique) events $a, b, c \in \mathcal E$ for which there exist pairwise assigned values $\sigma[ \, a, b \, ]$ and $\sigma[ \, a, c \, ]$, and $\sigma[ \, b, c \, ]$, and which are straight wrt. each other, i.e. (by Heron's formula)
$ 2 \, \sigma[ \, a, b \, ] \, \sigma[ \, a, c \, ] + 2 \, \sigma[ \, a, b \, ] \, \sigma[ \, b, c \, ] + 2 \, \sigma[ \, a, c \, ] \, \sigma[ \, b, c \, ] = $
$ \qquad, \qquad \qquad (\sigma[ \, a, b \, ])^2 + (\sigma[ \, a, c \, ])^2 + (\sigma[ \, b, c \, ])^2$,there is a familiy $\{ \mathcal F_i \}$ of sets of events, each of which satisfies the following conditions:
$a, b, c \in \mathcal F_i$
and
- $ \forall e \in \mathcal F_i : $
$ \exists \, p, q \in \mathcal F_i | e \not\equiv p$ and $e \not\equiv q$ and $p \not\equiv q$ and $\exists \, \sigma[ \, e, p \, ]$ and $\exists \, \sigma[ \, e, q \, ]$ and $\exists \, \sigma[ \, p, q \, ]$
and
- $ \forall j, k, n \in \mathcal F_i : $
if $ \exists \, \sigma[ \, j, k \, ], \sigma[ \, j, n \, ], \sigma[ \, k, n \, ]$
then $ 2 \, \sigma[ \, j, k \, ] \, \sigma[ \, j, n \, ] + 2 \, \sigma[ \, j, k \, ] \, \sigma[ \, k, n \, ] + 2 \, \sigma[ \, j, n \, ] \, \sigma[ \, k, n \, ] = $
$ \qquad \qquad \qquad (\sigma[ \, j, k \, ])^2 + (\sigma[ \, j, n \, ])^2 + (\sigma[ \, k, n \, ])^2$
and
$ \forall x, y \in \mathcal F_i : $
if $x \not\equiv y$ and $\exists \, \sigma[ \, x, y \, ]$ and $\exists \, z$ and $\exists \, \sigma[ \, x, z \, ]$ and $\exists \, \sigma[ \, y, z \, ]$ and $ 2 \, \sigma[ \, x, y \, ] \, \sigma[ \, x, z \, ] + 2 \, \sigma[ \, x, y \, ] \, \sigma[ \, y, z \, ] + 2 \, \sigma[ \, x, z \, ] \, \sigma[ \, y, z \, ] = $
$ \qquad \qquad \qquad (\sigma[ \, x, y \, ])^2 + (\sigma[ \, x, z \, ])^2 + (\sigma[ \, y, z \, ])^2$
then $z \in \mathcal F_i$.Finally: $ \mathcal E := \text{Intersection}[ \, \{ \mathcal F_i \} \, ]$
?
