Considering a set $\mathcal S$ of events such that for each pair $p, q \in \mathcal S$ Synge's world function $\sigma$ is defined and the corresponding value $\sigma[ ~ p, q ~ ]$ is given, and such that a subset $\mathcal D_0 \subseteq \mathcal S$ constitutes a causal diamond, then surely the values of curvature invariants, such as the Ricci scalar $R$, the Kretschmann scalar $K_1$, among others, are in an abstract sense defined; "at", or "referring to", each event $x$ of (at least: the interior of) causal diamond $\mathcal D_0$.
My question:
Are there specific expressions for values of curvature invariants, at event $x \in (\mathcal D_0 \setminus \partial \mathcal D_0)$, explicitly in terms of values of Synge's world function $\sigma[ ~ p, q ~ ]$ of (some, or all) event pairs $p, q \in \mathcal D_0$, and (surely) in particular involving values $\sigma[ ~ p, x ~ ]$ of (some, or all) events $p \in \mathcal D_0$ and (the "target event") $x$ ? (And if so: Which are these expressions?, of course.)
Presumably, the sought expressions would be "in the limit of vanishing size of causal diamond $\mathcal D_0$", i.e. symbolically
$$ {\large \underset{\text{Size}[ ~ \mathcal D_0 ~ ] \rightarrow 0 }{\text{Lim}}\Big[ ~ ... ~ \Big] }$$
where $\text{Size}[ ~ \mathcal D_0 ~ ]$ as "the size of causal diamond $\mathcal D_0$" would itself be explicitly expressed as
$$ {\large \text{Size}[ ~ \mathcal D_0 ~ ] := \underset{p, q ~ \in ~ \mathcal D_0}{\text{Sup}}\Big[ \{ \sqrt{ 2 ~ \text{Abs}[ ~ \sigma[ ~ p, q ~ ] ~ ] } \} ~ \Big] }.$$
