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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

$$ {\large \text{Size}[ ~ \mathcal D_0 ~ ] := \underset{p, q ~ \in ~ \mathcal D_0}{\text{Sup}}\Big[ \{ \sqrt{ 2 ~ \text{Abs}[ ~ \sigma[ ~ p, q ~ ] ~ ] } \} ~ \Big] }.$$

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  • $\begingroup$ I've noticed the (1) immediate "drive-by downvote" to my OP question; canonically indicating that it "does not show any research effort; it is unclear, or not useful". I took that as motivation to right away do a corresponding web search (which, admittedly, I hadn't done in a while). And, indeed, we find prominently: arxiv.org/abs/2304.00088 (A. Nasiri, "Synge's world function applied to causal diamonds and causal sets"). Quote: "Of great importance is the recovery of curvature." $\endgroup$
    – user12262
    Sep 28, 2023 at 20:40
  • $\begingroup$ As motivational example for my question consider timelike path segment $\gamma \subset \mathcal D_0$ with event $x \in \gamma$. Then the expression $$ \Large \underset{\sigma[~ p,q ~] \rightarrow 0; ~~ x \text{ between } p,q,\in \gamma}{\text{Lim}}\Big[ \\ \left( \frac{\sigma[~ p,q ~]}{\sigma[~ p,x ~] ~ \sigma[~ x,q ~]} + \frac{\sigma[~ p,x ~]}{\sigma[~ p,q ~] ~ \sigma[~ x,q ~]} + \frac{\sigma[~ x,q ~]}{\sigma[~ p,x ~] ~ \sigma[~ p,q ~]} -\frac{2}{\sigma[~ p,q ~]} -\frac{2}{\sigma[~ p,x ~]} -\frac{2}{\sigma[~ x,q ~]}\right)^{(-1/2)}~ \Big] $$ is a.k.a. curvature radius $\rho_{\gamma}$ at $x$. $\endgroup$
    – user12262
    Sep 29, 2023 at 7:01

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