Consider participant $P$ having taken part in a (gap-less) sequence $\mathcal P$ of (distinct) events;
among them events $\varepsilon_{AP}, \varepsilon_{BP}, \varepsilon_{FP}, \varepsilon_{JP}, \varepsilon_{KP} \in \mathcal P$.
Let the values of Lorentzian distances $\ell : \mathcal P \times \mathcal P \rightarrow \mathbb R^+$ be given for each pair of events in set $\mathcal P$.
How to express the value of Frenet-Serret parameter $\kappa$ of curve $\mathcal P$ at event $\varepsilon_{FP}$ in terms of the values $\ell$?
Without loss of generality, the events which had been visited by $P$ shall be ordered such that $\ell^2[ \, \varepsilon_{BP}, \, \varepsilon_{AP} \, ] \ne 0, \qquad \ell^2[ \, \varepsilon_{FP}, \, \varepsilon_{BP} \, ] \ne 0, \qquad $ and $\qquad \ell^4[ \, \varepsilon_{FP}, \, \varepsilon_{AP} \, ] \gt \ell^4[ \, \varepsilon_{FP}, \, \varepsilon_{BP} \, ] + \ell^4[ \, \varepsilon_{BP}, \, \varepsilon_{AP} \, ],$ and so on.
Denote by $\tilde \varepsilon_{\Phi P} \in \mathcal P$ any variable event between events $\varepsilon_{BP}$ and $\varepsilon_{FP}$; and denote by $\tilde \varepsilon_{\Gamma P} \in \mathcal P$ any variable event between events $\varepsilon_{FP}$ and $\varepsilon_{JP}$. Consequently, event $\varepsilon_{FP}$ is between any $\tilde \varepsilon_{\Phi P}$ and any $\tilde \varepsilon_{\Gamma P}$.
The value of Frenet-Serret parameter $\kappa$ of curve $\mathcal P$ at event $\varepsilon_{FP}$ is then
$ \kappa_{\mathcal P}[ \, \varepsilon_{FP} \, ] := $ $$\Large \matrix{ \text{lim}_{\left\{ \!\left(\frac{\ell^2[ \, \tilde \varepsilon_{\Gamma P}, \, \tilde \varepsilon_{\Phi P} \, ]}{\ell^2[ \, \varepsilon_{JP}, \, \varepsilon_{BP} \, ]} \right) \rightarrow 0 \right\}} \! \! \Big[ \! \Big( \frac{\begin{vmatrix} 0 & \ell^2[ \, \varepsilon_{FP}, \, \tilde \varepsilon_{\Phi P} \, ] & \ell^2[ \, \tilde \varepsilon_{\Gamma P}, \, \varepsilon_{FP} \, ] & 1 \cr \ell^2[ \, \varepsilon_{FP}, \, \tilde \varepsilon_{\Phi P} \, ] & 0 & \ell^2[ \, \tilde \varepsilon_{\Gamma P}, \, \tilde \varepsilon_{\Phi P} \, ] & 1 \cr \ell^2[ \, \tilde \varepsilon_{\Gamma P}, \, \varepsilon_{FP} \, ] & \ell^2[ \, \tilde \varepsilon_{\Gamma P}, \, \tilde \varepsilon_{\Phi P} \, ] & 0 & 1 \cr 1 & 1 & 1 & 0 \end{vmatrix}}{ \ell^2[ \, \varepsilon_{FP}, \, \tilde \varepsilon_{\Phi P} \, ] \, \ell^2[ \, \tilde \varepsilon_{\Gamma P}, \, \varepsilon_{FP} \, ] \, \ell^2[ \, \tilde \varepsilon_{\Gamma P}, \, \tilde \varepsilon_{\Phi P} \, ] } \Big)^{\large (1/2)} \Big] . }$$
(A proof of this statement, which is not explicitly provided here for the time being, would proceed along analogous algebraic steps as the proof of the similar statement in the setting of Euclidean space.)
On dimensional grounds, the value $c \, \kappa_{\mathcal P}[ \, \varepsilon_{FP} \, ]$ is identified as magnitude of acceleration of $P$ at event $\varepsilon_{FP}$, where $c$ denotes signal front speed.