From mostly recent papers such as these, and their common reference to papers by J. R. Letaw and J. D. Pfausch from the early 1980s, we can learn that
[...] a trajectory in (3 + 1)-dimensional Minkowski spacetime [...] can be characterized by three geometric invariants: the curvature $a[ ~ \tau ~ ]$, which represents the magnitude of proper acceleration, the first torsion $a[ ~ \tau ~ ]$, and the second torsion (also known as hypertorsion) $\nu[ ~ \tau ~ ]$ of the worldline. The [first and second] torsions $b[ ~ \tau ~ ]$ and $\nu[ ~ \tau ~ ]$ correspond to the proper angular velocities in a given tetrad frame
Now, for the curvature and the torsion of (the image of) a curve in 3-dimensional Euclidean space we have (more or less well-known) corresponding metric expressions, i.e. in terms of (extrinsic) distances between points of a given (image of) a curve $\gamma$, at any point $P \in \gamma$:
- as Menger curvature: $\kappa_M[ ~ \gamma, P ~ ] := $
$$ {\large \underset{J, Q ~ \in ~ \gamma; JP ~ \lt ~ JQ, PQ ~ \lt ~ JQ \rightarrow 0 }{\underset{~}{\Large \text{Lim}}}\left[ ~ \frac{ \normalsize \left( (-) \begin{bmatrix} 0 & 1 & 1 & 1 \cr 1 & 0 & JP^2 & JQ^2 \cr 1 & JP^2 & 0 & PQ^2 \cr 1 & JQ^2 & PQ^2 & 0 \end{bmatrix} \right)^{\left(\frac{1}{2}\right)} }{\underset{JP^2 ~ PQ^2 ~ PQ^2}{~}} ~ \right] },$$
where the determinant in the denominator is of course known as a Cayley-Menger determinant;
cmp. eqs. (1.2.16) and (1.2.17) of this very new reference
and
- the Blumenthal metric torsion which is expressed in terms of several Cayley-Menger determinants involving four points; cmp. eqs. (1.3.8), (1.3.15), (1.3.17).
The appeal of these metric expressions is twofold, namely:
to be readily adaptable to the case of (timelike) (images of) curves in 3+1 dimensional flat spacetime, namely simply through replacement of the squares of distance values with values $s^2$ of (timelike) spacetime intervals; and moreover being readily adaptable to the case of (timelike) (images of) curves in 3+1 dimensional general spacetime, namely simply through replacement of the squares of distance values with the squares of (non-zero) Lorentzian distance values $\ell^2$; and
being manifestly independent of any curve parametrizations, coordinate assignments, definition and selection of any frames, or somesuch.
Therefore
My question:
How exactly is hypertorsion $\nu[ ~ \gamma, P ~ ]$ of a (suitable) (image of) a timelike curve $\gamma$ (a.k.a. timelike worldline), in 3+1 dimensional flat spacetime, in point (a.k.a. spacetime event) $P \in \gamma$ explicitly expressed in terms of values $s^2$ of (timelike) spacetime intervals of pairs of events which make up this worldline ?