general relativity - Given the Wikipedia notion of "arc length", how is its manifestly real "signed variant" to be called and denoted?

I am dissatisfied with the presentation (not to say "definition") of "arc length", in its "Generalization to (pseudo-)Riemannian manifolds", as given in Wikipedia. (Who isn't?. But I'll sketch it here as a starting point anyways.) Namely:

[arc] length of curve $\gamma$ as $$\ell[~\gamma~] := \int_0^1 dt~\sqrt{\pm~g[~\gamma'[~t~],\gamma'[~t~]~]},$$

where [...] the sign ["$\pm$"] in the square root is chosen once for a given curve, to ensure that the square root is a real number.

In contrast, more useful I find the following definition variant (which is broadly similar to the above, but different in some decisive details): $$\int_0^1 dt~(\pm)[~t~]~\lvert\sqrt{(\pm)[~t~]~g[~\gamma'[~t~],\gamma'[~t~]~]}~\rvert,$$ where the sign "$\pm$" is chosen separately for each individual value $t$,
or in other words, the sign "$(\pm)[~t~]$" is chosen as a function of "the variable $t$",
to ensure that the square root is a real number for each individual value $t$
(and with all other symbols the same as in the Wikipedia presentation above).

Is this latter definition variant already known by some particular name and notation in the literature ?

And vice versa: Has the name "signed arc length" and/or the symbol "$s[~\gamma~]$" been used in any other sense (inconsistent with this latter definition variant); at least within the context of discussing pseudo-Riemannian manifolds ?

Note on notation:

The symbol "$(\pm)[~t~]$" for denoting "the appropriate sign as a function of the variable $t$" has been used above in order to mimic the symbol "$\pm$" which appears (presently) in the Wikipedia article. A more explicit and perhaps more established notation for this function would be "$\text{sgn}[~g[~\gamma'[~t~],\gamma'[~t~]~]~]$".

Documentation of prior research (in response to a deleted answer):

As of recently, Google searches for "signed arc length" or "signed arclength" seem to yield fewer than 100 distinct results, several of which even dealing with general relativity (and hence with spacetime, and/or pseudo-Riemannian manifolds as models of spacetime), but none of them (except this PSE question) presenting in this context anything resembling the sought particular expression.^*

My attempts at a web search for this particular expression didn't seem to bring up any relevant results either; even considering several different choices of notation.

(*: In order to make this determination I've been trying to match symbols or items of the given notations of the documents I had found to the following notions (here in my specific, but generally of course arbitrary notation):

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  spacetime, as set of events $\mathcal S$,

  
  


• 
  
  

  a strictly ordered subset of spacetime, $\Gamma \subset \mathcal S$, and

  
  


• 
  

  two signed measures $\mu_s$ and $\mu_g$ for which

  
  

$$\forall x \in \Gamma : \lim_{ A \rightarrow x }~\left( |~\mu_s[~A~]~| ~ \mu_s[~A~] - \mu_g[~A~] \right) = 0,$$

• or at least one signed measure $\mu_s$ together with real numbers $g[~x, \Gamma, \mathcal S~]$ which are not necessarily positive, for which

$$\forall x \in \Gamma : \lim_{ A \rightarrow x }~\left( |~\mu_s[~A~]~| ~ \mu_s[~A~] \right) = g$$ ).

Answer

I don't remember having seen the specific expression of the proposed "signed arc length" either (anywhere but related to the OP question), nor anything resembling ⁽¹⁾ the more abstract expression for determining the sought resemblance.

For naming this proposed functional from the set of curves (or rather, arcs) into the set of real numbers (incl. $\mathbb R_{-}$) the choice "signed arc length" seems reasonable, but not very specific ⁽²⁾.

However I can think of several names which on first sight may seem reasonable and more evocative, but which refer⁽³⁾ instead to (largely) inapplicable notions; for instance:

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  not "Lorentzian arc length", not "sub-Lorentzian arc length", not even "signed Lorentzian (arc) length";

  
  


• 
  
  

  not "(Synge's) World (arc) function", not "Minkowski arc length";

  
  


• 
  

  not "pseudo-Riemannian arc length";

  
  

even though, on the other hand, the definition of "pseudo-(arc)-length", (eq. 11.59) itself may well allow an interpretation which includes the proposed functional.

But if relying on other's interpretation is not an option then, as far as I know, it remains only to call the (definition, and any accordingly determined value) of the proposed functional "proper arc length".