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

Question

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):

• 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 1

The "signed arc-length" is not used in relativity and I give reasons why. You are free to call and denote it in any way you like.

$s$ and $\ell$ are interchangeably used to denote arc-length of space-like paths $g(\gamma',\gamma')>0$ in relativity and $\tau[\gamma]$ is used for "proper time" of $g(\gamma',\gamma')<0$ time-like paths but the notation is not entirely strict and if you write your text carefully, you should reach no collision.

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To start the explanation, one should ask: what is an arc length good for? The answer is

1. Measuring distances between two points
2. Measuring the change in proper time of an observer passing through two events
3. As a Lagrangian giving a variational principle

There is no usual sense in which 1. and 2. could have a varying signature of $g(\gamma',\gamma')$. But if you want to extend beyond that, I feel it most natural to consider a different formalism: take ${\rm d}s = \sqrt{g(\gamma'(p),\gamma'(p))}{\rm d}p$ for any curve, be it time-like or space-like. This will be a relativistic invariant, the imaginary part will give you the proper-time change of an observer on the curve and the real part will give you the "superluminal shift", the distance which could not have been traveled by a physical observer. This way you do not need to worry about the signs and the real and imaginary part clearly separate the physical meaning of the contributions.

The problem with the "signed arc-length" also is that it is much more degenerate than the "usual arc-length". The "usual arc-length" $\sqrt{g(\gamma',\gamma')}$ demi-definitely grows in the positive imaginary direction or the positive real direction for every additional piece of the curve (semi-definitely because there is the possibility of $\sqrt{g(\gamma',\gamma')}=0$). On the other hand, the "signed arc-length" allows for a plethora of curves which do not increment the arc-length by taking a time-like piece $\sqrt{g(\gamma',\gamma')}<0$ and compensating it with a space-like piece $\sqrt{g(\gamma',\gamma')}>0$. This means that the unique characterization of a null-geodesic or a path followed by a lightray $\ell(\gamma)=0$ is suddenly smeared out somewhere in $s[\gamma]=0$.

As for 3., the variational principle does not really care what is the phase of the Lagrangian and you can use either your own formalism or the one I propose - the resulting geodesics will be the same. Do note that the extremum of the action will always be on a geodesic with one fixed sign of $g(\gamma',\gamma')$.

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However, the fact is that in relativity there is a lot of "fudge" in the formalism we use. For instance, take the function $$\psi(x) = \exp(-\frac{1}{(x-x_1)^2} -\frac{1}{(x-x_2)^2}),\, x \in (x_1,x_2),\, \psi(x)=0 \; \rm otherwise$$ and make a coordinate transform $x \to x'=x + \psi(x)$ (note that $\psi(x)$ is $C^\infty$ and so is the transform). This will change the metric and every object inside the interval $x \in (x_1,x_2)$ but not the physics. This is due to diffeomorphism invariance.

I.e. the formal objects of relativity such as coordinate curves $\gamma^\mu(p)$ coordinates $x^\mu$ or even the metric $g^{\mu \nu}$ are not the core of the "physics". So what is? When you get down to it, you realize that the core of physics is the sum of time-like and null geodesics and their relations - which can be locally described by space-like geodesics.

You can now see that non-geodesics are a part of the "fudge" which is only a formal tool to get to the "core". The curious curves you propose cannot be geodesics, hence they are a part of the non-physical fudge - at least from a strictly relativist point of view. This means that no major relativist textbook spends any time to define this unnecessarily complicated convention because it has no physical significance.

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The only possible context where I can think of the physical relevance of meddling with signs and phases of the arc-length is the Path integral quantization of a relativistic particle. There you have the propagator of a free particle given as a path integral $$\int \mathcal{D}[\gamma] \exp(i\int \sqrt{-g(\gamma',\gamma')} {\rm d}p)$$ and the phase of the argument of the exponential most certainly matters. Even though you might be interested only in physical particles with time-like propagation, the propagator obtained from the non-path-integral relativistic quantum mechanics is non-zero even for space-like separations and you thus should consider even (partly) space-like paths.

This is a final argument against the convention you propose which I am too lazy to put in a completely rigorous derivation: The quantum propagator is non-zero for space-like separations, but exponentially decaying. If $S[\gamma]=\int \sqrt{-g(...)} {\rm d}p$ has a positive imaginary part, $e^{iS}$ is exponentially small. That is, the path integral quantization will give the correct result when the space-like path has a positive imaginary part. Thus, the correct sign convention is to use $L = \sqrt{-g(...)}$ ($-+++$ signature of the metric) without any sign modifications.

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I can think of beyond-standard theories which could use the convention you propose to formulate new physics in a legitimate way. For instance, your convention defines an "average space/time-likeness" of the trajectory which could be used in Lorentz-violating theories.

But I do not know about any beyond-standard theory using this formalism and the short review I give shows that this is certainly not a useful (and thus used) convention in standard theoretical physics.

Answer 2

I have never seen that second definition before. The first definition is standard. For a Riemannian manifold the metric tensor is positive definite, that is $$g(u,u)>0\quad\forall u\ne0$$ We have the standard relation (tensor product omitted) $$\mathrm{d}s^2=g_{ij}\mathrm{d}x^i \mathrm{d}x^j$$ Let $t\in\mathbb{R}$ be a curve parameter and $\gamma:[0,1]\rightarrow M$ be a curve. Then the functional $$\ell[\gamma]=\int_\gamma \mathrm{d}s=\int_0^1\sqrt{g(\dot\gamma,\dot\gamma)}\,\mathrm{d}t$$ defines arc length. For $M$ a physical spacetime, we no longer have a positive definite metric. It is simple to check that $g(\dot\gamma,\dot\gamma)<0$ for a timelike curve in $(-++\,+)$. Thus we require a minus in the square root: $$\ell[\gamma]=\int\sqrt{-g(\dot\gamma,\dot\gamma)}\,\mathrm{d}\tau$$ I see no reason to change this definition. What purpose would an overall sign serve?

Answer 3

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:

• 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".