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Recently it has been affirmed here (again) that the quantity called "interval (also 'spacetime interval' or 'invariant interval')" is referring to two (in general distinct) events as arguments, such as $\varepsilon_A$ and $\varepsilon_B$, and is denoted (as shorthand) by $$ \Delta s^2.$$

Consequently, writing out the concrete arguments explicitly, the "interval between" (also called "interval of") events $\varepsilon_A$ and $\varepsilon_B$ is denoted as $$ \Delta s^2[~\varepsilon_A, \varepsilon_B~],$$

where by definition $$ \Delta s^2[~\varepsilon_A, \varepsilon_B~] \equiv \Delta s^2[~\varepsilon_B, \varepsilon_A~].$$

If two events are "timelike separated" (such as events $\varepsilon_{(\mathsf P \mathsf H )}$ and $\varepsilon_{(\mathsf P \mathsf J )}$ where participant $\mathsf P$ took part in both, having met and passed participant $\mathsf H$, and separately met and passed participant $\mathsf J$) then the "interval between" these two events is of opposite sign than the "interval between" any two events which are "spacelike separated".

Following the convention set out in Wikipedia (even though it is just inverse from the convention adopted in the answer referred to above) therefore explicitly: $$\Delta s^2[~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf J )}~] \lt 0.$$

With this convention, considering three events which are (pairwise) "timelike separated" from each other (such as events $\varepsilon_{(\mathsf P \mathsf H )}$, $\varepsilon_{(\mathsf P \mathsf J )}$, and $\varepsilon_{(\mathsf P \mathsf K )}$, where participant $\mathsf P$'s meeting/passing of participant $\mathsf J$ had been between participant $\mathsf P$'s meetings/passings of participants $\mathsf H$ and $\mathsf K$, respectively) therefore holds

$$ \Delta s^2[~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf K )}~] \lt \Delta s^2[~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf J )}~],$$

$$ \Delta s^2[~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf K )}~] \lt \Delta s^2[~\varepsilon_{(\mathsf P \mathsf J )}, \varepsilon_{(\mathsf P \mathsf K )}~],$$

and the "inverse triangle inequality":

$$\sqrt{-\Delta s^2[~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf J )}~]} + \sqrt{-\Delta s^2[~\varepsilon_{(\mathsf P \mathsf J )}, \varepsilon_{(\mathsf P \mathsf K )}~]} \le \sqrt{-\Delta s^2[~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf K )}~]}.$$

Note the square roots, "$\sqrt ~$", operating on each term of the latter inequality. Correspondingly, the (any) "interval, $\Delta s^2$", is considered a "squared quantity" (and appropriately its name involves an exponent: "$~^2$").

My question:

Is there a name for the corresponding linear quantity "$\Delta \overline s$" ?,
for which (written in shorthand) $$ \text{sgn}[~\Delta \overline s~] \equiv \text{sgn}[~\Delta s^2~]$$

and

$$\Delta \overline s \equiv \text{sgn}[~\Delta s^2~] \times \sqrt{ \text{sgn}[~\Delta s^2~] \times \Delta s^2~]},$$

and therefore in turn

$$ \Delta s^2 \equiv \text{sgn}[~\Delta \overline s~] \times \Delta \overline s \times \Delta \overline s,$$

and such that (again following the sign convention of Wikipedia) the "inverse triangle inequality" relating three pairwise "timelike separated" events is (simply, linearly)

$$\overline s [~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf K )}~] \le \overline s [~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf J )}~] + \overline s [~\varepsilon_{(\mathsf P \mathsf J )}, \varepsilon_{(\mathsf P \mathsf K )}~], $$

provided

$$\overline s [~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf K )}~] \lt \overline s [~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf J )}~] \lt 0$$

and

$$\overline s [~\varepsilon_{(\mathsf P \mathsf H )}, \varepsilon_{(\mathsf P \mathsf K )}~] \lt \overline s [~\varepsilon_{(\mathsf P \mathsf J )}, \varepsilon_{(\mathsf P \mathsf K )}~] \lt 0.$$

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  • What you are defining is a "signed" version (in the sense that it takes into account the time-like and space-like nature) of the arclength of a curve in Minkowski spacetime (that is a pseudo-Riemannian manifold). – yuggib May 30 '15 at 11:26
  • @yuggib: "[...] a "signed" version [...] of the arclength of a curve in Minkowski ST" -- What "curve" ?? The OP question only mentioned discrete events; and only few of those. "a pseudo-Riemannian manifold" -- The section you linked (actually called "Arclength and the line element") mentions "the first fundamental form $ds^2$" and the corresponding "(signed) line element $ds$", btw. However, these don't seem to take two explicit arguments; and therefore have little to do with the OP question. – user12262 May 30 '15 at 16:12
  • The curve is the path in spacetime taken between the two events. This path is written as an integral, since space-time is, last time I checked, *continuous* and not discrete. If you look at the arclength formula, you will see that the initial and final spacetime events are the extrema of integration in calculating the length. Thinking about the discrete difference between coordinates is not so useful, since it does not take into account that between the events it would be possible, in principle, to violate the timelike condition even if the extrema are timelike separated. – yuggib May 30 '15 at 16:50
  • @yuggib: "The curve is the path in ST taken between the two events." -- But which path ?? Or do you mean any path; incl. even paths containing elements (i.e. events) such that some pairs of them are timelike related, and other pairs spacelike? "If you look at the arclength formula, [...]" -- Well, I generally think of "the arclength formula" in this form ... "[...] discrete difference between coordinates is not so useful, since [...]" -- The OP question doesn't mention coordinates. (And yes, coordinates aren't by themselves useful.) – user12262 May 30 '15 at 17:46
  • Usually $ds$ is considered a length element, and its not-infinitesimal version is in some sense the total (relativistic) length. The interval $\Delta s^2$ as you intend it as a mere discrete difference is useful to give the space-like/light-like/time-like character of the relativistic distance between two points; but if you want to use it to do things as triangular inequalities (as you would do with a distance in maths) – yuggib May 30 '15 at 18:10
  • you should look at the version that takes into account the path taken, if else you cannot take track of the causal condition you imply (the event in the middle happens "between the other two") – yuggib May 30 '15 at 18:13
  • @yuggib: "[...] the total (relativistic) length." -- Well: if I squint really hard (and you squint as well) and we read your suggestion for the name I asked about rather as "the (linear, relativistic) separation", and if you expand your comment (interpreted this way) into an answer, then I'd be happy to accept it. (And even Wikipedia may eventually articulate it, too.) "[...] the version that takes into account the path taken" -- But even the "interval" $s^2[~\varepsilon_A, \varepsilon_B~]$ is unique, for given events $\varepsilon_A, \varepsilon_B$; so why not $\overline s$ ?? – user12262 May 30 '15 at 18:52
  • Well, I would say that my interpretation is far from being established, so maybe it is better to wait and see if someone else comes up with a more common terminology ;-) If else I will expand the comments in an answer. And anyways, $\bar{s}$ surely is unique as you define it, but the causal connections are meaningful only if you take into account the spacetime path taken, otherwise you cannot know if two events are actually happening to one observer one after another. – yuggib May 31 '15 at 08:24
  • @yuggib: "that my interpretation is far from being established" -- IMHO it's almost past due to establish it (or sth. sufficiently similar). "[...] know if two events are actually happening to one observer one after another" -- The question how to establish "between-ness" is of course interesting. (I had assigned it a while ago as homework.) – user12262 May 31 '15 at 10:50

1 Answers1

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The sought quantity is called "signed separation" of a given pair of events.

p.s.
A note on the relation between signed separation and signed arclength:

Considering two events $\varepsilon_A, \varepsilon_B$ and a path $\Gamma$ "from $\varepsilon_A$ to $\varepsilon_B$" such that

  • for any element $\varepsilon_{\xi} \in \Gamma$ holds

$$ \overline s[~\varepsilon_A, \varepsilon_{\xi}~] + \overline s[~\varepsilon_{\xi}, \varepsilon_B~] = \overline s[~\varepsilon_A, \varepsilon_B~] $$

  • and any three elements $\varepsilon_{\phi}, \varepsilon_{\kappa}, \varepsilon_{\xi} \in \Gamma$ are straight wrt. each other, i.e.

$$ 0 = (\overline s[~\varepsilon_{\phi}, \varepsilon_{\kappa}~] + \overline s[~\varepsilon_{\kappa}, \varepsilon_{\xi}~] + \overline s[~\varepsilon_{\xi}, \varepsilon_{\phi}~]) \times (\overline s[~\varepsilon_{\phi}, \varepsilon_{\kappa}~] + \overline s[~\varepsilon_{\kappa}, \varepsilon_{\xi}~] - \overline s[~\varepsilon_{\xi}, \varepsilon_{\phi}~]) \times (\overline s[~\varepsilon_{\phi}, \varepsilon_{\kappa}~] - \overline s[~\varepsilon_{\kappa}, \varepsilon_{\xi}~] + \overline s[~\varepsilon_{\xi}, \varepsilon_{\phi}~]) \times (\overline s[~\varepsilon_{\phi}, \varepsilon_{\kappa}~] - \overline s[~\varepsilon_{\kappa}, \varepsilon_{\xi}~] - \overline s[~\varepsilon_{\xi}, \varepsilon_{\phi}~]),$$

then the signed separation of $\varepsilon_A$ and $\varepsilon_B$ equals the signed arclength of path $\Gamma$ from $\varepsilon_A$ to $\varepsilon_B$:

$$\overline s[~\varepsilon_A, \varepsilon_B~] = \int_{\Gamma} ds.$$

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