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Thanks for reading.

The invariant interval between two events is defined as...

$S^2=(ct)^2-(x)^2$

...where $t$ is the time between the events and $x$ is the distance between the events.

When its positive, that means that we can find an observer that's at the location of the first event and at the location of the second event, just as the events happen. In that case, $S$ would be $c \tau$, the proper time that passes for that observer between the two events multiplied by the velocity of light, aka the distance that a light-beam shot by him just as the first event happens would travel.

In case the above gives trouble, the following video explains it pretty well:

https://www.youtube.com/watch?v=VWx2WCsuoic

However, some books define the invariant interval like so:

$S^2=(x)^2-(ct)^2$

This invariant interval is positive when the other is negative...but, what does it represent? It can no longer be interpreted as proper time...so, what's its interpretation?

I can see that it gives equations of horizontal hyperbolas in a Minkowski Diagram...but, I'm not sure how to interpret those (as for the vertical hyperbolas, I know to interpret them as each joining all events of constant proper time for an observer moving relative to the stationary frame in the diagram).

Is it...proper length? Length between events for the "stationary" observer? Hmm...can we make a similar light-experiment that shows what it is, the same way the one in the linked video shows how the invariant interval can be the proper time (multiplied by the speed of light)?

Thanks!

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The sign is just a convention. Originally distance$^2$ naturally had positive sign, and time appeared in relativistic equations in the form $ict$, which would give a negative sign to time$^2$. Then it was realised that one could get rid of the imaginary numbers by using a metric signature $(-1,1,1,1)$. This convention is often still used, but many physicists prefer to use the signature $(1,-1,-1,-1)$, because time is considered more fundamental than distance and because it is often seems more natural to use formulae with energy$^2$ positive.

It does not actually matter what sign convention is, so long as one chooses a convention and sticks to it, and so long as one knows what convention is being used.

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In the timelike case, as you said, it is possible to select an inertial observer such that it is present at the locations of both events. And $\tau$ is a time (its proper time), measured by the observer, between them.

In the spacelike case, it is possible to select an observer such that the events are simultaneous in its frame. And $S$ is the distance between the locations as measured by that observer.

The limiting case is the speed of light $S = \tau = 0$, where there is no possible frame to select.

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