Proper time interval and arbitrary intervals

Question

I feel like this question is something very elementary, while I sense it can be very enlightening. Anyway, when discussing proper time, we define it as the time measured by a clock that follows a timelike world line. Naturally, we define the proper time interval as the change in proper time.A common didactic way to present it is with the notion of observer, as follows: An observer travels from an $x_1$ position, where an Event $1$ occurs, and reaches an $x_2$ position with the same clock, in time to register an Event $2$. The measured time interval is the proper time interval between these events. Of course, this definition depends on the fact that the observer has enough time to "reach" the other event. Well, clearly, the range of situations whereupon an pair of events would not have a proper time would be enormous, just think of situations in which we have events separated by very large distances, one occurring shortly after the other. We quickly see that there would be an extensive range of situations where we should have speeds exceeding the speed of light.

I see no reason to believe that the "problem" would stop there, I believe that this can be an interesting motivation to better introduce the metric of space-time, when studying the subject initially. So, my question is: Mathematically, how does the topology or algebra of space time, or both, "circumvent" this simple difficulty, so that proper time can be well defined?

Answer 1

The algebra of spacetime does not 'circumvent the difficulty'. Indeed, it is your question which assumes there is a difficulty to circumvent. A proper time interval cannot be assigned to a pair of space-like separated events- that is not a difficulty but a description of fact.

An alternative way to consider your didactic way of explaining proper time, in which a clock moves between two events, is to think of a proper time interval as being one between two events that happen at the origin in the frame of reference of the clock. The point about space-like separated events is that there is no physical reference frame in which they happen at the same location- a clock would have to move faster than the speed of light to be present at both of them.

Answer 2

To me, it's not clear what you are asking or really interested in,
possibly because some terms are being confused and because some terms are restricted in their use.

The following might be useful to distinguish the terms.

"Proper-time" is a property of a future-directed timelike worldline segment, not of the endpoint-events. It's akin to a distance along a path. Given a pair of timelike-related events, there are many future-directed timelike worldline segments, each with their own proper time. This is essentially the Clock Effect (which is related to the Twin Paradox).

Among all worldlines joining two timelike related events, the inertial worldline segment has the longest proper time. Sometimes this is referred to as the "interval between the two timelike-related events". This is akin to the magnitude of the timelike-displacement vector joining the two events.
[Maybe this is what you are really interested in?... The displacement vector, and not the path?]

"Proper time" is not defined for lightlike-related or spacelike-related events, as others have noted.

However, there is the general notion of "signed square-interval between any two events", which could be associated with the square-norm of the vector displacement from one event to the other. For a spacelike displacement, this could be realized directly by (say) a meterstick (rather than a clock).
[Again, maybe this is what you are really interested in?... The displacement vector, and not the path?]

So, as others have noted, there is a problem in trying to realize such arc-lengths or displacements--directly--with the "proper time" of a clock along the arc or displacement.

(Note, however, a distant inertial clock and pairs of light signals [radar signals] can measure all of these quantities indirectly, using the "radar method"... so all measurements are read off of a clock along an inertial worldline.)