Why does the proper time between two events depend on the worldline between them?

On Wikipedia, it is stated that:

Proper time along a timelike world line is defined as the time as measured by a clock following that line.

and also

The proper time interval between two events depends not only on the events but also the world line connecting them, and hence on the motion of the clock between the events.

Unfortunately, both of these statements make sense to me, but how they can both be correct does not. If the proper time interval between two events is the time as measured by a clock following that line, then surely that measurement is independent of path? If I watch someone travel (through space) slowly, but directly between two events and measure a proper time $$\tau_1$$, and someone else travels quickly, but indirectly and measures a proper time $$\tau_2$$, how can these measurements be different?

From both of their perspectives, the events happened at the same point, separated by a time interval $$\tau$$, neither of them is aware they have moved at all, but as an outside observer I have seen that their wordlines have different lengths.

• "If the proper time interval between two events is the time as measured by a clock following that line...". Which line? Apr 7 '20 at 17:00
• the proper time is the time experienced by observers moving along the worldline. There is an invariant spacetime distance between two points given by the Minkowski metric, but that equates the proper time only experienced by an inertial observer moving between the two points Apr 7 '20 at 17:01
• Google the twin paradox. Your two paths could correspond to the two twins. They will disagree on the proper time when they meet. Apr 7 '20 at 17:13
• If you and I and Alice and Bob meet up at a certain restaurant for breakfast, that meeting is an event. If you and I and Alice and Bob meet at another restaurant for dinner, that's another event. The four different paths that we each follow in-between those two events are represented by our world-lines in 4D spacetime. Four different paths--four different world lines--connecting the same pair of events. If Alice happens to pop over to Jupiter and back at lunch time, at relativistic speed, then her watch, which followed a very different path from yours or mine, will show less elapsed time. Apr 7 '20 at 17:27
• Possibly useful: physics.stackexchange.com/questions/508931/… Apr 7 '20 at 22:32

If the proper time interval between two events is the time as measured by a clock following that line, then surely that measurement is independent of path?

I think the problem is that you are not understanding the intended use of the words. The path here is the path in 4-D spacetime, and is exactly the same thing as the world line. The difference in world line describes the difference in motion.

From both of their perspectives, the events happened at the same point, separated by a time interval $$\tau$$, neither of them is aware they have moved at all

but one of them measured $$\tau_1$$ and the other measured $$\tau_2$$, so they both described the event where they met with different time coordinate, reflecting the fact that they followed different world lines.

The point is that proper time as measured by a clock only strictly applies at the location of the clock. Outside of that immediate neighbourhood we have only conventional definitions of synchroneity. This is a diagram I use in special relativity showing time coordinates for the twin paradox using Einstein synchroneity.

• Ok I had a think about this and I think I get it, I'd prefer to describe it mathematically first and then layer on the physics: (Strictly staying within inertial frames) I take two events in spacetime, which is the equivalent of taking two points in a 4D space with a non-Euclidean metric, $diag(-1,1,1,1)$, and I draw a "straight" line between them. The difference between two Lorentz frames is equivalent purely mathematically to two sets of basis vectors defined on this space. And "changing frame" is just changing your basis vectors with a transformation defined by the Lorentz matrix... Apr 7 '20 at 18:22
• ...If I then boost into a frame in which the two events exist at the same spacial point, assuming they are timelike separated and this can be done, what I am doing here is changing my basis vectors such that the straight worldline points only along my "time" basis vector. So even though the length of the worldline separating the two events hasn't changed, I no longer see them as being separated by any spacial component. And so in this special basis set the "proper time" is just the length of the time component as measured by that "observer". Apr 7 '20 at 18:25
• Yes. Of course, staying strictly with inertial motions in flat spacetime, the two world lines in the original question cannot meet at a second event. Apr 7 '20 at 18:26
• yes. proper time always means the length of time as measured by a particular observer. Apr 7 '20 at 18:29
• Yes, although personally I hate calling them "rotations". You are transforming the basis, and it is represented using a 4x4 matrix which shares some properties of rotations in 3D. But sin and cos get replaced by sinh and cosh, so really it is not rotation. I never think about things that way (I don't find it helpful) so I won't say more because I will probably say something wrong. Apr 7 '20 at 18:35

1) Suppose the events $$E(x_1, t_1)$$ and $$E(x_2,t_2)$$ occur at the same $$x_1=x_2$$ in some Lorentz frame. Then, the interval $$\tau=t_2-t_1$$ in THAT frame is the proper time; it is the time elapsed between events measured at the same spatial coordinate. To make this more obvious, recall $$ds^2$$ vs $$d\tau^2$$.

2) More properly, it is the time interval measured by a single clock at the (common) location of the events, as opposed to two different clocks at different locations $$x_1,x_2$$.

3) It can be conveniently rephrased by saying that there is an observer traversing between the two events, and he therefore measures the time between the events on the same clock-the one he carries with himself. This is a little slippery, because in SR, observers are entire coordinate systems and not somebody sitting at the origin, but by 'traversing' we mean that the clock at the origin has coincided with the events at different times.

4) Now, this 'traversal' is characterised by the trajectory of the observer, which has some arclength -$$ds^2$$, and this is by definition an invariant quantity. The proper time, therefore is independent of the observer traversing, and is an invariant too.

5) It is important to realise that it is not always possible to find a frame where the two events are at the same $$x$$-these are 'timelike seperate events'. If $$E(x_2,t_2)$$ lies outside the light cone at $$E(x_2,t_2)$$, then no observer can 'traverse' between the events-not even light can! This is essentially saying that only those events are accessible to an observer at some $$E$$, which lie in the light cone at $$E$$. I would have included a diagram, but the answer by @CharlesFrancis already has a good illustration of the idea.