Spacetime
Spacetime Mechanics
3D Diagrams
Zero Intervals
Proper Time
The Two Towers
Spacetime & Reality
Proper Time
Proper time is a name given to a "timelike" spacetime interval between two events. The corresponding name given to a "spacelike" spacetime interval is "proper distance". The proper time, or distance, between any two events is frame invariant.
However, because of the way that the time lapse and spatial distance are combined in a given spacetime interval, observers at rest in different frames of reference would measure the time interval and spatial distance between any two events to be different, even though the spacetime interval is constant.
So the time interval and spatial distance between two events, that would be measured by an observer in any arbitrary frame of reference, are called the "coordinate time" and the "coordinate distance", to distinguish them from the "proper" time or distance.
This leads to the question: How can you measure the "proper" time interval or distance that corresponds to a given spacetime interval?:
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A "proper" time interval is measured in the frame of reference where the events are co-located spatially. This is because, in that circumstance, the spatial distance is zero so the time interval matches the spacetime interval.
- A "proper" distance is measured in the frame of reference where the events are simultaneous. This is because, in that circumstance, the time interval is zero so the spatial distance matches the spacetime interval.
It is said that clocks tell the proper time of their rest frame, but as the following example shows, that is not always the case...
Clocks and Proper Time Example
If we have two synchronized clocks, A and B, mutually at rest some distance apart, then both will tell the "proper" time of that rest frame provided that we compare two displays of the same clock. E.g. If we compare the 0th second to the 5th second displayed on clock A, the time interval of 5 seconds matches the proper time. The same is true for clock B.
But what if we want to compare the 0th second displayed on clock A to the 5th second displayed on clock B? The time interval between the clock displays is still 5 seconds, but the proper time between these events will vary. This is because the spacetime interval is a combination of that time interval and the spatial distance between the clocks. Thus:
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If the spatial distance between the clocks is 3 light seconds, then, as is explained here, the spacetime interval (and hence the proper time) is 4 seconds.
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If the spatial distance between the clocks is 5 light seconds, then the proper time is zero seconds (because the time interval and spatial distance are of equal magnitude).
- If the spatial distance between the clocks is 10 light seconds, then the proper distance is 8.66 light seconds (because the spatial distance is larger than the time interval).
I.e. As soon as we introduce a spatial distance between the events, a clock no longer tells the proper time interval between them.
Conclusions
It is evident from the above example that clocks do not always tell the proper time. Indeed, they actually tell the coordinate time. So rather than saying that "a clock tells the proper time of its rest frame" it is more accurate to say that "a clock tells the coordinate time of its rest frame, which matches the proper time for spatially co-located events".
Furthermore, as coordinate time is what is displayed by clocks, it follows that coordinate time is "real" in the sense that it correlates to the actual passage of time that an observer would experience in a given frame of reference. Whereas proper time only correlates to the actual time passing when the proper time matches the coordinate time.
A more technical introduction to Proper Time may be found here.
