Under Special Relativity, all motion is relative and relative motion gives rise to the dilation of time (as described by the Lorentz transformation). If the time dilation is taken to be symmetrical, it leads to the impression that there is a paradox, as is illustrated by the "Twin Paradox" thought experiment...
There are two twins, one of whom stays on the Earth. The other twin travels in a spaceship at near-light velocity to a distant planet orbiting another sun. The spaceship then turns round and travels back to the Earth. From the point of view of the twin on Earth, less time would pass for the traveler than for himself. So when he returned, the traveler would be younger than the twin on Earth. Indeed if the traveler traveled at 0.866 c (86.6% of the velocity of light) he would experience half the time elapse of his twin. So if they were 20 years old at the start, and the traveler was 50 when he returned, the twin who stayed on the Earth would be 80 years old.
However, that only tells us what takes place from the point of view of the twin who stays on the Earth. If the time dilation is symmetrical, then, from the point of view of the traveller, he is stationary in his rest frame and the Earth, and his twin, recede very fast from him, then return. So, from the point of view of the traveler, the twin on Earth should experience less time passing than the traveler. I.e. the traveler should be 80 and the twin on the Earth only 50 years old!
It is clear that the assumption that the time dilation is symmetrical leads to an impossible situation, hence the paradox. It is this assumption that Paul Langevin addressed in 1911. He suggested that, as the traveler underwent acceleration at the beginning, turnaround and end of the journey, his rest frame was not inertial. Whereas, the twin on Earth did not undergo acceleration, therefore his rest frame was inertial. Thus the situation is not symmetrical. In effect, this meant that the traveler was actually moving with respect to the Earth.
Note: The suggestion that the traveler is actually moving is not intended to imply or require the existence of an absolute state of rest against which the motion occurs. It merely describes an asymmetry between the motion of the Earth and that of the traveler with respect to the Earth.
The following description is taken from Wikipedia - Proper Time:
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events. The twin paradox is an example of this effect.
The dark blue vertical line represents an inertial observer measuring a coordinate time interval t between events E1 and E2. The red curve represents a clock measuring its proper time T between the same two events.
In terms of four-dimensional spacetime, proper time is analogous to arc length in three-dimensional (Euclidean) space. By convention, proper time is usually represented by the Greek letter T (tau) to distinguish it from coordinate time represented by t.
By contrast, coordinate time is the time between two events as measured by a distant observer using that observer's own method of assigning a time to an event. In the special case of an inertial observer in special relativity, the time is measured using the observer's clock and the observer's definition of simultaneity.
I would add that, as coordinate time is frame dependent, the coordinate time interval mentioned in the example is only one of many possible coordinate time intervals between the events. Indeed, the "proper time" mentioned is the coordinate time measured by a clock that is co-located with both events, but follows a non-linear trajectory between them. Proper time is therefore a subset of coordinate time that deals with events that are co-located.
Further details of the twin paradox, including a worked example, may be found here.
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