Active Reading Solution: Coordinate Time, Proper Time
The airplane plays a very special role in this scenario, by representing the reference frame in which both events happen at the same place. This means, among other things, that...
- The time registered on the airplane clock is both a coordinate time and a proper time. It is the proper time of the airplane: that is, if that time says thirty seconds have elapsed, then exactly thirty seconds have elapsed for anyone on the airplane. It is also the coordinate time in the reference frame of the airplane. (Note that the airplaneunlike, say, Emma the twinremains in one inertial reference frame throughout its journey.)
- The time registered on the mountain clocks is a coordinate time but it is not a proper time. It is the time between these two events in the inertial reference frame of the mountains. But there is no object in that reference frame that experiences both of those events, so it is not a proper time.
Asher plays the same role in the twin paradox, since the two events (Emma's departure and her arrival) occur at the same place in his reference frame.
- The airplane's reference frame is the only frame that can measure both a coordinate and a proper time between these two events.
- Any other coordinate time will be longer than the airplane's; any other proper time will be shorter than the airplane's. All of this follows from the dictum that "moving clocks run slow."
- The time the airplane measures is also the spacetime interval between the two events (a topic we will introduce a few sections hence), if you use relativistic units (c=1).