Question 4

  Suppose you are standing at a railroad crossing, watching a train go by. Both you and a passenger in the train are looking at a clock on the train. (a) Which of you measures the proper time interval? (b) Who measures the proper length of the train car? (c) Who measures the proper distance between the railroad ties under the track? Justify your answers.

SOLUTION:
(a) To answer this, you need to know the definition of proper time interval. The proper time interval between two events (clock ticks, in this case) is the time interval measured by an observer at rest with respect to the events, so that the observer sees the events as taking place at the same point in space. Clearly, the passenger on the train is in the same inertial reference frame as the clock on the train, so the passenger measures the proper time interval.
(b) Again, you need to know the definition of proper length. The proper length (or distance between points) is the length measured by an observer at rest with respect to the two points, so that the observer sees the events (measurement of both points) as taking place at the same point in time. The passenger is, once more, at rest with respect to the train car, and hence measures its proper length.
(c) You are standing on the ground and are at rest with respect to the railroad ties, so you measure the proper distance between them.