Introduction to Classical Mechanics · Problem 11.58
Relativity (Kinematics) — Morin
Physics — Mechanics · time-dilation-simultaneity
Problem
Time dilation and Lv/c2 **
A person walks very slowly at speed u from the back of a train of proper length L to the front. The time-dilation effect in the train frame can be made arbitrarily small by picking u to be sufficiently small (because the effect is second order in u). Therefore, if the person's watch agrees with a clock at the back of the train when he starts, then it also (essentially) agrees with a clock at the front when he finishes.
Now consider this setup in the ground frame, where the train moves at speed v. The rear clock reads Lv/c2 more than the front, so in view of the preceding paragraph, the time gained by the person's watch during the process must be Lv/c2 less than the time gained by the front clock. By working in the ground frame, explain why this is the case. Assume u≪v.