This section introduces the reciprocity of time dilation. Elapsed time of a moving clock is slower when measured by a stationary observer; and elapsed time of a stationary clock is slower when measured by a moving observer.
Time dilation is reciprocal because moving between two observers is reciprocal.
So the following two statements are both true:
Elapsed time a moving clock is dilated (longer)
when measured by a stationary observer.
Elapsed time a stationary clock is dilated (longer)
when measured by a moving observer.
The first statement was demonstrated by the thought experiment presented in previous sections.
The second statement can be demonstrated by modify that experiment slightly as below:
Put the carriage on the ground and let Bob performs the experiment on the ground.
Now considering Amy on the moving train, she will see the carriage and the ground
is moving in the opposite direction.
If we follow the same thought process used in previous sections, we will
arrive the same conclusion in the opposite way.
That is T = gamma*T', where T is the elapsed observed by Amy and T' is the elapsed time observed by Bob.
So, because of reciprocity of time dilation,
Amy and Bob both will claim the other clock is slower (or dilated):
Bob on the ground will observe Amy's clock is slower.
Amy on the train will observe Bob's clock is slower.