Relativity - Relativity of Time Intervals

Joe is sitting in a train car moving at a speed, u. Bob is stationary on a platform being passed by the train car. The seats of the train car face inward so Joe is facing Bob when the train passes. A single pulse of light is emitted from the wall of the train car between Joe and Bob. There is a mirror on the opposite wall of the car. The distance from the light source to the mirror is d.

A seemingly simple question is, how long does it take for the source of light to strike the mirror then return to the source? The answer is different for both Joe and Bob.

For Joe, the calculation is very simple. The time it takes for the light to travel two d, our distance from the source to the mirror, is equal to 2d divided by the speed of light.

For Bob, the original point from which the pulse of light was emitted has moved a certain distance. This distance is equal to the speed of the train times the amount of time it takes the light to hit the mirror and return. Recall that "u" is the speed of the train. So the total distance traveled by the source is u times time (or t). For Bob, the light makes an upside down V shape in its complete motion. If we connect the ends of the upside down V we make a triangle. Cut this triangle in half vertically. We can now conclude that the distance (L) from the source to the mirror is equal to the square root of [d squared + (u times t over 2) squared].

Bob concludes that the total time from the source to mirror, back to the source is equal to 2L over the speed of light. If we plug in our formula for L, we have:
Total time = (2/speed of light) times the square root of [d squared + (u times t divided by 2) squared]

We want this relationship to be independent of d. To do this, lets take our equation for time that Joe uses; time = 2d divided by speed of light. So d = c times time divided by 2. Our equation for Bob's time now reads:

t = (2/speed of light) times the square root of [(speed of light times Joe's time divided by 2) squared + (u times Bob's time divided by 2) squared]

Square this entire formula and solve for Bob's time. We get:

Bob's time = Joe's time divided by {The square root of [1 - (u squared divided by the speed of light squared)]}

Since [1 - (u squared divided by the speed of light squared)]} is less than 1, Bob's time is greater than Joe's time. Therefore Bob measures a longer round trip than Joe and both are correct.