Crossing the brachistochrone- Einstein’s train

[http://www.schoolphysics.co.uk/age16-19/Relativity/text/Time_dilation/index.html]

In Einstein’s original thought experiment on relativity, a passenger (Alice) is on a train. As she moves, she sends a beam of light (a photon) at a mirror on the roof of the train. Because Alice stands still with respect to the train, the photon is reflected vertically back to Alice. On the other hand, Bob, an observer outside the train, sees the light signals displaced because of the motion of the train, relative to him. Thus the distance the photon travels is different for the two observers. But because the speed of light is constant, the time elapsed for the two observers will also be different.

The problem can be illustrated with the following triangle:

If the vertical distance between Alice (at point O΄) and the mirror is O΄D, and it takes some time Δt΄, according to Alice, for the light to travel back and forth this distance, then it will be Ο΄D=cΔt΄/2, where c is the speed of light. On the other hand, if v is the speed of the train, then at some time Δt, according to Bob (at point O), the train will have moved a distance OO΄=vΔt/2, while the light signals will have traveled a distance ΟD=cΔt/2. It is supposed that at the moment Alice emitted the signal, the train passed in front of Bob (so that originally the points O and O΄ coincided).

The relationship between the times, according to the two different observers, is given by applying the Pythagorean theorem on the previous triangle:

The relativistic factor γ_L is Lorentz factor. This factor goes from 1 to infinity, as the speed v goes from 0 to the speed of light c. Thus γ_L can be defined only if v<c.