Kinematics of Relativity: Deriving Dilation Time

[gema]

I'm about to ask derivation of dilation time in terms of Special Relativity.
I saw explanations in Introduction to Classical Mechanics by David Morin that dilation time is formed by assumption that light speed is absolute refers to all inertial reference. He derived it by comparing 2 reference; A who sitting on the train with constant speed v refers to the ground, and B who at rest on the ground. They are to see the light traveling on the train, which has long h.

He got that A's time to see the light traveling from end to end , in transverse direction of light traveling, of the train is tA=2h/c
And B's time to see the light traveling on the train tB=2h/(c^2-v^2)^1/2, by simply using pythagoras.
Then, he got that tB=\gammatA, and this formula is generally used.
My question, How can this formula be general, since this are just derived by assuming that train is traveling in transverse direction of light speed.

 

[Doc Al]

My question, How can this formula be general, since this are just derived by assuming that train is traveling in transverse direction of light speed.

The 'light clock' is oriented transverse to the direction of train travel just to make the derivation easy. Of course, the behavior of a light clock does not depend on its orientation.

(If you put the light clock parallel to the direction of motion, you'd have to worry about length contraction when analyzing it from another frame.)