There are actually two right-triangles in this lecture: (1) One in which the hypoteuse = D and the opposite = L (but the adjacent is not specified). (2) One in which the hypotenuse = c, the adjacent = c.cos(theta), and the opposite = c.sin(theta). Prof. Greene shifts from the first triangle to the second then back to the first without making these shifts clear. Not only this is confusing, it also involves additional steps of derivation that could have been avoided if there were only one triangle. Other writers show the derivation of gamma from only one triangle using only Pythagoras theorem.
Just to note that always what happens is that WE SEE that other clock is faster or slower than ours, but not that the clock is faster or slower, not that the time goes slower or faster, only that we see one clock different form the other. Since there is not fix and unique reference system it can not be affirm that one clock is faster or slower or the time is faster or slower, only that we measure it differently. The light does not take an horizontal speed when the box move, we SEE that trajectory because our relative movement. Since there is not an absolute space we can not affirm that the light have this or that trajectory, only that in relation to us we measure that trajectoy in our reference.
I know this is implicity in the explanation.
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