First, no real light-clock will concern itself with the one-way trip. It will always measure time based on the round-trip of the pulse, and for the round trip the simplified gamma equation gives the correct answer regardless of the orientation of the clock.

Second, if you want a general formula that will give the actual one-way trip conversion for any orientation, there is one, and I gave it before:

The x and t values in that equation will depend on the orientation and length of the clock.

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SeanF

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Here's a copy of the original paper. You'll note that there's no perpendicular light clock involved in the derivation of what we call gamma (Einstein used beta for this term). Perhaps you can instead tell us what you think is wrong with the derivation Einstein actually used.

Good luck, I asked that a week ago. Maybe you'll get an answer.

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Some try to tell me, thoughts they cannot defend,... - Moody Blues.

Neptune- The original Dark Matter.

The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth

There is a very nice derivation of the Lorentz transformation that makes very few assumptions other than those relating to the basic symmetries of a very general flat spacetime, e.g. translational symmetry. At the end, you are left with a constant that determines if the spacetime is Galilean (the constant is infinite), Euclidean (the constant is positive), or Minkowski (the constant is negative). You can then determine the value of the constant from experiment and show that this means that spacetime possesses the geometry of Minkowski and the transformation from one inertial frame to another is given by the Lorentz transformation.

I'll try and track down a copy.