# Inference of relativistic time dilation wrong?

One popular example of relativistic time dilation uses the idea of a Light clock where time is measured in terms of cycles the light between the two mirrors (which are at distance L from each other) completes.

However, isn't this only true under the assumption that the distance L of the two mirrors does not change , i.e. that dimensions orthogonal to the direction of motion aren't subject to (Lorentz-) contraction? If this is the case, this example is a nice illustration of time-dilation while it can by no means be used to infer time dilation purely by the constancy of the speed of light.

The 'real' argument is to just derive the Lorentz transformations from first principles, then note that $$y' = y$$ for a boost along the $$x$$ direction. But this isn't pedagogically useful, because usually people want to see proofs of time dilation and length contraction individually first, so they can accept the Lorentz transformations in the first place. So we shouldn't use this powerful tool.
I quietly slipped in an extra assumption above: that by symmetry, the transverse length contraction due to velocity $$v$$ should be the same as that due to velocity $$-v$$. Relaxing this gives generalizations of the Lorentz transformations described here where transverse length contraction does exist.