special relativity - What if a light clock travels perpendicular to mirrors that make up the clock?

I'm guessing you're all familiar with the classic intuitive way of explaining time dilation: with a light clock traveling at velocity v directed at a parallel direction to the mirrors that make up the light clock.

Now, what happens if we have a light clock that goes upwards at v? That is to say, what if its velocity is in a direction perpendicular to the mirrors' orientations? As far as I can tell, this situation wouldn't present any time dilation. Thoughts?

Answer

In that case, time dilation still occur, of course. In order to show this using t=d/v, you'd have to take into account the space contraction in the direction of motion. Mathematically, if d is the height of the clock, then the time taken from a photon at the bottom to reach the top of the clock isn't $\frac{d+vt}{c}$ but $ \frac{d/\gamma+vt}{ c}$. When you calculate the time taken for that photon to get back to the bottom of the clock and add it up to the time previously calculated, you get the exact same time dilation than the clock moving parallelly to the mirrors.