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In Einstein's thought experiment involving light moving on a train, the mirrors the light is bouncing between are perpendicular to the train, and it is these mirrors that the formula for time dilation is based on. But if the mirrors were positioned differently - for example at angle of 30 degrees to the side of the train or in the direction of the train's motion - wouldn't the formula be different? My calculations tell me it would be. So I don't understand how this formula applies to everything in the train no matter what direction it's moving in.

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Only if mirrors have zero distance one to other — in the direction of moving train — we know reliably the magnitude of it, because only zero multiplied by arbitrary factor is still zero.

In all other cases their distance is affected by relativistic effect, which we (or Einstein) wanted by this “clock” calculate.

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The original thought experiment is valid and it tells you how the clocks on the train appear to be running slow. It also tells you (because by symmetry static and moving observers agree on the speed of the train) how the moving measuring rods appear to shorten in the direction of travel.

That preferential shortening in one direction also changes the apparent angle of your mirror in your enhanced experiment.

As a result you cannot know the path the light takes in the static frame of reference for your experiment without making an assumption. If you get a different answer for the time dilation your assumption contradicts the original experiment.

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wouldn't the formula be different?

No, it's the same. The time dilation formula is independent of the direction of motion or the light clock inclination. I am in the middle of my second book in which a section entitled Einstein's Oblique Light Clock is somehow devoted to your question. However, I used this oblique light clock to derive the relativistic reflection law. In other words, when the oblique light clock moves, the incident $(\hat{i}')$ and reflected $(\hat{r}')$ angles are no longer equal. The calculations are slightly complicated, however, I provide you with the relevant figure that may help you to derive the right equations. Remember that you need to prove that:

$$\frac{\tau''_1+\tau''_2}{\tau'_1+\tau'_2}=\sqrt{1-\frac{v^2}{c^2}}$$

For simplicity, you can either assume $\theta''=\phi''$ or $\psi''=0$.

Einstein's Oblique Light Clock

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