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Light Clock Example in Time Dilation?

When researching time dilation this morning he most common example was the 'mirrored light clock' example. I understand the basics of dilation (The speed of light is not "plussesable" in the fact that if you travelled at 0.6c and shot a beam of light, it would not be travelling at 0.4c, therefore time would have to slow down.)
However then I dont get why when the light mirror clock example is used that the light beam appears to be diagonal from someone who is stationary.
Could someone help me understand this please.

It does not appear to be moving on a diagonal, but you know it is moving away from you, and up and down at a fixed rate, so when you do a calculation of how far it must be travelling, it is the combination of the up and down motion and the moving away that gives you a sort of hypotenuse of these two perpendicular movements.

But let me point out a very common misunderstanding of Special Relativity. There is no such thing as 'stationary'. All movement is simply relative to something else. So, to the observer in the spaceship, the object/observer you call stationary - let us say the Earth - appears to be moving away from him. And the observer in the spaceship WILL observe time dilation of the Earth. And neither observer's observations are more or less valid than the other.

The more you think about it, the more confusing it gets! But we have been able to verify this in many tens of thousands of experiments, so this is not just some odd theory.

But hey, we accept that when you watch me walk away from you, you see that I get smaller, and I see that you get smaller. Which one is 'right'?

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