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So I just learnt about relativity and time dilation. One of the visualisations used to explain time dilation is an apparatus contaning a beam of light moving between two mirrors. When the apparatus is stationary, the light moves a certain distance d, say in one second between the mirrors. But when the apparatus is say, fixed to a spaceship, the light beam moves in a zig zag fashion, covering a long distance $D$. The speed of light is constant. Therefore the time for light must slow down, because the distance being covered by light is greater. I have a link below which explains the visualisation. https://youtu.be/TgH9KXEQ0YU

This is what we were explained. However, why is it that the beam of light can't cover that greater distance in a greater time $t$? Why should time slow down for the beam of light?

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It is one of the postulates of special relativity that the speed of light $c$ is the same for all non-inertial observers. Being a postulate, this means there is no known cause for it - we know that $c$ is constant, but we don't know why (For some mor informtion on this, you may read Why and how is the speed of light in vacuum constant, i.e., independent of reference frame?).

However, a constant speed of light means that time must slow down for observers with relative velocity $v$ "in order to keep $c$ the same for them". This explanation is of course a bit sloppy, but I think it covers some intuition and in addition, you apparently learned special relativity in more depth.

So why does time slow down for light? This formulation seems a bit misleading to me. There is no reference frame in which a photon is at rest. One could go even further and say that if one were to move at the speed of light, time wouldn't really "exist" or it would stand still. But, you can say that in your reference frame (for example earth), the time of the spaceship is slowed down.

So the short, somewhat unsatisfactory answer is "we don't really know, but we know that it is the case".

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For the relationship $$\text{speed}=\frac{\text{distance}}{\text{time}}$$ to hold true, if the distance changes either the time or the speed must change (or both). In the case of light we can in other experiments measure that its speed is always constant. Thus, it must be time which is changing.

Why can time change in this manner? Actually it shouldn't be odd at all that time can vary (slow down i.e.); you only expect it not to change because of what you are used to. Why would you expect time to be different in kind than, say, speed or distance? We have no physical ground to assume so. This is the abstract intuition-breaking claim Albert Einstein brought forward in his theory of special relativity.

What is odd is the fact that the speed of light is constant in all frames (that speed has an upper limit in our world). That is not something we've been able to explain yet - we can just measure that it is the case.

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