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If I understand it correctly:

  • given 2 bodies travelling at different relative speeds, time will appear to run faster for the faster moving of the two, from the slower objects perspective

  • a single light photon will be seen to be moving at the same speed for both because although the 'distance travelled by light' value will be greater for the faster body, the 'time it took to travel' value will also be greater, resulting in the same speed measurement (x/y = 2x/2y)

However, if both bodies are moving in opposite directions, as we have no fixed point of reference, who is to say which is travelling faster?

(The next part of the question was about being able to work out the rate of expansion of the universe based on a "fixed" point relative to light, but I'm not actually sure how to translate that from my brain into words! I may come back to that...)

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If you're standing on earth and your friend is traveling at $0.9c$ in a rocket (relative to the earth), you may think that the rocket is traveling faster. However, from your friend's perspective, you are the one traveling $0.9c$ in the other direction. You should only be concerned with the relative motion between the two frames of reference when examining time dilation, as there is no "absolute motion".

Basically, if you are both holding clocks, both observers will measure the other clock to go slower. It may seem like a paradox, but it's not. It does however affect simultaneity and other things, which the observers will not agree on.

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Saying 'time will be increased" is not really the way to look at it, it kinda implies an absolute time, which is wrong. Each observer in his reference frame will not notice any change in time passing, it will all seem normal but just to him. The observer in each different reference frame will think it's the other guys in their frames that have a "problem" with time.

They will however, all agree that the speed of light is the same, but they will need to reconcile their measurements.

Read more on this at Lorentz Transformation.

If both bodies are moving in opposite directions, as we have no fixed point of reference, who is to say which is travelling faster?

There is no absolute standard of time or space, this is just the same as the above remarks, each of the observers will think it's the other guy who is moving away at high speed.

Next part of the question was about being able to work out the rate of expansion of the universe based on a "fixed" point relative to light, but I'm not actually sure how to translate that from my brain into words! I may come back to that...)

Before you do this, look through the site, this question has been addressed in many ways already, I think. On a universal scale you are dealing with GR, not the special case of SR.

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    $\begingroup$ You are welcome, books you might already have that I found useful were Hartle, Relativity Demystified by McMahon, (this guy just goes through worked example after worked example, it's very informative, and the typos keep you on your toes:) and Sean Carroll's free PDF lecture notes, finally Susskind on YouTube. $\endgroup$ – user154420 Jul 3 '17 at 13:43

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