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I am quite new to special relativity, and I wanted to get my head around how length contraction might work observing 2 or more objects moving within a particular frame.

Suppose an observer at rest on earth can see two spaceships A & B flying towards each other. From the Earth, spaceship A is moving at 0.8 c to the right and spaceship B -0.6 c to the left. By the relativistic velocity addition formula from each spaceship's point of view the relative velocity is 0.946 c.

Now consider an observer on spaceship A. He sees spaceship B coming towards him at 0.946 c and I guess he therefore sees the spaceship length contracted by $\gamma$ = 3.08 times shorter?

If I am correct here then my question is what does he then see Earth and the rest of space (e.g. distance from the Earth to Sun etc.) contracted by? Is this also contracted by 3.08 times? Or does he instead view the Earth travelling towards him at 0.8 c and therefore view the contraction of Earth and space by a $\gamma$ factor of 1.66 instead? The second answer would seem to make more sense to me but it is ok for there be two or more different length contractions going on like this in a single frame? Is it even ok to think about the problem in this way (I know I have to try and rid myself of many of the expectations of classical mechanics!)?

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  • $\begingroup$ Welcome to PSE! No need to apologise for asking questions here, that is what this site is for :) $\endgroup$ Jan 1, 2021 at 13:32

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it is ok for there be two or more different length contractions going on like this in a single frame?

Yes, this is totally fine (it is what actually happens). If the observed length contraction was the same for all objects an observer sees, then this would suggest that there is an absolute frame of reference. For example, if an observer saw all objects contracted by $0.5$, then they could say "ah, I am in the frame where everything is contracted by $0.5$."

Another easy counterexample to consider is any object at rest with respect to some observer will not be length contracted at all.

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The second answer would seem to make more sense to me but it is ok for there be two or more different length contractions going on like this in a single frame?

Yes, the second answer is correct and it is fine to have different objects contracted differently.

Length contraction, in a geometrical sense, is similar to having an object at an arbitrary angle and measuring the width say on a north-south line. For instance, if I had a 3 m by 4 m table and I line it up right I would get that it is 4 m long in the north-south direction, but if I line a second identical table just right it could be 5 m long in the north-south direction. Both measurements are OK and can be correct together.

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