# Length contraction looking at 2 moving objects in a frame

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!)?

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

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 $$0.5$$."