I, standing still with respect to myself, can see how fast A) my son is moving and B) my daughter is moving. But because spacetime is not absolute, I don’t know C) how fast my son is moving with respect to my daughter (and vice versa) from his (or her) point of view. That is, if he is moving at 0.5c and she is moving at .866c , how fast are they moving with respect to one another? Hint: it is not 0.866-0.5=.366c.
After a lot of searching online and through books I have come to the conclusion that the best explanation of this is on page 27 (plus the appendix, which fleshes things out algebraically) of David Mermin’s course notes. Careful study of this, and working out of the maths by hand, will be worth the roughly 1-hour investment of time that will be required.
This can be expanded infinitely many times by the same formula as 4.16 just above:
Online Explanations of Einstein Velocity Addition
- Wikipedia: Starts out easy, becomes quite technical
- Hyperphysics: just presents the math, in a digestible fashion, but with no physical explanation.
- Khan Academy: as usual, very clear on the math, almost nothing on the deep metaphysics at play
- David Mermin’s course notes, chapter 4 (p.21-35) has a very detailed logical derivation of velocity addition. He has published the same idea in the American Journal of Physics.
- Ben Yu-Kuang Hu elaborates on Mermin’s model (arXiv link here) applying it to perpendicular, rather than parallel, rays of light, PDF download here.