Adding Relative Velocities

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This is a quite simple question, and hopefully a simple answer. I know the basics of SR and GR, but I'm bothered by this one. It seems that by adding speeds you can get speeds greater than c.

Say we have 4 observers on 4 different bodies. Each observer believes his body to be a rest. We will say that all observers line up to form a straight line progressing from observer 1 -> 4.

Observer 1 views observer 2's body to be moving away at 1/2 c. Oberserver 2 views observer 3 to be moving away at 1/2 c. And observer 3 sees observer 4 to be moving away at 1/2 c. This would mean that observer 4 was moving away from observer 1 at 1 1/2 c. How is the universal maximum c not broken here?
 

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  • #3
JesseM
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This is a quite simple question, and hopefully a simple answer. I know the basics of SR and GR, but I'm bothered by this one. It seems that by adding speeds you can get speeds greater than c.

Say we have 4 observers on 4 different bodies. Each observer believes his body to be a rest. We will say that all observers line up to form a straight line progressing from observer 1 -> 4.

Observer 1 views observer 2's body to be moving away at 1/2 c. Oberserver 2 views observer 3 to be moving away at 1/2 c. And observer 3 sees observer 4 to be moving away at 1/2 c. This would mean that observer 4 was moving away from observer 1 at 1 1/2 c. How is the universal maximum c not broken here?
According to the Lorentz transformation velocities don't add in the way they do in classical physics, instead you have to use the relativistic velocity addition formula. For example, if observer 2 sees observer 3 moving at 0.5c, and observer 1 sees observer 2 moving at 0.5c in the same direction, then observer 1 will see observer 3 moving at (0.5c + 0.5c)/(1 + 0.5*0.5) = 1c/1.25 = 0.8c. Then if observer 3 sees observer 4 moving at 0.5c and observer 1 sees observer 3 moving at 0.8c, that means observer 1 sees observer 4 moving at (0.5c + 0.8c)/(1 + 0.5*0.8) = 1.3c/1.4 = 0.93c.

The fact that velocities don't add in the same way as they do in classical physics has to do with the fact that each observer measures velocity in terms of distance/time on a set of rulers and synchronized clocks at rest relative to themselves, but each observer sees the measurements of rulers and clocks of other observers distorted by length contraction, time dilation, and the relativity of simultaneity (which says that clocks synchronized in one frame will be out-of-sync in other frames).
 
  • #4
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Thanks Jesse for the quick response. That makes sense to me, I think I actually knew that at one time and forgot it. I hate when that happens. I don't use this stuff in my field so it fades out of memory fast.
 

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