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So my textbook says the following - roughly translated - in the context of Special Relativity:

"Assume we have two observers, A and B, moving relative to each other. Observer A measures a velocity v for observer B. Because of the symmetry of the situation, observer B measures the same velocity for observer A. If this isn't clear, note that A and B can be replaced by one another. If A and B measured a different velocity, then either of the two would be in a 'special reference frame'. This is in contradiction with the postulates of special relativity."

Alright, so I get this intuitively, of course. However, I can't follow their line of reasoning. How would we know that the laws of physics are different for A and B, if they don't measure the same velocity? Velocity is already relative, so how do we know for sure there would have to be a different set of laws for, say, A to get this result?

I'm familiar with the following argument: We've already deduced that a light clock yields different time intervals between its ticks for different observers. Assume there'd be another 'invariant' clock that would tick the same for any observer (a supposed 'universal time' clock). Then any observer would be able to measure their absolute speed, by comparing the time interval measured by the universal clock and a licht clock (that are assumed to not be movingg relative to one another). This is a contradiction, for velocity can't be absolute by our postulates.

Is it perhaps possible to give the same kind of argument with the symmetry problem? Would it somehow be possible for either of the observers to deduce an 'absolute velocity', if they don't measure the same velocity (which would yield the desired contradiction).

I'm hoping someone could help me out with either of the arguments!

Sha Vuklia
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  • Well, intuitively I'd say, their frame of reference is different, so in theory it's possible to measure a different velocity, because velocity is relative. But I get what you're saying with the law/rule thing. If they measured a different velocity, we would have to be able to deduce that from some law of rule, but they are the same, so it's impossible. I'm still not getting it entirely, but this is helpful. – Sha Vuklia Jan 04 '17 at 18:48
  • I think the argument as it stands is untenable. With the same reasoning one also concludes that the velocities have the same sign. And this is not true. – Valter Moretti Jan 04 '17 at 18:49
  • @AlbertAspect Question though. Why would we need such a law (which we can't think of, obviously) for the two observers to measure a different velocity? Why does the fact that we don't have a law for it a good argument that it can't be true? Or is this a case of; as long as it's not proven wrong, we can assume it's true? – Sha Vuklia Jan 04 '17 at 19:00
  • @ValterMoretti Good point. You could even be using different units. –  Jan 04 '17 at 19:01
  • @ValterMoretti well, if left unchanged my argument also fails. –  Jan 04 '17 at 19:01
  • At most one may conclude that, with a suitable choice of units and orientations the velocities result identical. But now considering three reference frames one may fall in troubles...there is as similar cohomological problem with Poincare' transforms... – Valter Moretti Jan 04 '17 at 19:04
  • @ShaVuklia "Velocity is relative" just means that two different observers in general obtain two different results when measuring a velocity of an object; the measured velocity is directly related to the velocity of the observer. The point is that observer A has the same right as observer B to consider itself at rest (because both of them are, in their own frame of reference), therefore when A measures the speed of B he gets the same result of B measuring the speed of B. Every other possibility leads to contradiction with the relativity postulate. – tomph Jan 04 '17 at 19:05
  • @tomph I get that, but what then is the exact contradiction? What is the step between what you said and concluding it's a contradiction? – Sha Vuklia Jan 04 '17 at 19:09
  • @ShaVuklia I erased my argument, I need to get rid of that loophole! –  Jan 04 '17 at 19:11
  • @ShaVuklia the symmetry breaks when A/B changes inertial frame and the other point of reference does not change inertial frame in a symmetrical manner. – Yogi DMT Jan 04 '17 at 19:16
  • @ShaVuklia I think the symmetry they are assuming is that universe is isotropic; therefore there is no preferred direction. Perhaps, one can just write down velocity transformations between the two frames and calculate the speeds explicitly. – Ali Jan 04 '17 at 19:25
  • @Ali Now that is helpful! :) Thank goodness. But we'd get v and -v right? We would have to correct by mirroring the reference frames, right? But hey, I think this is an amazing solution! Thank you! – Sha Vuklia Jan 04 '17 at 19:29
  • @ShaVuklia Depends on how each of them defines their coordinates. If they both define their coordinates in the same directions then you are absolutely right. On the other hand, if there is nothing else in the universe, an intuitive coordinate system for A could be pointing towards B's initial position, and vice versa, for B it could be pointing towards A's initial position. This way, they will both measure the same velocity. – Ali Jan 04 '17 at 19:38
  • @Ali Yea, exactly, got it! (that's what I meant by mirroring btw, but the way you put it is much clearer!) Anyhow, would the following also be a reasonable argument (when considering Galileo transformations, not Lorentz): relative velocity is the first time derivative of their distance, and distance is considered an invariant quantity, so relative velocity is also an invariant quantity. Qed? – Sha Vuklia Jan 04 '17 at 19:40
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    @ShaVuklia In Galilean transformations, you need the invariance of time differences as well as the invariance of distances to make that argument. – Ali Jan 04 '17 at 19:43
  • Related ? : Is measure relative velocity the same for both observer n particle – Frobenius Jan 05 '17 at 21:05

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Alright, I got a more intuitive/physical answer (instead of just plugging in values in Galileo/Lorentz transformations!).

We are going to assume two things: object A and object B are moving relative to each other (acceleration=0). Now, we can fix object A and make A face in the direction of B. Assume A tells us that B moves with speed v. Now, we maintain a completely identical situation if we fix B instead of A. In that case we have an object (B, in this case) that is facing another object that moves relative to it [in the same way as above]. This is literally the same situation as above, so if A gives us a value v for B, then B must give us the same value for A.

In short: Situation I: An object (A) is moving relative to another (B). Situation II: An object (B) is moving relative to another (A), in the same way. Conclusion: Those situations are identical, so any results concocted in situation I must also apply to situation II.

I know this is an overly cumbersome explanation, but this is the only way I believe I truly get it. :)

Sha Vuklia
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  • So for instance can you also conclude that if A finds B red coloured also B finds A red coloured? It does not work... – Valter Moretti Jan 05 '17 at 08:23
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    Moreover the discussed issue has not to do with inertial reference frames, Galileian invariance and all that which concern dynamics not kinematics. Indeed, the fact that the absolute value of the relative velocity is equal is valid also when A is inertial and B is not. More generally it holds regardless the relative acceleration. A better starting point in classical physics, in my view, is to notice or to assume as a basic experimental fact that the distance between A and B is symmetric. If there is a shared notion of time, all that leads to the wanted statement about relative velocities. – Valter Moretti Jan 05 '17 at 08:45
  • @ValterMoretti In response to your first comment; no, I don't think that's how one could go about it. You cannot deduce the colour of an object directly from its relative motion, so you would need more information, which is not given in this case. It is however possible - in theory - to express their relative motion in terms of their relative velocity. So, if B is able to measure A's velocity, he will give us the same value as A does, because of the symmetry of the situation. – Sha Vuklia Jan 05 '17 at 11:02
  • Your claim is untenable, sorry, why colours do not enter the game? What you are saying is completely arbitrary, you are assuming lots of physical suppositions about kinematical relations. This is bad physics. Please be more careful this is not the right way to learn physics. – Valter Moretti Jan 05 '17 at 11:05
  • @ValterMoretti In response to your second comment: of course the discussion has to do with inertial reference frames, because, as I've already stated clearly, I'm using a symmetry argument. So we cannot have an inertial and a non-inertial reference frame... – Sha Vuklia Jan 05 '17 at 11:05
  • @ValterMoretti "You cannot deduce the colour of an object directly from its relative motion, so you would need more information, which is not given in this case." – Sha Vuklia Jan 05 '17 at 11:06