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!