Why do two moving inertial frames of reference atribute the same velocity to each other?

Question

In both relativity and Newtonian mechanics, it is a self-evident axiom according to which two inertial observers who move relative to each other measure the same velocity of, say, $v$. I want to know why this is so. One may say that our daily experience, easily verifies this postulate, however, to me, it is not convincing.

Assume that two objects $A$ & $B$ are at rest with respect to each other. Then, $B$ decides to approach $A$ at a constant velocity after undergoing an arbitrary acceleration. If $A$ measures the velocity of $B$ as $v_{BA}$, I can easily claim that $B$ measures the velocity of $v_{AB}$ for $A$ so that for non-relativistic velocities we always have: $$v_{BA} \approx v_{AB}=v$$
However, which experiments have verified the above equation thus far for when the velocities are relativistic?

For better compatibility with experiment, assume that $B$, who has already undergone an acceleration, can never exceed the speed of light as measured by $A$. ($v_{BA}<c$) Now, assume that $v_{AB}$ complies with the challenging equation below: $$v_{AB}=\gamma_{v_{BA}}v_{BA}$$ where $\gamma_{v_{BA}}$ is the traditional Lorentz factor. It is clear that if $c/{\sqrt{2}}<v_{BA}<c$, $v_{AB}$ exceeds the speed of light so that for $v_{BA}\approx c$, $v_{AB}$ tends to infinity. If we want to disprove the odd equation above, we have to investigate the viewpoint of observer $B$ for when his velocity is a considerable portion of the speed of light as measured by $A$. According to our current technology, it seems impossible to set a human being in motion at a significant portion of the speed of light, and want him/her to tell us about their measurements as to the relative velocity of the surrounding objects especially the other observer ($A$)!

Although this deduction upsets the symmetry of special relativity, I am curious to know if there are some experiments against this idea.

Answer 1

There is no need to devise new experiments to disprove your formula- the formula is clearly already incompatible with the existing experimental evidence for special relativity.

In any case, your understanding of the experimental requirements is wrong. You do not have to accelerate a human to a speed approaching c in order to test relativistic effects- you just need sufficiently accurate clocks and measuring devices. Modern atomic clocks are capable of measuring time dilation at everyday speeds.

Answer 2

In addition to being intuitive and also required by the principle of relativity, this is actually fairly easy to derive directly from the Lorentz transform. For simplicity of notation I will suppress the y and z direction and do a Lorentz transform in matrix form using units where c=1. So if frame B is moving at velocity v with respect to frame A then the transform from A to B is $$\Lambda(v)= \left(\begin{array}{rr} {\gamma} & -{\gamma} v \\ -{\gamma} v & {\gamma} \end{array}\right)$$ where $\gamma=(1-v^2)^{-1/2}$. Now, the transform from B to A is $\Lambda(u)$ such that $$\Lambda(u)\Lambda(v)= \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right)$$ The solution to this is $\Lambda(u)=\Lambda(v)^{-1}=\Lambda(-v)$. So the velocity of A with respect to B is equal and opposite of the velocity of B with respect to A.