Addition of Velocities in Special Relativity?

The formula for relativistic addition of two (1d constant) velocities $u$ and $v$ in Einstein's special theory of relativity SR, reads

• $\frac{u+v}{1+uv}$    (1)

to be compared with addition in Galilean relativity:

• $u+v$.

To give meaning and prove (1) two inertial frames $(x,t)$ and $(x^\prime ,t^\prime )$ are introduced connected by the Lorentz transformation

• $x^\prime =\gamma (x+vt)$, $t^\prime =\gamma (t+vx)$,  (2)
• $\gamma =\frac{1}{\sqrt{1-v^2}}$, 

where $v$ with $\vert v\vert \lt 1$ is the relative velocity between the two systems, $u=\frac{dx}{dt}$ is the velocity of an object $O$ as measured by an observer $X$ in the unprimed system, while (1) is supposed to give the velocity of $O$ for an observer $X^\prime$ in the primed system as the result of relativistic addition of velocities.

To prove (1) the standard argument is to differentiate (2) with respect to $t$ to get  

• $dx^\prime =\gamma (\frac{dx}{dt}+v)dt$, 
• $dt^\prime=\gamma (1+v\frac{dx}{dt})dt$,

and so by division with $u=\frac{dx}{dt}$ obtain 

• $\frac{dx^\prime }{dt^\prime}=\frac{u+v}{1+uv}$,

which is (1). 

We thus see that the Galilean sum $u+v$, which is valid in any chosen single inertial frame,  in SR is replaced by $\frac{u+v}{1+uv}$ as the result of introducing two inertial systems connected by the Lorentz transformation. 

The key point is that the sum of two velocities $u$ and $v$ in SR is made in two steps with $u$ the velocity of an object $O$ with respect to the unprimed system (as viewed by $X$) and $v$ the relative velocity between the systems, with the sum supposed to be the velocity in the primed system (as viewed by $X^\prime$). 

In particular, $X^\prime$ is denied the possibility of directly observing the object $O$ in the primed system and determining its velocity in that system. One may ask why? In any case, $X^\prime$ is instead referred to an observation of $O$ made by $X$ in the unprimed system, which is then transferred to the primed system by the Lorentz transformation and finally combined with the motion between the systems to serve as the observation by $X^\prime$. 

The two step procedure is used to bring in two inertial systems connected by the Lorentz transformation. Of course, to describe the motion of an object it suffices in principle with just one inertial system, where the motion is followed in space and time. But in a SR with just one inertial system there is no Lorentz transformation and thus no content beyond what can be given to any inertial system such as Galilean relativity.  

This connects to the previous post making the point that SR in just one chosen frame without any Lorentz transformation appears to be empty of content. 

The above analysis poses more questions concerning the physics of the Lorentz transformation: 

• Why cannot $X^\prime$ observe $O$ in the $(x^\prime ,t^\prime )$-system?
• Why can the motion of $O$ as viewed by $X$ be transferred to be the view of $X^\prime$ by the Lorentz transformation, when the transformation was derived to match light signals and not general motion? 

We compare with addition of velocities in MMR given by (modulo signs)

• $u+v+uv$, 

as a result of composite Doppler shifts in a case where $X^\prime$ cannot directly observe $O$ and so is referred to a Doppler shift observation by $X$, which is transferred by another Doppler shift to $X^\prime$ as a consequence composite shifts with  

•  $\frac{1}{1+u}\frac{1}{1+v}=\frac{1}{1+u+v+uv}$.  

There may thus be some rationale to view addition of velocities as composition of observations in  

different inertial systems by composite Doppler effects, but building the composition on the Lorentz transformation lacks physics since the Lorentz transformation itself so does.