Mathematically, I didn't see anything wrong or inconsistent with p=g(V)p'(1+V/u'). Of course, while you can't SET u'=0 in the expression, you can take the limit of the whole expression as u' --> 0. That is what "would have to be explained".
For me, it's always helpful to think geometrically.
In terms of rapidities, the equations translate to
(1) p=gp'(1+V/u')
p=\cosh\theta p' (1 + \frac{c \tanh\theta}{c\tanh \phi'})
(2) E=gE'(1+Vu'/c^2)
E=\cosh\theta E' (1 + \frac{c\tanh\theta\ c\tanh \phi'}{c^2})
which don't look very recognizable [in terms of an analogy to Euclidean geometry] like (upon minor rearrangement)
E=\cosh\theta E' + \sinh\theta(cp')
p=\sinh\theta(E'/c) + \cosh\theta p'
which, of course, follows Minkowski's viewpoint and interpretation as a spacetime geometry. I would think that composing boosts would be rather tedious in the proposed equations (1)-(2).
The " setting of u' = c " corresponds to the " setting of \tanh\phi' = 1[/tex] ".<br />
<br />
The " setting of u' = 0 " corresponds to the " setting of \tanh\phi&#039; = 0[/tex] &quot;, which has a problem in (1) which needs to be addressed by taking a limit involving p&#039;/u&#039; .Here are some thoughts on the tardyon to photon &quot;limit&quot;.<br />
<ul>
<li data-xf-list-type="ul"> For a timelike 4-momentum-vector (representing a tardyon, a massive particle) (E/c) &gt; |p|.<br />
For a lightlike 4-momentum-vector (representing a photon, a massless particle) (E/c) = |p|.</li>
<li data-xf-list-type="ul">There is no finite set of boosts that will make a tardyon into a photon.<br />
Geometrically, you can&#039;t boost a timelike-vector (whose tip is on a hyperbola asymptotic to the light cone) into a [null] lightlike-one (whose tip is on that light cone)... The rapidity (the Minkowski angle) would tend to inifinty, without ever reaching it.</li>
<li data-xf-list-type="ul">In the spatial-velocity representation where V=c tanh(theta), the V=c limit appears as a mere finite endpoint, which [by our Galilean intuition] naively suggests that getting from-0.98c-to-0.99c is probably about as easy from getting from-0.99c-1.00c.</li>
<li data-xf-list-type="ul">It seems the two limiting procedures are different. In one case, we boost, taking the momentum 4-vector up the hyperbola asymptotically to the light cone. (This involves V and theta.) In the other case, we slide the tip of the momentum 4-vector onto the light cone in such a way which continuously but asymmetrically changes its temporal and spatial components while discontinuously changing its invariant norm. (This involves u&#039;.)</li>
</ul><br />
Pedagogically, I think that the necessary explanation of the u&#039;--&gt;0 limit and the breaking of the functional symmetry between E and p are too high a price to emphasize the interpretation sought. So, IMHO, while it might be an interesting interpretation [i.e. side comment], I can&#039;t see it forming the foundations of a pedagogical presentation of relativity.