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Consider the law of addition of velocities for a particle moving in the x-y plane:

[itex]u_x=\frac{u'_x+v}{1+u'_xv/c^2},\, u_y=\frac{u'_y}{\gamma(1+u'_xv/c^2)}[/itex]

In the book by Szekeres on mathematical physics on p.238 it is said that if u'=c, then it follows from the above formulae that u=c, i.e. invariance of the speed of light under the given Lorentz boost, which is of course exactly what we wish, since a Lorentz boost must preserve the null cone.

The weird thing is that when I start with [itex]c^2=(u')^2=(u'_x)^2+(u'_y)^2[/itex] and try to apply the above formulae to simply get [itex](u'_x)^2+(u'_y)^2=(u_x)^2+(u_y)^2[/itex], i.e. working backwards towards the invariance of the null cone, I get quickly lost in the actual computation, because it seems to be leading nowhere…nothing cancels out. It doesn't matter which boost one takes…it doesn't seem to work (at the moment).

It can't be that hard. I don't know what I'm doing wrong, and can't imagine that I missed some concept. Did anybody already see how the computation goes? Is there anything one should pay attention to? Thanks