Many people maintain that the Lorentz transformation is derived
mathematically consistently and that there is therefore no way to
challenge SR on internal consistency issues. Is this really so? Let's
for example have a look at Einsteins own derivation (from his book
'Relativity: The Special and General Theory') given at
http://www.bartleby.com/173/a1.html which seems to be a very elegant
way of deriving the Lorentz transformation.

It is only necessary here to examine the initial equations for this,
which describe the 'equations of motion of a light signal' in the
unprimed and primed reference frames, i.e.

(1)  x-ct=0
(2)  x'-ct'=0
where c is the speed of light (which obviously has to be a constant >0)

In the same way, the propagation of a signal in the opposite direction
yields
(3) x+ct=0
(4) x'+ct'=0
(note that these equations are not written explicitly in Einstein's
derivation).

>From equations (1)-(4), the Lorentz transformation is then derived by
some algebraic manipulations.

But are the above equations mathematically consistent at all? Let's
subtract equation (1) from (3), which yields
(5) 2ct=0
which means that for any time t>0
(6) c=0,
in contradiction to the requirement that c>0.

This shows that the equations used to derive the Lorentz transformation
are mathematically inconsistent. The fact that the Lorentz
transformation itself seems to be mathematically consistent only
demonstrates that the 'length contractions' and 'time dilations'
involved in the completion of the derivation are not ony physically
unacceptable (as argued on my page
http://www.physicsmyths.org.uk/lightspeed.htm ) but also mathematically
inconsistent as they contradict the initial definitions.

Thomas