contradictions in SR
May 17 2009 at 3:49 PM Anonymous
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Some textbooks and web resources try to avoid the problems inherent with Einstein's 'two-way' method of deriving the Lorentz transformation and use the so called 'light sphere' constraint
(1) x2+y2+z2=c2t2
(2) x'2+y'2+z'2=c2t'2
instead, apparently not noticing however that the quadratic form of the equations leads here to exactly the same inconsistency with the sign of the propagation direction as with Einstein's approach:
Restricting the problem to a motion along the x-axis and points on the latter, the constraint would be
(3) x2=c2t2
(4) x'2=c2t'2 .
Now (3) has the solutions
(3a) x+=ct
(3b) x -=-ct ,
whereas (4) has the solutions
(4a) x'+=ct'
(4b) x' -=-ct' ,
Now any pair of these solutions for (3) and (4) respectively satisfies the original quadratic constraint, for instance also (3a) in combination with (4b), which is however not possible as this would mean that in the primed frame the light pulse would travel in the opposite direction to the one in the unprimed frame. It is therefore obvious that only (3a),(4a) and (3b),(4b) are acceptable pairs here, which thus rules out the use of the quadratic form.
Adding an index L (for light) to the coordinates, and the set of equations thus reads
(5a) xL+=ct
(5b) xL-=-ct
and
(6a) x'L+=ct'
(6b) x'L-=-ct' ,
and we have thus
(7) xL-= -xL+
(8) x'L-= -x'L+.
Now again, since the primed signal can not travel into the opposite direction to the unprimed signal, this implies that whenever the unprimed coordinate of the light signal changes sign, so must the primed one. This condition is violated by the Lorentz transformation because if one has
x'L+ = ã(xL+-vt)
and replaces xL+ by -xL+ , one won't get -x'L+ unless v=0.
shorten version of
http://www.physicsmyths.org.uk/lorentz2.htm