Regarding the 'Light Sphere' Derivation of the Lorentz Transformation

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. By subtracting the two equations, they get (under the assumption y'=y and z'=z) (3) x2-x'2 = c2 (t2-t'2) 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' ,

(where I have used indices + and - rather than 1 and 2 as on my page Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation).

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.

Furthermore, to be even more rigorous, one should also formally indicate that (1a)-(2b) gives the coordinates of a light signal, nothing more and nothing less. So one should add an index L (for light) to the coordinates, and the set of equations thus reads

(5a)       xL+=ct
(5b)       xL-=-ct


(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+ = xL+/γ -vt'

and replaces xL+ by -xL+ , one won't get -x'L+ unless v=0.

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Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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