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
^{2}-x'^{2} = c^{2 (}t^{2}-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
*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
_{L+} by -x_{L+} , one won't get -x'_{L+} unless v=0.

(1) x^{2}+y^{2}+z^{2}=c^{2}t^{2}

(2) x'^{2}+y'^{2}+z'^{2}=c^{2}t'^{2}

(3) x^{2}=c^{2}t^{2}

(4) x'^{2}=c^{2}t'^{2} .

(3a) x_{+}=ct

(3b) x_{ -}=-ct ,

(4a) x'_{+}=ct'

(4b) x'_{ -}=-ct' ,

(5a) x_{L+}=ct

(5b) x_{L-}=-ct

(6a) x'_{L+}=ct'

(6b) x'_{L-}=-ct' ,

(7) x_{L-}= -x_{L+}

(8) x'_{L-}= -x'_{L+}.

x'_{L+} = x_{L+}/γ -vt'

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

email:

email:

See also my sister site https://www.plasmaphysics.org.uk