Lorentz Transformation Dilemma

My issue with special relativity is mathematical. The Lorentz Transformation produces a completely degenerate solution by simply rearranging a standard derivation of it. This means that an equation A x^2 + B xt + C t^2 = 0 is obtained where A = B = C = 0 assuming a constant speed of light, standard linear relations between x’, t’, x, and t, and a spherical wave. The lefthand side of the equation always equals zero regardless of the values of x and t.

Yes, scientists write r^2c^2*t^2=R^2c^2*T^2=0. Respecting both equalities gives Galilean transform and no time dilation. Ignoring the last equality gives Lorentz transform with time dilation.
JohnErik

To make sure I’m understanding your notation, is the r and ct in one reference frame such as the S frame and the R and cT in another reference frame such as the S’ frame?

Perhaps we have different starting points because I cannot derive either the Lorentz or Galilean transformation from the equations without additional information. The equations, at least for the Lorentz transformation, can be either for a circle where r^2 = x^2 + y^2 or for a sphere where r^2 = x^2 + y^2 + z^2. Assuming a constant speed of light c where
x’ = α<sub>1</sub>
x + α<sub>2</sub> t, y’ = y, z’ = zt’ = β<sub>1</sub>
x + β<sub>2</sub> t,I arrive at the equation
(α<sub>1</sub><sup>2</sup>
– β<sub>1</sub><sup>2</sup> c<sup>2</sup> – 1) x<sup>2</sup> – 2 (β<sub>1</sub> β<sub>2</sub> c<sup>2</sup> + α<sub>1</sub><sup>2</sup>
v) xt + (c<sup>2</sup> – β<sub>2</sub><sup>2</sup>
c<sup>2</sup> + α<sub>1</sub><sup>2</sup> v<sup>2</sup>) t<sup>2</sup> = 0where v is the speed between reference frames. This equation is of the form A x^2 + B xt + C t^2 = 0 and cannot be solved without more information.
On the other hand, the values for the components from the Lorentz transformation are known as
α<sub style=”backgroundcolor: transparent; fontfamily: inherit;”>1</sub>
= β<sub style=”backgroundcolor: transparent; fontfamily: inherit;”>2</sub> = γ and β<sub style=”backgroundcolor: transparent; fontfamily: inherit;”>1</sub>
= – γ v/c<sup style=”backgroundcolor: transparent; fontfamily: inherit;”>2</sup>where γ is the Lorentz factor. Substituting these values for the components produces
(0) x<sup>2</sup> – 2 (0) xt + (0) t<sup>2</sup> = 0
which is degenerate.

Sorry, the message didn’t post as it was written. Is there any way in this forum to post standard mathematical equations?
The message is posted below without HTML markups.
Perhaps we have different starting points because I cannot derive either the Lorentz or Galilean transformation from the equations without additional information. The equations, at least for the Lorentz transformation, can be either for a circle where r^2 = x^2 + y^2 or for a sphere where r^2 = x^2 + y^2 + z^2. Assuming a constant speed of light c where
x’ = α1 x + α2 t, y’ = y, z’ = z
t’ = β1 x + β2 t,
I arrive at the equation
(α1^2 – β1^2 c^2 – 1) x^2 – 2 (β1 β2 c^2 + α1^2 v) xt + (c^2 – β2^2 c^2 + α1^2 v^2) t^2 = 0
where v is the speed between reference frames. This equation is of the form A x^2 + B xt + C t^2 = 0 and cannot be solved without more information.
On the other hand, the values for the components from the Lorentz transformation are known as
α1 = β2 = γ and β1 = – γ v/c^2
where γ is the Lorentz factor. Substituting these values for the components produces
(0) x^2 – 2 (0) xt + (0) t^2 = 0
which is degenerate.


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