Lorentz Transformation Dilemma

  • Lorentz Transformation Dilemma

     Randy updated 1 month, 3 weeks ago 2 Members · 6 Posts
  • Randy

    Member
    July 27, 2021 at 11:29 pm

    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 left-hand side of the equation always equals zero regardless of the values of x and t.

  • John-Erik

    Member
    July 28, 2021 at 1:19 pm

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

    John-Erik

    • Randy

      Member
      July 28, 2021 at 8:00 pm

      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?

      • John-Erik

        Member
        July 29, 2021 at 6:46 pm

        Yes

    • Randy

      Member
      July 31, 2021 at 1:32 pm

      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’ = z

      t’ = 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> = 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

      α<sub style=”background-color: transparent; font-family: inherit;”>1</sub>
      = β
      <sub style=”background-color: transparent; font-family: inherit;”>2</sub> = γ and β<sub style=”background-color: transparent; font-family: inherit;”>1</sub>
      = – γ v/c
      <sup style=”background-color: transparent; font-family: 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.


      • Randy

        Member
        July 31, 2021 at 1:45 pm

        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.

Viewing 1 - 2 of 2 replies

Log in to reply.

Original Post
0 of 0 posts June 2018
Now