[John-Erik]

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]

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]

Yes

[Randy]

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

(α₁²
– β₁² c² – 1) x² – 2 (β₁ β₂ c² + α₁²
v) xt + (c² – β₂²
c² + α₁² v²) t² = 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) x² – 2 (0) xt + (0) t² = 0

which is degenerate.

[Randy]

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.

[Jerry]

Hi. What type of observations does the Lorentz Transformation present to physical reality? A main idea I’ve heard is that when the Lorentz Transformation is applied to “faster than light” velocities of objects, that it always prevents travel at or over c.

If possible, what could we describe accurately in words about this, with the least amount of math and technical terminology?

Thanks!

• This reply was modified 2 years, 3 months ago by  Jerry.