# In deriving Special Relativity equation, from math point of view, why don’t we chose length perpendicular to the line of motion to be modified?

In deriving Special Relativity equation, we modify the length parallel to motion in a way that outside observer would agree that light moving in perpendicular and in parallel with motion would meet in the same time and spot (the same event). But the same effect could be achieved if instead we make the length perpendicular to motion becoming longer and set the length parallel to motion fixed.

I’m referring to this derivation from Feyman Lectures on Physics’ The Special Theory of Relativity (https://www.feynmanlectures.caltech.edu/I_15.html): The first equation in question is:

$$t_1 + t_2 = \frac{2L/c} {1 - u^2 / c^2}$$ (Eq. 1)

which is the time needed for light to go from $$B$$ to $$E^\prime$$ and coming back to $$B^\prime$$.

The second equation is:

$$2 t_3 = \frac{2L/c} {\sqrt{1 - u^2 / c^2}}$$ (Eq. 2)

which is the time needed for light to go from $$B$$ to $$C^\prime$$ and coming back to $$B^\prime$$.

In order to make it equal so the light would arrive at $$B^\prime$$ in the same time, we choose to shorten the length $$L$$ parallel to line of motion to $$L \sqrt{1 - u^2/c^2}$$, so Eq. 1 would become:

$$t_1 + t_2 = \frac{2L/c} {\sqrt{1 - u^2 / c^2}}$$ (Eq. 3)

where $$t_1 + t_2$$ would equal with $$2 t_3$$ from Eq. 2.

But the same effect could be achieved by lengthening line $$L$$ perpendicular with line of motion to $$L / \sqrt{1 - u^2/c^2}$$ so Eq. 2 would become:

$$2 t_3 = \frac{2L/c} {1 - u^2 / c^2}$$ (Eq. 4)

where again $$t_3$$ would equal with $$t_1 + t_2$$ from Eq. 1.

If we are doing that, what I know immediately from the equation is the time in the other observer would be slowed down by a factor of $$\frac{1}{1 - u^2 / c^2}$$ instead of $$\frac{1}{\sqrt{1 - u^2 / c^2}}$$.

I know there would be paradoxes when it does happen, and it would not be invariant with Maxwell’s equation, but just from Special Relativity point of view, what is wrong with this? Why don't we choose this solution?

And please do not just vote down this question, but also give reason when it's considered stupid. I have searched and no answer to find.

• I'm voting to close this question as off-topic because it’s an open ended “What if ____ happened” question that is explicitly off topic s as described in help center – Kyle Kanos Dec 1 '19 at 12:07
• @KyleKanos I have modified the question, would you mind to take a look at it again? – Ari Royce Dec 1 '19 at 12:25
• Griffith specifically discussed this question in his textbook ("Introduction to Electrodynamics", third edition, pg492, last paragraph); Schwartz too ("Principles of Electrodynamics", pg 110, last paragraph). I believe it is in every special relativity books. – verdelite Dec 1 '19 at 13:34
• @verdelite I’am aware of that, e.g. Maxwell’s Equations would not be invariant if we choose that, but I want to know, just from Special Relativity point of view alone and mathematically, could we conclude that the length to be modified must be the length parallel to line of motion? – Ari Royce Dec 1 '19 at 15:30
• Yes, I'm aware of that, just want to know whether it could be concluded just from SR point of view alone. Anw it has been answered well by @Dale – Ari Royce Dec 2 '19 at 15:27

## 2 Answers

just from Special Relativity point of view, what is wrong with this? Why don't we choose this solution?

This transform is called the Voigt transform. It may have actually been developed prior to the development of the Lorentz transform. While the Voigt transform does respect the second postulate, the invariance of c, it does not respect the first, the principle of relativity.

Specifically, the principle of relativity requires that $$\Lambda^{-1}(v)=\Lambda(-v)$$ where $$\Lambda$$ is the transformation matrix between inertial frames. But with the Voigt transform the inverse of the Voigt transform is not even a Voigt transform. This means that two frames would be distinguishable from one another since one would transform differently than the other.

As you have pointed out, your new transformation culminates in paradoxes like this SE question to which you also gave an answer. If it were not paradoxical, it could be a challenging point of view.

I have published an article regarding this matter in a dissenting journal, though a small part of the article is devoted to this matter. (To download the article, please click on "Complete Issue (.pdf)" for "Summer 2018. Volume 29. Special Issue 3" in this link, and see p. 45. Sec. 4.1.)

• I'am aware of that, but I'm looking for what is wrong mathematically if we choose instead to lengthen the line $L$ perpendicular to line of motion. – Ari Royce Dec 1 '19 at 15:01
• @Ari Royce Hidayat What do you exactly mean by mathematically wrong? I do not think there is any mathematical deficiency, but rather there is something physically wrong with your calculations in that they result in paradox. – Mohammad Javanshiry Dec 1 '19 at 15:21
• In fact not, as it has been answered well by @Dale. That's what I'm looking for. – Ari Royce Dec 1 '19 at 17:58