The Crux of Special Relativity: Deriving the Lorentz Transformation (#SoME2)

In this video, we derive the Lorentz Transformation, which is like the foundation on which special relativity is built. I highly encourage you to "Pause and Ponder", and follow along with the proof, doing the algebra in your notebook alongside the video.

For the graphs in this video, I used the Desmos Graphing Calculator.

[MatheFysyk]

The peer review of SoME2 brought me here, and I congratulate you for the quality of this video ! Keep it up, that's great !

[marceleza]

You got my like before the first 30 seconds in the video. That's pretty fast! Love it! Keep them coming!

[angeldude]

This is a great companion to sudgelacmoe's submission (though missing the hashtag). That video assumes knowledge of the Lorentz boost formulas and then manipulates them into a nicer form, where as this video derives said formulas from nothing more than the basic postulates. The fact that they can be derived so intuitively just from those postulates is honestly incredible. Both videos help show how Special Relativity really isn't as bizarre as it's usually treated and can be very simple when you understand what it's actually saying.

[curtjaimungal]

Hi there, is there a way I can contact you personally (for example, a DM on Twitter or an email address)? Great job.

[massimilianodellaguzzo]

Hi Quick QEDs, nice video! (... waiting for your future videos, they will surely be interesting
)
I really like the Lorentz Transformations, and in this (long) comment I would like to tell you about " the correspondence principle ".



The correspondence principle is related to Lorentz Transformations. (and to Galileo's Transformations)
Maybe you know this principle, ... and there is something strange.

The two main Lorentz transformations are::
a) x '= gamma * (x - v * t)
b) x = gamma * (x '+ v * t ' )

The other two Lorentz Transformations:
c) t '= gamma * (t - vx/c^2)
and
d) t = gamma * (t '+ vx '/c^2)
are obtained from a) and b)

In this case it is enough to consider the two Transformations a) and b), because c) and d) depend on a) and b)
At low speeds the Lorentz factor (gamma) is a number very close to 1,
and so the two Lorentz transformations a) and b) become:
a_1) x '= x - v * t
b_1) x = x ' + v * t '

Substituting a_1 in b_1 we obtain:
x = x - v * t + v * t '
v * t ' = v * t
t ' = t

" THE CORRESPONDENCE PRINCIPLE " IS SATISFIED:
x '= x - v * t
t ' = t
(At low speeds, GALILEO'S TRANSFORMATIONS ARE OBTAINED)

And it is not the same, if we consider the two Lorentz Transformations a) and c)
a) x '= gamma * ( x - v * t )
c) t '= gamma * (t - vx/c^2)
At low speeds, the two Lorentz transformations a) and c) become:

a_1) x ' = x - v * t
c_1) t ' = t - vx/c^2

But if we consider large values of x, then t ' is not equal to t. (AND GALILEO'S TRANSFORMATIONS ARE NOT OBTAINED)
Also in this case it is enough to consider two Lorentz transformations, because b) and d) depend on a) and c)

It's too weird:
1) if we consider a) and b) the correspondence principle is satisfied
2) if we consider a) and c) the correspondence principle is not satisfied
... And if we consider a) and b) the Andromeda paradox (at low speeds) makes no sense. (because t '= t)



I think about the Lorentz Transformation c) t '= gamma * (t - vx/c^2) ...
... If t ' = 0 then t = vx/c^2, it's really "STRANGE" !
If instead we consider x = v * t, then c) t '= gamma * (t - vx/c^2) becomes: t ' = t/gamma.
(... And the correspondence principle applies!)

If you are interested in the subject, here is the link to a video by Roger Anderton: (about correspondence principle)