User:Crdrost/Relativity derivation

Some history that might be useful
In the late 1800s, physics was building up to a boiling point. In the mid 1800s, Maxwell had been working with others like Thomson on coming to an understanding of electromagnetism. They both understood that there was something peculiarly geometric about this new physics, mostly because a magnet would create lines in iron filings. Starting from these magnetic lines and the work that had been done before him, Maxwell in November 1854 had already written a letter to Thomson containing an expression of what we today incorrectly call "Ampère's law," that today we'd say was like $$\nabla\times B = \mu_0 J.$$ In addition Faraday's law of electromagnetic induction was known to take various forms that were essentially equivalent to its modern form $$\nabla\times E = -\dot B,$$ and Coulomb's law was known, today we'd phrase it as $$\nabla\times E = \rho/\epsilon_0$$, and it's probably safe to say that Maxwell was not concerned with magnetic monopoles: or, as we'd now state, $$\nabla \cdot B = 0.$$

Well, there is a subtle problem with these equations, and it's that they do not respect the continuity equation $$\dot \rho + \nabla \cdot J = 0$$. Maxwell does not seem to have realized this directly, but as he tried to grapple with a physical analogy he had the idea that perhaps there was an analogy of the fields to some system of a vast number of tiny little cogs permeating space. Because cogs could not rotate if they were touching, Maxwell imagined that perhaps little particles of charge were flowing between them, keeping them separate, and indirectly came to the necessity for the so-called "displacement current" $$\nabla \cdot B = \mu_0 J + \mu_0 \epsilon_0 \dot E.$$ This effect is not trivial: it has the magnetic fields jiggling the electric fields which then jiggle the magnetic fields again. With this his micro-mechanical model allowed him to calculate the speed of transverse waves moving through these gears, and had he done the calculation correct he would have gotten $$1/(2\sqrt{\epsilon_0 \mu_0})$$, but he didn't understand the prefactor 1/2 and instead just got $$1/\sqrt{\epsilon_0\mu_0}$$, which turned out to be within 1% of the then-known value for $$c$$, at which point he postulated, correctly, that light was an electromagnetic wave. Though today we interpret just about every single equation differently from his interpretation, nevertheless we call these the "Maxwell Equations" because Maxwell's work took us most of the way towards them.

In vacuum, in the modern interpretation, the prediction from theory is fairly clear: the equations become,


 * $$\begin{array}{rlrl}\nabla\cdot E ~=&0& \nabla\times E ~=& - \dot B\\\nabla\cdot B~=&0&\nabla\times B~=&\mu_0 \epsilon_0 \dot E\end{array}$$

and taking a curl of a curl with a vector identity we can find


 * $$\nabla \times (\nabla \times B) = \nabla (\nabla \cdot B) - \nabla^2 B = \mu_0 \epsilon_0 \nabla\times \dot E = -\mu_0\epsilon_0\ddot B.$$

and similarly $$\mu_0 \epsilon_0 \ddot E - \nabla^2 E = 0.$$ These equations admit of solutions $B_0(\vec r - c~\hat e~t)$ for any unit vector $\hat e,$ if $c = 1/\sqrt{\mu_0 \epsilon_0}.$ And that is where our crisis begins. Notice that the waves travel at speed $$c$$ in every direction, not $$c + v$$ in one direction and $$c - v$$ in the opposite, corresponding to the light being carried in a medium moving with speed $v$.

What I really wanted to get to
The Galilean transform is x' = x - v t, t' = t. The Lorentz transform is a lot more complicated. However the Lorentz transform is to first order in velocity, x' = x - b w, w' = w - b x, showing *only* the effect of relativity of simultaneity. I want to tell people that this is the *whole* Lorentz transform.

To do this, look at the matrix involved,


 * $$L = \begin{bmatrix}1&-\beta\\-beta&1\end{bmatrix}$$

Diagonalizing it we realize that its eigenvectors are trivially [1, 1] and [1, -1] with eigenvalues $1-\beta$ and $1+\beta.$ So therefore it becomes:


 * $$L(\beta) = C \hat L C^{-1} = \sqrt{\frac12}\begin{bmatrix}1&1\\1&-1\end{bmatrix} \begin{bmatrix}1-\beta&0\\0&1+\beta\end{bmatrix} \sqrt{\frac12}\begin{bmatrix}1&1\\1&-1\end{bmatrix}.$$

Now imagine that we try to do $N$ boosts by some velocity $v/N$ in the $x$-direction, we have


 * $$[L(\phi/N)]^N = \sqrt{\frac12}\begin{bmatrix}1&1\\1&-1\end{bmatrix} \begin{bmatrix}\left(1-\frac\phi N\right)^N&0\\0&\left(1+\frac\phi N\right)^N\end{bmatrix} \sqrt{\frac12}\begin{bmatrix}1&1\\1&-1\end{bmatrix}.$$

By a well-known limit as $N \to \infty$ this is simply


 * $$\lim_{N\to\infty} [L(\phi/N)]^N = \sqrt{\frac12}\begin{bmatrix}1&1\\1&-1\end{bmatrix} \begin{bmatrix}e^{-\phi}&0\\0&e^\phi\end{bmatrix} \sqrt{\frac12}\begin{bmatrix}1&1\\1&-1\end{bmatrix}

= \begin{bmatrix}\cosh\phi&-\sinh\phi\\-\sinh\phi&\cosh\phi\end{bmatrix}.$$

So we see that the general form of the Lorentz transform is actually not the core physics. The core physics is simply "as you accelerate forwards by a speed $v$ if a series of clocks *were* in sync with you, they no longer are. Instead the ones a distance $L$ ahead of you appear to be earlier by a time $v L /c^2,$ when you faithfully work backwards to figure out what time is actually on them, from what time is actually showing."