I read that Maxwell equations are covariant under Lorentz transformations, but I can't find a proof. Or at least a proof understandable by someone that doesn't know higher mathematics (please don't start writing hieroglyphics in tensor notation because I can't understand them: and note that because of that request this question is not a duplicate). I tried to do simple calculus finding what follow. Let's consider the two standard systems $S$ and $S'$ (in motion towards positive $x$) we use in relativity. Let's suppose in $S'$ Maxwell equations work:
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle{
\nabla' \cdot \mathbf{E}' = \frac{\rho'}{\varepsilon_0}}\\
\displaystyle{
\nabla' \cdot \mathbf{B}' = 0 }\\
\displaystyle{
\nabla' \times \mathbf{E}' = -\frac{\partial \mathbf{B}'}{\partial t'} }\\
\displaystyle{
\nabla' \times \mathbf{B}' = \mu_0 \mathbf{J}' + \varepsilon_0 \mu_0 \frac{\partial \mathbf{E}'}{\partial t'}}
\end{array}
\right.
\end{equation}
Then I expect that doing substitutions
\begin{equation}
\mathbf{E}' = (E_x , \gamma (E_y – v B_z) , \gamma (E_z + v B_y)
\end{equation}
\begin{equation}
\mathbf{B}' = \left(B_x , \gamma \left(B_y + \frac{v}{c^2} E_z \right) , \gamma \left(B_z – \frac{v}{c^2} E_y \right) \right)
\end{equation}
\begin{equation}
\rho'=\left(1-\frac{u_x v}{c^2} \right) \gamma \rho
\end{equation}
\begin{equation}
\frac{\partial}{\partial x'} = \gamma \left( \frac{\partial}{\partial x } + \frac{v}{c^2} \frac{\partial}{\partial t } \right)
\end{equation}
\begin{equation}
\frac{\partial}{\partial y'} = \frac{\partial}{\partial y}
\end{equation}
\begin{equation}
\frac{\partial}{\partial z'} = \frac{\partial}{\partial z}
\end{equation}
\begin{equation}
\frac{\partial}{\partial t'} = \gamma \left( \frac{\partial}{\partial t } + v \frac{\partial}{\partial x } \right)
\end{equation}
\begin{equation}
u_x' = \frac{u_x – v}{1-\frac{u_x v}{c^2}}
\end{equation}
\begin{equation}
u_y' = \frac{u_y}{\gamma \left( 1-\frac{u_x v}{c^2} \right)}
\end{equation}
\begin{equation}
u_z' = \frac{u_z}{\gamma \left( 1-\frac{u_x v}{c^2} \right)}
\end{equation}
I should obtain
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle{
\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}}\\
\displaystyle{
\nabla \cdot \mathbf{B} = 0 }\\
\displaystyle{
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} }\\
\displaystyle{
\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \varepsilon_0 \mu_0 \frac{\partial \mathbf{E}}{\partial t}}
\end{array}
\right.
\end{equation}
But if you do that you will see that this works only for transverse component ($y$ and $z$) of vector equations (3rd and 4th, the ones with curl). Let's focus, for example, on the simpler Maxwell equation, the 2nd one, that in $S
'$ is: $\nabla
' \cdot
\mathbf
{B}' = 0$. Doing substitutions this transforms in $\nabla
\cdot
\mathbf
{B} = 0 $ only if the $x$
component of $ \nabla
\times \mathbf
{E} + \frac
{\partial \mathbf
{B}}{\partial t} $ is zero. At first glance you could say "where is the problem? This works, because of 3rd Maxwell equations". The problem is that this is not primed! My starting hypothesis where the primed Maxwell equations: the not primed Maxwell equations are what I'm trying to proof. If in some way you can proof that the longitudinal component of primed 3rd Maxwell equation transform correctly in not primed one, then the proof would work, but if you try to transform $\nabla
' \times \mathbf
{E}
' = -\frac
{\partial \mathbf
{B}
'}
{\partial t
'} $ you will find $\nabla
\times \mathbf
{E} = -\frac
{\partial \mathbf
{B}}{\partial t} $ only if you suppose $\nabla
\cdot
\mathbf
{B}=0$. In other words
- primed 2nd equation became not primed 2nd equation only if longitudinal component of not primed 3rd equation works
- longitudinal component of primed 3rd equation became longitudinal component of not primed 3rd equation only if not primed 2nd equation works
This impasse is frustrating. Doing calculus you can see that things go in a similarly whit the other two equations
- primed 1st equation became not primed 1st equation only if longitudinal component of not primed 4th equation works
- longitudinal component of primed 4th equation became longitudinal component of not primed 4th equation only if 1st equation works
Even more compactly I can write (here numbers refers to Maxwell equations in the order I used above)
- $2'$ to $2$ if $3x$
- $3x'$ to $3x$ if $2$
- $1'$ to $1$ if $4x$
- $4x'$ to $4x$ if $1$
In this way you can see at glance that we are in a no exit street! I could add
- $3y'$ to $3y$ with no problems
- $3z'$ to $3z$ with no problems
- $4y'$ to $4y$ with no problems
- $4z'$ to $4z$ with no problems
Maybe I wouldn't have problems using potential formulation of Maxwell equations? (it doesn't look a prohibitive difficulty, like tensor approach, but I didn't try this way) Anyway reading Resnick "Introduzione alla relatività ristretta" make me think that field formulation should be good for this proof, but he does calculus explicitly only for the $y$ component of $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$, which is one of the special cases in which this proof works! I can't believe that to prove the invariance we need a change in formalism, surely it is possible reach the goal with some cunning trickery I can't see. But which one?
Best Answer
It seems that you've actually proven invariance: Given that the full set of Maxwell's equations are true in one frame, they're true in another frame.
But you don't like that, because you want somehow to do that independently for one of the equations at a time. That's not possible, because it's only the set of Maxwell's equations that together have the invariance property. That's the Really Deep Point of all this: Coulomb's law, or Faraday's law or any other part by itself is a frame-dependent thing. Only with Maxwell's unification (including the displacement term) do you get a unified, invariant set. And that invariance requires special relativity, not Newtonian relativity.