[Special Relativity]Questions on Maxwell Equations' Derivation

[genxium]

In Einstein's paper section 6 (I'm reading an English version online: https://www.fourmilab.ch/etexts/einstein/specrel/www/), it's said that one of the Maxwell Equations in frame $K$
$$\frac{1}{c}\frac{\partial X}{\partial t} = \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z}$$, where $<X, Y, Z>$ denotes the vector of the electric force and $<L, M, N>$ that of the magnetic force, can be "transformed" into frame $K'$ which is moving at a constant speed $v$ on the $x-axis$ with respect to $K$,
$$\frac{1}{c}\frac{\partial X}{\partial \tau} = \frac{\partial}{\partial \eta} \{ \beta \cdot (N - \frac{v}{c}Y) \} - \frac{\partial}{\partial \zeta} \{ \beta \cdot (M + \frac{v}{c}Z) \}$$, where $\beta = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

I'm confused by the existence of terms $\frac{\partial Y}{\partial \eta}$ and $\frac{\partial Z}{\partial \zeta}$, because it's derived in section 3 that

$$\tau = \beta \cdot (t - \frac{vx}{c^2})$$
$$\xi = \beta \cdot (x - vt)$$
$$\eta = y$$
$$\zeta = z$$

I can't find a way to apply partial derivative operations to make $\frac{\partial Y}{\partial \eta}$ and $\frac{\partial Z}{\partial \zeta}$ come out. Could anyone give me some tips? Any help is appreciated.

 

[michael879]

I'm not sure where your confusion is.. η=y and ζ=z, as you showed so the two derivatives are just w.r.t. y and z. If your confusion is about how the electric field terms end up on the right side of the equation, I think you're looking in the wrong place. Lorentz transformations don't only occur on the coordinate system, they also transform 4-vectors and 4-tensors. The electromagnetic field strength tensor ($F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$) transforms as a rank 2 4-tensor. So when you switch reference frames you actually need to apply a Lorentz transformation to each index of F, in addition to evaluating F at the transformed coordinates. That is where the mixing of the E and B fields comes from, its not some trick found by manipulating the coordinate system.

 

[genxium]

The electromagnetic field strength tensor ($F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$) transforms as a rank 2 4-tensor. So when you switch reference frames you actually need to apply a Lorentz transformation to each index of F, in addition to evaluating F at the transformed coordinates.

Thanks a lot michael879! I'm afraid I can't understand the tensor part immediately, but it looks like a right way to go after searching for some information about it.

I'm totally new to Special Relativity, your answer is really helpful :p

 

[Orodruin]

If you are new to SR I would suggest starting somewhere different from Einstein's original paper. It is very seldom that the way science has progressed historically is the most pedagogical one. A better place to start would be a modern textbook appropriate to your level of understanding.

 

[michael879]

If you are new to SR I would suggest starting somewhere different from Einstein's original paper. It is very seldom that the way science has progressed historically is the most pedagogical one. A better place to start would be a modern textbook appropriate to your level of understanding.

^ very good advice :P As for tensor's, they're a little strange intuitively at first, but they really are just a generalization of vectors. Once you understand 4-vectors (which are rank 1 tensors) tensors are pretty straight forward.

Also, I would recommend learning and using Einstein notation (also known as tensor notation). It makes understanding special relativity infinitely easier, as 3-vectors just end up being awkward

 

[genxium]

Thank you for the advice @Orodruin :)

Learning SR is really getting me into a depth-first-search like recursion :p

 

[michael879]

Learning SR is really getting me into a depth-first-search like recursion :p

Lmao, I rly like that analogy. I don't think physics ever stops being like that