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[Special Relativity]Questions on Maxwell Equations' Derivation

  1. Sep 21, 2014 #1
    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 [itex]K[/itex]
    [tex]\frac{1}{c}\frac{\partial X}{\partial t} = \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z}[/tex], where [itex]<X, Y, Z>[/itex] denotes the vector of the electric force and [itex]<L, M, N>[/itex] that of the magnetic force, can be "transformed" into frame [itex]K'[/itex] which is moving at a constant speed [itex]v[/itex] on the [itex]x-axis[/itex] with respect to [itex]K[/itex],
    [tex]\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) \} [/tex], where [itex]\beta = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}[/itex]

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

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

    I can't find a way to apply partial derivative operations to make [itex]\frac{\partial Y}{\partial \eta}[/itex] and [itex]\frac{\partial Z}{\partial \zeta}[/itex] come out. Could anyone give me some tips? Any help is appreciated.
     
  2. jcsd
  3. Sep 22, 2014 #2
    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 ([itex]F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu[/itex]) 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.
     
  4. Sep 22, 2014 #3
    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
     
  5. Sep 22, 2014 #4

    Orodruin

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    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.
     
  6. Sep 22, 2014 #5
    ^ 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
     
  7. Sep 22, 2014 #6
    Thank you for the advice @Orodruin :)

    Learning SR is really getting me into a depth-first-search like recursion :p
     
  8. Sep 23, 2014 #7
    Lmao, I rly like that analogy. I don't think physics ever stops being like that
     
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