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In the Paper Generalization of Coulomb's law to Maxwell's equations using special relativity by Donald H. Kobe, he tries to derive Maxwell's equations by trying to find covariant laws between tensor objects that satisfy coulombs law. A similar deduction is given in a thesis by one of his students, the paragraph in question begins at equation (43).

His assumptions are that the electric field components are components of a tensor (he doesn't specify which one at this point). An argument from Appendix A shows why this tensor can only be a second rank tensor (which I find to be convincing). Since we know how the electric field behaves when one rotates in space, it follows that the $E$ field components have to be components with only one spatial index: $E_i = F^{i0}$. The next thing he does is look at Gauss' law for the electric field: \begin{align} \partial_i E_i = \frac{4 \pi}{c} j^0 \end{align}

Although he doesn't say it, I interpret it the way he requires the law to hold in all inertial frames.

The part in question comes now:

He then argues that since $j^0$ behaves like the 0-component of a 4-tensor, $\partial_i E_i$ has to behave like a 0-component as well. But $\partial_i E_i = \partial_{\mu}F^{\mu 0} + \partial_0 F^{00}$. Since $\partial_0 F^{00}$ behaves like the 000-Component of a third-rank tensor, he concludes that $\partial_0 F^{00}$ has to be $0$, and that the covariant formulation of the law must be \begin{align} \partial_{\mu} F^{\mu 0} = \frac{4 \pi}{c} j^0 \end{align}

I didn't discover a flaw in this logic, this seems to be a reasonable deduction to me. However - the author of this paper, claims that this generalization is simply "guessing" He writes (page 4):

[...] At crucial points, however, the author performs generalization of Gauss' law to a Lorentz-covariant law, which is a weak point of theoretical arguemntations since generealizing means simply guessing. Similar methods are presented in the papers of Krefetz and Kobe [...]

Is the author right, and the derivation that I showed contains either flawed arguments or additional assumptions that I didn't see?

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Both authors may be right, although Kobe's argument for $E$ being a part of second-rand tensor isn't clear to me and seems artificial. The second author just points out that the first author didn't derive Maxwell's equations from nothing, but that he actually assumes things that are not directly based in experience - such as the Lorentz covariance of the equation $$ \nabla \cdot \mathbf E = \rho/\epsilon_0. $$ This does not follow from Coulomb's law. Hence "guessing".

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  • $\begingroup$ It makes sense to see it like that. Would you agree with my statement, that given the Lorentz covariance of the equation, everything else I stated follows? $\endgroup$ – Quantumwhisp Aug 13 '18 at 20:38
  • $\begingroup$ I am not sure, I do not get Kobe's argument for why $E$ is part of second-rank tensor; he does not seem to use any physical definition of E, besides the Gauss law. The actual standard argument for E,B forming F is much easier to understand: transformation formulae of E and B follow from the Lorentz invariance of the Lorentz force formula and the transformation formulae for 3-force. Then, it can be explicitly shown that E,B transform as 2-nd rank tensor. $\endgroup$ – Ján Lalinský Aug 13 '18 at 22:18
  • $\begingroup$ Do you have an explicit link to this "standard argument" or some way to examine it? $\endgroup$ – Quantumwhisp Aug 14 '18 at 14:44
  • $\begingroup$ Well, I probably overstated that with 'standard'. It is not easy to find an exposition where that is fully done. But the idea is simple and most natural, if one wants to find transformation properties of E,B without any assumption about Maxwell's equations: assume $\mathbf f = q\mathbf E + q\mathbf v \times \mathbf B$ is the same in all frames and use transformation equations for $\mathbf f$ and velocity of test particle $\mathbf v$ to find how $\mathbf E,\mathbf B$ transform. It is a little work but not difficult; the result is the standard transformation equations for E,B. $\endgroup$ – Ján Lalinský Aug 14 '18 at 23:41

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