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This has been bugging me for quite a while now. My question is essentially about how one shows that Maxwell equations are invariant under Lorentz transforms. Writing them in index notation, it is usually appealed to that all terms involved are Lorentz tensors (or contractions thereof), and therefore the covariance is manifest. This does, however, assume that we know the transformation properties of the objects in the equation i.e that they are tensorial.

For example, writing the source equations as:

d_u F^(uv) = j^v

we do indeed know that d_u transforms as a covariant 4-vector, directly from the definition of LT. We can also construct an argument to should that j^v (which has as components the charge density and current density) is a contravariant 4-vector, by considering how densities change under LT (i.e. length contraction) etc. However, the question still remains that we need to first show that F^(uv) is a tensor in order to state that Maxwell's equations are indeed invariant under LT. This problem reduces to showing that the E and B fields transform in certain ways, but how does one derive these transformation properties? All I can find in books and on the internet is people deriving the E and B field transformation properties FROM the tensorial nature of F^(uv) itself, which is viciously circular if you are trying to show that Maxwell's equations.

I have found a few specific arguments of people seeming to find these E and B transformation properties for specific physical situations, but nothing too convincing. I looked at Einstein's paper on SR (not the original 1905 one, but one a few years later), and in it he in fact essentially says "Maxwell's equations are invariant under LT if we assume that E and B transform as the following...". I cannot see how, without some other verification (theoretical or experimental) for those transformation laws, one can state that Maxwell's equations are invariant under LT.

Any help with this would be much appreciated - it's been bugging me for so long!

P.S. Please let me know if I should post this in the "Electrodynamics" section instead!

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# Covariance of Maxwell Eqn. - conceptual question

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