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An example shows that Maxwell Equations do not need Galilean invariance, therefore Lorentz Invariance is not needed for Maxwell Equations. The example: In a train moving at a constant speed relative the earth, there are electrical charges static relative to the train. In the train, an observer can only observe the electrical field, no magnetic field can be observed. Out of the train, on the nearby ground, an observer can observe the electrical field and magnetic field produced by the electrical charges in the train. This example showed that in different related inertia systems, the same electric and magnetic experiments can show different results. These different results are not mere space dimension difference caused by the relative movements of the inertial frames. These difference is physical electromagnetic property difference. Maxwell Equations do not need to have Galilean invariance which is a space dimension invariance. Maxwell's equations need to be different in different related inertial frames in order to correctly describe a same electromagnetic experiment observed in the different inertial frames. Therefore Lorentz Invariance is not needed for Maxwell Equations. For the same reason the Relativity theory is not needed. Because Lorentz transform tries to use space dimension change to mask the physical electromagnetic property difference in Maxwell Equations.

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    $\begingroup$ the maxwell equations have travelling waves as a solution. these waves have a velocity. let's call it $c$ (weird choice). in which reference frame do I observe $c$ as the velocity of this wave? $\endgroup$ – Bort Feb 11 '16 at 13:30
  • $\begingroup$ Whichever frame you choose to apply Maxwell equations. $\endgroup$ – Charlie Jiang Feb 12 '16 at 11:50
  • $\begingroup$ so how are velocities added then? I observe that in my everyday life velocities are fairly additive, but a light pulse in a moving zeppelin (not a train, for extra flavour) moves with c regardless of the velocity. btw: this is my last comment on this matter, meant to give your thinking guidance. For what its worth: 1) equality of c suffices to establish the lorentz transformation 2) the maxwell equations were derived without lorentz invariance but are lorentz invariant. you can treat that as a weird coincidence or try to learn something from it $\endgroup$ – Bort Feb 12 '16 at 12:28
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There's a misunderstanding here. Maxwell equations are not invariant under galilean transformations. Physics needs to be the same in any inertial frame, thus we need to define a new set of transformations. This is why we introduce the Lorentz transformation : it leaves Maxwell's equations the same in any inertial frame. Or, if you want, we need all the terms of the equations we write to be Lorentz covariant.

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  • $\begingroup$ Dimitri: "Physics needs to be the same in any inertial frame", this need is artificial, not required by nature. As you can see from my original example, in different inertial frames the observed difference in not only the space dimension difference caused by the relative movement of the inertial frames. It is electromagnetic property caused physical electromagnetic property difference. Lorentz transformation tries to use space dimension transform to mask the physical electromagnetic property difference, which is not needed and wrong. Same as Relativity theory. $\endgroup$ – Charlie Jiang Feb 12 '16 at 11:25
  • $\begingroup$ This is not an artificial need. What yould be the point of doing physics if no one worked with the same laws ? The fields indeed change from one reference frame to another, but it doesn't alter physics. For instance, the Lorentz force takes the same form in every inertial frame. To convince yourself, you might want to redo the calculations of the first few paragraphs of this article. $\endgroup$ – Dimitri Feb 12 '16 at 13:43

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