1 charge $Q$ is moving with velocity $v$, to a lab observer is has a magnetic field, and an electric field with curl due to the changing magnetic field around it

But from the charges perspective, itself is stationary and has no magnetic field which means that it has no curl and simply follows coulombs law.

But apparently curl is an invariant quantity under a lorentz transformation so why is the curl different in both cases?

  • 2
    $\begingroup$ The fields themselves are not invariant under Lorentz transformations, and neither are their curls. $\endgroup$
    – Javier
    Nov 8, 2020 at 17:10

1 Answer 1


"curl is an invariant quantity under a Lorentz transformation" is not correct. It depends on curl of what. More fully, the curl should be thought of as part of a 4-vector type of differential operation which can be written $$ \frac{\partial A^a}{\partial x^b} - \frac{\partial A^b}{\partial x^a} $$ where $A^a$ is a 4-vector. Once you know this you can also learn the effect of a Lorentz transformation on it, but I am guessing you have not learned this area to this extent yet so I won't go into it.

In the case of electromagnetic fields, both the electric and magnetic fields transform from one reference frame to another: they are not invariant. In fact a nice way to find both the electric and the magnetic field around a charge moving at constant velocity is to start from the purely electric field of a stationary charge, and then change reference frame. But of course to use this method you need to already know how the fields change. Google will soon lead you to more information on that if you wish to explore further.

  • $\begingroup$ although what motivated me to do this was a google search for of divergence is invariant under a lorentz transformation as gauss law is derived from the electroSTATIC coulombs law so was wondering what math could prove its valid for changing electric fields. $\endgroup$ Nov 8, 2020 at 19:54
  • $\begingroup$ Can anyone here answer if this logic is correct : in the coulomb gauge of the potential formulation of maxwells equations. for static foeld the divergence is :div e= -(laplace(phi)) and the div of a changing e field is div e = -laplace(phi) - d(div a)/dt . Because in the coulomb gauge( which is equally as valid) it seems that the div e of both a static and changing e field are the same. as the second terms is zero in this gauge. does this mean that gauss law must apply for both a changing field as this is identicle to a static version? $\endgroup$ Nov 8, 2020 at 19:55

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