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?

• The fields themselves are not invariant under Lorentz transformations, and neither are their curls. Nov 8, 2020 at 17:10

"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.