I'm trying to understand how one derives the relativistic treatment of the electromagnetic interaction from the classical one and which are the extra postulates made. We can start from Maxwell's equations and the Lorentz force. From the Galilean invariance of Newton's second law of motion [itex]F=\frac{dp}{dt}=q(E+v\times B)[/itex] one can derive (ref) how electric and magnetic field change when transforming to another inertial frame of reference. From this we realize (same ref) that Maxwell's equations don't have the same form in all inertial frames. Therefore the Lorentz force is now Galilean invariant, meaning that it must be described in another space (where inertial frames of reference transform in a different way). But how do we proceed and end up with the relativistic treatment? Let's say we take the Minkowskian geometry of space-time as postulated. Do we need any more postulates to end up at [itex]F=\frac{dp_{rel}}{dt}=q(E+v\times B)[/itex] where [itex]p_{rel}=m\frac{dx}{d\tau}=m\gamma v[/itex] the relativistic momentum instead of the classical [itex]p=m\frac{dx}{dt}=mv[/itex]?(adsbygoogle = window.adsbygoogle || []).push({});

Additionally, can we derive Lorentz invariance of the Maxwell equations from [itex]F=\frac{dp_{rel}}{dt}=q(E+v\times B)[/itex] in the same way as we (same ref) tried showing their Galilean invariance? I'm not sure how to handle the effect of 4D Lorentz transformations on 3D vector fields.

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# How does special relativity follow from classical electrodynamics

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