So I know that magnetism can be obtained from the combination of electric fields and special relativity. I am familiar with the way one can derive the magnetic field of a current carrying wire felt by a moving charge from length contraction.
Now: we all know that if a magnet falls through a coil, the magnetic flux through its cross sectional area will change with time and this gives rise to an induced current.
I would like to know if we can understand it solely from relativity, without having to assume that a change in magnetic flux implies an induced electric field.
Let's consider the situation in the frame of reference of the magnet: the magnet sees the coil moving upwards; the coil is made of a conducting material so there are lots of delocalised (free to move) electrons.
There is a stationary magnetic field (produced by the magnet) and the delocalised electrons are moving upwards (with the coil) in it. The magnetic field lines are in the same direction as the upwards velocity of the electrons, so the v $\times$ B term in the Lorentz force is 0.
I tried to summarise it in this diagram (it's very simplistic and not very well drawn, sorry):
How do we explain the induced current (i.e. the flow of electrons) in the coil without referring to Faraday's/Lenz's law?