Faraday's law of induction and special relativity According to Faraday's law of induction, a time-varying magnetic field will induce an electric field. If a magnet is moving with uniform velocity relative to a coil of copper wire, an eddy current will be induced in the wire. To see this phenomenon, we can connect a LED to the wire and should observe that the LED lights up while the magnet slows down.
However, special relativity ensures that casual events must not happen simultaneously (the interval between any 2 non spacelike events must be >= 0 for all inertia frames of reference). We pick the same inertia frame as the coil, so the kinetic energy of the magnet is transferred to the LED. To investigate this further, we make a switch on the wire and it's initially open. Once we found the magnet is approaching the coil, then we turn on the switch, and now the causality flows like this
  Magnet is getting closer to the coil.
  The observer sees the magnet coming.
  The observer turns on the switch.
  The magnet slows down.
  The LED lights up.

Now the problem is this, if the switch remains open at all time, the magnet will just pass through the coil and nothing happens.
How do we send a message to the magnet by closing the switch (by what form and what mechanism)? And what is the speed of this
message? After the magnet receives this message, it returns another message to the coil. But by what form and what mechanism?
And what is the speed of this returned message?
 A: 
How do we send a message to the magnet by closing the switch (by what form and what mechanism)? And what is the speed of this message?

When we close the switch current begins to flow through the coil. This current then produces a retarded potential by $$\vec A ( t,\vec r)=\frac{\mu_0}{4\pi}\int\frac{\vec J(t_r,\vec r')}{|\vec r-\vec r'|}d^3 \vec r'$$ where $t_r=t-\frac{|\vec r-\vec r'|}{c}$ is the retarded time. The use of the retarded time in this equation means that the speed of the message about the current travels at $c$.

After the magnet receives this message, it returns another message to the coil. But by what form and what mechanism? And what is the speed of this returned message?

The magnet is a permanent magnet and its field has already been affecting the loop for some time in advance of throwing the switch. So there is actually not much of a message*. It is important to remember that the energy that drives the current and powers the LED does not come directly from the KE of the magnet. It comes indirectly, through the field of the magnet. So the current and LED are driven due to the magnet's field that is already present before the switch is thrown. It does not need to wait for a message to come back from the magnet.
*There is a small message. As the magnet gradually decelerates there is a small radiation field. This communicates the additional change in the magnetic field due to the deceleration, specifically the difference between the field at the new decelerated velocity vs the field that it would have produced due to continued uniform motion. This message is not what powers the light bulb, and in this scenario actually reduces the power delivered.
