Electrodynamics - relativistic generalization of a formula

[Jerbearrrrrr]

[PLAIN]http://img46.imageshack.us/img46/1699/dsfadsfas.png

The formula holds in the rest frame.
Why is this particular extension 'the' extension?
(Context is Electrodynamics/SR. It's just a piddly 16 lecture course that outlines Electrodynamics and SR and a bit of Quantum phenomena at the end. Ohm's law hasn't been mentioned and I don't think you're expected to know a lot about Ohm's law itself)

[edit]
Oh.
If you use a "taylor series in U" as in,
$$\sigma F^{ab}U_b = J^a + \lambda U^a$$
and solve for lambda, it does turn out as what it's meant to be.

But why is this the correct thing to substitute in? Or is it a naughty question that demands us to be psychic?

 

[gabbagabbahey]

Why not just use the first equation you are given (supposed to show)? In the rest frame of the ohmic medium, you have:

$$\sigma F^{ab}U_{b}=(0,\sigma\textbf{E})=(0,\textbf{j})$$

What do you get when you apply a Lorentz boost to this 4-vector?