# Trouble with obtaining Ohm's law in proper frame

A naive generalisation of Ohm's law is as follows:

$$J^{\alpha}=\frac{\sigma}{c} F^{\alpha\beta}U_{\beta}$$

It is known that the above will only work in the proper frame of the conductor. However, I do not seem to be able to obtain the expression $$J=\sigma E$$ from the above equation. In the frame of the conductor, choosing the spatial index, I will have

$$J^i=\frac{\sigma}{c}E^i$$

which is off by a factor of $$1/c$$.

I encountered the same problem with the correct generalisation of Ohm's law. Again choosing the spatial index, I will have from

$$J^{\alpha}=\frac{\sigma}{c} F^{\alpha\beta}U_{\beta}-\frac{1}{c^2}\left(U_{\nu}J^{\nu}\right)U^{\alpha}$$

$$J^{i}=\frac{\sigma}{c}E^i$$

which is not Ohm's law in the proper frame. This time round, it is expected since the first part of the RHS is actually the expression in the first equation.

How do I remedy this issue?

• One usually defines $U^\mu= \gamma( c, {\bf v})$, so $U^0=c$ in the proper frame. Does this cancel your unwanted factor of $c$?. – mike stone Mar 27 at 13:01
• It turns out that the mistake was in writing out $F^{i0}$ where I had carelessly included an extra $1/c$ term. – Thomas Mar 27 at 13:21
• @Thomas You are allowed to answer your own question – BioPhysicist Mar 27 at 13:31

It turns out that the mistake was in writing out $$F^{i0}$$ where I had carelessly included an extra 1/c term.