Maxwell's eqn in invariant form

[mathfeel]

Maxwell's eqn, in invariant form reads:

$$F^{\mu \nu}{}_{;\nu} = J^{\mu}$$

and

$$F_{\alpha \beta ;\gamma} + F_{\beta \gamma ;\alpha}+F_{\gamma \alpha; \beta} = 0$$

Can someone give Maxwell's eqn if there is magnetic charge and current? I do not believe the form (matrix element) of F change, however, if it does, please state that as well.

 

[DW]

Originally posted by mathfeel
Maxwell's eqn, in invariant form reads:

$$F^{\mu \nu}{}_{;\nu} = J^{\mu}$$

and

$$F_{\alpha \beta ;\gamma} + F_{\beta \gamma ;\alpha}+F_{\gamma \alpha; \beta} = 0$$

Can someone give Maxwell's eqn if there is magnetic charge and current? I do not believe the form (matrix element) of F change, however, if it does, please state that as well.

The second can be written in terms of the electromagnetic duel tensor $$D^{\mu \nu}$$ as
$$D^{\mu \nu}{}_{;\nu} = 0$$
Instead of setting that equal to zero try setting it proportional to your hypothetical magnetic four current $$M^\mu$$ like:
$$D^{\mu \nu}{}_{;\nu} = kM^{\mu}$$
(Normally I would explicitely put in the constants determied by your system of units for both sets of equations)
I haven't checked into this, but off the top of my head I think this would work. Of course your next job will be to go out and find a magnetic monopole in order to justify having done this.

 

[mathfeel]

How do you define the dual tensor?

 

[DW]

Originally posted by mathfeel
How do you define the dual tensor?

The electromagnetic duel tensor $$D_{\mu\nu}$$ is related to the electromagnetic tensor $$F^{\mu\nu}$$ and the rank 4 Levi-Civita tensor $$\epsilon_{\alpha\beta\mu\nu}$$ by
$$D_{\mu\nu} = \frac{1}{2}F^{\alpha\beta}\epsilon_{\alpha\beta\mu\nu}$$.