# Covariant macroscopic electromagnetism

1. Aug 29, 2014

### Staff: Mentor

I wondered if anyone had a good online reference on the covariant formulation of Maxwell's macroscopic equations and the other equations of classical electromagnetism?

The wikipedia article talks about constituitive equations in vacuum, which doesn't make a lot of sense to me since M and P are 0 by definition.

2. Aug 29, 2014

### ShayanJ

I had the same problem but I couldn't find the answer in online sources.
I only can suggest you "Classical electromagnetic theory" by Jack Vanderlinde.

3. Aug 29, 2014

### Meir Achuz

Which Wikipedia article?

4. Aug 30, 2014

### Staff: Mentor

5. Aug 30, 2014

### vanhees71

Hm, but it's given just in the next section of the Wikipedia article, referring to Minkowski's model for the linear-response consitutive relations.

6. Aug 30, 2014

### Staff: Mentor

Yes, but I don't want to assume linear constitutive relationships. The Wikipedia treatment is also just too disjointed for me to follow well. I am looking for a more systematic and general covariant presentation.

7. Aug 30, 2014

### ShayanJ

OK, this is from the book I mentioned:
$H_{\mu \nu}$ is the anti-symmetric tensor containing H and D fields. It has the same dimensions as the H field and the D and H components located in it and what are their signs can be derived by analogy with $F_{\mu \nu}$.
$P_{\mu \nu}$ is the anti-symmetric tensor containing M and P fields. It has the same dimensions as the M field and, again, where the D and H components located in it and what are their signs can be derived by analogy with $F_{\mu \nu}$.
Now we have the equations below:
$H_{\mu\nu}=\varepsilon_0c^2F_{\mu\nu}+P_{\mu\nu}$
$\partial_\mu H^{\mu\nu}=J^\nu$
EDIT: Looks like the Wikipedia page contains the things I wrote in this post. So what else do you need DaleSpam?

Last edited: Aug 30, 2014
8. Aug 30, 2014

### robphy

Last edited: Aug 30, 2014
9. Aug 30, 2014

### atyy

10. Aug 31, 2014

### robphy

The covariant constitutive equation relates E with D and B with H, even in the vacuum case (e.g., $\vec D=\epsilon_0 \vec E$).

From a more abstract viewpoint, that example is actually more than a proportionality relation between two vector fields. The vector E is the metric-dual of a one-form, and (in (3+1)-dimensions) the vector D is the metric-dual of the Hodge-dual of a twisted two-form. (The latter is similar to how a cross-product of two vectors in 3D can be thought of as a [pseudo-]vector.)

11. Aug 31, 2014

### Staff: Mentor

Thanks. I am still going through these.

Hehl's "gentle introduction" paper made an interesting comment that I had never considered. After introducing equation 8.2 he says "there is no physical exchange between the bound and the external charges". Since each is separately conserved he is correct, but I guess that means that this approach cannot model something like dielectric breakdown where the bound charge becomes so strong that it turns into a free current.

12. Sep 4, 2014

### DrDu

A covariant definition of the polarization-magnetization 2-form P is
dP=*j
where j is the charge-current form for the microscopic charges in the medium and * is the dual.
Obviously P is defined only up to the differential dA of a 1-form C as ddC=0. In optics one usually choses C so as to make magnetization vanish while in static situations this leads to a divergenge as ω→0.
Note the close analogy to the definition of the vector potential A in terms of the electromagentic field strength
dA=E
where A is also defined only up to a total derivative df of a gauge function f.
Macroscopic equations are obtained by considering the components of P with low wavenumber.
Covariant constituitive relations are obtained when the induced charge-current density j due to E is calculated covariantly.