Covariant macroscopic electromagnetism

[Dale]

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.

 

[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.

 

[Meir Achuz]

Which Wikipedia article?

 

[Dale]

Which Wikipedia article?

This one: https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism#Vacuum

 

[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.

 

[Dale]

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.

 

[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?

 

[robphy]

Check out various works by Fred Hehl:
http://arxiv.org/abs/physics/9907046
http://arxiv.org/abs/physics/0005084
http://arxiv.org/abs/0807.4249
arxiv.org/find/physics/1/au:+Hehl_F/0/1/0/all/0/1
...and his references.

Do a google search for these three:

• van dantzig electromagnetism
• post electromagnetics
• ingarden jamiolkowski electrodynamics

(I am interested in premetric formulations.)

 

[atyy]

I haven't read this, but googling "covariant nonlinear optics" suggests http://dx.doi.org/10.1364/OE.20.008982.

 

[robphy]

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.

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.)

 

[Dale]

Check out various works by Fred Hehl:
http://arxiv.org/abs/physics/9907046
http://arxiv.org/abs/physics/0005084
http://arxiv.org/abs/0807.4249
arxiv.org/find/physics/1/au:+Hehl_F/0/1/0/all/0/1
...and his references.

Do a google search for these three:

• van dantzig electromagnetism
• post electromagnetics
• ingarden jamiolkowski electrodynamics

(I am interested in premetric formulations.)

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.

 

[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.