Talk:Covariant formulation of classical electromagnetism

Finally ...

... an article on EM that at least tries to present it as a coherent whole AND adherent to the even more fundamental priciples of relativity.

I have a couple of suggestions that I might attempt if it is to your liking:

1.) Very early on a short explanation of what "manifestly" means and a mention of the fact that the Maxwell eqns (3-dim versions) ARE Lorentz-covariant, but not manifestly so.

2.) A brief motivation of the field tensor. This would basicaally say that the Lorentz force is an experimental fact. A consequence of this is that the order of some of the definitions would be reversed.

If these very small changes sound good to you, I can post a more detailed suggestion here. It would be very small changes in content, my main source of reasoning would be Landau.

3.) A mantion in the end of the remarkable accuracy of classical EM, the failure(!) of EM to be "perfect", via, e.g. the stability of the atom, concluding with a link to QED. YohanN7 (talk) 01:05, 16 July 2009 (UTC)

Maxwell's equations

In Jackson's book, page 557, Maxwell's equations are stated as follows:

$\partial _{\alpha }F^{\alpha \beta }=\mu _{0}J^{\beta }$ .

The upper indices for tensor F is exact reverse of what's stated in this article. F is an antisymmetric tensor. So, there is a difference in sign. Can someone please explain what's going on?— 老陳 (talk) 07:46, 6 March 2010 (UTC)

I think this and other trivial sign errors were caused by different group of people with different Minkowski sign convention editing the same article. I have tried to make these self-consistent within the article. If you find other problems, please help make corrections.-老陳 (talk) 00:37, 19 April 2010 (UTC)

Still looks like in the first part of the article the metric is (+---) while in the last section it changes to (-+++)...--159.149.107.42 (talk) 14:27, 10 October 2011 (UTC)

This article needs to use a consistent metric so the equations make sense with respect to each other. Otherwise, the metric must be explicitly stated in each section so there's no confusion. Fastman99 (talk) 01:41, 17 June 2013 (UTC)

Well, I think to answer the first post, going from upper indexed tensor to the lower indexed tensor you will get the correct signs as in Jackson using the +--- signature. But, the magnetic stress tensor does look like -+++ was used. I would have to look it up if anyone wants to check on that. There really needs to be a consensus on whether or not +--- was used consistently or not. I recommend using equations elusively from Jackson as well as using the same tensor indexes. — Preceding unsigned comment added by JFrech14 (talk • contribs) 03:35, 26 September 2015 (UTC)

Dubious section on materials

The expression in the section on materials is problematic, because the equations given are not covariant (just because you write things with upper and lower indices doesn't mean it is correct under transformations). To get covariant equations with polarizable materials (where the material itself sets a preferred rest frame), you need an additional velocity-dependent term that couples the electric and magnetic fields. (e.g. this extra coupling term gives rise to the Sagnac effect, not that you'd know it from the WP article on the subject.) If you include this term and define things correctly, then the equations can indeed be covariant (although they simplify in the rest frame of the materials). This is describe e.g. in Landau & Lifshitz, which I don't have in front of me now so I won't try to write the equations from memory.

— Steven G. Johnson (talk) 03:36, 27 March 2010 (UTC)

According to D.J. Griffiths' electrodynamics (3rd edn);

${\mathcal {D}}^{\mu \nu }u_{\nu }=c^{2}\epsilon F^{\mu \nu }u_{\nu }$

${\mathcal {H}}^{\mu \nu }u_{\nu }={\frac {1}{\mu }}G^{\mu \nu }u_{\nu }$

where u = 4-velocity of material, ε and μ are the proper permittivity and permeability of the material (i.e. in rest frame fo material), ${\mathcal {H}}$ is dual to ${\mathcal {D}}$ and G dual to F.

Also, the Ampere-Gauss equation is indeed:

${J^{\mu }}_{\text{free}}=\partial _{\nu }{\mathcal {D}}^{\mu \nu }$

since ${\mathcal {D}}$ is antisymmetric: Griffiths uses the (−+++) Minkowski metric, this article uses (+−−−). Maschen (talk) 17:14, 21 September 2012 (UTC)

Right, the velocity dependence I had recalled is in the constitutive equations. — Steven G. Johnson (talk) 18:44, 21 September 2012 (UTC)

I removed the "dubious" tag, now that sources have been added and not challanged. Maschen (talk) 15:28, 20 October 2012 (UTC)

Units?

Probably stupidly, having just been through the article trying to clean up the units, I now ask:

1. should we include them throughout for every quantity like the first half of the article already does (i.e. for the more complex quantities like the M-P and D-H tensors)?
2. delete all the units already included?
3. convert them to their symbols, and execute #1?

It's not really a big problem as it is, just thought to ask in case... Maschen (talk) 22:02, 4 September 2012 (UTC)

Option 1: I would prefer to include SI units for all quantities. This makes it easier for the reader to relate the variables in this article to quantities with which he is familiar. JRSpriggs (talk) 02:50, 5 September 2012 (UTC)

Option 1 or 3, as per JRSpriggs. — Quondum 05:11, 5 September 2012 (UTC)

Ok - 1 it is (also words may be quicker for a reader to take in). Maschen (talk) 07:11, 5 September 2012 (UTC)

About the capitalized units, I was lazy in simply loading up the article and copy-pasting the titles. Sorry to cause extra work for nothing... Maschen (talk) 08:22, 5 September 2012 (UTC)

All the equations presented are independent of units. You can use seconds or hours or angstroms or lightyears. What you are trying to do is relate what physical quantity is represented. Instead, name it and use that to link to the article on the subject. Kerdek (Tell me if I screw up) 03:09, 26 September 2015 (UTC)

Manifestly covariant?

The concept of "manifestly covariant" (in explicit quotes) in the lead and the article appears to be a very different concept from manifest covariance. In particular, even when restricted to the case of special relativity (i.e. in a flat Minkowski space), the equations provided are not covariant in general. Further unstated restrictions are required, specifically that the coordinate system be straight and uniform/linear (admittedly partially implied/assumed by the retrospective reference to an inertial coordinate system), and that the basis be holomorphic. I think that rather than grossly redefining an existing term (manifestly covariant/manifest covariance) and to do justice to the name of the article, the article should stick to the standard definition. In particular, this would require replacing the partial derivative with the covariant derivative, and removing non-tensors from the equations. The simpler (non-covariant) equations of the more restricted context can readily be obtained from the covariant equations by anyone wishing to do so. Reactions? — Quondum 05:59, 5 September 2012 (UTC)

This was raised in the archive. Sounds like a good plan although I'd best to leave it to those who know better than me... Maschen (talk) 07:11, 5 September 2012 (UTC)

I'll wait for comment from others before tackling this (though I make no suggestion that I am one of "those who know better than" you). I recognise that covariant is less onerous than manifestly covariant, and thus I may get resistance from some saying that some equations are covariant (at least in a holomorphic basis) even if they are not manifestly covariant. I trust that such opposition will be expressed here, prior to me making changes in the direction of covariant expressions (and indeed manifestly covariant expressions). — Quondum 08:12, 5 September 2012 (UTC)

Do not confuse this article (EM in special relativity) with Maxwell's equations in curved spacetime (EM in general relativity). If you need to replace "manifestly covariant" with "invariant under Lorentz transformations" (or some such), then do so. JRSpriggs (talk) 08:55, 5 September 2012 (UTC)

On a related note the section Electromagnetism in general relativity (in this article) reads nearly word-for-word with the Summary section in Maxwell's equations in curved spacetime. It's not a problem to have overlap but would it help to define the boundary between EM in SR (this article) and GR (that article) by simply deleting the section in this article and ensuring the link to Maxwell's equations in curved spacetime? Maschen (talk) 09:02, 5 September 2012 (UTC)

Special relativity does not require that a rectilinear inertial frame be used, though this is used in most treatments. It is merely a special case of general relativity in which the manifold is Minkowski space. Thus this is not really the distinction. I have no problem with an article that confines itself to rectilinear inertial coordinates (as seems to be the case in this article). As JRSpriggs suggests, this might be resolved by a suitable framing in the lead, with my only quibble being about the use of the term "covariant" in the title, and even then it could be argued that it is being used in the restricted framing. The section on general relativity does not directly relate to this, since it is describing a generalization "beyond" the scope of the article. I'll consider a rewording of the lead as suggested. — Quondum 09:44, 5 September 2012 (UTC)

Ok - just thought that the section doesn't really do much for this article, readers will click the link anyway and find all the detail there. Thanks for yours and JRSpriggs' edits. I'll stay out of this article now (aside from very minor linking). Maschen (talk) 10:00, 5 September 2012 (UTC)

You make a good point, and I agree with you: repeating the formulae is a bit over the top here. A simple mention that the equations may be adapted to curved spacetime with the link is all that's needed, or probably only a "see also" link. — Quondum 10:14, 5 September 2012 (UTC)

I've made my change mentioned above. I does strike me that within the restrictions (where the partial derivative and covariant derivative are indistinguishable, and tensor densities are indisinguishable from tensors), the equations do exhibit a form of "manifest covariance", and that this is partly the point of the article (perhaps more hidden after my edit). If this form of "restricted manifest covariance" is defined and explained in an introduction, it may be useful. I wouldn't know whether sources use this kind of terminology, though. — Quondum 10:14, 5 September 2012 (UTC)

I think the terminology used is exactly as JRSpriggs says above (invariance under Lorentz transformations). Your edit doesn't seem to hide that, it's mentioned upfront! Maschen (talk) 10:39, 5 September 2012 (UTC)

I meant that it hides the fact that equations can be written in "restricted manifestly covariant" form – that is to say, that one can tell at a glance that it is Lorentz invariant within the coordinate/basis constraint. There are many ways of writing equations that are Lorentz invariant, where the invariance is rather obscure. Put another way: when replacing a manifestly covariant equation with restricted equivalents (partial for covariant derivative, tensor densities for tensors where desired etc.), the result would be what I'm calling "restricted manifestly covariant". — Quondum 11:13, 5 September 2012 (UTC)

Ah, now that was a masterful touch, inserting "manifestly" in the lead in just the right spot. That pretty much takes care of my reservations. Thanks, JRSpriggs! — Quondum 16:48, 5 September 2012 (UTC)

Covariant form of Maxwell's monopole equations?

See also: talk:magnetic monopole § Covariant form of Maxwell's monopole equations?

Here [1] is a paper (and here [2] are more related) on the covariant form of Maxwell's equations including monopoles (it's not hard to imagine a monopole 4-current and find a second inhomogeneous equation from the Faraday and electric Gauss equations for monopoles, though obviously OR without citations). The equations are:

$\partial _{\mu }F^{\mu \nu }=\beta J_{\mathrm {electric} }^{\nu }$

$\partial _{\mu }\star F^{\mu \nu }={\frac {\beta }{\gamma c}}J_{\mathrm {magnetic} }^{\nu }$

$\star F^{\mu \nu }=G^{\mu \nu }$

in more detail the vector set is:

$\nabla \cdot \mathbf {E} =\alpha \rho _{\mathrm {electric} }$

$\nabla \cdot \mathbf {B} ={\frac {\beta }{\gamma }}\rho _{\mathrm {magnetic} }$

$\nabla \times \mathbf {E} +\gamma {\frac {\partial \mathbf {B} }{\partial t}}=-\beta \mathbf {J} _{\mathrm {magnetic} }$

$\nabla \times \mathbf {B} -{\frac {\beta }{\alpha }}{\frac {\partial \mathbf {E} }{\partial t}}=\beta \mathbf {J} _{\mathrm {electric} }$

where:

Units	α	β	γ
SI	1/ε0	μ0	1
Gaussian	4π	4π/c	1/c
Heaviside-Lorentz	1	1/c	1/c

Any objections to inclusion (aside from those who think monopoles are "impossible"!)? Of course we can change the notation for α, β, γ to something less confusing with notation for the Lorentz factor... Maschen (talk) 20:33, 21 September 2012 (UTC)

I think that this should be in an article on magnetic monopoles, not here. JRSpriggs (talk) 06:09, 22 September 2012 (UTC)

I'll take it to talk:Magnetic monopole. It was placed here because people at the monopole article would probably point to here... Maschen (talk) 06:41, 22 September 2012 (UTC)

Four-gradient index conventions

In this article the four-gradient is introduced as $\partial ^{\nu }={\frac {\partial }{\partial x_{\nu }}}$, but then written as $\partial _{\nu }$, which leads to strange-looking equations like ${{{\partial }_{\sigma }}{{F}^{\mu \nu }}+{{\partial }_{\mu }}{{F}^{\nu \sigma }}+{{\partial }_{\nu }}{{F}^{\sigma \mu }}=0}$ ... --88.68.132.83 (talk) 15:48, 30 March 2021 (UTC)

That's because the four-gradient can be covariant or contravariant like any tensor. The next expression is simple wrong in that expression the variance of the derivatives and the tensor needs to be the same. 201.185.160.227 (talk) 00:56, 29 December 2022 (UTC)

Lorentz tensors in a Polarisable Medium

Reference 5 is not a reference at all. More importantly, it claims that there is no Lorenz covariance in a polarisable medium. This must be a mistake. I will remove it in a day or two unless valid counter arguments are produced. Aoosten (talk) 12:26, 7 April 2024 (UTC)