Talk:Covariant formulation of classical electromagnetism/Archive 1

Archive 1

Suggestions

I think this needs to be completely reorganized and rewritten with better notation. Time permitting I'll be more specific, but for now let me just say that there are many missed opportunities to explain logical relations between the various expressions which are stated. For example, instead of just writing out the stress-energy tensor, explain how it arises from a well-known machinery which applies in other situations. This can be linked to articles on Noether's theorem in differential equations. It would also be valuable to explain how the Killing vectors of Minkowski spacetime are related to electromagnetic fields, and to seque into discussion of null and non-null fields. ---CH 19:22, 14 April 2006 (UTC)

Yip, totally agree with that; just added the cleanup tag. MP (talk) 16:17, 21 April 2006 (UTC)

Four-vector notation

I was trying to figure out the notation, like for starters, what are the superscripts a and b? So I went to four-vector, where it is explained "for a = 0, 1, 2, 3". Now what the heck does that mean? Do these four small integers have some special significance that someone could just tell us please? Dicklyon 20:39, 12 July 2006 (UTC)

The superscripts are a shorthand labelling convention. So, for example, $U^{a}$ (a=0,1,2,3) means that if you put a=0, you get $U^{0}$, a=1, you get $U^{1}$ etc.... In more detail:

$(U^{a})=(U^{0},U^{1},U^{2},U^{3})$

The 'four-vector' business, just means, roughly, that the $U^{a}$ thingy is a vector with 4 components - $U^{0}$ is the 0th component, $U^{1}$ is the 1st component etc.... Hope that helps. MP (talk) 20:44, 12 July 2006 (UTC)

OK, but then the symbol with superscript can be either a whole 4-vector, or a component, depending on context or what? Seems awkward. Dicklyon 06:09, 13 July 2006 (UTC)

$U^{a}$ for (a=0,1,2,3) is a 4-vector, and $U^{0},U^{1},U^{2},U^{3}$ are it's components. Each component has a single scalar value while the vector contains all four scalar values (see Einstein notation). In relativity, (0,1,2,3) are typically taken to be (t,x,y,z). Because time and space as we know them from day-to-day experience are only a special case in relativity, numerical indices are used for generality. Also, "for (a=t,x,y,z)" just looks rediculous.

Question

If time speeds up, would we even be able to make that observation?

Opinion re tidy up

This page contains some important information not captured elsewhere on Wikipedia. For example, it is the only page I could find which mentions the contravariant and covariant electromagnetic tensor being equal, owing to the form of the Minowski metric. It's also the most compact page describing some other underlying features of the electromagnetic tensor, and has the most explicit description of the relationship between wave equations and Maxwell's equations. I agree that the page is a bit of a mess, but it is useful for teaching purposes, despite its messiness, so certainly shouldn't be deleted or massively reformed without finding more appropriate places for its information.

Concern:

In the Charge conservation paragraph, the 4-current is not balanced dimensionally. The spatial components of the vector have dimension of [Q/s/m^3] and the time component (the 0-component) has dimension of [m.Q/s/m^3]. Something is not right here. —The preceding unsigned comment was added by 72.67.64.205 (talk) 03:19, 3 May 2007 (UTC).

The convention used earlier on this page is that $x^{0}=ct$, not t; this should take care of the problem. --ScottAlanHill 20:59, 16 August 2007 (UTC)

Notation

Please define μ₀. Brews ohare (talk) 21:41, 10 March 2008 (UTC)

Rename?

What would people think of renaming this article Covariant formulation of classical electromagnetism? (Or Manifestly covariant formulation of classical electromagnetism?) First, this article's scope is beyond just Maxwell's equations, since the Lorentz force is included. Second, there are things relevant to the "Formulation of Maxwell's equations in special relativity" that seem not to be in the scope of this article -- in particular, how E and B relativistically transform (explicitly in vector notation, as opposed to implicitly through the rules for F). (Likewise, the Lorentz force is only given in 4-vector notation, not 3-vector.)

Likewise, what would people think of a new article incorporating relativistic transformation rules of E and B and such things in both vector and four-vector notation? It could also have some qualitative stuff about how electric fields in one frame can be magnetic fields in another, and how aspects of magnetism can be derived from aspects of electricity assuming Lorentz invariance. Of course, it would link to this article and any other relevant ones. The perfect title would be Relativistic electromagnetism, but unfortunately that article has defined itself to be a particular pedagogical argument, as opposed to an encyclopedic exposition, so we would need a different title, maybe Electromagnetism and special relativity or something. (PS: Is there already an article like this that I don't know about?) --Steve (talk) 01:49, 29 March 2008 (UTC)

Hi there. I agree on a name change. I think some of the electromagnetism articles need a big overhaul. There seems to be too missing content, lack of direction etc. in many such articles. I created Mathematical descriptions of the electromagnetic field some time ago as a 'store-house' to dump some common mathematical details that were in articles (but which made the articles overly complex). You may find something useful there. Perhaps part of that article can be merged into this one. For example, the relativistic transformation for E and B are there too (using a general Lorentz transformation); no need for a separate article (at least not yet). As for a name change, I think Electromagnetism and special relativity is a good suggestion. Hope this info. is useful. Thanks. MP ^{(talk•contribs)} 10:27, 29 March 2008 (UTC)

I'd like to see transformations expressed in the form of Maxwell's equations#Transformation of fields from Maxwell's equations as well as in four-vector notation, which is pretty opaque to the uninitiated. I'd also like to be able to avoid the Relativistic electromagnetism article, which is pretty flaky in my opinion - mainly intended to tease the imagination, and not really an encyclopedia article. Maybe Relativistic electrodynamics would serve as a title if you want to include Lorentz force? Brews ohare (talk) 11:56, 29 March 2008 (UTC)

Well, I did the rename, and also reorganized this article to better reflect its scope and be easier to read. I ended up switching some things to cgs, not for any real good reason, but because the top of the article said it was in cgs and I was just going along with that. I'd like to see SI and cgs both in the article, maybe as in the Four-potential article. (What do other people think about cgs-versus-SI?) I may also have introduced errors and typos, and anyone who sees one is greatly encouraged to fix it. :-) --Steve (talk) 18:56, 1 April 2008 (UTC)

Done and done. See classical electromagnetism and special relativity. A bit of a mouthful, but might as well be specific. --Steve (talk) 01:33, 10 April 2008 (UTC)

Vacuum vs. materials

What the current page calls "Maxwell's equations" are only Maxwell's equations in vacuum, i.e. the microscopic Maxwell equations. This needs to be made explicit.

Furthermore, one can actually make a nice covariant representation of the full macroscopic Maxwell equations (i.e. including dielectric/magnetic materials, at least for linear materials); e.g. see the presentation in Landau. (These are important, because they automatically give you Doppler shifts, "ether drag", and other stuff, although it's true that the basics of these can be derived without working out the full Maxwell equations. Also, it's nice not to be stuck with vacuum only.)

—Steven G. Johnson (talk) 18:28, 8 September 2008 (UTC)

Some clarifications made.

There were a few points where notation was used before definition, and I've attempted to fix some of that up. There's still an issue with implicit use of the index lowered four vector $x_{\alpha }$ which is used implicitly in the $\partial ^{\alpha }$ operator that defines the tensor. Some shuffling of the content in this article is required to really fix things up so that they are displayed in a logical fashion.

There is also a fair amount of overlap between this article and Electromagnetic_tensor. These articles also do not appear consistent in their definition of the field tensor despite both claiming a -+++ metric (signs of the electric field coordinates are inverted ... see note in the talk page Talk:Electromagnetic_tensor for comments on this). I think it would be reasonable to merge the two articles, so perhaps the shuffling required here for only using terms after definition (or at least close to that) is best deferred to such a merge.

Peeter.joot (talk) 14:48, 9 September 2008 (UTC)

I don't see the case for a merge. Would four-potential be merged into this article too? What about electromagnetic stress-energy tensor? And four-current? And Lorenz gauge condition??

Of course there's overlap, every Wikipedia article overlaps the articles for its sub-topics, super-topics, and related topics. Overlap is not in itself a reason for merging. Both articles have well-defined (and distinct) scopes, purposes, and directions for future expansion/improvement.

Correcting errors in any article is always great. I'm glad you found that inconsistency. If you don't correct it, I will (eventually), but it will take me a while to get to it. :-) --Steve (talk) 19:30, 9 September 2008 (UTC)

I'm not sure enough of myself to correct the tensor inconsistency (I think the other page is wrong or there is a notational subtlety I didn't grasp). I get results consistent with this article in my own calcuations based on trying to reconsile the clifford algebra and tensor notations: http://www.geocities.com/peeter_joot/geometric_algebra/maxwell_to_tensor.pdf . Bo Thide's online book http://www.plasma.uu.se/CED/Book/ also agrees with this article. Peeter.joot (talk) 04:46, 12 September 2008 (UTC)

Covariant or ordinary derivative in GR?

Is the section "In general relativity" consistent with Maxwell's equations in curved spacetime? Shouldn't we be using the covariant derivative, not the ordinary partial derivative? --Michael C. Price ^talk 09:21, 30 September 2008 (UTC)

Try replacing the partial derivatives with covariant derivatives and then expand the expressions. You will find that the extra terms involving the Christoffel symbols all cancel out. In other words, the formulas are invariant as they stand. This is one of the advantages of writing the equations in this form. JRSpriggs (talk) 16:38, 30 September 2008 (UTC)

Only in a torsion free system. I.e. a system without spin. --Michael C. Price ^talk 03:34, 1 October 2008 (UTC)

They do not include torsion, but they are still generally invariant! If you do not believe it, then apply the coordinate transformation and observe that the second derivatives of the coordinates cancel out. JRSpriggs (talk) 16:44, 1 October 2008 (UTC)

I understand that you are concerned about how spin can be coupled to the electromagnetic field. I do not know enough about quantum mechanics to address specifically quantum mechanical issues. However, if one considers Dirac's equation for the electron as a classical field, then one could link them by letting the electric current, $J^{\alpha }\,,$ generated by the electron be a linear combination (see Dirac equation#Dirac bilinears) of: ${\bar {\psi }}\gamma ^{\mu }\psi \,,$ ${\bar {\psi }}\gamma ^{\mu }\gamma ^{5}\psi \,,$ and $\partial _{\nu }({\bar {\psi }}\sigma ^{\mu \nu }\psi )\,.$ The last term may be regarded as the current corresponding to the polarization/magnetization tensor of the electron which is where its spin could enter. JRSpriggs (talk) 13:41, 4 October 2008 (UTC)

signs in Lagrangian

It is written:

${\mathcal {L}}\,=\,{\mathcal {L}}_{\mathrm {field} }+{\mathcal {L}}_{\mathrm {int} }=-{\frac {1}{4\mu _{0}}}F^{\alpha \beta }F_{\alpha \beta }+A_{\alpha }J^{\alpha }\,.$

a personal calculation with this Lagrangian, I get an off by -1 sign error, so I initially came to the conclusion that this should be:

${\mathcal {L}}\,=\,{\mathcal {L}}_{\mathrm {field} }+{\mathcal {L}}_{\mathrm {int} }={\frac {1}{4\mu _{0}}}F^{\alpha \beta }F_{\alpha \beta }+A_{\alpha }J^{\alpha }\,.$

Searching online I find disagreement as well:

http://www.wooster.edu/physics/lindner/Ph377Spring2003HW/HW2.pdf http://quantummechanics.ucsd.edu/ph130a/130_notes/node453.html

If there is a difference in convention that leads to these sorts of differences these should be clarified. —Preceding unsigned comment added by Peeter.joot (talk • contribs) 17:18, 29 December 2008 (UTC)

I do not see any way that the plus sign can be justified. It must be minus. JRSpriggs (talk) 07:02, 30 December 2008 (UTC)

To make it simple to see, ignore relativistic effects including magnetism and the Lagrangian for one charged particle together with the voltage field is

$L={mv^{2} \over 2}-q\phi +{\epsilon _{0} \over 2}(-{\vec {\nabla }}\phi )^{2}\,$

where the last term can be written as

${\epsilon _{0} \over 2}(-{\vec {\nabla }}\phi )^{2}={\epsilon _{0} \over 2}{\vec {E}}^{2}=-{\frac {1}{4\mu _{0}}}F^{\alpha \beta }F_{\alpha \beta }\,.$

OK? JRSpriggs (talk) 16:48, 30 December 2008 (UTC)

I understand the differences now (discussed this in PF: http://www.physicsforums.com/showthread.php?t=281937 ). The sign of the dot product $A^{\mu }J_{\mu }$ is metric dependent. For the time negative metric implied by this article this Lagrangian makes sense. Peeter.joot (talk) 15:19, 1 January 2009 (UTC)