Maxwell's equations in higher dimensions.

[muppet]

I'm not sure if this should go here, or in electrodynamics, or in relativity, but never mind. I'm given to understand that Maxwell's equations are completely compatible with the theory of relativity, and apply over all distance scales. I've also heard of Kaluza-Klein theories in which adding a dimension to space-time results in the equations emerging naturally from the equations of general relativity.
My question is, how do maxwell's equations work in four (or higher)-dimensional space-time when two of them are formulated in terms of curl, which is only defined on R^3?
I guess the same goes for 10+D string theories, although I suspect there the answer has something to do with QFT and the MEs being a classical theory :S

 

[muppet]

Any ideas?

Admins, could this thread be more appropriate in another forum perhaps?

 

[CompuChip]

In special relativity, one defines a tensor
$$F_{\mu\nu} = \left( \begin{matrix} 0 & \frac{-E_x}{c} & \frac{-E_y}{c} & \frac{-E_z}{c} \\ \frac{E_x}{c} & 0 & -B_z & B_y \\ \frac{E_y}{c} &B_z & 0 & -B_x \\ \frac{E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)$$
and then Maxwell's equations can be written as derivatives of this *,
$$\partial_\nu F^{\mu\nu} \propto J^\mu$$
where J is the four-current (the time-component contains the electric charge distribution $\rho$ and the spatial component is the current vector $\vec J$).

In general relativity there is a similar expression, but there the curvature of the space-time has to be taken into account (there are factors of $\sqrt{|g|}$ or something like that, but the idea stays the same).

String theories are currently way beyond me, so I can't tell you anything about that.

*) Disclaimer: this is about the right form, didn't check it so might not be entirely correct. For the exact formula's etc. consult a good book, or perhaps Wikipedia.

 

[muppet]

I'll come back to that when I'm a little further in my analysis in my variables course I think... :P
Thanks CompuChip (and to the admin who moved it!)

 

[robphy]

One could define the (n+1)-dimensional Maxwell Equations independent of dimensionality using the "curl" of a vector-potential: $$\nabla_{[a}A_{b]}$$. In my opinion, this is an appropriate question for this S&GR forum.

If you instead ask how the usual (3+1)-dimensional Maxwell Equations arise out of an (n+1)-dimensional theory, this is not really an S&GR question any more [since some additional foreign structures would have to be postulated]... and is better suited to BSM https://www.physicsforums.com/forumdisplay.php?f=66 .

 

[Stingray]

Generalizing Maxwell's equations is very direct if you write them in the notation of differential forms. The electromagnetic field is then represented by the Faraday tensor. This is a closed 2-form (i.e. $\mathrm{d} F = 0$) satisfying
$$\mathrm{d} \star F = J .$$
Here, the current J is a closed 3-form in the usual 4 dimensional spacetime. It is dual to the 1-form (or covector) current that you're probably more used to.

All of these statements remain remain meaningful when adding dimensions. The only difference is that J becomes a closed (n-1)-form. It is still dual to a covector current, though.

I don't know if this makes any sense to you. If not, you can read up on differential forms at some point. Making Maxwell's equations work in higher dimensions does not require any Kaluza-Klein constructions. It also has nothing to do with QFT.

 

[geoffc]

If you're interested, try the first few chapters of Zwiebach's "First Course in String Theory" which generalizes (among other things) Gauss' law in higher dimensions. Definitely worth looking at.

 

[geoffc]

Sorry - in terms of your question concerning the curl, I believe the mathematical device that you're looking for is the wedge product which reduces to the ordinary curl in three dimension. Try here for a better discussion.

http://en.wikipedia.org/wiki/Exterior_algebra