How Does Kaluza-Klein Theory Link Electromagnetism and General Relativity in 5D?

[kleinwolf]

I read only the overview that Kaluza Klein is a 5D theory, in which EM and GR are linked...but can somebody tells me how this is done, since from EM laws, you can, i thought make them covariant by replacing derivatives by covariant ones...(the ones including elements of the metric tensor)...so i think it should be something like : EM energy should be taken into account as itself modifiyng the metric through the GR field equ, but it's not clear (at least for me), how E and B (or the covariant EM tensor), can be put into only one more dimesion...do you know if the corresponding metric in 5D is singular, since I don't understand how you can displace your self in the EM field, which is of another nature than the space-time dimensions...

 

[pervect]

Wikipedia has an exposition - if you understand fibre bundles, that is. (Unfortunately, I don't, at least not yet :-().

http://en.wikipedia.org/wiki/Kaluza-Klein_theory

 

[hellfire]

Kaluza proposed a 5-metric in which the 4-spacetime metric was in the 0-3 part and the electromagnetic vector and scalar potentials were the in the i4 and 4j parts but also in the 0-3 part. The formalism is then the same as for general relativity, but with one dimension more. The action is analogue the Einstein-Hilbert action. However, to get the correct equations for gravity and electromagnetism there must be no dependence of the 5-metric on the 4 coordinate. To explain this Klein proposed the that the 5th dimension was compactified. This is all I can tell you about this. Try pages 13, 14 and 15 of http://arxiv.org/gr-qc/9805018 .

 

[samalkhaiat]

I read only the overview that Kaluza Klein is a 5D theory, in which EM and GR are linked...but can somebody tells me how this is done, since from EM laws, you can, i thought make them covariant by replacing derivatives by covariant ones...(the ones including elements of the metric tensor)...so i think it should be something like : EM energy should be taken into account as itself modifiyng the metric through the GR field equ, but it's not clear (at least for me), how E and B (or the covariant EM tensor), can be put into only one more dimesion...do you know if the corresponding metric in 5D is singular, since I don't understand how you can displace your self in the EM field, which is of another nature than the space-time dimensions...

Read my post (#2) in thread called "Fifth Dimensional physics", in General physics forum, dated 09-29-2005.

Well done hellfire I give you 7 out 10 for your reply.

regards

sam

 

[dextercioby]

Wikipedia has an exposition - if you understand fibre bundles, that is. (Unfortunately, I don't, at least not yet :-().

http://en.wikipedia.org/wiki/Kaluza-Klein_theory

It took me a while (a matter of years ) to realize that ALL in physics must be put in the elegant language of differential geometry, that is bundle theory... But I'm happy i did.

As for Maxwell's equations in vacuum, the most elegant form form them is

dF=0

and

\delta F=\mu_{0} j

,where d is the space-time Cartan exterior differential and \delta is its adjoint which one can prove it to be a codifferential...

Daniel.

 

[pervect]

It took me a while (a matter of years ) to realize that ALL in physics must be put in the elegant language of differential geometry, that is bundle theory... But I'm happy i did.
As for Maxwell's equations in vacuum, the most elegant form form them is
dF=0
and
\delta F=\mu_{0} j
,where d is the space-time Cartan exterior differential and \delta is its adjoint which one can prove it to be a codifferential...
Daniel.

Well, I can understand that much even without fibre bundles, as long as \delta F means d * F, where * is the Hodges dual.

Though frankly I'm more comfortable with covariant derivatives than exterior derivatives.

Unfortunately, I'm not even sure what books/articles I should read if I wanted to understand the Wikipedia article on KK theory (I don't think it would take an extrodinary amount of reading to understand fibre bundles from where I'm at, but I really don't know where to start).

I think there was some stuff on KK theory in MTW at a level I could follow, but I'm not quite sure where it was. Much like the OP, KK theory is on my list of interesting things I want to find out more about sometime.

 

[kleinwolf]

So the extra dimension in KK is a space-like one...but what are degenerate dimension, in a degenerate metric ?...i still have not caught that...because those dimensions do not change the space-time interval on which relativity is based...but it can maybe have influence, because there is one more parameter in the metric elements for example...

 

[robphy]

Although I haven't read through it yet, this looks useful:
http://arxiv.org/abs/hep-ph/9810524
Early History of Gauge Theories and Kaluza-Klein Theories, with a Glance at Recent Developments
Authors: Lochlain O'Raifeartaigh (Dublin Institute for Advanced Studies), Norbert Straumann (University of Zuerich)
Comments: Revised and slightly extended version. Accepted for publication in Reviews of Modern Physics

http://www.iop.org/EJ/abstract/0034-4885/50/9/001
Kaluza-Klein theories
D Bailin et al 1987 Rep. Prog. Phys. 50 1087-1170



I also stumbled upon http://www.vttoth.com/kaluza.htm while googling. I haven't read through it.

 

[masudr]

There is a good chapter in these notes that describes the Kaluza-Klein theory http://www.unine.ch/phys/string/lecturesGR.ps.gz

 

[dextercioby]

On a flat spacetime manifold where one can choose a metric \eta_{\mu\nu}= \mbox{diag} \left(+1, -1, -1, -1\right)

\delta= \star d \star

Daniel.

P.S. A good survey on differential geometry & the em field on a flat spacetime is found in Bjo/rn Felsager's book: "Geometry, particles and fields".