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EM+GR equiv. 5D?

  1. Nov 23, 2005 #1
    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 someting 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...
     
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  3. Nov 23, 2005 #2

    pervect

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  4. Nov 23, 2005 #3

    hellfire

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    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 this paper.
     
  5. Nov 23, 2005 #4

    samalkhaiat

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    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
     
  6. Nov 25, 2005 #5

    dextercioby

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

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

    [tex] dF=0 [/tex]

    and

    [tex] \delta F=\mu_{0} j [/tex]

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

    Daniel.
     
  7. Nov 25, 2005 #6

    pervect

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    Well, I can understand that much even without fibre bundles, as long as [itex] \delta F [/itex] means [itex] d * F[/itex], 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.
     
  8. Nov 25, 2005 #7
    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...
     
  9. Nov 26, 2005 #8

    robphy

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    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.
     
  10. Nov 26, 2005 #9
  11. Nov 30, 2005 #10

    dextercioby

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

    [tex] \delta= \star d \star [/tex]

    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".
     
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