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I Line element from Kaluza-Klein for Kids

  1. Jul 14, 2017 #1
    In http://vixra.org/abs/1406.0172, the five-dimensional Kaluza-Klein line element d˜s^2 is given by,

    upload_2017-7-14_18-53-32.png

    Does this look correct? Thanks!
     

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  2. jcsd
  3. Jul 15, 2017 #2

    haushofer

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    That depends. You could add an extra scalar in g_55. There is no a priori reason why that element equals 1 as in your line element.
     
  4. Jul 15, 2017 #3
    Thank you. Is there a reasonable argument that one could make to set it to 1 or if you set it to 1 what does that imply? Thanks!
     
  5. Jul 15, 2017 #4

    Urs Schreiber

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    That scalar is famous as the dilaton or radion, because it gives the size of the circle fibers on which the Kaluza-Kein compactification takes place. The key subtlety of Kaluza-Klein theory is in this scalar.

    Namely for Kaluza-Klein compactification to give an effective Einstein-Maxwell theory (gravity coupled to electromagnetism) in 4d from a pure Einstein theory (pure gravity) in 5d, the dilaton must be small and approximately constant. But since in 5d the dilaton is part of the dynamical field of gravity, it generally evolves in time. In fact in pure gravity a KK-compactification with a small dilaton will collapse to a singularity in short time (Penrose 03, section 10.3). This is the reason why, after an initial excitement about KK-theory as a unified field theory in the 1920s, people eventually gave up on it.

    This changed when it was discovered that when gravity is embedded into string theory then the KK-dilaton and similar "moduli" fields may be stable (have approximately constant value) due to extra fields and effects present in the theory. This is called moduli stabilization.
     
  6. Jul 15, 2017 #5

    haushofer

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    A nice playground without having to worry about strings is to compactify 6D GR on a torus or a spherical surface. You can see that if you add a Maxwell field to the theory and electromagnetic flux on the compact space, the moduli of the torus or sphere are stabilized. You'll also see how the topology of the compactified space plays a role in the stabilization.
     
  7. Jul 15, 2017 #6

    Urs Schreiber

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    True, Freund-Rubin-type flux compactifications are the archetype of all moduli stabilization mechanisms. I suppose that 6d case which you have in mind is the popular one due to

    S. Randjbar-Daemi, A. Salam and J. A. Strathdee,
    "Spontaneous Compactification In Six-Dimensional Einstein-Maxwell Theory",
    Nucl. Phys. B 214, 491 (1983).
     
  8. Jul 15, 2017 #7
    Thank you for your help! Does string theory then have something like a line element that might in some proper limit look like the line element above. I know K,K. theory is a dead end but wonder in what way it overlaps, if at all, with better theories such as string theory. Edit, is the overlap of string theory and K.K. theory the idea of adding extra dimensions to the four we know and love?
     
    Last edited: Jul 15, 2017
  9. Jul 15, 2017 #8

    Urs Schreiber

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    You might like the 11d Kaluza-Klein monopole solution to 11d supergravity. Its line element is of the form that you like to see, but for a spatially non-constant value of the dilaton.

    Consider on a manifold of the form ##(\mathbb{R}^{0,1} \times \mathbb{R}^3 \times S^1) \times \mathbb{R}^6## the line element

    $$
    d s_{11}^2 =
    - d t^2
    + (1+\mu/r) d s_{\mathbb{R}^3}^2
    + (1+ \mu/r)^{-1} (d x^{11} - A_i d x^i)^2
    + d s_{\mathbb{R}^6}^2
    \,,
    $$

    where ##\mu## is some positive real constant (called the charge of the KK-monopole) and where ##r## denotes the distance in the ##\mathbb{R}^3##-factor from its origin. This means that the KK-circle is collapsed to zero size at the origin of the ##\mathbb{R}^3##-factor.

    TaubNUT.png

    Here the factor ##\mathbb{R}^{0,1} \times \mathbb{R}^6## is the "worldvolume of the KK-monopole", which from the 10d perspective is the worldvolume of a D6-brane. If we think of this, in turn, as compactified (say wrapping a tiny Calabi-Yau) then the 5d part of the above geometry is

    $$
    d s_{5}^2 =
    - d t^2
    + (1+\mu/r) d s_{\mathbb{R}^3}^2
    + (1+ \mu/r)^{-1} (d x^{11} - A_i d x^i)^2
    \,,
    $$

    Far away from the locus of the monopole, hence for ##r \to \infty## this approaches

    $$
    - d t^2
    + d s_{\mathbb{R}^3}^2
    + (d x^{11} - A_i d x^i)^2
    \,,
    $$

    which is the expression you were after in your first message (for flat spatial metric and with ##x^{11}## denoting what you denoted ##x^5##)
     
  10. Jul 19, 2017 at 7:12 AM #9
    Learning for me takes place a quantum jump at a time. I have Googled much on K.K. theory but your reply and in particular the above caused a quantum jump in my understanding, it makes a bit more sense now. Thanks!
     
  11. Jul 20, 2017 at 3:52 AM #10

    Demystifier

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    viXra? Why is this paper in viXra? viXra is usually considered to be something like a crackpot version of arXiv. The paper above looks too good for viXra.
     
  12. Jul 21, 2017 at 7:45 AM #11

    MathematicalPhysicist

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    There are also crackpots articles in arxiv; nowadays everyone can post something in the net.

    Just because everyone can post to vixra doesn't mean there aren't diamonds in it.
     
  13. Jul 21, 2017 at 7:48 AM #12

    Demystifier

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    Fine, but if a serious scientist can choose to publish either in arXiv or viXra, what would make him to choose viXra and not arXiv? I cannot imagine any good reason.
     
  14. Jul 21, 2017 at 7:53 AM #13

    MathematicalPhysicist

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    I cannot either, perhaps he or she is not affiliated to any university (when you first register to the arxiv you need to write which university you are affiliated with).

    I remember a poster, called Kea (or was it Marni something) that stopped being affiliated to a university.
     
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