Line element from Kaluza-Klein for Kids

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  • #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.
 
  • #3
Spinnor
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... There is no a priori reason why that element equals 1 as in your line element.
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!
 
  • #4
Urs Schreiber
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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?
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.
 
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  • #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.
 
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  • #6
Urs Schreiber
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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.
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).
 
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  • #7
Spinnor
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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?
 
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  • #8
Urs Schreiber
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Does string theory then have something like a line element that might in some proper limit look like the line element above.
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##)
 
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  • #9
Spinnor
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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).
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
MathematicalPhysicist
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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.
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.
 
  • #12
Demystifier
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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.
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.
 
  • #13
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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.
 
  • #14
Aufbauwerk 2045
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.
I may be mistaken, but if memory serves, arXiv in the old days did minimal screening. Now it has the "endorsement" system but still no peer review. Does this perhaps make it a kind of "old boys club?"

As far as serious scientists, I think Prof. Winterberg certainly qualifies -- last surviving student of Heisenberg, instrumental in the development of GPS, and a highly regarded expert in nuclear technology and space propulsion. He has been a rather courageous champion of freedom of information in nuclear science. He has published some papers on viXra. These papers may sometimes raise questions about string theory, which I suppose makes them anathema to many. So what? No one is forced to read them. By the way, there is some minimal screening on viXra. It is not a complete free-for-all.

In general, I am against any website being promoted as some kind of ultimate gatekeeper on the Internet. This is an extremely dangerous tendency. Just on that basis alone, I hope more scientists publish on viXra. Beware of those, whether Internet tycoons or distinguished professors, who want to control information access and stifle dissent.
 
  • #15
Aufbauwerk 2045
P.S. here is an interesting article about an objection to the arXiv process. I think it speaks for itself. So I will just conclude by saying that, in my opinion, it would be a horrible thing if scientists accepted the idea that all their important papers had to go through arXiv's moderation process. Not that I'm saying that has happened. Just that it would be horrible.

https://www.nature.com/news/arxiv-rejections-lead-to-spat-over-screening-process-1.19267
 
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