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I was first considering to post this in the GR section on the forum, but as I've understood it, compactification is essential in string theory, so I thought that perhaps you guys know the subject better also in Kaluza-Klein theory.
Compactification in Kaluza-Klein theory as I understand it is the statement that all fields become periodic with trespect to the fifth dimension and that the topology of ##M^5## goes to ##M^4\times S^1##.
Klein introduced the idea in order to explain why we did not observe the extra dimension by making the circle 'small' - and since this also implies that the fields on the manifold are periodic, it provides a plausible explanation of the so called 'cylinder condition.. But what other consequences are there of this compactification?
In the following video lecture from Perimeter Institute (http://pirsa.org/displayFlash.php?id=13020069 at about 48 minutes ) the lecturer states that the compactification prevents us from 'gauging away' electromagnetism by a coordinate transformation due to the presence of a preferred frame. But why is this so?
As is noted here:http://staff.science.uva.nl/~jpschaar/report/node12.html
a general (infinitesimal) coordinate transformation ##(x^\mu \to x^\mu + \xi^\mu)## implies that the Kaluza-Klein vector potential changes to ##A_\mu \to A_\mu + \delta A_\mu## where
$$\delta A_\mu = A_\rho (\partial_\mu \xi^\rho) + \xi^\rho(\partial_\rho A_\mu) + \partial_\mu \xi^5$$
which implies a change in the maxwell field strength of ##F_{\mu\nu} \to F_{\mu\nu} + \delta F_{\mu\nu}## where
$$\delta F_{\mu\nu} = (\partial_\mu \xi^\rho) F_{\rho\nu} + (\partial_\nu \xi^\rho) F_{\mu\rho} +\xi^\rho \partial_\rho F_{\mu \nu} $$
Now what is stopping me from picking a ##\xi^\mu## that satisfies
$$F_{\mu\nu} + \delta F_{\mu \nu} = 0$$
where I would have found a frame with no electromagnetism? Is this somehow impossible due to compactification?
.
At the moment I'm also reading a paper on gravitational waves in a Kaluza-Klein space (found here: http://arxiv.org/abs/gr-qc/0411028) where one is perturbing a KK vacuum solution ##j_{AB}##
$$g_{AB} = j_{AB} + h_{AB}$$
The author then states that
"When spontaneous compactification takes place, the universe acquires a kaluza-klein structure (##M^5 \to M^4 \times S^1##) and the 5D local Poincare group is spontaneousy broken into a 4d local Poincare group and a ##U(1)## local gauge group. The wave, originally a 5d object now feels the effect of compactification and it's components transform in a different way under 4d coordinate transformations"
Later the author also states that
"compactification implies that the general covariance is lost and 4d fields contained in the 5d perturbation acquire a different behaviour under 4d coordinate transformations becoming distinct 4-d dynamical fields."
Firs of all, could someone explain the first quote in terms that does not require a lot of group theory? Secondly, if I'm not entirely misunderstanding, it clearly seems like this author claims that the relevant fields get different transformation properties under compactification. This could possibly be related to my first question regarding the transformation properties of the field tensor, but the question is - how and why are these new transformation properties acquired? Do they somehow go from being non-physical to physical after compactification?
Compactification in Kaluza-Klein theory as I understand it is the statement that all fields become periodic with trespect to the fifth dimension and that the topology of ##M^5## goes to ##M^4\times S^1##.
Klein introduced the idea in order to explain why we did not observe the extra dimension by making the circle 'small' - and since this also implies that the fields on the manifold are periodic, it provides a plausible explanation of the so called 'cylinder condition.. But what other consequences are there of this compactification?
In the following video lecture from Perimeter Institute (http://pirsa.org/displayFlash.php?id=13020069 at about 48 minutes ) the lecturer states that the compactification prevents us from 'gauging away' electromagnetism by a coordinate transformation due to the presence of a preferred frame. But why is this so?
As is noted here:http://staff.science.uva.nl/~jpschaar/report/node12.html
a general (infinitesimal) coordinate transformation ##(x^\mu \to x^\mu + \xi^\mu)## implies that the Kaluza-Klein vector potential changes to ##A_\mu \to A_\mu + \delta A_\mu## where
$$\delta A_\mu = A_\rho (\partial_\mu \xi^\rho) + \xi^\rho(\partial_\rho A_\mu) + \partial_\mu \xi^5$$
which implies a change in the maxwell field strength of ##F_{\mu\nu} \to F_{\mu\nu} + \delta F_{\mu\nu}## where
$$\delta F_{\mu\nu} = (\partial_\mu \xi^\rho) F_{\rho\nu} + (\partial_\nu \xi^\rho) F_{\mu\rho} +\xi^\rho \partial_\rho F_{\mu \nu} $$
Now what is stopping me from picking a ##\xi^\mu## that satisfies
$$F_{\mu\nu} + \delta F_{\mu \nu} = 0$$
where I would have found a frame with no electromagnetism? Is this somehow impossible due to compactification?
.
At the moment I'm also reading a paper on gravitational waves in a Kaluza-Klein space (found here: http://arxiv.org/abs/gr-qc/0411028) where one is perturbing a KK vacuum solution ##j_{AB}##
$$g_{AB} = j_{AB} + h_{AB}$$
The author then states that
"When spontaneous compactification takes place, the universe acquires a kaluza-klein structure (##M^5 \to M^4 \times S^1##) and the 5D local Poincare group is spontaneousy broken into a 4d local Poincare group and a ##U(1)## local gauge group. The wave, originally a 5d object now feels the effect of compactification and it's components transform in a different way under 4d coordinate transformations"
Later the author also states that
"compactification implies that the general covariance is lost and 4d fields contained in the 5d perturbation acquire a different behaviour under 4d coordinate transformations becoming distinct 4-d dynamical fields."
Firs of all, could someone explain the first quote in terms that does not require a lot of group theory? Secondly, if I'm not entirely misunderstanding, it clearly seems like this author claims that the relevant fields get different transformation properties under compactification. This could possibly be related to my first question regarding the transformation properties of the field tensor, but the question is - how and why are these new transformation properties acquired? Do they somehow go from being non-physical to physical after compactification?
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