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Five-dimensional Kaluza-Klein theory is well-known to predict that the electromagnetic field can be described as a curled additional dimension over four-dimensional spacetime. That is, you only need General Relativity, and an additional curled circular dimension to obtain a vector massless field that fits the bill to describe electromagnetism

For instance, if we can describe local KK spacetime as a fiber $U(1) \times M^4$, and if we imagine pictorically the $M^4$ as a hose length, then the $U(1)$ fiber is like the hose thickness

Something that I'm curious is, under this theory, what would be the expected behaviour of simple defects on the topology.

One defect In particular that I want to consider is the hose pictorial image, but joining a new hose leg to an existing hose, basically producing a hose with 3 legs (or a 3-legged pant, if you prefer). In this case, the fiber on top of $M^4$ would be $U(1)$ only in the region far from the defect. Another way to depict this is imagining a T, and our $M^4$ universe is the upper arm of the T.

How would such defect look in our side of the universe? How would the electromagnetic field look near such a defect that significantly alters the simple $U(1)$ topology?

Qmechanic
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diffeomorphism
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    Just being picky: to ensure you just have electromagnetism and gravitation, the dilaton must be set to a constant as well. – JamalS Feb 27 '15 at 20:59
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    Can you write this as $X\times M^4$, where $X$ is something one-dimensional? I'm not really sure why you are calling this hose with three legs (usually known as a pair of pants or the three-holed sphere in the 2D case, by the way) a "defect of the $\mathrm{U}(1)$ topology", otherwise. Could you elaborate? – ACuriousMind Feb 27 '15 at 21:01
  • @CuriousMind Ok, I meant that the $X$ manifold is $U(1)$ far from the defect so that you recover electromagnetism in the asymptotic region. Agreed that 3-legged pant is a better pictorial description which is consistent with other textbooks on cobordism and such. – diffeomorphism Feb 27 '15 at 21:58
  • @JamalS, true.. Although if I remember, the dilaton behaves just as a neutral scalar field? or does it couple to the electromagnetic field? – diffeomorphism Feb 27 '15 at 22:00
  • @diffeomorphism It couples to the electromagentic field. You can see this by going through the computation with the ansatz yourself. I have also done the calculation here: http://physics.stackexchange.com/q/64735/ – JamalS Feb 27 '15 at 22:08

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As I understand it your question implies that the "hose" is at some region in $M^4$, this picture is allowed if you consider your $X\times M^4$ embedded in 6-dimensional space, however it doesn't make sense to say the topology of a loop (1 compact dimension) can turn from one loop into two...

You can have local changes in the radius of this loop, this is due to a non-constant dilaton (fluctuations thereof called "radion"). If this loop is not circular anymore then you loose your $U(1)$ isometry, and with it your massless gauge field.

However let's entertain this possibility, that there is some embedding into 6-dimensional space and the "hose" is splitting. This would physically mean that the 4-dimensional universe is splitting into two in regions of (pre-split) space-time where this split happens. Near this split, the compact extra dimension cannot be circular anymore, so that I would guess that the $U(1)$ gauge field acquires a mass, soon after to loose it once well into the new universe and circularity is resotred.

Note: discussion with ACuriousMind helped refine the answer, see comments.

Ali Moh
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  • The disjoint union of two circles is still a one-dimensional manifold - it's just disconnected, so "it doesn't make sense to say the topology of a loop (1 compact dimension) can turn from one loop into two" is an overstatement - yes, the topology is different, but the disjoint union of two circle is still compact and one-dimensional. Or is there something that forces the 5D space to be globally $X\times M^4$ with the same $X$? – ACuriousMind Feb 28 '15 at 00:02
  • True, mine was not a rigorous mathematical statement.. But what I meant was if you had two disjoint loops, and ask you to give me your 5 dimensional coordinates, how would you do so? you either have to specify two numbers, one for each loop, then it is actually $X\times X\times M^4$. Or you will have to say that you have two copies of $M^4$ universe, where in each copy I can specify one of the two extra coordinates, so to specify them simultaneously I'll have to be talking about two points in two parallel universes so to speak – Ali Moh Feb 28 '15 at 00:28
  • No, two disjoint circles are still a one-dimensional manifold described by exactly one coordinate. Take the coordinate $\theta\in [0,4\pi)$ and say that the values in $[0,2\pi)$ describe one circle (as the obvious angle coordinate $\theta$) and the values in $[2\pi,4\pi)$ describe the other circle (as the angle $\theta-2\pi$, naturally). This is a perfectly good coordinate. Disjoint union dos, in contrast to products, not increase the dimension. (Note that $S^1\times S^1$ is not the union of two circles $S^1 \cup S^1$, but the torus $T^2$, which is truly two-dimensional) – ACuriousMind Feb 28 '15 at 00:31
  • Well then in this case case two is realized. For then what is the physical difference between a point $(t_1,\vec{x}_1,\theta_1)$ and $(t_1,\vec{x}_1,\theta_1 + 2\pi)$? If there is no difference then this picture is redundant. If they are (as they should be) two different points, then we have two parallel universes – Ali Moh Feb 28 '15 at 00:37
  • To be more physically transparent. If all scattering amplitudes are invariant under $\theta\rightarrow\theta+2\pi$ then these two universes are redundant. If some of them is not, then how do you know given your 4-d coordinates which amplitude is realized? you must know if you are in the universe $\theta\in[0,2\pi]$ or in $\theta\in[2\pi,4\pi]$... Of course this is assuming equal radius.. if radii are not equal, then clearly each universe is specified by its dilation VEV – Ali Moh Feb 28 '15 at 00:40
  • I think you should add that (the "physical transparency") to your answer. I had hoped you indeed had an argument why we cannot get such a structure from 5D Kaluza-Klein, but even without this, it's an answer. – ACuriousMind Feb 28 '15 at 00:43
  • "it doesn't make sense to say the topology of a loop (1 compact dimension) can turn from one loop into two". The hose (or pant leg) becomes a T-shaped hose (or a T-shaped pant). The upper _ segment is supposed to be our full $M^4$, and the | of the T is supposed to be a four-dimensional manifold that connects with some compact region of the $M^4$. I'm just trying to get some rough predictions of how electromagnetism would distort near such defect – diffeomorphism Feb 28 '15 at 00:53
  • One thing for sure is that a smooth transition along this junction would definitely destroy the circular isometry which gave a massless gauge boson... maybe it would have an effective position dependent mass? – Ali Moh Feb 28 '15 at 02:21
  • That is to say there will be appreciable interactions between photons and radions – Ali Moh Feb 28 '15 at 02:51
  • If there were objects like these in the universe, how do I detect them? – diffeomorphism Feb 28 '15 at 03:11
  • as in, electromagnetically.. how would the electromagnetic field behave near the defect? – diffeomorphism Mar 01 '15 at 19:11
  • photons would be massive => short range, and v<c.. They would also couple to gravitons in an unconventional way... In other words they no longer resemble photons in any way – Ali Moh Mar 01 '15 at 19:31