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To say it right away, I am not an expert in string theory, but I know well General Relativity. So I wonder how the curling up of extra-dimensions which is assumed in many "Kaluza-Klein" like theories (string theory, Randall-Sundrum etc.) is actually created. The extra dimensions in these theories are apparently warped, and therefore their geometry must be characterized by a non-zero curvature (Riemann tensor). According to General Relativity the latter one should be generated by a non-zero energy-momentum tensor which must have its origin in a significant matter Lagrangian $L_{matter}$. $$T_{\mu\nu}= -\frac{2}{\sqrt{-g}}\frac{\delta (L_{matter})}{\delta(g_{\mu\nu})}.$$ This finally would mean that the matter as origin of this warping must be significant in order to produce this warping of extra dimensions on an extremely small scale. Is this conclusion correct or not?

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  • $\begingroup$ It is correct according to the axioms you assume. The quantum theories postulating the extra dimensions are free to have their own axioms and postulates, like some dimensions curled and independent and not contributing to the limit of GR. Another problem is that general relativity has not been definitevely quantized to be relevant to the quantum models. $\endgroup$ – anna v Jun 7 '18 at 10:46
  • $\begingroup$ @John Renie I've just checked on WP that the Gaussian curvature of a 2-dim cylinder is zero, i.e. its Riemann tensor is also zero. No non-zero energy-momentum tensor is necessary. In that respect I follow your argumentation. Nevertheless, it is not intuitive that a cylinder is supposed to have zero curvature. $\endgroup$ – Frederic Thomas Jun 7 '18 at 11:16
  • $\begingroup$ @FredericThomas the confusion between intrinsic and extrinsic curvature is very common. The curvature in GR is intrinsic. $\endgroup$ – John Rennie Jun 7 '18 at 11:43
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The compact extra dimensions are not curved, or more precisely they are not necessarily curved.

As an analogy consider starting with a flat sheet of paper and rolling it up to form a cylinder. The intrinsic curvature is still zero i.e. a flatlander confined to the surface would find that the geometry was still Euclidean. The cylinder looks curved to us because of the way it has been embedded in our 3D space.

The rolling up of the extra spatial dimensions is analogous to this though the surface they form is obviously more complicated than a cylinder. It has long been assumed that they would form a Calabi-Yau manifold, though with the failure to find low energy supersymmetry I believe this is now starting to be questioned. Anyhow the Calabi-Yau manifold can be intrinsically flat just like a cylinder or perhaps a better example would be a six-torus.

This doesn't mean the curvature in the extra dimensions must be zero, just that it can be zero.

As for how the compact dimensions are stabilised, the first proposal I know of for this was the KKLT mechanism.

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It's worth pointing out first that the standard model relies upon exotic geometries - that is if one thinks about them geometrically. Take for instance the strong force. Before it's quantised it's described as an $SU(3)$ bundle over space time. This just means spacetime has an 'internal' geometry that looks like $SU(3)$, and this is a group, the group of special unitary transformations in 3 complex dimensions; this is hard to visualise, but there is a nicer topological description, it's $S^3$, the 3-sphere that is the surface of the 4d solid ball.

So here we already have a picture of spacetime with an exotic geometry. If you can imagine it, a spacetime atom is not 4d, but 7d; and these atoms, unlike real atoms, are not distinct, but continuously change from one point to another. The curvature of the bundle manifests itself as the field strength.

Electromagnetism is similar; in fact it was the direct precursor to the theories of the weak and strong force and so this is not suprising; here, the geometry of the point is $U(1)$. This is topologically the same as a circle; so we need to add another dimension for this. This makes the dimension of a space time point 8d. The weak force adds $SU(2)$, and this is topologically speaking the double cover of $SO(3)$, the rotation group in 3d. This has 3 degrees of freedom. So altogether we have 11 dimensions at each space time point; or another way of saying the same thing, 11 degrees of freedom.

Now, whilst bosonic string theory is 26d; the main contenders, the five superstring theories are 10d; M-theory had 11d. So in these latter theories we are not far of the actual degrees of freedom at a spacetime point.

Now, it's the curvature of the electromagnetism bundle that manifests itself as the electromagnetic field strength; locally, after you choose a basis this splits up into the electric and magnetic fields; a similar story can be said for the other Yang-Mill fields - the weak and strong force.

All this before the theories are actually quantised though.

It's probably worth adding at this point, that it's was the geometrisation of gravity by Einstein that inspired Weyl to introduce a similar scheme for electromagnetism - the gauge principle - and which, over half a century later led to the current geometric picture of gauge theories. The question arises as to whether this geometric picture is anything more than a mathematical convenience that helps physicists and mathematicians grasp and work with the theory better. A case in point is gravity, it posits the curvature of spacetime. We have now direct evidence of that through gravitational lensing. We can actually see the curvature in action.

Since geometry changes in the very large, far from our ordinary and everyday experience right here on this earth, one can then be led to the idea that geometry may also change in the very small. Probably one of the first pieces of evidence that pointed towards this is the discovery of spinors. Michael Atiyah said that in some sense they are 'the square root of geometry'; another piece of evidence, is the Ahranov-Bohm experiment that suggests that it is not the electric or magnetic field that is fundamental but the electromagnetic potential which was originally considered to be a conveniant way of talking physical problems with electromagnetism. This again is an expression of curvature and enters directly into the coupling of the electromagnetic field with the electron in Diracs equation.

The difference with this and string theory is that the extra dimensions are posited as actual space; one can take it from a purely pragmatic point of view, in the way particle theorists take gauge theory and think of it as merely another way to think about the physics in order to get a fully unified picture. After all, that kind of spatial picture has already proven its worth in the Hilbert space of quantum states which is directly inspired by our own 3d Euclidean space.

As to your other question about what generates this curvature. I'm not sure that this is a question that is tackled at all in the theory. The compact dimensions are given by fiat, in the same way we have them in classical field theory underlying the standard model. The main question here is to get a phenomenological my accurate description of the physics we all know and love.

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It seems that the OP is asking for "causality" not technique for compactification.

The proposal that extra dimensions exists takes us down a path that (1) provides degrees of freedom for expressing Gauge fields using differential geometry and (2) leads to a naturalist problem. Namely, what mechanism justifies the compactification. Theorists do not always care to answer that question.

It should be noted that in the earliest days of unification theorists also played with extra dimensions that were not compact, i.e. extra copies of R^1. This accomplishes the task of extra fields. The emergence of a geometric description of the EM field and other fields was attractive but also introduced extra fields that no one knew what to do with. At first they "wished them away". This led to the result that EM is embedded in a higher dimensional geometric theory but we can't really figure out what to make of the "extra fields". Not only were these extra fields disconcerting to theorists but it turns out that "wishing them away" leads to a configuration that is not a solution to the field equations. This is an even bigger problem, inconsistency. Michael Duff (not 100% sure of the name) revived these theories by showing that one can build up a higher dimensional theory that have vanishing extra fields that is consistent with the field equations (a big deal). Using compact extra dimensions and taking the limit as the radius goes to zero makes terms legitimately vanish. This means that the physics we know can be extracted from these theories in some limit. It also means that if those dimensions loosen up we would "see" those extra fields, i.e. they may get excited and have a measurable effect in 4-dim. But nothing explained why these extra spatial degrees of freedom are the way they are, it is an assumed configuration that does not violate logical consistency.

These very same ideas are connected to modern Gauge theory, where one couples an internal symmetry group to the manifold. The difference here is that the manifold of the internal symmetry group is "frozen". It cannot breath, expand or contract. It dies not have its own dynamics. In Kaluza-Klein theories since the extra portions of the manifold are due to the geometry of a larger dimensional space-time, and the extra dimensions can expand and contract according to a higher dimensional version of Einstein's equations. This should allow one to develop a theory that explains "WHY" the dimensions contract and shrink. Whether or not anyone has explicitly developed such a model I cannot say (haven't kept up with the literature on this). But it is possible to explain why some dimensions contract and others expand. Most theorists aren't that interested in such a detail, only in showing that the assumption of contraction is not inconsistent with the field equations.

But as a final note to assume that the extra dimensions are open or closed (a line or plane, or a circle or sphere or other manifold) places some constraint on topology, not geometry. I do not believe it is possible for a material stress energy tensor to actually change topology, i.e. to somehow make an infinite line close up into a circle. What these theories are saying is that if we start with the assumption that the extra dimensions are closed (i.e. a compact manifold) we have a chance of shrinking by some mechanism.

If you're really interested in learning how these techniques work and have evolved over time I recommend:

Introduction to the Theory of Relativity by Peter G. Bergmann

The Dawning of Gauge Theory by Lochlainn O'Raifeartaigh

These may be old books but they are very good. The Appendix of Bergmann's book has a derivation of KK theory with one extra dimension.

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