Editing Kaluza–Klein theory (section)

== Geometric interpretation ==
The Kaluza–Klein theory has a particularly elegant presentation in terms of geometry. In a certain sense, it looks just like ordinary gravity in [[free space]], except that it is phrased in five dimensions instead of four.

=== Einstein equations ===
The equations governing ordinary gravity in free space can be obtained from an [[action (physics)|action]], by applying the [[variational principle]] to a certain [[action (physics)|action]]. Let ''M'' be a ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian manifold]], which may be taken as the [[spacetime]] of [[general relativity]]. If ''g'' is the [[Metric (mathematics)|metric]] on this manifold, one defines the [[action (physics)|action]] ''S''(''g'') as
: <math>S(g) = \int_M R(g) \operatorname{vol}(g),</math>
where ''R''(''g'') is the [[scalar curvature]], and vol(''g'') is the [[volume element]]. By applying the [[variational principle]] to the action
: <math>\frac{\delta S(g)}{\delta g} = 0,</math>
one obtains precisely the [[Einstein equation]]s for free space:
: <math>R_{ij} - \frac{1}{2} g_{ij} R = 0,</math>
where ''R''<sub>''ij''</sub> is the [[Ricci tensor]].

=== Maxwell equations ===
By contrast, the [[Maxwell equation]]s describing [[electromagnetism]] can be understood to be the [[de Rham cohomology|Hodge equations]] of a [[principal bundle|principal U(1)-bundle]] or [[circle bundle]] <math>\pi:P\to M</math> with fiber [[U(1)]].  That is, the [[electromagnetic field]] <math>F</math> is a [[harmonic form|harmonic 2-form]] in the space <math>\Omega^2(M)</math> of differentiable [[2-form]]s on the manifold <math>M</math>. In the absence of charges and currents, the free-field Maxwell equations are
: <math>\mathrm{d}F = 0 \quad\text{and}\quad \mathrm{d}{\star}F = 0,</math>
where <math>\star</math> is the [[Hodge star operator]].

=== Kaluza–Klein geometry ===
To build the Kaluza–Klein theory, one picks an invariant metric on the circle <math>S^1</math> that is the fiber of the U(1)-bundle of electromagnetism. In this discussion, an ''invariant metric'' is simply one that is invariant under rotations of the circle. Suppose that this metric gives the circle a total length <math>\Lambda</math>. One then considers metrics <math>\widehat{g}</math> on the bundle <math>P</math> that are consistent with both the fiber metric, and the metric on the underlying manifold <math>M</math>. The consistency conditions are:
* The projection of <math>\widehat{g}</math> to the [[vertical bundle|vertical subspace]] <math>\operatorname{Vert}_p P \subset T_p P</math> needs to agree with metric on the fiber over a point in the manifold <math>M</math>.
* The projection of <math>\widehat{g}</math> to the [[horizontal bundle|horizontal subspace]] <math>\operatorname{Hor}_p P \subset T_p P</math> of the [[tangent space]] at point <math>p \in P</math> must be isomorphic to the metric <math>g</math> on <math>M</math> at <math>\pi(P)</math>.

The Kaluza–Klein action for such a metric is given by
: <math>S(\widehat{g}) = \int_P R(\widehat{g}) \operatorname{vol}(\widehat{g}).</math>

The scalar curvature, written in components, then expands to
: <math>R(\widehat{g}) = \pi^*\left(R(g) - \frac{\Lambda^2}{2} |F|^2\right),</math>
where <math>\pi^*</math> is the [[pullback (differential geometry)|pullback]] of the fiber bundle projection <math>\pi: P \to M</math>. The connection <math>A</math> on the fiber bundle is related to the electromagnetic field strength as

:<math>\pi^*F = dA.</math>

That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from [[homology (mathematics)|homology]] and specifically, [[K-theory]]. Applying [[Fubini's theorem]] and integrating on the fiber, one gets
: <math>S(\widehat{g}) = \Lambda \int_M \left(R(g) - \frac{1}{\Lambda^2} |F|^2\right) \operatorname{vol}(g).</math>

Varying the action with respect to the component <math>A</math>, one regains the Maxwell equations. Applying the variational principle to the base metric <math>g</math>, one gets the Einstein equations
: <math>R_{ij} - \frac{1}{2} g_{ij} R = \frac{1}{\Lambda^2} T_{ij}</math>
with the [[electromagnetic stress–energy tensor]] being given by
: <math>T^{ij} = F^{ik} F^{jl} g_{kl} - \frac{1}{4} g^{ij} |F|^2.</math>

The original theory identifies <math>\Lambda</math> with the fiber metric <math>g_{55}</math> and allows <math>\Lambda</math> to vary from fiber to fiber. In this case, the coupling between gravity and the electromagnetic field is not constant, but has its own dynamical field, the [[Radion (physics)|radion]].

=== Generalizations ===
In the above, the size of the loop <math>\Lambda</math> acts as a coupling constant between the gravitational field and the electromagnetic field.  If the base manifold is four-dimensional, the Kaluza–Klein manifold ''P'' is five-dimensional. The fifth dimension is a [[compact space]] and is called the '''compact dimension'''. The technique of introducing compact dimensions to obtain a higher-dimensional manifold is referred to as [[compactification (physics)|compactification]]. Compactification does not produce group actions on [[Chirality (physics)|chiral]] [[fermions]] except in very specific cases: the dimension of the total space must be 2 mod 8, and the G-index of the Dirac operator of the compact space must be nonzero.<ref>L. Castellani et al., Supergravity and superstrings, vol.&nbsp;2, ch.&nbsp;V.11.</ref>

The above development generalizes in a more-or-less straightforward fashion to general [[principal G-bundle|principal ''G''-bundles]] for some arbitrary [[Lie group]] ''G'' taking the place of [[U(1)]]. In such a case, the theory is often referred to as a [[Yang–Mills theory]] and is sometimes taken to be synonymous. If the underlying manifold is [[supersymmetric]], the resulting theory is a super-symmetric Yang–Mills theory.