Editing Kaluza–Klein theory (section)

=== 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]].