The Circle Fibre: Hopf, the Monopole, and the Third Rung

So far the fifth dimension has been a trivial circle attached to every point of spacetime. It does not have to be trivial. A circle can twist as you move around the base space, and the twisting is exactly what an electromagnetic field strength measures.

The simplest twisted circle

The Hopf fibration writes the three-sphere S³ as a circle bundle over the two-sphere S². Each point of S² has a whole circle of S³ above it, and the circles cannot be chosen consistently over all of S²: the bundle is not a product. It is the basic example of a nontrivial circle bundle, and S³ is also the standard example of a contact three-manifold, with the Hopf circles as the orbits of the Reeb flow of its standard contact form. That second fact is textbook contact geometry. I have not checked that the contact form used in Book 4 Chapter 3 for the dm³ system is this standard one, so the link to that chapter is a pointer for the reader, not a claim.

A monopole made of geometry

Van Dongen (arXiv:gr-qc/0009087, §5.3.1) describes a Kaluza–Klein soliton built from the Euclidean Taub–NUT metric. Its fifth coordinate carries a term of the form dx⁵ + 4m(1 − cos θ)dφ, and the field read off from it is a magnetic monopole. The coefficient (1 − cos θ) is singular at one pole, as for the Dirac string, and the singularity is removed by using two coordinate patches whose fifth coordinates differ by a shift proportional to φ. For that to be consistent the fifth coordinate must be periodic, with period 16πm in his conventions, and the monopole charge is quantized. In his words, the regularity condition forces the fifth dimension to be compact: the bundle is nontrivial, and the fibre has the structure of a circle group U(1).

He adds an important caution. Classically, without the de Broglie relation that Klein used, the monopole says nothing about the quantization of electric charge. Putting in de Broglie's relation gives the Dirac quantization condition and so ties the magnetic monopole to quantized electric charge.

Why this belongs on the ladder

This is the one place where all three pieces meet. The nontrivial circle bundle over a sphere is a Kaluza–Klein space. Its odd-dimensional total space carries a connection one-form η, as in the contact chapter, and in the Hopf case η is the contact form of S³. The ladder's third rung and the fifth Kaluza dimension are two different readings of the same kind of object: a circle bundle over a space with a curvature two-form.

Analogy and shared form, not a claimed shared invariant.