Kaluza–Klein theory

Remarks

Under a change of coordinate $y \mapsto y + k \lambda(x)$, the one-form changes as $A_\mu \mapsto A_\mu - \partial_\mu \lambda$, which is a gauge transformation of electromagnetism. So $A_\mu$ behaves as an electromagnetic potential, with field strength $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$, the electromagnetic tensor. Example: with $\phi \equiv 1$ and $A = 0$, the metric $g_{\mu\nu} dx^\mu dx^\nu + dy^2$ with $g$ the Minkowski metric is flat, so it solves the five-dimensional vacuum equations. Reducing the five-dimensional Einstein–Hilbert action over $S^1$ gives the four-dimensional Einstein–Hilbert action, the Maxwell action $-\frac14 \phi^3 F_{\mu\nu} F^{\mu\nu}$ (with $k$ chosen suitably) and terms for the scalar $\phi$. Non-example: a metric whose components depend on $y$ fails the cylinder condition; such fields describe the massive Kaluza–Klein modes, whose masses are multiples of $1/r$.