Editing Kaluza–Klein theory (section)

== Kaluza hypothesis ==
In his 1921 article,<ref name="kal"/> Kaluza established all the elements of the classical five-dimensional theory: the [[Kaluza–Klein metric]], the [[Kaluza–Klein–Einstein field equations]], the equations of motion, the stress–energy tensor, and the cylinder condition. With no [[free parameter]]s, it merely extends general relativity to five dimensions. One starts by hypothesizing a form of the five-dimensional Kaluza–Klein metric <math> \widetilde{g}_{ab}</math>, where Latin indices span five dimensions. Let one also introduce the four-dimensional spacetime metric <math> {g}_{\mu\nu}</math>, where Greek indices span the usual four dimensions of space and time; a 4-vector <math> A^\mu </math> identified with the electromagnetic vector potential; and a scalar field <math>\phi</math>. Then decompose the 5D metric so that the 4D metric is framed by the electromagnetic vector potential, with the scalar field at the fifth diagonal. This can be visualized as
: <math>
\widetilde{g}_{ab} \equiv \begin{bmatrix}
 g_{\mu\nu} + \phi^2 A_\mu A_\nu & \phi^2 A_\mu \\
 \phi^2 A_\nu & \phi^2\end{bmatrix}.
</math>

One can write more precisely
: <math>
\widetilde{g}_{\mu\nu} \equiv g_{\mu\nu} + \phi^2 A_\mu A_\nu, \qquad
\widetilde{g}_{5\nu} \equiv \widetilde{g}_{\nu 5} \equiv \phi^2 A_\nu, \qquad
\widetilde{g}_{55} \equiv \phi^2,
</math>
where the index <math>5</math> indicates the fifth coordinate by convention, even though the first four coordinates are indexed with 0, 1, 2, and 3. The associated inverse metric is
: <math>
\widetilde{g}^{ab} \equiv \begin{bmatrix}
 g^{\mu\nu} & -A^\mu \\
 -A^\nu & g_{\alpha\beta} A^\alpha A^\beta + \frac{1}{\phi^2}
\end{bmatrix}.
</math>

This decomposition is quite general, and all terms are dimensionless. Kaluza then applies the machinery of standard [[general relativity]] to this metric. The field equations are obtained from five-dimensional [[Einstein equations]], and the equations of motion from the five-dimensional geodesic hypothesis. The resulting field equations provide both the equations of general relativity and of electrodynamics; the equations of motion provide the four-dimensional [[geodesic equation]] and the [[Lorentz force law]], and one finds that electric charge is identified with motion in the fifth dimension.

The hypothesis for the metric implies an invariant five-dimensional length element <math>ds</math>:
: <math>
ds^2 \equiv \widetilde{g}_{ab}\,dx^a\,dx^b = g_{\mu\nu}\,dx^\mu\,dx^\nu + \phi^2 (A_\nu\,dx^\nu + dx^5)^2.
</math>