Editing Kaluza–Klein theory (section)

== Field equations from the Kaluza hypothesis ==
The [[Kaluza–Klein–Einstein field equations]] of the five-dimensional theory were never adequately provided by Kaluza or Klein because they ignored the [[scalar field]]. The full Kaluza field equations are generally attributed to Thiry,<ref name="thry"/> who obtained vacuum field equations, although Kaluza<ref name=kal/> originally provided a stress–energy tensor for his theory, and Thiry included a stress–energy tensor in his thesis. But as described by Gonner,<ref name="gon"/> several independent groups worked on the field equations in the 1940s and earlier. Thiry is perhaps best known only because an English translation was provided by Applequist, Chodos, & Freund in their review book.<ref>{{cite book |title=Modern Kaluza–Klein Theories |last=Appelquist |first=Thomas |author2=Chodos, Alan |author3=Freund, Peter G. O. |date=1987 |publisher=Addison–Wesley |location=Menlo Park, Cal. |isbn=978-0-201-09829-7 }}</ref> Applequist et al. also provided an English translation of Kaluza's article. Translations of the three (1946, 1947, 1948) Jordan articles can be found on the [[ResearchGate]] and [[Academia.edu]] archives.<ref name="jor1"/><ref name="jor2"/><ref name="jor3"/> The first correct English-language Kaluza field equations, including the scalar field, were provided by Williams.<ref name=LLW />

To obtain the 5D Kaluza–Klein–Einstein field equations, the 5D [[Kaluza–Klein–Christoffel symbol|Kaluza–Klein–Christoffel symbols]] <math>\widetilde{\Gamma}^a_{bc}</math> are calculated from the 5D Kaluza–Klein metric <math>\widetilde{g}_{ab}</math>, and the 5D [[Kaluza–Klein–Ricci tensor]] <math>\widetilde{R}_{ab}</math> is calculated from the 5D [[Connection (mathematics)|connections]].

The classic results of Thiry and other authors presume the cylinder condition:
: <math>\frac{\partial \widetilde{g}_{ab}}{\partial x^5} = 0.</math>
Without this assumption, the field equations become much more complex, providing many more degrees of freedom that can be identified with various new fields. Paul Wesson and colleagues have pursued relaxation of the cylinder condition to gain extra terms that can be identified with the matter fields,<ref>{{cite book |title=Space–Time–Matter, Modern Kaluza–Klein Theory |last=Wesson |first=Paul S. |date=1999 |publisher=World Scientific |location=Singapore |isbn=978-981-02-3588-8 |url-access=registration |url=https://archive.org/details/spacetimematterm0000wess }}</ref> for which Kaluza<ref name=kal /> otherwise inserted a stress–energy tensor by hand.

It has been an objection to the original Kaluza hypothesis to invoke the fifth dimension only to negate its dynamics. But Thiry argued<ref name=gon/> that the interpretation of the Lorentz force law in terms of a five-dimensional [[geodesic]] militates strongly for a fifth dimension irrespective of the cylinder condition. Most authors have therefore employed the cylinder condition in deriving the field equations. Furthermore, vacuum equations are typically assumed for which
: <math>\widetilde{R}_{ab} = 0,</math>
where
: <math>\widetilde{R}_{ab} \equiv \partial_c \widetilde{\Gamma}^c_{ab} - \partial_b \widetilde{\Gamma}^c_{ca} + \widetilde{\Gamma}^c_{cd}\widetilde{\Gamma}^d_{ab} - \widetilde{\Gamma}^c_{bd}\widetilde{\Gamma}^d_{ac}</math>
and
: <math>\widetilde{\Gamma}^a_{bc} \equiv \frac{1}{2} \widetilde{g}^{ad}(\partial_b \widetilde{g}_{dc} + \partial_c \widetilde{g}_{db} - \partial_d \widetilde{g}_{bc}).</math>
The vacuum field equations obtained in this way by Thiry<ref name=thry/> and Jordan's group<ref name=jor1 /><ref name=jor2 /><ref name=jor3 /> are as follows.

The field equation for <math>\phi</math> is obtained from
: <math>\widetilde{R}_{55} = 0 \Rightarrow \Box \phi = \frac{1}{4} \phi^3 F^{\alpha\beta} F_{\alpha\beta},</math>
where <math>F_{\alpha\beta} \equiv \partial_\alpha A_\beta - \partial_\beta A_\alpha,</math> <math>\Box \equiv g^{\mu\nu} \nabla_\mu \nabla_\nu,</math> and <math>\nabla_\mu</math> is a standard, 4D [[covariant derivative]]. It shows that the electromagnetic field is a source for the [[scalar field]]. Note that the scalar field cannot be set to a constant without constraining the electromagnetic field. The earlier treatments by Kaluza and Klein did not have an adequate description of the scalar field and did not realize the implied constraint on the electromagnetic field by assuming the scalar field to be constant.

The field equation for <math>A^\nu</math> is obtained from
: <math>\widetilde{R}_{5\alpha} = 0 = \frac{1}{2\phi} g^{\beta\mu} \nabla_\mu(\phi^3 F_{\alpha\beta}) - A_\alpha\phi\Box\phi.</math>
It has the form of the vacuum Maxwell equations if the scalar field is constant.

The field equation for the 4D Ricci tensor <math>R_{\mu\nu}</math> is obtained from
: <math>\begin{align}
  \widetilde{R}_{\mu\nu} - \frac{1}{2} \widetilde{g}_{\mu\nu} \widetilde{R} &= 0 \Rightarrow \\
  R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R &= \frac{1}{2} \phi^2 \left(g^{\alpha\beta} F_{\mu\alpha} F_{\nu\beta} - \frac{1}{4} g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta}\right) + \frac{1}{\phi} (\nabla_\mu \nabla_\nu \phi - g_{\mu\nu} \Box\phi),
\end{align}</math>
where <math>R</math> is the standard 4D Ricci scalar.

This equation shows the remarkable result, called the "Kaluza miracle", that the precise form for the [[electromagnetic stress–energy tensor]] emerges from the 5D vacuum equations as a source in the 4D equations: field from the vacuum. This relation allows the definitive identification of <math>A^\mu</math> with the electromagnetic vector potential. Therefore, the field needs to be rescaled with a conversion constant <math>k</math> such that <math>A^\mu \to kA^\mu</math>.

The relation above shows that we must have
: <math>\frac{k^2}{2} = \frac{8\pi G}{c^4} \frac{1}{\mu_0} = \frac{2G}{c^2} 4\pi\epsilon_0,</math>
where <math>G</math> is the [[gravitational constant]], and <math>\mu_0</math> is the [[permeability of free space]]. In the Kaluza theory, the gravitational constant can be understood as an electromagnetic coupling constant in the metric. There is also a stress–energy tensor for the scalar field. The scalar field behaves like a variable gravitational constant, in terms of modulating the coupling of electromagnetic stress–energy to spacetime curvature. The sign of <math>\phi^2</math> in the metric is fixed by correspondence with 4D theory so that electromagnetic energy densities are positive. It is often assumed that the fifth coordinate is spacelike in its signature in the metric.

In the presence of matter, the 5D vacuum condition cannot be assumed. Indeed, Kaluza did not assume it. The full field equations require evaluation of the 5D [[Kaluza–Klein–Einstein tensor]]
: <math>\widetilde{G}_{ab} \equiv \widetilde{R}_{ab} - \frac{1}{2} \widetilde{g}_{ab}\widetilde{R},</math>
as seen in the recovery of the electromagnetic stress–energy tensor above. The 5D curvature tensors are complex, and most English-language reviews contain errors in either <math>\widetilde{G}_{ab}</math> or <math>\widetilde{R}_{ab}</math>, as does the English translation of Thiry.<ref name=thry /> In 2015, a complete set of 5D curvature tensors under the cylinder condition, evaluated using tensor-algebra software, was produced.<ref name="LLW"/>