Editing Kaluza–Klein theory (section)

== Equations of motion from the Kaluza hypothesis ==
The equations of motion are obtained from the five-dimensional geodesic hypothesis<ref name=kal/> in terms of a 5-velocity <math>\widetilde{U}^a \equiv dx^a/ds</math>:
: <math>
\widetilde{U}^b \widetilde{\nabla}_b \widetilde{U}^a = \frac{d\widetilde{U}^a}{ds} + \widetilde{\Gamma}^a_{bc} \widetilde{U}^b \widetilde{U}^c = 0.
</math>

This equation can be recast in several ways, and it has been studied in various forms by authors including Kaluza,<ref name=kal/> Pauli,<ref>{{cite book |last1=Pauli |first1=Wolfgang |title=Theory of Relativity |url=https://archive.org/details/theoryofrelativi00paul |url-access=registration |date=1958 |publisher=Pergamon Press |location=New York |pages=Supplement 23 |edition=translated by George Field}}</ref> Gross & Perry,<ref>{{cite journal |last1=Gross |first1=D. J. |last2=Perry |first2=M. J. |journal=Nucl. Phys. B |date=1983 |volume=226 |issue=1 |pages=29–48 |doi=10.1016/0550-3213(83)90462-5 |title=Magnetic monopoles in Kaluza–Klein theories |bibcode = 1983NuPhB.226...29G }}</ref> Gegenberg & Kunstatter,<ref>{{cite journal |last1=Gegenberg |first1=J. |last2=Kunstatter |title=The motion of charged particles in Kaluza–Klein space–time |first2=G. |journal=Phys. Lett. |date=1984 |volume=106A |issue=9 |page=410 |bibcode=1984PhLA..106..410G |doi=10.1016/0375-9601(84)90980-0}}</ref> and Wesson & Ponce de Leon,<ref>{{cite journal |last1=Wesson |first1=P. S. |last2=Ponce de Leon |first2=J. |title=The equation of motion in Kaluza–Klein cosmology and its implications for astrophysics |journal=Astronomy and Astrophysics |date=1995 |volume=294 |page=1 |bibcode=1995A&A...294....1W }}</ref> but it is instructive to convert it back to the usual 4-dimensional length element <math>c^2\,d\tau^2 \equiv g_{\mu\nu}\,dx^\mu\,dx^\nu</math>, which is related to the 5-dimensional length element <math>ds</math> as given above:
: <math>
ds^2 = c^2\,d\tau^2 + \phi^2 (kA_\nu\,dx^\nu + dx^5)^2.
</math>

Then the 5D geodesic equation can be written<ref>{{cite conference |last=Williams |first=Lance L. |conference=48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 30 July 2012 – 01 August 2012. Atlanta, Georgia |chapter=Physics of the Electromagnetic Control of Spacetime and Gravity |date=2012 |title=Proceedings of 48th AIAA Joint Propulsion Conference |volume=AIAA 2012-3916 |doi=10.2514/6.2012-3916 |isbn=978-1-60086-935-8 |s2cid=122586403 }}</ref> for the spacetime components of the 4-velocity:
:<math>
  U^\nu \equiv \frac{dx^\nu}{d\tau},
</math>
:<math>
  \frac{dU^\nu}{d\tau} + \widetilde{\Gamma}^\mu_{\alpha\beta} U^\alpha U^\beta + 2 \widetilde{\Gamma}^\mu_{5\alpha} U^\alpha U^5 + \widetilde{\Gamma}^\mu_{55} (U^5)^2 + U^\mu \frac{d}{d\tau} \ln \frac{c\,d\tau}{ds} = 0.
</math>

The term quadratic in <math>U^\nu</math> provides the 4D [[Geodesics in general relativity|geodesic equation]] plus some electromagnetic terms:
: <math>
\widetilde{\Gamma}^\mu_{\alpha\beta} = \Gamma^\mu_{\alpha\beta} + \frac{1}{2} g^{\mu\nu} k^2 \phi^2 (A_\alpha F_{\beta\nu} + A_\beta F_{\alpha\nu} - A_\alpha A_\beta \partial_\nu \ln \phi^2).
</math>

The term linear in <math>U^\nu</math> provides the [[Lorentz force law]]:
: <math>
\widetilde{\Gamma}^\mu_{5\alpha} = \frac{1}{2} g^{\mu\nu} k \phi^2 (F_{\alpha\nu} - A_\alpha \partial_\nu \ln \phi^2).
</math>

This is another expression of the "Kaluza miracle". The same hypothesis for the 5D metric that provides electromagnetic stress–energy in the Einstein equations, also provides the Lorentz force law in the equation of motions along with the 4D geodesic equation. Yet correspondence with the Lorentz force law requires that we identify the component of 5-velocity along the fifth dimension with electric charge:
:<math>
kU^5 = k \frac{dx^5}{d\tau} \to \frac{q}{mc},
</math>
where <math>m</math> is particle mass, and <math>q</math> is particle electric charge. Thus electric charge is understood as motion along the fifth dimension. The fact that the Lorentz force law could be understood as a geodesic in five dimensions was to Kaluza a primary motivation for considering the five-dimensional hypothesis, even in the presence of the aesthetically unpleasing cylinder condition.

Yet there is a problem: the term quadratic in <math>U^5</math>,
: <math>
\widetilde{\Gamma}^\mu_{55} = -\frac{1}{2} g^{\mu\alpha} \partial_\alpha \phi^2.
</math>
If there is no gradient in the scalar field, the term quadratic in <math>U^5</math> vanishes. But otherwise the expression above implies
: <math>U^5 \sim c \frac{q/m}{G^{1/2}}.
</math>
For elementary particles, <math>U^5 > 10^{20} c</math>. The term quadratic in <math>U^5</math> should dominate the equation, perhaps in contradiction to experience. This was the main shortfall of the five-dimensional theory as Kaluza saw it,<ref name=kal/> and he gives it some discussion in his original article.{{clarify|date=September 2023}}

The equation of motion for <math>U^5</math> is particularly simple under the cylinder condition. Start with the alternate form of the geodesic equation, written for the covariant 5-velocity:
: <math>
\frac{d\widetilde{U}_a}{ds} = \frac{1}{2} \widetilde{U}^b \widetilde{U}^c \frac{\partial\widetilde{g}_{bc}}{\partial x^a}.
</math>

This means that under the cylinder condition, <math>\widetilde{U}_5</math> is a constant of the five-dimensional motion:
: <math>
\widetilde{U}_5 = \widetilde{g}_{5a} \widetilde{U}^a = \phi^2 \frac{c\,d\tau}{ds} (kA_\nu U^\nu + U^5) = \text{constant}.
</math>