Kaluza–Klein–Christoffel symbol

In Kaluza–Klein theory, a unification of general relativity and electromagnetism, the five-fimensional Kaluza–Klein–Christoffel symbol is the generalization of the four-dimensional Christoffel symbol. They directly appear in the geodesic equations of Kaluza–Klein theory and indirectly through the Kaluza–Klein–Riemann curvature tensor also appear in the Kaluza–Klein–Einstein field equations.

The Kaluza–Klein–Christoffel symbols are named after Theodor Kaluza, Oskar Klein and Elwin Bruno Christoffel.

Definition

Let ${\widetilde {g}}_{ab}$ be the Kaluza–Klein metric. The Kaluza–Klein–Christoffel symbols are given by:^[1]

{\widetilde {\Gamma }}_{ab}^{c}:={\frac {1}{2}}{\widetilde {g}}^{cd}(\partial _{a}{\widetilde {g}}_{bd}+\partial _{b}{\widetilde {g}}_{ad}-\partial _{d}{\widetilde {g}}_{ab}).

For non-compactified spacetime indices, the Kaluza–Klein–Christoffel symbols don't reduce to the ordinary Christoffel symbols. Instead one has:
${\widetilde {\Gamma }}_{\mu \nu }^{\sigma }=\Gamma _{\mu \nu }^{\sigma }-{\frac {1}{2}}A^{\sigma }\left(\partial _{\mu }\left(\phi ^{2}A_{\nu }\right)+\partial _{\nu }\left(\phi ^{2}A_{\mu }\right)\right).$
For compactified spacetime indices, the Kaluza–Klein–Christoffel symbols simplify with the cylinder condition:
${\widetilde {\Gamma }}_{44}^{\mu }=-{\frac {1}{2}}{\widetilde {g}}^{\mu d}\partial _{d}{\widetilde {g}}_{44}=-{\frac {1}{2}}{\widetilde {g}}^{\mu \nu }\partial _{\nu }{\widetilde {g}}_{44}=-\phi g^{\mu \nu }\partial _{\nu }\phi ,$

${\widetilde {\Gamma }}_{44}^{4}=-{\frac {1}{2}}{\widetilde {g}}^{4d}\partial _{d}{\widetilde {g}}_{44}=-{\frac {1}{2}}{\widetilde {g}}^{4\nu }\partial _{\nu }{\widetilde {g}}_{44}=\phi A^{\nu }\partial _{\nu }\phi .$
Analogous to the ordinary Christoffel symbol, but additionally using the relation ${\widetilde {g}}=\phi ^{2}g$ ,^[2] one has:
${\widetilde {\Gamma }}_{ab}^{a}=\partial _{b}\ln {\sqrt {-{\widetilde {g}}}}=\partial _{b}\ln \left(\phi {\sqrt {-g}}\right).$