Kaluza–Klein–Riemann curvature tensor

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In Kaluza–Klein theory, a unification of general relativity and electromagnetism, the five-fimensional Kaluza–Klein–Riemann curvature tensor (or Kaluza–Klein–Riemann–Christoffel curvature tensor) is the generalization of the four-dimensional Riemann curvature tensor (or Riemann–Christoffel curvature tensor). Its contraction with itself is the Kaluza–Klein–Ricci tensor, a generalization of the Ricci tensor. Its contraction with the Kaluza–Klein metric is the Kaluza–Klein–Ricci scalar, a generalization of the Ricci scalar.

The Kaluza–Klein–Riemann curvature tensor, Kaluza–Klein–Ricci tensor and scalar are namend after Theodor Kaluza, Oskar Klein, Bernhard Riemann and Gregorio Ricci-Curbastro.

Let ${\displaystyle {\widetilde {g}}_{ab}}$ be the Kaluza–Klein metric and ${\displaystyle {\widetilde {\Gamma }}_{ab}^{c}}$ be the Kaluza–Klein–Christoffel symbols. The Kaluza–Klein–Riemann curvature tensor is given by:

${\displaystyle {\widetilde {R}}_{acb}^{d}:=\partial _{c}{\widetilde {\Gamma }}_{ab}^{d}-\partial _{b}{\widetilde {\Gamma }}_{ac}^{d}+{\widetilde {\Gamma }}_{ce}^{d}{\widetilde {\Gamma }}_{ab}^{e}-{\widetilde {\Gamma }}_{be}^{d}{\widetilde {\Gamma }}_{ac}^{e}.}$

The Kaluza–Klein–Ricci tensor and scalar are given by:^[1]

${\displaystyle {\widetilde {R}}_{ab}:={\widetilde {R}}_{acb}^{c}=\partial _{c}{\widetilde {\Gamma }}_{ab}^{c}-\partial _{b}{\widetilde {\Gamma }}_{ac}^{c}+{\widetilde {\Gamma }}_{cd}^{c}{\widetilde {\Gamma }}_{ab}^{d}-{\widetilde {\Gamma }}_{bd}^{c}{\widetilde {\Gamma }}_{ac}^{d},}$

${\displaystyle {\widetilde {R}}:={\widetilde {g}}^{ab}{\widetilde {R}}_{ab}.}$

• Overduin, J. M.; Wesson, P. S. (1997). "Kaluza–Klein Gravity". Physics Reports. 283 (5): 303–378. arXiv:gr-qc/9805018. Bibcode:1997PhR...283..303O. doi:10.1016/S0370-1573(96)00046-4. S2CID 119087814.