Kaluza–Klein–Riemann curvature tensor

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. Both are required in the Kaluza–Klein Einstein field equations.

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, which includes a graviphoton (or gravivector) ${\displaystyle A^{a}}$ and a graviscalar (or radion) ${\displaystyle \phi }$, and ${\displaystyle {\widetilde {\Gamma }}_{ab}^{c}}$ be the Kaluza–Klein–Christoffel symbols. The Kaluza–Klein–Riemann curvature tensor is given by:^[1]

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

The Kaluza–Klein–Ricci tensor and scalar are given by:^[2][3]

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

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

Both formulas can be related to the ordinary Ricci tensor and Ricci scalar.^[4]

• The Kaluza–Klein Ricci tensor is given by:^[5]

  ${\displaystyle {\widetilde {R}}_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}f_{4,\nu }^{a}f_{\mu a}^{4}-{\frac {1}{2}}\phi ^{-1}\nabla _{\mu }\nabla _{\nu }\phi +{\frac {1}{4}}\phi ^{-2}\partial _{\mu }\partial _{\nu }\phi ;}$

  ${\displaystyle {\widetilde {R}}_{\mu 4}={\frac {1}{2}}\nabla _{a}f_{4,\mu }^{a}+{\frac {1}{4}}f_{4,\mu }^{a}\phi ^{-1}\partial _{a}\phi ;}$

  ${\displaystyle {\widetilde {R}}_{44}={\frac {1}{2}}(f_{4,ab}f_{4}^{ab}+\phi ^{-1}\partial _{a}\phi \partial ^{a}\phi -2\nabla _{a}\nabla ^{a}\phi ).}$

• Using the inverse Kaluza–Klein metric,^[6] the Kaluza–Klein Ricci scalar is given by:

  ${\displaystyle {\begin{aligned}{\widetilde {R}}={\widetilde {g}}^{ab}{\widetilde {R}}_{ab}&=g^{\mu \nu }{\widetilde {R}}_{\mu \nu }+2g^{\mu 4}{\widetilde {R}}_{\mu 4}+g^{44}{\widetilde {R}}_{44}\\&=R-{\frac {1}{2}}g^{\mu \nu }f_{4,\nu }^{a}f_{\mu 4}^{4}-{\frac {1}{2}}\phi ^{-1}\square _{\nabla }\phi +{\frac {1}{4}}\phi ^{-2}\square _{\partial }\phi -2A^{\mu }{\widetilde {R}}_{\mu 4}+\left(g_{\mu \nu }A^{\mu }A^{\nu }+\phi ^{-2}\right){\widetilde {R}}_{44}.\end{aligned}}}$

• Steven, Weinberg (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (PDF). ISBN 978-0471925675.
• 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.
• Sabine Hossenfelder (2003). Schwarze Löcher in Extra-Dimensionen: Eigenschaften und Nachweis (PDF) (PhD thesis) (in German). Frankfurt am Main. Retrieved 2026-01-18.
• Choquet-Bruhat, Yvonne (2008). General Relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press. ISBN 978-0-19-923072-3. ISSN 0964-9174.