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.

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.

Definition

Let $\widetilde_$ be the Kaluza–Klein metric and $\widetilde_^c$ be the Kaluza–Klein–Christoffel symbols. The ''Kaluza–Klein–Riemann curvature tensor'' is given by:

: $\widetilde_^d :=\partial_c\widetilde_^d -\partial_b\widetilde_^d +\widetilde_^d\widetilde_^e -\widetilde_^d\widetilde_^e.$

The ''Kaluza–Klein–Ricci tensor'' and ''scalar'' are given by:Overduin & Wesson 1997, Equation (4)

: $\widetilde_ :=\widetilde_^c =\partial_c\widetilde_^c -\partial_b\widetilde_^c +\widetilde_^c\widetilde_^d -\widetilde_^c\widetilde_^d,$
: $\widetilde :=\widetilde^\widetilde_.$

Literature

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References

{{DEFAULTSORT:Kaluza-Klein-Riemann Curvature Tensor
Theories of gravity
Particle physics
Physical cosmology
String theory
Physics beyond the Standard Model