Kaluza–Klein–Christoffel Symbol
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In
Kaluza–Klein theory In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to ...
, a unification of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
and
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, 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 In Kaluza–Klein theory, a speculative unification of general relativity and electromagnetism, the five-dimensional Kaluza–Klein–Einstein field equations are created by adding a hypothetical dimension to the four-dimensional Einstein field equ ...
. The Kaluza–Klein–Christoffel symbols are named after
Theodor Kaluza Theodor Franz Eduard Kaluza (; 9 November 1885 – 19 January 1954) was a German mathematician and physicist known for the Kaluza–Klein theory, involving field equations in five-dimensional space-time. His idea that fundamental forces can b ...
,
Oskar Klein Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physics, theoretical physicist. Oskar Klein is known for his work on Kaluza–Klein theory, which is partially named after him. Biography Klein was born ...
and
Elwin Bruno Christoffel Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provid ...
.


Definition

Let \widetilde_ be the Kaluza–Klein metric. The ''Kaluza–Klein–Christoffel symbols'' are given by:Overduin & Wesson 1997, Equation (4) : \widetilde_^c :=\frac\widetilde^(\partial_a\widetilde_+\partial_b\widetilde_-\partial_d\widetilde_).


Properties

* For non-compactified spacetime indices, the Kaluza–Klein–Christoffel symbols don't reduce to the ordinary Christoffel symbols. Instead one has: *: \widetilde_^\sigma =\Gamma_^\sigma -\fracA^\sigma\left(\partial_\mu\left(\phi^2A_\nu\right)+\partial_\nu\left(\phi^2A_\mu\right)\right). * For compactified spacetime indices, the Kaluza–Klein–Christoffel symbols simplify with the cylinder condition: *: \widetilde_^\mu =-\frac\widetilde^\partial_d\widetilde_ =-\frac\widetilde^\partial_\nu\widetilde_ =-\phi g^\partial_\nu\phi, *: \widetilde_^4 =-\frac\widetilde^\partial_d\widetilde_ =-\frac\widetilde^\partial_\nu\widetilde_ =\phi A^\nu\partial_\nu\phi. * Analogous to the ordinary Christoffel symbol, but additionally using the relation \widetilde=\phi^2g,Pope, Equation (1.14) one has: *: \widetilde_^a =\partial_b\ln\sqrt =\partial_b\ln\left(\phi\sqrt\right). *:


Literature

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References

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