In Kaluza–Klein theory, a unification of general relativity and electromagnetism, the five-dimensional Kaluza–Klein metric is the generalization of the four-dimensional metric tensor. It additionally includes a scalar field called graviscalar (or radion) and a vector field called graviphoton (or gravivector), which correspond to hypothetical particles. The Kaluza–Klein metric further leads to the Kaluza–Klein Christoffel symbols, Kaluza–Klein Riemann and Ricci curvature tensor as well as the Kaluza–Klein Einstein field equations.
The Kaluza–Klein metric is named after Theodor Kaluza and Oskar Klein.
Definition The Kaluza–Klein metric is given by:Weinberg 72, Equations (2.1) to (2.3)Witten 81, Equation (3)Duff 1994, Equation (2)Overduin & Wesson 1997, Equation (5)Hossenfelder 03, Equation (4.2)Choquet-Bruhat 08; Chapter XIV, Equations (2.1)-(2.3) and (7.1)-(7.3); Appendix VII, Equation (3.4)Pope, Equation (1.8)
Its inverse matrix is given by:Hossenfelder 03, Equation (4.21)
Defining an extended gravivector shortens the definition to:
which also shows that the radion cannot vanish as this would make the metric singular.
Properties
A contraction directly shows the passing from four to five dimensions:
If is the four-dimensional and is the five-dimensional line element,Duff 1994, Equation (1) then there is the following relation resembling the Lorentz factor from special relativity:Hossenfelder 03, Equation (4.3)Pope, Equation (1.7)
The determinants and are connected by:Pope, Equation (1.14)
Although the above expression fits the structure of the matrix determinant lemma, it cannot be applied since the former term is singular.
Analogous to the metric tensor, but additionally using the above relation , one has:
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.
Definition Let be the Kaluza–Klein metric, which includes a graviphoton (or gravivector) and a graviscalar (or radion) , and be the Kaluza–Klein–Christoffel symbols. The Kaluza–Klein–Riemann curvature tensor is given by:Choquet-Bruhat 08; Appendix VII, Equation (4.1)
The Kaluza–Klein–Ricci tensor and scalar are given by:Overduin & Wesson 1997, Equation (4)Choquet-Bruhat 08; Appendix VII, Equation (5.1)
Both formulas can be related to the ordinary Ricci tensor and Ricci scalar.Choquet-Bruhat 08, Chapter XIV, Equations (3.2)-(3.4) and Chapter XVI, Equations (3.1)-(3.3)
Properties
The Kaluza–Klein Ricci tensor is given by:Choquet-Bruhat 08, Equations (2.16) to (2.18)
Using the inverse Kaluza–Klein metric,Hossenfelder 03, Equation (4.21) the Kaluza–Klein Ricci scalar is given by:
References
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