Kaluza–Klein Metric

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 is named after Theodor Kaluza and Oskar Klein.

Definition

The ''Kaluza–Klein metric'' is given by:

: $\widetilde_ :=\begin g_+\phi^2A_\mu A_\nu & \phi^2A_\mu \\ \phi^2A_\nu & \phi^2 \end.$

Its inverse matrix is given by:

: $\widetilde^ =\begin g^ & -A^\mu \\ -A^\nu & g_A^\mu A^\nu+\phi^ \end.$

Defining an extended gravivector $A_a=(A_\mu,1)$ shortens the definition to:

: $\widetilde_ =\operatorname(g_,0) +\phi^2A_aA_b,$

which also shows that the radion $\phi$ cannot vanish as this would make the metric singular.

Properties

* A contraction directly shows the passing from four to five dimensions:
*: $g^g_=4,$
*: $\widetilde^\widetilde_=5.$
* If $\mathrms^2 =g_\mathrmx^\mu\mathrmx^\nu$ is the four-dimensional and $\mathrm\widetilde^2 =\widetilde_\mathrm\widetilde^a\mathrm\widetilde^b$ is the five-dimensional line element, then there is the following relation resembling the Lorentz factor from special relativity:Pope, Equation (1.7)
*: $\frac =\sqrt.$
* The determinants $\widetilde:=\det(\widetilde_)$ and $g:=\det(g_)$ are connected by:Pope, Equation (1.14)
*: $\widetilde =\phi^2g \Leftrightarrow \sqrt =\phi\sqrt.$

: Although the above expression $\widetilde_ =\operatorname(g_,0) +\phi^2A_aA_b$ 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 $\widetilde=\phi^2g$, one has:
*: $\widetilde^\partial_c\widetilde_ =\partial_c\ln(-\widetilde) =\partial_c\ln(-\phi^2g).$

Literature

*
*
*
*

References

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