Kaluza–Klein metric

The Kaluza–Klein metric is given by:Witten 81, Equation (3)Duff 1994, Equation (2)Overduin & Wesson 1997, Equation (5)Pope, Equation (1.8)

: $\backslash widetilde\{g\}\_\{ab\}$

:=\begin{bmatrix}

g_{\mu\nu}+\phi^2A_\mu A_\nu & \phi^2A_\mu \\

\phi^2A_\nu & \phi^2

\end{bmatrix}.

Its inverse matrix is given by:

: $\backslash widetilde\{g\}^\{ab\}$

=\begin{bmatrix}

g^{\mu\nu} & -A^\mu \\

-A^\nu & g_{\mu\nu}A^\mu A^\nu+\phi^{-2}

\end{bmatrix}.

Defining an extended gravivector $A\_a=(A\_\backslash mu,1)$ shortens the definition to:

: $\backslash widetilde\{g\}\_\{ab\}$

=\operatorname{diag}(g_{\mu\nu},0)

+\phi^2A_aA_b,

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

• A contraction directly shows the passing from four to five dimensions:
• : $g^\{\backslash mu\backslash nu\}g\_\{\backslash mu\backslash nu\}=4,$
• : $\backslash widetilde\{g\}^\{ab\}\backslash widetilde\{g\}\_\{ab\}=5.$
• If $\backslash mathrm\{d\}s^2$

=g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu is the four-dimensional and $\backslash mathrm\{d\}\backslash widetilde\{s\}^2$

=\widetilde{g}_{ab}\mathrm{d}\widetilde{x}^a\mathrm{d}\widetilde{x}^b is the five-dimensional line element,Duff 1994, Equation (1) then there is the following relation resembling the Lorentz factor from special relativity:Pope, Equation (1.7)

• : $\backslash frac\{\backslash mathrm\{d\}\backslash widetilde\{s\}\}\{\backslash mathrm\{d\}s\}$

=\sqrt{1+\phi^2\left(A_a\frac{\mathrm{d}x^a}{\mathrm{d}s}\right)^2}.

• The determinants $\backslash widetilde\{g\}:=\backslash det(\backslash widetilde\{g\}\_\{ab\})$ and $g:=\backslash det(g\_\{\backslash mu\backslash nu\})$ are connected by:Pope, Equation (1.14)
• : $\backslash widetilde\{g\}$

=\phi^2g

\Leftrightarrow

\sqrt{-\widetilde{g}}

=\phi\sqrt{-g}.

: Although the above expression $\backslash widetilde\{g\}\_\{ab\}$

=\operatorname{diag}(g_{\mu\nu},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 $\backslash widetilde\{g\}=\backslash phi^2g$, one has:
• : $$

\widetilde{g}^{ab}\partial_c\widetilde{g}_{ab}

=\partial_c\ln(-\widetilde{g})

=\partial_c\ln(-\phi^2g).

• {{cite journal |last=Witten |first=Edward |author-link=Edward Witten |date=1981 |title=Search for a realistic Kaluza–Klein theory |journal=Nuclear Physics B |volume=186 |issue=3 |pages=412–428 |bibcode=1981NuPhB.186..412W |doi=10.1016/0550-3213(81)90021-3}}
• {{cite arxiv |arxiv=hep-th/9410046 |first=M. J. |last=Duff |title=Kaluza-Klein Theory in Perspective |date=1994-10-07 |language=en}}
• {{cite journal |last=Overduin |first=J. M. |author2=Wesson, P. S. |date=1997 |title=Kaluza–Klein Gravity |journal=Physics Reports |volume=283 |issue=5 |pages=303–378 |arxiv=gr-qc/9805018 |bibcode=1997PhR...283..303O |doi=10.1016/S0370-1573(96)00046-4 |s2cid=119087814}}
• {{cite web |last=Pope |first=Chris |title=Kaluza–Klein Theory |url=https://people.tamu.edu/~c-pope/ihplec.pdf}}

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