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:^[1][2][3][4]

${\displaystyle {\widetilde {g}}_{ab}:={\begin{bmatrix}g_{\mu \nu }+\phi ^{2}A_{\mu }A_{\nu }&\phi ^{2}A_{\mu }\\\phi ^{2}A_{\nu }&\phi ^{2}\end{bmatrix}}.}$

Its inverse matrix is given by:

${\displaystyle {\widetilde {g}}^{ab}={\begin{bmatrix}g^{\mu \nu }&-A^{\mu }\\-A^{\nu }&g_{\mu \nu }A^{\mu }A^{\nu }+\phi ^{-2}\end{bmatrix}}.}$

Defining an extended gravivector ${\displaystyle A_{a}=(A_{\mu },1)}$ shortens the definition to:

${\displaystyle {\widetilde {g}}_{ab}=\operatorname {diag} (g_{\mu \nu },0)+\phi ^{2}A_{a}A_{b},}$

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

Properties

• A contraction directly shows the passing from four to five dimensions:

  ${\displaystyle g^{\mu \nu }g_{\mu \nu }=4,}$

  ${\displaystyle {\widetilde {g}}^{ab}{\widetilde {g}}_{ab}=5.}$

• If ${\displaystyle \mathrm {d} s^{2}=g_{\mu \nu }\mathrm {d} x^{\mu }\mathrm {d} x^{\nu }}$ is the four-dimensional and ${\displaystyle \mathrm {d} {\widetilde {s}}^{2}={\widetilde {g}}_{ab}\mathrm {d} {\widetilde {x}}^{a}\mathrm {d} {\widetilde {x}}^{b}}$ is the five-dimensional line element,^[5] then there is the following relation resembling the Lorentz factor from special relativity:^[6]

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

• The determinants ${\displaystyle {\widetilde {g}}:=\det({\widetilde {g}}_{ab})}$ and ${\displaystyle g:=\det(g_{\mu \nu })}$ are connected by:^[7]

  ${\displaystyle {\widetilde {g}}=\phi ^{2}g\Leftrightarrow {\sqrt {-{\widetilde {g}}}}=\phi {\sqrt {-g}}.}$

Although the above expression ${\displaystyle {\widetilde {g}}_{ab}=\operatorname {diag} (g_{\mu \nu },0)+\phi ^{2}A_{a}A_{b}}$ 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 ${\displaystyle {\widetilde {g}}=\phi ^{2}g}$,^[7] one has:

  ${\displaystyle {\widetilde {g}}^{ab}\partial _{c}{\widetilde {g}}_{ab}=\partial _{c}\ln(-{\widetilde {g}})=\partial _{c}\ln(-\phi ^{2}g).}$

Literature

• Witten, Edward (1981). "Search for a realistic Kaluza–Klein theory". Nuclear Physics B. 186 (3): 412–428. Bibcode:1981NuPhB.186..412W. doi:10.1016/0550-3213(81)90021-3.
• Duff, M. J. (1994-10-07). "Kaluza-Klein Theory in Perspective". arXiv:hep-th/9410046.
• Overduin, J. M.; Wesson, P. S. (1997). "Kaluza–Klein Gravity". Physics Reports. 283 (5): 303–378. arXiv:gr-qc/9805018. Bibcode:1997PhR...283..303O. doi:10.1016/S0370-1573(96)00046-4. S2CID 119087814.
• Pope, Chris. "Kaluza–Klein Theory" (PDF).