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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 string theory. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.^[1]

The five-dimensional (5D) theory developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Einstein in 1919^[2] and published them in 1921.^[3] Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components. Ten components are identified with the 4D spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called the "radion" or the "dilaton". Correspondingly, the 5D Einstein equations yield the 4D Einstein field equations, the Maxwell equations for the electromagnetic field, and an equation for the scalar field. Kaluza also introduced the "cylinder condition" hypothesis, that no component of the five-dimensional metric depends on the fifth dimension. Without this restriction, terms are introduced that involve derivatives of the fields with respect to the fifth coordinate, and this extra degree of freedom makes the mathematics of the fully variable 5D relativity enormously complex. Standard 4D physics seems to manifest this "cylinder condition" and, along with it, simpler mathematics.

In 1926, Oskar Klein gave Kaluza's classical five-dimensional theory a quantum interpretation,^[4][5] to accord with the then-recent discoveries of Heisenberg and Schrödinger. Klein introduced the hypothesis that the fifth dimension was curled up and microscopic, to explain the cylinder condition. Klein suggested that the geometry of the extra fifth dimension could take the form of a circle, with the radius of 10⁻³⁰ cm. More precisely, the radius of the circular dimension is 23 times the Planck length, which in turn is of the order of 10⁻³³ cm.^[5] Klein also made a contribution to the classical theory by providing a properly normalized 5D metric.^[4] Work continued on the Kaluza field theory during the 1930s by Einstein and colleagues at Princeton.

In the 1940s, the classical theory was completed, and the full field equations including the scalar field were obtained by three independent research groups:^[6] Thiry,^[7][8][9] working in France on his dissertation under Lichnerowicz; Jordan, Ludwig, and Müller in Germany,^{[10][11][12][13][14]} with critical input from Pauli and Fierz; and Scherrer^[15][16][17] working alone in Switzerland. Jordan's work led to the scalar–tensor theory of Brans–Dicke;^[18] Brans and Dicke were apparently unaware of Thiry or Scherrer. The full Kaluza equations under the cylinder condition are quite complex, and most English-language reviews, as well as the English translations of Thiry, contain some errors. The curvature tensors for the complete Kaluza equations were evaluated using tensor-algebra software in 2015,^[19] verifying results of Ferrari^[20] and Coquereaux & Esposito-Farese.^[21] The 5D covariant form of the energy–momentum source terms is treated by Williams.^[22]

Kaluza hypothesis

In his 1921 article,^[3] Kaluza established all the elements of the classical five-dimensional theory: the metric, the field equations, the equations of motion, the stress–energy tensor, and the cylinder condition. With no free parameters, it merely extends general relativity to five dimensions. One starts by hypothesizing a form of the five-dimensional metric ${\displaystyle {\widetilde {g}}_{ab}}$, where Latin indices span five dimensions. Let one also introduce the four-dimensional spacetime metric ${\displaystyle {g}_{\mu \nu }}$, where Greek indices span the usual four dimensions of space and time; a 4-vector ${\displaystyle A^{\mu }}$ identified with the electromagnetic vector potential; and a scalar field ${\displaystyle \phi }$. Then decompose the 5D metric so that the 4D metric is framed by the electromagnetic vector potential, with the scalar field at the fifth diagonal. This can be visualized as

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

One can write more precisely

${\displaystyle {\widetilde {g}}_{\mu \nu }\equiv g_{\mu \nu }+\phi ^{2}A_{\mu }A_{\nu },\qquad {\widetilde {g}}_{5\nu }\equiv {\widetilde {g}}_{\nu 5}\equiv \phi ^{2}A_{\nu },\qquad {\widetilde {g}}_{55}\equiv \phi ^{2},}$

where the index ${\displaystyle 5}$ indicates the fifth coordinate by convention, even though the first four coordinates are indexed with 0, 1, 2, and 3. The associated inverse metric is

${\displaystyle {\widetilde {g}}^{ab}\equiv {\begin{bmatrix}g^{\mu \nu }&-A^{\mu }\\-A^{\nu }&g_{\alpha \beta }A^{\alpha }A^{\beta }+{\frac {1}{\phi ^{2}}}\end{bmatrix}}.}$

This decomposition is quite general, and all terms are dimensionless. Kaluza then applies the machinery of standard general relativity to this metric. The field equations are obtained from five-dimensional Einstein equations, and the equations of motion from the five-dimensional geodesic hypothesis. The resulting field equations provide both the equations of general relativity and of electrodynamics; the equations of motion provide the four-dimensional geodesic equation and the Lorentz force law, and one finds that electric charge is identified with motion in the fifth dimension.

The hypothesis for the metric implies an invariant five-dimensional length element ${\displaystyle ds}$:

${\displaystyle ds^{2}\equiv {\widetilde {g}}_{ab}\,dx^{a}\,dx^{b}=g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }+\phi ^{2}(A_{\nu }\,dx^{\nu }+dx^{5})^{2}.}$

Field equations from the Kaluza hypothesis

The field equations of the five-dimensional theory were never adequately provided by Kaluza or Klein because they ignored the scalar field. The full Kaluza field equations are generally attributed to Thiry,^[8] who obtained vacuum field equations, although Kaluza^[3] originally provided a stress–energy tensor for his theory, and Thiry included a stress–energy tensor in his thesis. But as described by Gonner,^[6] several independent groups worked on the field equations in the 1940s and earlier. Thiry is perhaps best known only because an English translation was provided by Applequist, Chodos, & Freund in their review book.^[23] Applequist et al. also provided an English translation of Kaluza's article. Translations of the three (1946, 1947, 1948) Jordan articles can be found on the ResearchGate and Academia.edu archives.^[10][11][13] The first correct English-language Kaluza field equations, including the scalar field, were provided by Williams.^[19]

To obtain the 5D field equations, the 5D connections ${\displaystyle {\widetilde {\Gamma }}_{bc}^{a}}$ are calculated from the 5D metric ${\displaystyle {}_{}}$Zdroj:https://en.wikipedia.org?pojem=Kaluza–Klein_theory
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