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In physics, Kaluza–Klein theory (KK theory) is a unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the usual four of space and time. It is considered to be an important precursor to string theory. The five-dimensional theory was developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Einstein in 1919, and published them in 1921. Kaluza's theory was a purely classical extension of general relativity to five dimensions. The 5-dimensional metric has 15 components. Ten components are identified with the 4-dimensional spacetime metric, 4 components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called the "radion" or the "dilaton". Correspondingly, the 5-dimensional Einstein equations yield the 4-dimensional Einstein field equations, the Maxwell equations for the electromagnetic field, and an equation for the scalar field. Kaluza also introduced the hypothesis known as the "cylinder condition", that no component of the 5-dimensional metric depends on the fifth dimension. Without this assumption, the field equations of 5-dimensional relativity are enormously more complex. Standard 4-dimensional physics seems to manifest the cylinder condition. Kaluza also set the scalar field equal to a constant, in which case standard general relativity and electrodynamics are recovered identically. In 1926, Oskar Klein gave Kaluza's classical 5-dimensional theory a quantum interpretation, 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 also calculated a scale for the fifth dimension based on the quantum of charge. It wasn't until the 1940s that the classical theory was completed, and the full field equations including the scalar field were obtained by three independent research groups: Thiry, working in France on his dissertation under Lichnerowicz; Jordan, Ludwig, and Müller in Germany, with critical input from Pauli and Fierz; and Scherrer working alone in Switzerland. Jordan's work led to the scalar-tensor theory of Brans & Dicke; 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 complete Kaluza equations were recently evaluated using tensor algebra software. == The Kaluza Hypothesis == In his 1921 paper,〔 Kaluza established all the elements of the classical 5-dimensional theory: the metric, the field equations, the equations of motion, the stress-energy tensor, and the cylinder condition. The theory has no free parameters; it merely extends general relativity to five dimensions. One starts by hypothesizing a form of the 5-dimensional metric , where roman indices span 5 dimensions. Let us also introduce the 4-dimensional spacetime metric , where Greek indices span the usual 4 dimensions of space and time; a 4-vector which will be identified with the electromagnetic vector potential; and a scalar field . 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: : More precisely, we can write : where the index 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 : So far, 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 5-dimensional Einstein equations, and the equations of motion are obtained from the 5-dimensional geodesic hypothesis. The resulting field equations provide both the equations of general relativity and of electrodynamics; the equations of motion provide the 4-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 5-dimensional length element : : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kaluza–Klein theory」の詳細全文を読む スポンサード リンク
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