Kaluza–Klein theory

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 usual four of space and time and considered an important precursor to string theory. 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 usual four of space and time and considered an important precursor to string theory. The five-dimensional theory developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Einstein in 1919, and published them in 1921, which detailed a purely classical extension of general relativity to five dimensions and includes 15 components. Ten components are identified with the four-dimensional 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 five-dimensional Einstein equations yield the four-dimensional 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 assumption, the field equations of five-dimensional relativity grow enormous in complexity. Standard four-dimensional physics seems to manifest the cylinder condition. In 1926, Oskar Klein gave Kaluza's classical five-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 suggested that the geometry of the extra fifth dimension could take the form of a circle, with the radius of 10−30 cm. Klein also calculated a scale for the fifth dimension based on the quantum of charge. In the 1940s 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 evaluated using tensor algebra software in 2015. In his 1921 paper, 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 g ~ a b {displaystyle {widetilde {g}}_{ab}} , where Latin indices span five dimensions. Let one also introduce the four-dimensional spacetime metric g μ ν {displaystyle {g}_{mu u }} , where Greek indices span the usual four dimensions of space and time; a 4-vector A μ {displaystyle A^{mu }} identified with the electromagnetic vector potential; and a scalar field ϕ {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: