Kaluza–Klein theory

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 five-dimensional metric has 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 hypothesis known as the "cylinder condition", that no component of the five-dimensional metric depends on the fifth dimension. Without this assumption, the field equations of five-dimensional relativity are enormously more complex. Standard four-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 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 also calculated a scale for the fifth dimension based on the quantum of charge.

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