Kaluza—Klein Theory

Abstract

Theodor Kaluza (1885–1954) set forth his idea of unifying gravitation and electromagnetism within five-dimensional space-time in a paper from the beginning of 1919. It was presented by Albert Einstein (1879–1955) to the Berlin Academy of Science and published in its Sitzungsberichte in 1921 under the title “Zum Unitatsproblem der Physik” [4]. Having received the manuscript from the author in April 1919, Einstein was so impressed with the idea of unifying the basic forces in a five-dimensional space that he used it himself up to the mid-1940s in eight of his own papers. Kaluza’s idea of unifying gravitation and electromagnetism goes back to David Hilbert’s (1862–1943) unification program and to the pioneering work of two of his pupils: Hilbert sought unification within a four-dimensional space by having electromagnetism come from gravitation in 1915 [2]. His pupil Gunnar Nordstrom (1881–1923) explored unification within a five-dimensional unwarped (Minkowskian) space in the foregoing year [7]. The unification attempt by Hermann Weyl (1885–1955) in 1918, finally, was to apply a gauge transformation within a four-dimensional space with a generalized non-Riemannian metric [10]. Although Nordstrom was the first to introduce a five-dimensional space, it was Kaluza’s theory from 1919 that proposed a realistic unification of the two interactions. Nordstrom’s theory predated the general theory of relativity (1915) so the gravitation was derived from electromagnetism within a space described by a Minkowskian flat metric. As a consequence, it could not explain phenomena like light deflection and was therefore condemned as a prerelativistic theory. Kaluza’s idea, which was to serve as the model for the design of all unified theories in higher-dimensional spaces, was as follows: Within a five-dimensional space (with a Riemannian metric) there exists a unique five-dimensional gravitational force that upon projection onto the four-dimensional space of our experience splits into two phenomena: our familiar natural forces, being four-dimensional Einsteinian gravitation (known from the general theory of relativity), and Maxwellian electromagnetism. Thus in Kaluza’s theory – as in all modern higher-dimensional unified theories, and unlike Nordstrom’s – the fundamental force is gravitation. It, according to Kaluza, is the originator of electromagnetism. (Modern-day higherdimensional unified theories attribute all the other forces to gravitation as well.) Electromagnetism is thus an effect of the fifth dimension. The fifth components of the metric tensor gμ5(μ = 1, 2, 3, 4) are identical to the Maxwellian electromagnetic field Aμ. Kaluza was able to show that the five-dimensional momentum p5