Kaluza–Klein theory
[3] Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components.
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.
In 1926, Oskar Klein gave Kaluza's classical five-dimensional theory a quantum interpretation,[4][5] to accord with the then-recent discoveries of Werner Heisenberg and Erwin Schrödinger.
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.
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] Yves Thiry,[7][8][9] working in France on his dissertation under André Lichnerowicz; Pascual Jordan, Günther Ludwig, and Claus Müller in Germany,[10][11][12][13][14] with critical input from Wolfgang Pauli and Markus Fierz; and Paul Scherrer[15][16][17] working alone in Switzerland.
Jordan's work led to the scalar–tensor theory of Brans–Dicke;[18] Carl H. Brans and Robert H. 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 J.
Paul Wesson and colleagues have pursued relaxation of the cylinder condition to gain extra terms that can be identified with the matter fields,[24] for which Kaluza[3] otherwise inserted a stress–energy tensor by hand.
But Thiry argued[6] that the interpretation of the Lorentz force law in terms of a five-dimensional geodesic militates strongly for a fifth dimension irrespective of the cylinder condition.
The scalar field behaves like a variable gravitational constant, in terms of modulating the coupling of electromagnetic stress–energy to spacetime curvature.
[8] In 2015, a complete set of 5D curvature tensors under the cylinder condition, evaluated using tensor-algebra software, was produced.
: This equation can be recast in several ways, and it has been studied in various forms by authors including Kaluza,[3] Pauli,[25] Gross & Perry,[26] Gegenberg & Kunstatter,[27] and Wesson & Ponce de Leon,[28] but it is instructive to convert it back to the usual 4-dimensional length element
The fact that the Lorentz force law could be understood as a geodesic in five dimensions was to Kaluza a primary motivation for considering the five-dimensional hypothesis, even in the presence of the aesthetically unpleasing cylinder condition.
Start with the alternate form of the geodesic equation, written for the covariant 5-velocity: This means that under the cylinder condition,
By the time of Klein's contribution, the discoveries of Heisenberg, Schrödinger, and Louis de Broglie were receiving a lot of attention.
The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum.
Klein's Zeitschrift für Physik article of the same year,[4] gave a more detailed treatment that explicitly invoked the techniques of Schrödinger and de Broglie.
In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began.
Computing the scalar curvature of this bundle metric, one finds that it is constant on each fiber: this is the "Kaluza miracle".
A variety of predictions, with real experimental consequences, can be made (in the case of large extra dimensions and warped models).
If a spatial extra dimension is of radius R, the invariant mass of such standing waves would be Mn = nh/Rc with n an integer, h being the Planck constant and c the speed of light.
[31] Examples of experimental pursuits include work by the CDF collaboration, which has re-analyzed particle collider data for the signature of effects associated with large extra dimensions/warped models.
In other words, the cylinder condition of the previous development is dropped, and the stress–energy now comes from the derivatives of the 5D metric with respect to the fifth coordinate.
By applying the variational principle to the action one obtains precisely the Einstein equations for free space: where Rij is the Ricci tensor.
An analysis of results from the LHC in December 2010 severely constrains theories with large extra dimensions.
[34] The observation of a Higgs-like boson at the LHC establishes a new empirical test which can be applied to the search for Kaluza–Klein resonances and supersymmetric particles.
Hence a measurement of any dramatic change to the H → γγ cross-section predicted by the Standard Model is crucial in probing the physics beyond it.
