In physics, Kaluza–Klein theory (KK theory) is an attempt at creating a unified field theory of gravitation and electromagnetism based on the idea of a fifth dimension of space beyond the conventional four-dimensional spacetime of general relativity. According to this proposal, there are three dimensions of space and one of time but with an additional spatial dimension in the shape of a tiny circle. The Kaluza-Klein theory is named after Theodor Kaluza and Oskar Klein. It is not supported by experiments, but is a precursor to supergravity and modern string theory (in eleven-dimensional spacetime).
Kaluza hypothesis In his 1921 article, Kaluza established all the elements of the classical five-dimensional theory: the Kaluza–Klein metric, the Kaluza–Klein–Einstein 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 Kaluza–Klein metric , where Latin indices span five dimensions. Let one also introduce the four-dimensional spacetime metric , where Greek indices span the usual four dimensions of space and time; a 4-vector 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
One can write more precisely
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
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 five-dimensional Einstein equations, and the equations of motion from the five-dimensional geodesic hypothesis. The resulting field equations provide both the equations of general relativity and of electrodynamics; the equations of motion provide the four-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 five-dimensional length element :
Field equations from the Kaluza hypothesis The Kaluza–Klein–Einstein field equations of the five-dimensional theory were never adequately provided by Kaluza or Klein because they ignored the scalar field. The full Kaluza field equations are generally attributed to Thiry, who obtained vacuum field equations, although Kaluza originally provided a stress–energy tensor for his theory, and Thiry included a stress–energy tensor in his thesis. But as described by Gonner, several independent groups worked on the field equations in the 1940s and earlier. Thiry is perhaps best known only because an English translation was provided by Applequist, Chodos, & Freund in their review book. Applequist et al. also provided an English translation of Kaluza's article. Translations of the three (1946, 1947, 1948) Jordan articles can be found on the ResearchGate and Academia.edu archives. The first correct English-language Kaluza field equations, including the scalar field, were provided by Williams.
To obtain the 5D Kaluza–Klein–Einstein field equations, the 5D Kaluza–Klein–Christoffel symbols are calculated from the 5D Kaluza–Klein metric , and the 5D Kaluza–Klein–Ricci tensor is calculated from the 5D connections.
The classic results of Thiry and other authors presume the cylinder condition:
Without this assumption, the field equations become much more complex, providing many more degrees of freedom that can be identified with various new fields. Paul Wesson and colleagues have pursued relaxation of the cylinder condition to gain extra terms that can be identified with the matter fields, for which Kaluza otherwise inserted a stress–energy tensor by hand.
It has been an objection to the original Kaluza hypothesis to invoke the fifth dimension only to negate its dynamics. But Thiry argued 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. Most authors have therefore employed the cylinder condition in deriving the field equations. Furthermore, vacuum equations are typically assumed for which
where
and
The vacuum field equations obtained in this way by Thiry and Jordan's group are as follows.
The field equation for is obtained from
where and is a standard, 4D covariant derivative. It shows that the electromagnetic field is a source for the scalar field. Note that the scalar field cannot be set to a constant without constraining the electromagnetic field. The earlier treatments by Kaluza and Klein did not have an adequate description of the scalar field and did not realize the implied constraint on the electromagnetic field by assuming the scalar field to be constant.
The field equation for is obtained from
It has the form of the vacuum Maxwell equations if the scalar field is constant.
The field equation for the 4D Ricci tensor is obtained from
where is the standard 4D Ricci scalar.
This equation shows the remarkable result, called the "Kaluza miracle", that the precise form for the electromagnetic stress–energy tensor emerges from the 5D vacuum equations as a source in the 4D equations: field from the vacuum. This relation allows the definitive identification of with the electromagnetic vector potential. Therefore, the field needs to be rescaled with a conversion constant such that .
The relation above shows that we must have
where is the gravitational constant, and is the permeability of free space. In the Kaluza theory, the gravitational constant can be understood as an electromagnetic coupling constant in the metric. There is also a stress–energy tensor for the scalar field. The scalar field behaves like a variable gravitational constant, in terms of modulating the coupling of electromagnetic stress–energy to spacetime curvature. The sign of in the metric is fixed by correspondence with 4D theory so that electromagnetic energy densities are positive. It is often assumed that the fifth coordinate is spacelike in its signature in the metric.
In the presence of matter, the 5D vacuum condition cannot be assumed. Indeed, Kaluza did not assume it. The full field equations require evaluation of the 5D Kaluza–Klein–Einstein tensor
as seen in the recovery of the electromagnetic stress–energy tensor above. The 5D curvature tensors are complex, and most English-language reviews contain errors in either or , as does the English translation of Thiry. In 2015, a complete set of 5D curvature tensors under the cylinder condition, evaluated using tensor-algebra software, was produced.
Equations of motion from the Kaluza hypothesis The equations of motion are obtained from the five-dimensional geodesic hypothesis in terms of a 5-velocity :
This equation can be recast in several ways, and it has been studied in various forms by authors including Kaluza, Pauli, Gross & Perry, Gegenberg & Kunstatter, and Wesson & Ponce de Leon, but it is instructive to convert it back to the usual 4-dimensional length element , which is related to the 5-dimensional length element as given above:
Then the 5D geodesic equation can be written for the spacetime components of the 4-velocity:
The term quadratic in provides the 4D geodesic equation plus some electromagnetic terms:
The term linear in provides the Lorentz force law:
This is another expression of the "Kaluza miracle". The same hypothesis for the 5D metric that provides electromagnetic stress–energy in the Einstein equations, also provides the Lorentz force law in the equation of motions along with the 4D geodesic equation. Yet correspondence with the Lorentz force law requires that we identify the component of 5-velocity along the fifth dimension with electric charge:
where is particle mass, and is particle electric charge. Thus electric charge is understood as motion along the fifth dimension. 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.
Yet there is a problem: the term quadratic in ,
If there is no gradient in the scalar field, the term quadratic in vanishes. But otherwise the expression above implies
For elementary particles, . The term quadratic in should dominate the equation, perhaps in contradiction to experience. This was the main shortfall of the five-dimensional theory as Kaluza saw it, and he gives it some discussion in his original article.
The equation of motion for is particularly simple under the cylinder condition. Start with the alternate form of the geodesic equation, written for the covariant 5-velocity:
This means that under the cylinder condition, is a constant of the five-dimensional motion:
Kaluza's hypothesis for the matter stress–energy tensor Kaluza proposed a five-dimensional matter stress tensor of the form
where is a density, and the length element is as defined above.
Then the spacetime component gives a typical "dust" stress–energy tensor:
The mixed component provides a 4-current source for the Maxwell equations:
Just as the five-dimensional metric comprises the four-dimensional metric framed by the electromagnetic vector potential, the five-dimensional stress–energy tensor comprises the four-dimensional stress–energy tensor framed by the vector 4-current.
Quantum interpretation of Klein Kaluza's original hypothesis was purely classical and extended discoveries of general relativity. By the time of Klein's contribution, the discoveries of Heisenberg, Schrödinger, and Louis de Broglie were receiving a lot of attention. Klein's Nature article suggested that the fifth dimension is closed and periodic, and that the identification of electric charge with motion in the fifth dimension can be interpreted as standing waves of wavelength , much like the electrons around a nucleus in the Bohr model of the atom. The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum. Combining the previous Kaluza result for in terms of electric charge, and a de Broglie relation for momentum , Klein obtained an expression for the 0th mode of such waves:
where is the Planck constant. Klein found that cm, and thereby an explanation for the cylinder condition in this small value.
Klein's Zeitschrift für Physik article of the same year, gave a more detailed treatment that explicitly invoked the techniques of Schrödinger and de Broglie. It recapitulated much of the classical theory of Kaluza described above, and then departed into Klein's quantum interpretation. Klein solved a Schrödinger-like wave equation using an expansion in terms of fifth-dimensional waves resonating in the closed, compact fifth dimension.
Group theory interpretation
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. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and construction of this compact dimension is referred to as compactification.