- Kaluza–Klein theory
physics, Kaluza–Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitationand electromagnetism. The theory was first published in 1921 and was discovered by the mathematician Theodor Kaluzawho extended general relativityto a five-dimensional spacetime. The resulting equations can be separated out into further sets of equations, one of which is equivalent to Einstein field equations, another set equivalent to Maxwell's equationsfor the electromagnetic fieldand the final part an extra scalar field now termed the "radion".
A splitting of five-dimensional
spacetimeinto the Einstein equations and Maxwell equations in four dimensions was first discovered by Gunnar Nordströmin 1914, in the context of his theory of gravity, but subsequently forgotten. In 1926, Oskar Kleinproposed that the fourth spatial dimension is curled up in a circleof 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 the phenomenon of having a space-time with compact dimensions is referred to as compactification.
In modern geometry, the extra fifth dimension can be understood to be the
circle group"U(1)", as electromagnetismcan essentially be formulated as a gauge theoryon a fiber bundle, the circle bundle, with gauge group"U"(1). Once this geometrical interpretation is understood, it is relatively straightforward to replace "U"(1) by a general Lie group. Such generalizations are often called Yang–Mills theories. If a distinction is drawn, then it is that Yang–Mills theories occur on a flat space-time, whereas Kaluza–Klein treats the more general case of curved spacetime. The base space of Kaluza–Klein theory need not be four-dimensional space-time; it can be any (pseudo-) Riemannian manifold, or even a supersymmetric manifold or orbifoldor even a noncommutative space.
As an approach to the unification of the forces, it is straightforward to apply the Kaluza-Klein theory in an attempt to unify gravity with the strong and
electroweakforces by using the symmetry group of the Standard Model, SU(3)× SU(2)× U(1). However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality founders on a number of issues, including the fact that the fermions must be introduced in an artificial way (in nonsupersymmetric models). Nonetheless, KK remains an important touchstonein theoretical physics and is often embedded in more sophisticated theories. It is studied in its own right as an object of geometric interest in K-theory.
Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the
experimental physicsand astrophysicscommunities. A variety of predictions, with real experimental consequences, can be made (in the case of large extra dimensions/ warped models). For example, on the simplest of principles, one might expect to have standing waves in the extra compactified dimension(s). If an extra dimension is of radius "R", the energyof such a standing wave would be with "n" an integer, "h" being Planck's constantand "c" the speed of light. This set of possible energy values is often called the Kaluza–Klein tower.
Examples of experimental pursuits include work by the CDF collaboration, which has re-analyzed
particle colliderdata for the signature of effects associated with large extra dimensions/ warped models.
Brandenberger and Vafa have speculated that in the early universe,
cosmic inflationcauses three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic.
One particular variant of Kaluza–Klein theory is
space- time- mattertheory or induced matter theory, chiefly promulgated by Paul Wessonand other members of the so-called [http://astro.uwaterloo.ca/~wesson/ Space-Time-Matter Consortium] . In this version of the theory, it is noted that solutions to the equation
with the five-dimensional
Ricci curvature, may be re-expressed so that in four dimensions, these solutions satisfy Einstein's equations
with the precise form of the following from the
Ricci-flat conditionon the five-dimensional space. Since the energy-momentum tensoris normally understood to be due to concentrations of matter in four-dimensional space, the above result is interpreted as saying that four-dimensional matter is induced from geometry in five-dimensional space.
In particular, the
solitonsolutions of can be shown to contain the Robertson-Walker metricin both radiation-dominated (early universe) and matter-dominated (later universe) forms. The general equations can be shown to be sufficiently consistent with classical tests of general relativityto be acceptable on physical principles, while still leaving considerable freedom to also provide interesting cosmological models.
The Kaluza–Klein theory is striking because it has a particularly elegant presentation in terms of geometry. In a certain sense, it looks just like ordinary gravity in
free space, except that it is phrased in five dimensions instead of four.
The Einstein equations
The equations governing ordinary gravity in free space can be obtained from an action, by applying the
variational principleto a certain action. Let "M" be a (pseudo-) Riemannian manifold, which may be taken as the spacetimeof general relativity. If "g" is the metric on this manifold, one defines the action as
where "R(g)" is the
scalar curvatureand vol("g") is the volume element. By applying the variational principleto the action
one obtains precisely the
Einstein equations for free space:
Here, is the
The Maxwell equations
By contrast, the
Maxwell equations describing electromagnetismcan be understood to be the Hodge equations of a principal "U"(1)-bundle or circle bundlewith fiber "U"(1). This is, the electromagnetic field"F" is a harmonic 2-form in the space of differentiable 2-forms on the manifold "M". In the absence of charges and currents, the free-field Maxwell equations are
where * is the
The Kaluza–Klein geometry
To build the Kaluza–Klein theory, one picks an invariant metric on the circle that is the fiber of the "U"(1)-bundle of electromagnetism. In this discussion, an "invariant metric" is simply one that is invariant under rotations of the circle. Suppose this metric gives the circle a total length of . One then considers metrics on the bundle "P" that are consistent with both the fiber metric, and the metric on the underlying manifold "M". The consistency conditions are:
* The projection of to the vertical subspace needs to agree with metric on the fiber over a point in the manifold "M".
* The projection of to the horizontal subspace of the
tangent spaceat point must be isomorphic to the metric "g" on "M" at .
The Kaluza–Klein action for such a metric is given by
The scalar curvature, written in components, then expands to
where is the pullback of the fiber bundle projection . The connection "A" on the fiber bundle is related to the electromagnetic field strength as
That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from homology and specifically,
K-theory. Applying Fubini's theoremand integrating on the fiber, one gets
Varying the action with respect to the component "A", one regains the Maxwell equations. Applying the variational principle to the base metric "g", one gets the Einstein equations
stress-energy tensorbeing given by
sometimes called the
Maxwell stress tensor.
The original theory identifies with the fiber metric , and allows to vary from fiber to fiber. In this case, the coupling between gravity and the electromagnetic field is not constant, but has its own dynamical field, the radion.
Commentary and generalizations
In the above, the size of the loop acts as a coupling constant between the gravitational field and the electromagnetic field. If the base manifold is four-dimensional, the Kaluza–Klein manifold "P" is five-dimensional. The fifth dimension is a
compact space, and is called the compact dimension. The phenomenon of having a higher-dimensional manifold where some of the dimensions are compact is referred to as compactification. Compactification does not produce group actions on chiral fermions except in very specific cases: the dimension of the total space must be 2 mod 8 and the G-index of the Dirac operator of the compact space must me nonzero [L. Castellani et al, Supergravity and superstrings, Vol 2, chapter V.11] .
The above development generalizes in a more-or-less straightforward fashion to general principal "G"-bundles for some arbitrary
Lie group"G" taking the place of "U"(1). In such a case, the theory is often referred to as a Yang-Mills theory, and is sometimes taken to be synonymous. If the underlying manifold is supersymmetric, the resulting theory is a supersymmetric Yang–Mills theory.
Classical theories of gravitation
Why 10 dimensions?
*cite journal |last=Nordström |first=Gunnar |authorlink= |coauthors= |year=1914 |month= |title=Über die Möglichkeit, das elektromagnetische Feld und das Gravitationsfeld zu vereinigen |journal=Physikalische Zeitschrift |volume=15 |issue= |pages=504–506 |oclc=1762351 |url= |accessdate= |quote=
*cite journal |last=Kaluza |first=Theodor |authorlink= |coauthors= |year=1921 |month= |title=Zum Unitätsproblem in der Physik |journal=Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) |volume=1921 |issue= |pages=966–972 |issn= |url= |accessdate= |quote=
*cite journal |last=Klein |first=Oskar |authorlink= |coauthors= |year=1926 |month= |title=Quantentheorie und fünfdimensionale Relativitätstheorie |journal=Zeitschrift für Physik A Hadrons and Nuclei |volume=37 |issue=12 |pages=895–906 |doi=10.1007/BF01397481 |url= |accessdate= |quote=
*cite journal |last=Witten |first=Edward |authorlink= |coauthors= |year=1981 |month= |title=Search for a realistic Kaluza-Klein theory |journal=Nuclear Physics B |volume=186 |issue=3 |pages=412–428 |doi=10.1016/0550-3213(81)90021-3 |url= |accessdate= |quote=
*cite book |title=Modern Kaluza-Klein Theories |last=Appelquist |first=Thomas |authorlink= |coauthors=Chodos, Alan; Freund, Peter G. O. |year=1987 |publisher=Addison-Wesley |location=Menlo Park, Cal. |isbn=0201098296 |pages= "(Includes reprints of the above articles as well as those of other important papers relating to Kaluza-Klein theory.)"
*cite journal |last=Brandenberger |first=Robert |authorlink= |coauthors=Vafa, Cumrun |year=1989 |month= |title=Superstrings in the early universe |journal=Nuclear Physics B |volume=316 |issue=2 |pages=391–410 |doi=10.1016/0550-3213(89)90037-0 |url= |accessdate= |quote=
*cite book |title=Proceedings of the Symposium ‘The Oskar Klein Centenary’ |chapter=Kaluza-Klein Theory in Perspective |last=Duff |first=M. J. |authorlink= |editor=Lindström, Ulf (ed.) |year=1994 |publisher=World Scientific |location=Singapore |isbn=9810223323 |pages=22–35
*cite journal |last=Overduin |first=J. M. |authorlink= |coauthors=Wesson, P. S. |year=1997 |month= |title=Kaluza-Klein Gravity |journal=Physics Reports |volume=283 |issue=5 |pages=303–378 |doi=10.1016/S0370-1573(96)00046-4 |url= |accessdate= |quote=
*cite book |title=Space-Time-Matter, Modern Kaluza-Klein Theory |last=Wesson |first=Paul S. |authorlink= |coauthors= |year=1999 |publisher=World Scientific |location=Singapore |isbn=9810235887 |pages=
*cite book |title=Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza-Klein Cosmology |last=Wesson |first=Paul S. |authorlink= |coauthors= |year=2006 |publisher=World Scientific |location=Singapore |isbn=9812566619 |pages=
* Kaku, Michio and Robert O'Keefe. "". New York:
Oxford University Press, 1994. ISBN 0192861891
* The CDF Collaboration, " [http://www-cdf.fnal.gov/PES/kkgrav/kkgrav.html Search for Extra Dimensions using Missing Energy at CDF] ", (2004) "(A simplified presentation of the search made for extra dimensions at the
Collider Detector at Fermilab(CDF) particle physics facility.)"
* John M. Pierre, " [http://www.sukidog.com/jpierre/strings/extradim.htm SUPERSTRINGS! Extra Dimensions] ", (2003).
* [http://uk.arxiv.org/abs/hep-ph/0002255 TeV scale gravity, mirror universe, and ... dinosaurs] Article from [http://th-www.if.uj.edu.pl/acta/ Acta Physica Polonica B] by Z.K. Silagadze.
* Chris Pope, " [http://faculty.physics.tamu.edu/pope/ihplec.pdf Lectures on Kaluza-Klein Theory] ".
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