- Kaluza–Klein theory
In

physics ,**Kaluza–Klein theory**(or**KK theory**, for short) is a model that seeks to unify the two fundamental forces ofgravitation andelectromagnetism . The theory was first published in 1921 and was discovered by the mathematicianTheodor Kaluza who extendedgeneral relativity to a five-dimensional spacetime. The resulting equations can be separated out into further sets of equations, one of which is equivalent toEinstein field equations , another set equivalent toMaxwell's equations for theelectromagnetic field and the final part an extra scalar field now termed the "radion".**Overview**A splitting of five-dimensional

spacetime into theEinstein equation s and Maxwell equations in four dimensions was first discovered byGunnar Nordström in 1914, in the context of his theory of gravity, but subsequently forgotten. In 1926,Oskar Klein proposed that the fourth spatial dimension is curled up in acircle of very smallradius , 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 acompact 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)", aselectromagnetism can essentially be formulated as agauge theory on afiber bundle , thecircle bundle , withgauge group "U"(1). Once this geometrical interpretation is understood, it is relatively straightforward to replace "U"(1) by a generalLie 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 ororbifold or even anoncommutative 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

electroweak forces by using the symmetry group of theStandard 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 thefermion s must be introduced in an artificial way (in nonsupersymmetric models). Nonetheless, KK remains an importanttouchstone in theoretical physics and is often embedded in more sophisticated theories. It is studied in its own right as an object of geometric interest inK-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 physics andastrophysics communities. A variety of predictions, with real experimental consequences, can be made (in the case oflarge extra dimension s/warped model s). For example, on the simplest of principles, one might expect to havestanding wave s in the extra compactified dimension(s). If an extra dimension is of radius "R", theenergy of such a standing wave would be $E=nhc/R$ with "n" aninteger , "h" beingPlanck's constant and "c" thespeed 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 collider data for the signature of effects associated with large extra dimensions/warped model s.Brandenberger and Vafa have speculated that in the early universe,

cosmic inflation causes three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic.**pace-time-matter theory**One particular variant of Kaluza–Klein theory is

orspace -time -matter theory**induced matter theory**, chiefly promulgated byPaul Wesson and 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:$R\_\{AB\}=0,$

with $R\_\{AB\}$ the five-dimensional

Ricci curvature , may be re-expressed so that in four dimensions, these solutions satisfyEinstein's equation s:$G\_\{mu\; u\}\; =\; 8pi\; T\_\{mu\; u\},$

with the precise form of the $T\_\{mu\; u\}$ following from the

Ricci-flat condition on the five-dimensional space. Since theenergy-momentum tensor $T\_\{mu\; u\}$ is 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

soliton solutions of $R\_\{AB\}=0$ can be shown to contain theRobertson-Walker metric in both radiation-dominated (early universe) and matter-dominated (later universe) forms. The general equations can be shown to be sufficiently consistent with classicaltests of general relativity to be acceptable on physical principles, while still leaving considerable freedom to also provide interestingcosmological model s.**Geometric interpretation**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 principle to a certain action. Let "M" be a (pseudo-)Riemannian manifold , which may be taken as thespacetime ofgeneral relativity . If "g" is the metric on this manifold, one defines the action $S(g)$ as:$S(g)=int\_M\; R(g)\; mbox\{vol\}(g),$

where "R(g)" is the

scalar curvature and vol("g") is thevolume element . By applying thevariational principle to the action:$frac\{delta\; S(g)\}\{delta\; g\}\; =\; 0$

one obtains precisely the

Einstein equation s for free space::$R\_\{ij\}\; -\; frac\{1\}\{2\}g\_\{ij\}R\; =\; 0$

Here, $R\_\{ij\}$ is the

Ricci tensor .**The Maxwell equations**By contrast, the

Maxwell equation s describingelectromagnetism can be understood to be the Hodge equations of a principal "U"(1)-bundle orcircle bundle $pi:P\; o\; M$ with fiber "U"(1). This is, theelectromagnetic field "F" is a harmonic 2-form in the space $Omega^2(M)$ of differentiable2-form s on the manifold "M". In the absence of charges and currents, the free-field Maxwell equations are:$mathrm\{d\}F=0,quad$ and $quad\; mathrm\{d\}*F=0,$

where * is the

Hodge star .**The Kaluza–Klein geometry**To build the Kaluza–Klein theory, one picks an invariant metric on the circle $S^1$ 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 $Lambda$. One then considers metrics $widehat\{g\}$ 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 $widehat\{g\}$ to the vertical subspace $mbox\{Vert\}\_pP\; subset\; T\_pP$ needs to agree with metric on the fiber over a point in the manifold "M".

* The projection of $widehat\{g\}$ to the horizontal subspace $mbox\{Hor\}\_pP\; subset\; T\_pP$ of the

tangent space at point $pin\; P$ must be isomorphic to the metric "g" on "M" at $pi(p)$.The Kaluza–Klein action for such a metric is given by

:$S(widehat\{g\})=int\_P\; R(widehat\{g\})\; ;mbox\{vol\}(widehat\{g\}),$

The scalar curvature, written in components, then expands to

:$R(widehat\{g\})\; =\; pi^*left(\; R(g)\; -\; frac\{Lambda^2\}\{2\}\; vert\; F\; vert^2\; ight)$

where $pi^*$ is the pullback of the fiber bundle projection $pi:P\; o\; M$. The connection "A" on the fiber bundle is related to the electromagnetic field strength as

:$pi^*F\; =\; mathrm\{d\}A$

That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from homology and specifically,

K-theory . ApplyingFubini's theorem and integrating on the fiber, one gets:$S(widehat\{g\})=Lambda\; int\_M\; left(\; R(g)\; -\; frac\{1\}\{Lambda^2\}\; vert\; F\; vert^2\; ight)\; ;mbox\{vol\}(g)$

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

:$R\_\{ij\}\; -\; frac\{1\}\{2\}g\_\{ij\}R\; =\; frac\{1\}\{Lambda^2\}\; T\_\{ij\}$

with the

stress-energy tensor being given by:$T^\{ij\}\; =\; F^\{ik\}F^\{jl\}g\_\{kl\}\; -\; frac\{1\}\{4\}g^\{ij\}\; vert\; F\; vert^2$,

sometimes called the

.Maxwell stress tensor The original theory identifies $Lambda$ with the fiber metric $g\_\{55\}$, and allows $Lambda$ 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 $Lambda$ 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 aYang-Mills theory , and is sometimes taken to be synonymous. If the underlying manifold issupersymmetric , the resulting theory is a supersymmetric Yang–Mills theory.**ee also***

Classical theories of gravitation

*DGP model

*Randall-Sundrum model

*Supergravity

*Superstring theory

*Why 10 dimensions? **References***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=**Further reading***

* 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 theCollider 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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**Kaluza-Klein theory**— noun a) A (no longer valid) theory developed by physicists Theodor Kaluza Oskar Klein that attempted to combine gravity electromagnetism by adding a fifth dimension to our 4 dimensional spacetime. b) Any of a number of theories that try to… … Wiktionary**Kaluza-Klein-Theorie**— Die Kaluza Klein Theorie war einer der ersten Versuche zur Vereinheitlichung der fundamentalen Wechselwirkungen Gravitation und Elektromagnetismus. 1921 erweiterte Theodor Kaluza die vierdimensionale Raumzeit der Allgemeinen Relativitätstheorie… … Deutsch Wikipedia**Theorie de Kaluza-Klein**— Théorie de Kaluza Klein En physique, la théorie de Kaluza Klein (encore appelée théorie de KK) est historiquement le premier modèle ayant tenté d unifier les deux interactions fondamentales que sont la gravitation et l électromagnétisme. En 1919… … Wikipédia en Français**Théorie de kaluza-klein**— En physique, la théorie de Kaluza Klein (encore appelée théorie de KK) est historiquement le premier modèle ayant tenté d unifier les deux interactions fondamentales que sont la gravitation et l électromagnétisme. En 1919 Theodor Kaluza proposa… … Wikipédia en Français**Théorie de Kaluza-Klein**— En physique, la théorie de Kaluza Klein (encore appelée théorie de KK) est historiquement le premier modèle ayant tenté d unifier les deux interactions fondamentales que sont la gravitation et l électromagnétisme. En 1919 Theodor Kaluza proposa… … Wikipédia en Français**Klein**— may refer to:People with the surname Klein: *Klein (surname)In places: *Klein, Montana, community in the United States *Klein, Texas, community in the United States *Klein Betschkerek, community in Romania *Klein Bonaire, island near Bonaire in… … Wikipedia**Kaluza , Theodor Franz Eduard**— (1885–1954) German mathematical physicist The son of a phonetician, Kaluza was born at Ratibor in Germany and educated at the University of Königsberg where he served (1902–29) as a privatdocent (a largely unpaid teaching assistant). On Einstein… … Scientists**Theory of everything**— A theory of everything (TOE) is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories.… … Wikipedia**Theory of Everything**— Eine Weltformel, oder eine Theorie von Allem (TOE, Theory Of Everything) ist eine hypothetische Theorie der theoretischen Physik und Mathematik, die zusammen alle bekannten physikalischen Phänomene gänzlich erklärt und verknüpft. Mit der Zeit ist … Deutsch Wikipedia**Klein–Gordon equation**— Quantum mechanics Uncertainty principle … Wikipedia