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| {{about|gravitation and electromagnetism|the mathematical generalization of [[K theory]]|KK-theory}}
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| {{Beyond the Standard Model|expanded=Theories}}
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| In [[physics]], '''Kaluza–Klein theory''' ('''KK theory''') is a model that seeks to unify the two [[fundamental interaction|fundamental forces]] of [[gravitation]] and [[electromagnetism]]. The theory was first published in 1921. It was proposed by the mathematician [[Theodor Kaluza]] who extended [[general relativity]] to a five-dimensional spacetime. The resulting equations can be separated into further sets of equations, one of which is equivalent to [[Einstein field equations]], another set equivalent to [[Maxwell's equations]] for the [[electromagnetic field]] and the final part an extra [[Scalar field theory|scalar field]] now termed the "[[Radion (physics)|radion]]".
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| == Overview ==
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| [[Image:Kaluza Klein compactification.svg|frame|left|The space ''M'' × ''C'' is compactified over the compact set ''C'', and after Kaluza–Klein decomposition we have an [[effective field theory]] over M.]] A splitting of [[fifth dimension (geometry)|five-dimensional]] [[spacetime]] into the [[Einstein equation]]s and Maxwell equations in four dimensions was first discovered by [[Gunnar Nordström]] in 1914, in the context of [[Nordström's theory of gravitation|his theory]] of gravity, but subsequently forgotten. Kaluza published his derivation in 1921 as an attempt to unify electromagnetism with Einstein's general relativity.
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| In 1926, [[Oskar Klein]] proposed that the fourth spatial dimension is curled up in a [[circle]] of a very small [[radius]], so that a [[Elementary particle|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 (physics)|compactification]].
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| In modern geometry, the extra fifth dimension can be understood to be the [[circle group]] [[U(1)]], as [[electromagnetism]] can essentially be formulated as a [[gauge theory]] on a [[fiber bundle]], the [[circle bundle]], with [[gauge group]] U(1). In Kaluza–Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions. 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 theory|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|pseudo-]])[[Riemannian manifold]], or even a [[supersymmetry|supersymmetric]] manifold or [[orbifold]] or even a [[noncommutative space]].
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| 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 force|strong]] and [[electroweak]] forces 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 flounders on a number of issues, including the fact that the [[fermion]]s must be introduced in an artificial way (in nonsupersymmetric models). Nonetheless, KK remains an important [[Touchstone (metaphor)|touchstone]] in 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]].
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| 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]] and [[astrophysics]] communities. A variety of predictions, with real experimental consequences, can be made (in the case of [[large extra dimension]]s/[[warped model]]s). For example, on the simplest of principles, one might expect to have [[standing wave]]s in the extra compactified dimension(s). If a spatial extra dimension is of radius ''R'', the invariant [[mass]] of such standing waves would be ''M''<sub>''n''</sub> = ''nh''/''Rc'' with ''n'' an [[integer]], ''h'' being [[Planck's constant]] and ''c'' the [[speed of light]]. This set of possible mass values is often called the '''Kaluza–Klein tower'''. Similarly, in [[Thermal quantum field theory]] a compactification of the euclidean time dimension leads to the [[Matsubara frequency|Matsubara frequencies]] and thus to a discretized thermal energy spectrum.
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| Examples of experimental pursuits include work by the [[Collider Detector at Fermilab|CDF]] collaboration, which has re-analyzed [[particle collider]] data for the signature of effects associated with large extra dimensions/[[warped model]]s.
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| 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.
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| == Space-time-matter theory ==
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| One particular variant of Kaluza–Klein theory is '''[[space]]-[[time]]-[[matter]] theory''' or '''induced matter theory''', chiefly promulgated by [[Paul S. Wesson|Paul Wesson]] and other members of the so-called Space-Time-Matter Consortium.<ref>[http://5dstm.org 5Dstm.org]</ref> In this version of the theory, it is noted that solutions to the equation
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| :<math>R_{AB}=0\,</math>
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| with ''R<sub>AB</sub>'' the five-dimensional [[Ricci curvature]], may be re-expressed so that in four dimensions, these solutions satisfy [[Einstein's equation]]s
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| :<math>G_{\mu\nu} = 8\pi T_{\mu\nu}\,</math>
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| with the precise form of the ''T''<sub>''μν''</sub> following from the [[Ricci-flat condition]] on the five-dimensional space. Since the [[energy-momentum tensor]] ''T''<sub>''μν''</sub> 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.
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| In particular, the [[soliton]] solutions of ''R''<sub>''AB''</sub> = 0 can be shown to contain the [[Friedmann–Lemaitre–Robertson–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 classical [[tests of general relativity]] to be acceptable on physical principles, while still leaving considerable freedom to also provide interesting [[cosmological model]]s.
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| == Geometric interpretation ==
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| 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.
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| === The Einstein equations ===
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| The equations governing ordinary gravity in free space can be obtained from an [[action (physics)|action]], by applying the [[variational principle]] to a certain [[action (physics)|action]]. Let ''M'' be a ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian manifold]], which may be taken as the [[spacetime]] of [[general relativity]]. If ''g'' is the [[Metric (mathematics)|metric]] on this manifold, one defines the [[action (physics)|action]] ''S''(''g'') as
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| :<math>S(g)=\int_M R(g) \mathrm{vol}(g)\,</math>
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| where ''R''(''g'') is the [[scalar curvature]] and vol(''g'') is the [[volume element]]. By applying the [[variational principle]] to the action | |
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| :<math>\frac{\delta S(g)}{\delta g} = 0</math>
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| one obtains precisely the [[Einstein equation]]s for free space:
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| :<math>R_{ij} - \frac{1}{2}g_{ij}R = 0</math>
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| Here, ''R''<sub>''ij''</sub> is the [[Ricci tensor]].
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| === The Maxwell equations ===
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| By contrast, the [[Maxwell equation]]s describing [[electromagnetism]] can be understood to be the [[de Rham cohomology|Hodge equations]] of a [[principal bundle|principal U(1)-bundle]] or [[circle bundle]] π: ''P'' → ''M'' with fiber [[U(1)]]. That is, the [[electromagnetic field]] ''F'' is a [[harmonic form|harmonic 2-form]] in the space Ω<sup>2</sup>(''M'') of differentiable [[2-form]]s on the manifold ''M''. In the absence of charges and currents, the free-field Maxwell equations are
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| :d''F'' = 0 and d*''F'' = 0.
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| where * is the [[Hodge star]].
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| === The Kaluza–Klein geometry ===
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| To build the Kaluza–Klein theory, one picks an invariant metric on the circle '''S'''<sup>1</sup> 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 Λ. One then considers metrics <math>\widehat{g}</math> on the bundle ''P'' that are consistent with both the fiber metric, and the metric on the underlying manifold ''M''. The consistency conditions are:
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| * The projection of <math>\widehat{g}</math> to the [[vertical bundle|vertical subspace]] <math>\mbox{Vert}_pP \subset T_pP</math> needs to agree with metric on the fiber over a point in the manifold ''M''.
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| * The projection of <math>\widehat{g}</math> to the [[horizontal bundle|horizontal subspace]] <math>\mbox{Hor}_pP \subset T_pP</math> of the [[tangent space]] at point ''p'' ∈ ''P'' must be isomorphic to the metric ''g'' on ''M'' at π(''p'').
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| The Kaluza–Klein action for such a metric is given by
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| :<math>S(\widehat{g})=\int_P R(\widehat{g}) \;\mbox{vol}(\widehat{g})\,</math>
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| The scalar curvature, written in components, then expands to
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| :<math>R(\widehat{g}) = \pi^*\left( R(g) - \frac{\Lambda^2}{2} \vert F \vert^2\right)</math>
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| where π* is the [[pullback (differential geometry)|pullback]] of the fiber bundle projection π: ''P'' → ''M''. The connection ''A'' on the fiber bundle is related to the electromagnetic field strength as
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| :<math>\pi^*F = \mathrm{d}A</math>
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| That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from [[homology (mathematics)|homology]] and specifically, [[K-theory]]. Applying [[Fubini's theorem]] and integrating on the fiber, one gets
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| :<math>S(\widehat{g})=\Lambda \int_M \left( R(g) - \frac{1}{\Lambda^2} \vert F \vert^2 \right) \;\mbox{vol}(g) </math>
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| 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
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| :<math>R_{ij} - \frac{1}{2}g_{ij}R = \frac{1}{\Lambda^2} T_{ij}</math>
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| with the [[stress-energy tensor]] being given by
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| :<math>T^{ij} = F^{ik}F^{jl}g_{kl}
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| - \frac{1}{4}g^{ij} \vert F \vert^2,</math>
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| sometimes called the '''[[Maxwell stress tensor]]'''.
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| The original theory identifies Λ with the fiber metric ''g''<sub>55</sub>, and allows Λ 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 (physics)|radion]].
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| === Generalizations ===
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| In the above, the size of the loop Λ 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 technique of introducing compact dimensions to obtain a higher-dimensional manifold is referred to as [[compactification (physics)|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 be nonzero.<ref>L. Castellani et al., Supergravity and superstrings, Vol 2, chapter V.11</ref>
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| The above development generalizes in a more-or-less straightforward fashion to general [[principal G-bundle|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 super-symmetric Yang–Mills theory.
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| ==Empirical tests==
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| Up to now, no experimental or observational signs of extra dimensions have been officially reported. Many theoretical search techniques for detecting Kaluza-Klein Resonances have been proposed using the mass couplings of such resonances with the [[top quark]], however until the [[Large Hadron Collider]] (LHC) reaches full operational power observation of such resonances are unlikely. An analysis of results from the LHC in December 2010 severely constrains theories with [[large extra dimensions]].<ref name="arxiv.org">CMS Collaoration, "Search for Microscopic Black Hole Signatures at the Large Hadron Collider," http://arxiv.org/abs/1012.3375</ref>
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| The Discovery of a new boson with Higgs-like decay channels measured experimentally to significance to 4.9 [[Standard deviation|sigma]] puts a brand new empirical test in the search for Kaluza–Klein Resonances and Supersymmetric Particles.
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| The loop [[Feynman diagram]]s that exist in the Higgs Interactions allow any particle with electric charge and mass to run in such a loop. Standard Model particles besides the [[top quark]] and [[W and Z bosons|W boson]] do not make big contributions to the cross-section observed in the H → γγ decay, but if there are new particles beyond the Standard Model, they could potentially change the ratio of the predicted Standard Model H → γγ cross-section to the experimentally observed cross-section. 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.
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| == See also ==
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| * [[Classical theories of gravitation]]
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| * [[DGP model]]
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| * [[Randall–Sundrum model]]
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| * [[Supergravity]]
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| * [[Superstring theory]]
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| * [[String theory]]
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| * [[Quantum gravity]]
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| == Notes ==
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| <references/>
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| == References ==
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| *{{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= }}
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| *{{cite journal |last=Kaluza |first=Theodor |authorlink= |coauthors= |year=1921 |month= |title=Zum Unitätsproblem in der Physik |journal=[[Prussian Academy of Sciences|Sitzungsber. Preuss. Akad. Wiss.]] Berlin. (Math. Phys.) |volume= |issue= |pages=966–972 |issn= |url= |accessdate= |quote= }} http://archive.org/details/sitzungsberichte1921preussi
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| *{{cite journal |last=Klein |first=Oskar |authorlink= |coauthors= |year=1926 |month= |title=Quantentheorie und fünfdimensionale Relativitätstheorie |journal=[[Zeitschrift für Physik A]] |volume=37 |issue=12 |pages=895–906 |doi=10.1007/BF01397481 |url= |accessdate= |quote= |bibcode = 1926ZPhy...37..895K }}
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| *{{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= |bibcode = 1981NuPhB.186..412W }}
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| *{{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=0-201-09829-6 |pages= }} ''(Includes reprints of the above articles as well as those of other important papers relating to Kaluza–Klein theory.)''
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| *{{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= |bibcode = 1989NuPhB.316..391B }}
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| *{{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=981-02-2332-3 |pages=22–35 }}
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| *{{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= |arxiv = gr-qc/9805018 |bibcode = 1997PhR...283..303O }}
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| *{{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=981-02-3588-7 |pages= }}
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| *{{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=981-256-661-9 |pages= }}
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| == Further reading ==
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| * {{cite book | last = Grøn | first = Øyvind |authorlink=Øyvind Grøn| coauthors = Hervik, Sigbjørn | title = Einstein's General Theory of Relativity | location = New York | publisher = Springer | year = 2007 | isbn = 978-0-387-69199-2}}
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| * Kaku, Michio and Robert O'Keefe. ''[[Hyperspace (book)|Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension]]''. New York: [[Oxford University Press]], 1994. ISBN 0-19-286189-1
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| * 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.)''
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| * John M. Pierre, ''[http://www.sukidog.com/jpierre/strings/extradim.htm SUPERSTRINGS! Extra Dimensions]'', (2003).
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| * [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.
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| * Chris Pope, ''[http://faculty.physics.tamu.edu/pope/ihplec.pdf Lectures on Kaluza–Klein Theory]''.
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| {{theories of gravitation}}
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| {{Relativity}}
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| [[Category:Theories of gravitation]]
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