kaluza-klein theory


  • Equations of motion from the Kaluza hypothesis The equations of motion are obtained from the five-dimensional geodesic hypothesis[3] 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,[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 , which is related to the 5-dimensional length element as given above: Then the 5D geodesic equation can be written[29] 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”.

  • [32] In this version of the theory, it is noted that solutions to the equation may be re-expressed so that in four dimensions, these solutions satisfy Einstein’s equations
    with the precise form of the Tμν following from the Ricci-flat condition on the five-dimensional space.

  • Thus, by applying a single idea: the principle of least action, to a single quantity: the scalar curvature on the bundle (as a whole), one obtains simultaneously all of the
    needed field equations, for both the spacetime and the gauge field.

  • Because the energy–momentum tensor 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.

  • 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[3] a five-dimensional matter stress tensor of the form where is a density, and the length element is as defined above.

  • 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.

  • 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 .

  • The hypothesis for the metric implies an invariant five-dimensional length element : Field equations from the Kaluza hypothesis The field equations of the five-dimensional
    theory were never adequately provided by Kaluza or Klein because they ignored the scalar field.

  • [22] Kaluza hypothesis In his 1921 article,[3] Kaluza established all the elements of the classical five-dimensional theory: the metric, the field equations, the equations
    of motion, the stress–energy tensor, and the cylinder condition.

  • But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations
    in five dimensions.

  • Space–time–matter theory One particular variant of Kaluza–Klein theory is space–time–matter theory or induced matter theory, chiefly promulgated by Paul Wesson and other members
    of the Space–Time–Matter Consortium.

  • Furthermore, vacuum equations are typically assumed for which where and The vacuum field equations obtained in this way by Thiry[8] and Jordan’s group[10][11][13] are as follows.

  • Klein’s Nature article[5] 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.

  • Kaluza also introduced the “cylinder condition” hypothesis, that no component of the five-dimensional metric depends on the fifth dimension.

  • Applying Fubini’s theorem and integrating on the fiber, one gets Varying the action with respect to the component , one regains the Maxwell equations.

  • 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 models.

  • The equations of motion, the Euler–Lagrange equations, can be then obtained by considering where the action is stationary with respect to variations of either the metric on
    the base manifold, or of the gauge connection.

  • The full Kaluza field equations are generally attributed to Thiry,[8] who obtained vacuum field equations, although Kaluza[3] originally provided a stress–energy tensor for
    his theory, and Thiry included a stress–energy tensor in his thesis.

  • 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.

  • However, Klein’s approach to a quantum theory is flawed[citation needed] and, for example, leads to a calculated electron mass in the order of magnitude of the Planck mass.

  • Variations with respect to the base metric gives the Einstein field equations on the base manifold, with the energy–momentum tensor given by the curvature (field strength)
    of the gauge connection.

  • In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common
    4D of space and time and considered an important precursor to string theory.

  • 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.

  • The connection 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.

  • 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 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.

  • With no free parameters, it merely extends general relativity to five dimensions.

  • 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.

  • 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.

  • Applying the variational principle to the base metric , one gets the Einstein equations with the stress–energy tensor being given by sometimes called the Maxwell stress tensor.

  • Hence a measurement of any dramatic change to the cross-section predicted by the Standard Model is crucial in probing the physics beyond it.

  • Einstein equations[edit] The equations governing ordinary gravity in free space can be obtained from an action, by applying the variational principle to a certain action.

  • However, the article does demonstrate that electromagnetism and gravity share the same number of dimensions, and this fact lends support to Kaluza–Klein theory; whether the
    number of dimensions is really or in fact is the subject of further debate.

  • 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).

  • Standard Model particles besides the top quark and W boson do not make big contributions to the cross-section observed in the decay, but if there are new particles beyond
    the Standard Model, they could potentially change the ratio of the predicted Standard Model cross-section to the experimentally observed cross-section.

  • 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.

  • Klein solved a Schroedinger-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.

  • 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.

  • Ten components are identified with the 4D spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field
    sometimes called the “radion” or the “dilaton”.

  • Klein suggested that the geometry of the extra fifth dimension could take the form of a circle, with the radius of .

  • [8] See Williams[19] for a complete set of 5D curvature tensors under the cylinder condition, evaluated using tensor-algebra software.

  • Note that the scalar field cannot be set to a constant without constraining the electromagnetic field.

  • 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

  • It is studied in its own right as an object of geometric interest in K-theory.

  • 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.

  • 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.

  • Combining the previous Kaluza result for in terms of electric charge, and a de Broglie relation for momentum , Klein obtained[5] an expression for the 0th mode of such waves:
    where is the Planck constant.

  • The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum.

  • By applying the variational principle to the action one obtains precisely the Einstein equations for free space: where Rij is the Ricci tensor.

  • In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle.

  • [1] The five-dimensional (5D) theory developed in three steps.

  • For example, on the simplest of principles, one might expect to have standing waves in the extra compactified dimension(s).

  • 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 fermions
    must be introduced in an artificial way (in nonsupersymmetric models).


Works Cited

[‘1. Nordström, Gunnar (1914). “Über die Möglichkeit, das elektromagnetische Feld und das Gravitationsfeld zu vereinigen” [On the possibility of unifying the gravitational and electromagnetic fields]. Physikalische Zeitschrift (in German). 15: 504.
2. ^
Pais, Abraham (1982). Subtle is the Lord …: The Science and the Life of Albert Einstein. Oxford: Oxford University Press. pp. 329–330.
3. ^ Jump up to:a b c d e f g h Kaluza, Theodor (1921). “Zum Unitätsproblem in der Physik”. Sitzungsber. Preuss.
Akad. Wiss. Berlin. (Math. Phys.) (in German): 966–972. Bibcode:1921SPAW…….966K.
4. ^ Jump up to:a b c Klein, Oskar (1926). “Quantentheorie und fünfdimensionale Relativitätstheorie”. Zeitschrift für Physik A (in German). 37 (12): 895–906. Bibcode:1926ZPhy…37..895K.
5. ^ Jump up to:a b c d Klein, Oskar (1926). “The Atomicity of Electricity as a Quantum Theory Law”. Nature. 118 (2971): 516. Bibcode:1926Natur.118..516K. doi:10.1038/118516a0. S2CID 4127863.
6. ^ Jump up to:a b c Goenner,
H. (2012). “Some remarks on the genesis of scalar–tensor theories”. General Relativity and Gravitation. 44 (8): 2077–2097. arXiv:1204.3455. Bibcode:2012GReGr..44.2077G. doi:10.1007/s10714-012-1378-8. S2CID 13399708.
7. ^ Lichnerowicz, A.; Thiry,
M. Y. (1947). “Problèmes de calcul des variations liés à la dynamique classique et à la théorie unitaire du champ”. Compt. Rend. Acad. Sci. Paris (in French). 224: 529–531.
8. ^ Jump up to:a b c d Thiry, M. Y. (1948). “Les équations de la théorie
unitaire de Kaluza”. Compt. Rend. Acad. Sci. Paris (in French). 226: 216–218.
9. ^ Thiry, M. Y. (1948). “Sur la régularité des champs gravitationnel et électromagnétique dans les théories unitaires”. Compt. Rend. Acad. Sci. Paris (in French). 226:
10. ^ Jump up to:a b c Jordan, P. (1946). “Relativistische Gravitationstheorie mit variabler Gravitationskonstante”. Naturwissenschaften (in German). 11 (8): 250–251. Bibcode:1946NW…..33..250J. doi:10.1007/BF01204481. S2CID 20091903.
11. ^
Jump up to:a b c Jordan, P.; Müller, C. (1947). “Über die Feldgleichungen der Gravitation bei variabler “Gravitationslonstante””. Z. Naturforsch. (in German). 2a (1): 1–2. Bibcode:1947ZNatA…2….1J. doi:10.1515/zna-1947-0102. S2CID 93849549.
12. ^
Ludwig, G. (1947). “Der Zusammenhang zwischen den Variationsprinzipien der projektiven und der vierdimensionalen Relativitätstheorie”. Z. Naturforsch. (in German). 2a (1): 3–5. Bibcode:1947ZNatA…2….3L. doi:10.1515/zna-1947-0103. S2CID 94454994.
13. ^
Jump up to:a b c Jordan, P. (1948). “Fünfdimensionale Kosmologie”. Astron. Nachr. (in German). 276 (5–6): 193–208. Bibcode:1948AN….276..193J. doi:10.1002/asna.19482760502.
14. ^ Ludwig, G.; Müller, C. (1948). “Ein Modell des Kosmos und der Sternentstehung”.
Annalen der Physik. 2 (6): 76–84. Bibcode:1948AnP…437…76L. doi:10.1002/andp.19484370106. S2CID 120176841.
15. ^ Scherrer, W. (1941). “Bemerkungen zu meiner Arbeit: “Ein Ansatz für die Wechselwirkung von Elementarteilchen””. Helv. Phys. Acta
(in German). 14 (2): 130.
16. ^ Scherrer, W. (1949). “Über den Einfluss des metrischen Feldes auf ein skalares Materiefeld”. Helv. Phys. Acta. 22: 537–551.
17. ^ Scherrer, W. (1950). “Über den Einfluss des metrischen Feldes auf ein skalares Materiefeld
(2. Mitteilung)”. Helv. Phys. Acta (in German). 23: 547–555.
18. ^ Brans, C. H.; Dicke, R. H. (November 1, 1961). “Mach’s Principle and a Relativistic Theory of Gravitation”. Physical Review. 124 (3): 925–935. Bibcode:1961PhRv..124..925B. doi:10.1103/PhysRev.124.925.
19. ^
Jump up to:a b c Williams, L. L. (2015). “Field Equations and Lagrangian for the Kaluza Metric Evaluated with Tensor Algebra Software” (PDF). Journal of Gravity. 2015: 901870. doi:10.1155/2015/901870.
20. ^ Ferrari, J. A. (1989). “On an approximate
solution for a charged object and the experimental evidence for the Kaluza-Klein theory”. Gen. Relativ. Gravit. 21 (7): 683. Bibcode:1989GReGr..21..683F. doi:10.1007/BF00759078. S2CID 121977988.
21. ^ Coquereaux, R.; Esposito-Farese, G. (1990).
“The theory of Kaluza–Klein–Jordan–Thiry revisited”. Annales de l’Institut Henri Poincaré. 52: 113.
22. ^ Williams, L. L. (2020). “Field Equations and Lagrangian of the Kaluza Energy-Momentum Tensor”. Advances in Mathematical Physics. 2020: 1263723.
23. ^ Appelquist, Thomas; Chodos, Alan; Freund, Peter G. O. (1987). Modern Kaluza–Klein Theories. Menlo Park, Cal.: Addison–Wesley. ISBN 978-0-201-09829-7.
24. ^ Wesson, Paul S. (1999). Space–Time–Matter, Modern Kaluza–Klein
Theory. Singapore: World Scientific. ISBN 978-981-02-3588-8.
25. ^ Pauli, Wolfgang (1958). Theory of Relativity (translated by George Field ed.). New York: Pergamon Press. pp. Supplement 23.
26. ^ Gross, D. J.; Perry, M. J. (1983). “Magnetic monopoles
in Kaluza–Klein theories”. Nucl. Phys. B. 226 (1): 29–48. Bibcode:1983NuPhB.226…29G. doi:10.1016/0550-3213(83)90462-5.
27. ^ Gegenberg, J.; Kunstatter, G. (1984). “The motion of charged particles in Kaluza–Klein space–time”. Phys. Lett. 106A (9):
410. Bibcode:1984PhLA..106..410G. doi:10.1016/0375-9601(84)90980-0.
28. ^ Wesson, P. S.; Ponce de Leon, J. (1995). “The equation of motion in Kaluza–Klein cosmology and its implications for astrophysics”. Astronomy and Astrophysics. 294: 1. Bibcode:1995A&A…294….1W.
29. ^
Williams, Lance L. (2012). “Physics of the Electromagnetic Control of Spacetime and Gravity”. Proceedings of 48th AIAA Joint Propulsion Conference. 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 30 July 2012 – 01 August 2012. Atlanta,
Georgia. Vol. AIAA 2012-3916. doi:10.2514/6.2012-3916. ISBN 978-1-60086-935-8. S2CID 122586403.
30. ^ David Bleecker, “Gauge Theory and Variational Principles Archived 2021-07-09 at the Wayback Machine” (1982) D. Reidel Publishing (See chapter 9)
31. ^
Ravndal, F., Oskar Klein and the fifth dimension, arXiv:1309.4113 [physics.hist-ph]
32. ^ 5Dstm.org
33. ^ L. Castellani et al., Supergravity and superstrings, vol. 2, ch. V.11.
34. ^ Khachatryan, V.; et al. (CMS Collaboration) (2011). “Search
for microscopic black hole signatures at the Large Hadron Collider”. Physics Letters B. 697 (5): 434–453. arXiv:1012.3375. Bibcode:2011PhLB..697..434C. doi:10.1016/j.physletb.2011.02.032. S2CID 122803232.
35. ^ Pardo, Kris; Fishbach, Maya; Holz,
Daniel E.; Spergel, David N. (2018). “Limits on the number of spacetime dimensions from GW170817”. Journal of Cosmology and Astroparticle Physics. 2018 (7): 048. arXiv:1801.08160. Bibcode:2018JCAP…07..048P. doi:10.1088/1475-7516/2018/07/048. S2CID
2. Kaluza, Theodor (1921). “Zum Unitätsproblem in der Physik”. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.): 966–972. Bibcode:1921SPAW…….966K. https://archive.org/details/sitzungsberichte1921preussi
3. Klein, Oskar (1926).
“Quantentheorie und fünfdimensionale Relativitätstheorie”. Zeitschrift für Physik A. 37 (12): 895–906. Bibcode:1926ZPhy…37..895K. doi:10.1007/BF01397481.
4. Witten, Edward (1981). “Search for a realistic Kaluza–Klein theory”. Nuclear Physics B.
186 (3): 412–428. Bibcode:1981NuPhB.186..412W. doi:10.1016/0550-3213(81)90021-3.
5. Appelquist, Thomas; Chodos, Alan; Freund, Peter G. O. (1987). Modern Kaluza–Klein Theories. Menlo Park, Cal.: Addison–Wesley. ISBN 978-0-201-09829-7. (Includes reprints
of the above articles as well as those of other important papers relating to Kaluza–Klein theory.)
6. Duff, M. J. (1994). “Kaluza–Klein Theory in Perspective”. In Lindström, Ulf (ed.). Proceedings of the Symposium ‘The Oskar Klein Centenary’.
Singapore: World Scientific. pp. 22–35. ISBN 978-981-02-2332-8.
7. Overduin, J. M.; Wesson, P. S. (1997). “Kaluza–Klein Gravity”. Physics Reports. 283 (5): 303–378. arXiv:gr-qc/9805018. Bibcode:1997PhR…283..303O. doi:10.1016/S0370-1573(96)00046-4.
S2CID 119087814.
8. Wesson, Paul S. (2006). Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza–Klein Cosmology. Singapore: World Scientific. Bibcode:2006fdpc.book…..W. ISBN 978-981-256-661-4.
Photo credit: https://www.flickr.com/photos/brentschmidt/5042785753/’]