Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 019, 15 pages      arXiv:0912.3021      https://doi.org/10.3842/SIGMA.2010.019
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Emergent Abelian Gauge Fields from Noncommutative Gravity

Allen Stern
Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Al 35487, USA

Received December 30, 2009, in final form February 14, 2010; Published online February 18, 2010

Abstract
We construct exact solutions to noncommutative gravity following the formulation of Chamseddine and show that they are in general accompanied by Abelian gauge fields which are first order in the noncommutative scale. This provides a mechanism for generating cosmological electromagnetic fields in an expanding space-time background, and also leads to multipole-like fields surrounding black holes. Exact solutions to noncommutative Einstein-Maxwell theory can give rise to first order corrections to the metric tensor, as well as to the electromagnetic fields. This leads to first order shifts in the horizons of charged black holes.

Key words: noncommutative gravity; Groenewold-Moyal star; exact solutions.

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References

  1. Doplicher S., Fredenhagen K., Roberts J.E., Spacetime quantization induced by classical gravity, Phys. Lett. B 331 (1994), 39-44.
  2. Stern A., Noncommutative point sources, Phys. Rev. Lett. 100 (2008), 061601, 4 pages, arXiv:0709.3831.
  3. Stern A., Particlelike solutions to classical noncommutative gauge theory, Phys. Rev. D 78 (2008), 065006, 11 pages, arXiv:0804.3121.
  4. Pinzul A., Stern A., Noncommutative AdS3 with quantized cosmological constant, Classical Quantum Gravity 23 (2006), 1009-1021, hep-th/0511071.
  5. Chaichian M., Tureanu A., Zet G., Corrections to Schwarzschild solution in noncommutative gauge theory of gravity, Phys. Lett. B 660 (2008), 573-578, arXiv:0710.2075.
    Chaichian M., Setare M.R., Tureanu A., Zet G., On black holes and cosmological constant in noncommutative gauge theory of gravity, J. High Energy Phys. 2008 (2008), no. 4, 064, 18 pages, arXiv:0711.4546.
  6. Fabi S., Harms B., Stern A., Noncommutative corrections to the Robertson-Walker metric, Phys. Rev. D 78 (2008), 065037, 7 pages, arXiv:0808.0943.
  7. Calmet X., Kobakhidze A., Second order noncommutative corrections to gravity, Phys. Rev. D 74 (2006), 047702, 3 pages, hep-th/0605275.
  8. Banerjee R., Mukherjee P., Samanta S., Lie algebraic noncommutative gravity, Phys. Rev. D 75 (2007), 125020, 7 pages, hep-th/0703128.
  9. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  10. Chamseddine A.H., Deforming Einstein's gravity, Phys. Lett. B 504 (2001), 33-37, hep-th/0009153.
  11. Chamseddine A.H., SL(2,C) gravity with a complex vierbein and its noncommutative extension, Phys. Rev. D 69 (2004), 024015, 8 pages, hep-th/0309166.
  12. Stern A., Particle classification and dynamics in GL(2,C) gravity, Phys. Rev. D 79 (2009), 105017, 16 pages, arXiv:0903.0882.
  13. Aschieri P., Blohmann C., Dimitrijevic M., Meyer F., Schupp P., Wess J., A gravity theory on noncommutative spaces, Classical Quantum Gravity 22 (2005), 3511-3532, hep-th/0504183.
  14. Schupp P., Solodukhin S., Exact black hole solutions in noncommutative gravity, arXiv:0906.2724.
  15. Ohl T., Schenkel A., Cosmological and black hole spacetimes in twisted noncommutative gravity, J. High Energy Phys. 2009 (2009), no. 10, 052, 12 pages, arXiv:0906.2730.
  16. Dolan B.P., Gupta K.S., Stern A., Noncommutative BTZ black hole and discrete time, Classical Quantum Gravity 24 (2007), 1647-1655, hep-th/0611233.
    Dolan B.P., Gupta K.S., Stern A., Noncommutativity and quantum structure of spacetime, J. Phys. Conf. Ser. 174 (2009), 012023, 6 pages.
  17. Groenewold H.J., On the principles of elementary quantum mechanics, Physica 12 (1946), 405-460.
  18. Moyal J.E., Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99-124.
  19. Utiyama R., Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956), 1597-1607.
  20. Kibble T.W.B., Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961), 212-221.
  21. Hehl F.W., Von Der Heyde P., Kerlick G.D., Nester J.M., General relativity with spin and torsion: foundations and prospects, Rev. Modern Phys. 48 (1976), 393-416.
  22. Chamseddine A.H., Applications of the gauge principle to gravitational interactions, Int. J. Geom. Methods Mod. Phys. 3 (2006), 149-176, hep-th/0511074.
  23. Isham C.J., Salam A., Strathdee J.A., SL6,C gauge invariance of Einstein like Lagrangians, Lett. Nuovo Cimento 5 (1972), 969-972.
  24. Jurco B., Schraml S., Schupp P., Wess J., Enveloping algebra valued gauge transformations for non-abelian gauge groups on non-commutative spaces, Eur. Phys. J. C Part. Fields 17 (2000), 521-526, hep-th/0006246.
  25. Bonora L., Schnabl M., Sheikh-Jabbari M.M., Tomasiello A., Noncommutative SO(n) and Sp(n) gauge theories, Nuclear Phys. B 589 (2000), 461-474, hep-th/0006091.
  26. Marculescu S., Ruiz Ruiz F., Seiberg-Witten maps for SO(1,3) gauge invariance and deformations of gravity, Phys. Rev. D 79 (2009), 025004, 18 pages, arXiv:0808.2066.
  27. Davis A.-C., Lilley M., Törnkvist O., Relaxing the bounds on primordial magnetic seed fields, Phys. Rev. D 60 (1999), 021301, 5 pages, astro-ph/9904022.
  28. Mazumdar A., Sheikh-Jabbari M.M., Noncommutativity in space and primordial magnetic field, Phys. Rev. Lett. 87 (2001), 011301, 4 pages, hep-ph/0012363.
  29. Nasseri F., Schwarzschild black hole in noncommutative spaces, Gen. Relativity Gravitation 37 (2005), 2223-2226, hep-th/0508051.
  30. Nozari K., Fazlpour B., Reissner-Nordström black hole thermodynamics in noncommutative spaces, Acta Phys. Polon. B 39 (2008), 1363-1374.
  31. Banerjee R., Majhi B.R., Samanta S., Noncommutative black hole thermodynamics, Phys. Rev. D 77 (2008), 124035, 8 pages, arXiv:0801.3583.
  32. Banerjee R., Majhi B.R., Modak S.K., Noncommutative Schwarzschild black hole and area law, Classical Quantum Gravity 26 (2009), 085010, 11 pages, arXiv:0802.2176.

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