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Geometrical origin of the cosmological constant

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Abstract

We show that the description of the space-time of general relativity as a diagonal four dimensional submanifold immersed in an eight dimensional hypercomplex manifold, in torsionless case, leads to a geometrical origin of the cosmological constant. The cosmological constant appears naturally in the new field equations and its expression is given as the norm of a four-vector U, i.e., Λ = 6g μν U μ U ν and where U can be determined from the Bianchi identities. Consequently, the cosmological constant is space-time dependent, a Lorentz invariant scalar, and may be positive, negative or null. The resulting energy momentum tensor of the dark energy depends on the cosmological constant and its first derivative with respect to the metric. As an application, we obtain the spherical solution for the field equations. In cosmology, the modified Friedmann equations are proposed and a condition on Λ for an accelerating universe is deduced. For a particular case of the vector U, we find a decaying cosmological constant \({\Lambda\propto a(t)^{-6\alpha}}\).

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References

  1. Riess A.G. et al.: Astron. J 116, 1009 (1998)

    Article  ADS  Google Scholar 

  2. Perlmutter S. et al.: Astrophys. J 517, 565 (1999)

    Article  ADS  Google Scholar 

  3. Knop R.A. et al.: Ap. J 598, 102 (2003)

    Article  ADS  Google Scholar 

  4. Riess A.G. et al.: Ap. J 607, 665 (2004)

    Article  ADS  Google Scholar 

  5. Bennet C.L. et al.: Ap. J. Suppl. Ser. 148, 1 (2003)

    Article  ADS  Google Scholar 

  6. Weinberg S.: Rev. Mod. Phys. 61, 1 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Carroll, S.M.: Living Rev. Relativ. 3, 1 (2001). http://www.livingreviews.org/lrr-2001-1

  8. Padmanabhan T.: Phys. Rep. 380, 235 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Peebles P.J.E., Ratra B.: Rev. Mod. Phys. 75, 559 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Nobbenhuis, S.: Found. Phys. 36, 613 (2006). arXiv:gr-qc/0411093v3

  11. Narlikar J.V., Pecker J.C., Vigier J.P.: J. Astrophys. Astr 12, 7 (1991)

    Article  ADS  Google Scholar 

  12. Tiwari R.N., Ray S.: Ind. J. Pure. Appl. Math. 27, 207 (1996)

    MathSciNet  Google Scholar 

  13. Tiwari R.N., Ray S., Bhadra S.: Ind. J. Pure. Appl. Math. 31, 1017 (2000)

    MATH  Google Scholar 

  14. Dymnikova I.: Class. Quant. Grav. 19, 725 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Dymnikova I.: Grav. Cosmol. Suppl. 8(N1), 131 (2002)

    MathSciNet  Google Scholar 

  16. Dymnikova I.: Class. Quant. Grav. 20, 3797 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Peebles P.J.E., Ratra B.: Astrophys. J 325, L17 (1988)

    Article  ADS  Google Scholar 

  18. Sakharov, A.D.: Doklady Akad. Nauk. SSSR, 177, 70 (Trans: 1968, Soviet Phys. Doklady, 12, 1040) (1968)

  19. Al-Rawaf A.S., Taha M.O.: Gen. Relativ. Gravit. 28, 8 (1996)

    Article  MathSciNet  Google Scholar 

  20. Al-Rawaf A.S., Taha M.O.: Phys. Lett. B 366, 69 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  21. Abdel-Rahman A.M.M.: Gen. Relativ. Gravit. 29, 10 (1997)

    Article  MathSciNet  Google Scholar 

  22. Majernik V.: Phys. Lett. A 282, 362 (2001)

    Article  ADS  MATH  Google Scholar 

  23. Majernik V.: Gen. Relativ. Gravit. 35, 6 (2003)

    Google Scholar 

  24. Crumeyrolle, A.: Ann. de la Fac. des Sciences de Toulouse, 4e série, 26, 105 (1962). http://www.numdam.org

  25. Crumeyrolle, A.: Ann. de la Fac. des Sciences de Toulouse, 4e série, 29, 53 (1965). http://www.numdam.org

  26. Crumeyrolle A.: Riv. Mat. Univ. Parma. 5(2), 85 (1964)

    MathSciNet  MATH  Google Scholar 

  27. Clerc, R.L.: Ann. de L’I.H.P. Section A 12(4), 343 (1970). http://www.numdam.org

  28. Clerc, R.L.: Ann. de L’I.H.P. Section A 17(3), 227 (1972). http://www.numdam.org

  29. Clerc, R.L.: Ann. de L’I.H.P. Section A 17(3), 259 (1972). http://www.numdam.org

  30. Kunstatter G., Moffat J.W., Malzan J.: J. Math. Phys. 24, 886 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Moffat J.W.: J. Math. Phys. 25, 347 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  32. Mann R.B., Moffat J.W.: Phys. Rev. D 31, 2488 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  33. Einstein A.: Ann. Math. 46, 578 (1945)

    Article  MATH  Google Scholar 

  34. Einstein A.: Rev. Mod. Phys. 20, 35 (1948)

    Article  ADS  Google Scholar 

  35. Moffat J.W.: Phys. Rev. D 19, 3554 (1979)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Schuller F., Wohlfarth M., Grimm T.W.: Class. Quant. Grav. 20, 4269 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. Hess, P.O., Greiner, W.: Int. J. Mod. Phys. E 16, 1643 (2007). arXiv:0705.1233v1 [hep-th]

  38. Lovelock D.: Annali di Matematica Pura 83, 43 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lovelock D.: Annali di Matematica Pura 83, 53 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  40. Chamseddine, A.H.: arXiv:hep-th/0610099v1

  41. Mantz, C., Prokopec, T.: arXiv:0804.0213v1 [gr-qc]

  42. Hess P.O., Greiner W.: Int. J. Mod. Phys. E 18, 51 (2009) . arXiv:0812.1738v1 [gr-qc]

    Article  ADS  Google Scholar 

  43. Hess, P.O., Maghlaoui, L., Greiner, W.: Int. J. Mod. Phys. E 19, 1315 (2010). arXiv:1001.5208v1 [gr-qc]

  44. Hucks J.: J. Math. Phys 34, 5986 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Yaglom I.M.: A simple non-Euclidean geometry and its physical basis. Springer, Berlin (1979)

    MATH  Google Scholar 

  46. Cruceanu, V., Gadea, P.M., Masqué, J.M.: Para-Hermitian and Para-Kahler manifolds. In: Cruceanu, V. (ed.) Selected Papers. Editura PIM, Iasi (2006)

  47. Kantor I.L., Solodovnikov A.S.: Hypercomplex Numbers, An Elementary Introduction to Algebra. Springer, Berlin (1989)

    Google Scholar 

  48. Ulrych S.: Phys. Lett. B 625, 313 (2005) arXiv:physics/0009079v3 [physics.gen-ph]

    Article  MathSciNet  ADS  Google Scholar 

  49. Ulrych S.: Phys. Lett. B 633, 631 (2006) arXiv:gr-qc/0602018v1

    Article  MathSciNet  ADS  Google Scholar 

  50. Fjelstad P.: Am. J. Phys 54, 416 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  51. Capelli P.F.: Bull. Math. Soc 47, 585 (1941)

    Article  MathSciNet  Google Scholar 

  52. Olariu, S.: arXiv:math/0008119v1 [math.CV]

  53. Antonuccio, F.: arXiv:hep-th/9812036v1

  54. Volick D.: Phys. Rev D 68, 068510 (2003)

    Article  Google Scholar 

  55. Punzi, R., Schuller, F., Wohlfarth, M.: JHEP 0702/030 (2007)

  56. Schuller F., Wohlfarth M.: JCAP 0712, 013 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  57. Schuller F.: Ann. Phys. (NY) 299, 174 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. Schuller F.: Phys. Lett. B 540, 119 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. Yano K.: Differential Geometry on Complex and Almost Complex Spaces. Pergamon Press, Oxford (1965)

    MATH  Google Scholar 

  60. Guth A.H.: Phys. Rev. D 23, 347 (1981)

    Article  ADS  Google Scholar 

  61. Ozer M., Taha M.O.: Nucl. Phys. B 287, 776 (1987)

    Article  ADS  Google Scholar 

  62. Chen W., Wu Y.: Phys. Rev. D 41, 695 (1990)

    Article  ADS  Google Scholar 

  63. Lopez J., Nanopoulos D.V.: Mod. Phys. Lett. A 11, 1 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. Ozer M., Taha M.O.: Mod. Phys. Lett. A 13, 571 (1998)

    Article  ADS  Google Scholar 

  65. John M.V., Joseph K.B.: Phys. Rev. D 61, 087304 (2000)

    Article  ADS  Google Scholar 

  66. Mbonye M.R.: Int. J. Mod. Phys. A 18, 811 (2003)

    Article  ADS  MATH  Google Scholar 

  67. Choi J.R.: Chin. Phys. C 35, 233 (2011)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Laboratory of Theoretical Physics, Department of Physics, University of Jijel, BP 98, Ouled Aissa, 18000, Jijel, Algeria

    H. Azri & A. Bounames

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  1. H. Azri
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  2. A. Bounames
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Correspondence to A. Bounames.

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Azri, H., Bounames, A. Geometrical origin of the cosmological constant. Gen Relativ Gravit 44, 2547–2561 (2012). https://doi.org/10.1007/s10714-012-1413-9

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