Skip to main content
SpringerLink
Log in

Cylindrically Symmetric Scalar Field and it’s Lyapunov stability in General Relativity

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In this paper we found an Exact solution for massless scalar field with cosmological constant. This exact solution generalized the Levi-Civita vacuum solution Levi-Civita (Rend. Acc. Lincei 27:183, 1917) to a massless scalar field, with a cosmological constant term.This solution in the absence of the Cosmological constant recovers the spacetime of a massless scalar field with cylindrical symmetry (Buchdahl metric (Buchdahl in Phys. Rev. 115:1325, 1959)). Also if the scalar field disappears, the spacetime will be a representation of de-Sitter space.We prove that the form of the metric’s function which was purposed in Momeni and Miraghaei (Int. J. Mod. Phys. A 24(31):5991, 2009) is valid even if we assume a general form. Furthermore we show that in which conditions this solution satisfies energy conditions. Finally the credibility of focusing theorem is proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Momeni, D., Miraghaei, H.: Int. J. Mod. Phys. A (IJMPA) 24(31), 5991–6000 (2009). doi:10.1142/S0217751X09046369. arXiv:0903.5171

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Buchdahl, H.A.: Phys. Rev. 115, 1325–1328 (1959)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Zofka, M., Bicak, J.: Class. Quantum Gravity 25, 015011 (2008)

    Article  MathSciNet  Google Scholar 

  4. Kandalkar, S.P., Gawande, S.P.: Astrophys. Space Sci. 318, 263–267 (2008)

    Article  ADS  Google Scholar 

  5. Senovilla, J.M.M.: Cylindrically symmetric dust spacetime. Class. Quantum Gravity 17, 2843–2846 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Vuille, C.: Gen. Relativ. Gravit. 39, 621–632 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Stephani, H., Kramer, D., MacCallum, M.A.H., Hoenselaers, C., Herlt, E.: Exact solutions of Einstein’s field equations, 2nd edn. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  8. Levi-Civita, T.: Rend. Acc. Lincei 27, 183 (1917)

    Google Scholar 

  9. Wang, A.Z.: Phys. Rev. D 68 064006 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  10. Frankel, T.: Gravitational Curvature, an Introduction to Einstein’s Theory. W.H. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  11. Linet, B.: J. Math. Phys. 27, 1817 (1986)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Tian, Q.: Phys. Rev. D 33, 3549 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  13. Visser, M., Barcelo, C.: Energy conditions and their cosmological implications. arXiv:gr-qc/0001099 (2000)

  14. Barcelo, C., Visser, M.: Twilight for the energy conditions? Int. J. Mod. Phys. D 11, 1553 (2002). arXiv:gr-qc/0205066

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Visser, M.: Lorentzian Wormholes: From Einstein to Hawking. AIP Press/Springer, New York (1995)

    Google Scholar 

  16. Santos, J., Alcaniz, J.S., Reboucas, M.J.: Energy conditions and supernovae observations. Phys. Rev. D 74, 067301 (2006). arXiv:astro-ph/0608031

    Article  ADS  Google Scholar 

  17. Astier, P., et al.: Astron. Astrophys. 447, 31 (2006). arXiv:astro-ph/0510447

    Article  ADS  Google Scholar 

  18. SNLS: Super Nova Legacy Survey. http://legacy.astro.utoronto.ca/ http://cfht.hawaii.edu/SNLS/. Electronic data available at http://snls.in2p3.fr/conf/papers/cosmo1/

  19. Riess, A.G., et al.: New Hubble space telescope discoveries of Type Ia Supernovae at z>1: narrowing constraints on the early behavior of dark energy. arXiv:astro-ph/0611572. Electronic data available at http://braeburn.pha.jhu.edu//ariess/R06/

  20. Santos, J., Alcaniz, J.S., Reboucas, M.J., Pires, N.: Lookback time bounds from energy conditions. Phys. Rev. D 76, 043519 (2007). arXiv:0706.1779 [astro-ph]

    Article  MathSciNet  ADS  Google Scholar 

  21. Santos, J., Alcaniz, J.S., Pires, N., Reboucas, M.J.: Energy conditions and cosmic acceleration. Phys. Rev. D 75, 083523 (2007). arXiv:astro-ph/0702728

    Article  MathSciNet  ADS  Google Scholar 

  22. Gong, Y.G., Wang, A.: Phys. Rev. D 75, 043520 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  23. Astier, P.: Phys. Lett. B 500, 8 (2001)

    Article  ADS  Google Scholar 

  24. Huterer, D., Turner, M.S.: Phys. Rev. D 64, 123527 (2001)

    Article  ADS  Google Scholar 

  25. Weller, J., Albrecht, A.: Phys. Rev. D 65, 103512 (2002)

    Article  ADS  Google Scholar 

  26. Alam, U., Sahni, V., Saini, T.D., Starobinsky, A.A.: Mon. Not. R. Astron. Soc. 354, 275 (2004)

    Article  ADS  Google Scholar 

  27. Gong, Y.G.: Class. Quantum Gravity 22, 2121 (2005)

    Article  MATH  ADS  Google Scholar 

  28. Gong, Y.G., Zhang, Y.Z.: Phys. Rev. D 72, 043518 (2005)

    Article  ADS  Google Scholar 

  29. Chevallier, M., Polarski, D.: Int. J. Mod. Phys. D 10, 213 (2001)

    Article  ADS  Google Scholar 

  30. Linder, E.V.: Phys. Rev. Lett. 90, 091301 (2003)

    Article  ADS  Google Scholar 

  31. Weller, J., Albrecht, A.: Phys. Rev. Lett. 86, 1939 (2001)

    Article  ADS  Google Scholar 

  32. Corasaniti, P.S., Copeland, E.J.: Phys. Rev. D 67, 063521 (2003)

    Article  ADS  Google Scholar 

  33. Lee, S.: Phys. Rev. D 71, 123528 (2005)

    Article  ADS  Google Scholar 

  34. Wetterich, C.: Phys. Lett. B 594, 17 (2004)

    Article  ADS  Google Scholar 

  35. Alam, U., Sahni, V., Starobinsky, A.A., Cosmol, J.: Astropart. Phys. 06, 008 (2004)

    Google Scholar 

  36. Gong, Y.G.: Int. J. Mod. Phys. D 14, 599 (2005)

    Article  MATH  ADS  Google Scholar 

  37. Szydlowski, M.: Int. J. Mod. Phys. A 20, 2443 (2005)

    Article  ADS  Google Scholar 

  38. Wang, B., Gong, Y.G., Su, R.-K.: Phys. Lett. B 605, 9 (2005)

    Article  ADS  Google Scholar 

  39. Easson, D.A.: Phys. Lett. B 0702, 004 (2007)

    Google Scholar 

  40. Easson, D.A.: Int. J. Mod. Phys. A 19, 5343 (2004)

    Article  MATH  ADS  Google Scholar 

  41. Li, H., et al.: astro-ph/0612060 (2006)

  42. Zhao, G.-Z., et al.: astro-ph/0612728 (2006)

  43. Zhang, X., Wu, F.-Q.: astro-ph/0701405 (2007)

  44. Xu, L.-I., Zhang, C.-W., Chang, B.-R., Liu, H.-Y.: astro-ph/0701519 (2007)

  45. Wei, H., Tang, N.-N., Zhang, S.N.: Phys. Rev. D 75, 043009 (2007)

    Article  ADS  Google Scholar 

  46. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Spacetime, pp. 88–96. Cambridge University Press, Cambridge (1973)

    Book  Google Scholar 

  47. Gong, Y.G., Wang, A.Z.: Phys. Lett. B 652, 63–68 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  48. Bertolami, O., Carvalho Sequeira, M.: Phys. Rev. D 79, 104010 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  49. Santos, J., Reboucas, M.J., Alcaniz, J.S.: Int. J. Mod. Phys. D 19, 1315–1321 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Setare, M.R., Momeni, D.: Geodesic stability for KS Black hole in Horava-Lifshitz gravity via Lyapunov exponents. arXiv:1001.3767v1 [physics.gen-ph] (2001)

  51. Cardoso, V., Miranda, A.S., Berti, E., Witek, H., Zanchin, V.T.: Phys. Rev. D 79, 064016 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  52. Mehrpooya, M., Momeni, D.: Spherically symmetric massive scalar fields in GR. Int. J. Mod. Phys. A 25(7) (2010)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Islamic Azad University, Karaj branch, P.O. Box 31485-313, Karaj, Iran

    H. R. Rezazadeh

Authors
  1. H. R. Rezazadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. R. Rezazadeh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rezazadeh, H.R. Cylindrically Symmetric Scalar Field and it’s Lyapunov stability in General Relativity. Int J Theor Phys 50, 208–217 (2011). https://doi.org/10.1007/s10773-010-0509-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-010-0509-9

Keywords

Search

Navigation