Skip to main content
SpringerLink
Log in

Geometric properties of the Kantowski-Sachs and Bianchi-type Killing algebra in relation to a Klein-Gordon equation

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract.

We study the geometric properties of generators for the Klein-Gordon equation in Kantowski-Sachs and certain Bianchi-type spaces. Several versions of the Klein-Gordon equation are derived from its dependence on a potential function. The criteria for different versions of the (1+3) Klein-Gordon equation originates from analyzing three sources, viz. through generators that are identically the Killing algebra, or with the Killing vector fields that are recast into linear combinations and thirdly, real sub-algebras within the conformal algebra. In turn, these equations admit a catalogue of infinitesimal symmetries that are equivalent to the corresponding Killing vector fields in Kantowski-Sachs, Bianchi type III, IX, VIII, VI0 and VII0 space-times, with the exception of a linear vector \(W=u\partial_{u}\) in every case. The sheer number of results are displayed in appropriate tables. Subsequently, in application, we derive some Noetherian conservation laws and identify some exact solutions by quadratures.

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. S. Jamal, A.H. Kara, R. Narain, J. Appl. Math. 2012, 1 (2012)

    Article  Google Scholar 

  2. S. Jamal, A.H. Kara, R. Narain, G. Shabbir, Indian J. Phys. 89, 411 (2015)

    Article  ADS  Google Scholar 

  3. S. Capozzielo, G. Marmo, C. Rubano, P. Scudellaro, Int. J. Mod. Phys. D 6, 491 (1997)

    Article  ADS  Google Scholar 

  4. A. Paliathanasis, L. Karpathopoulos, A. Wojnar, S. Capozziello, Eur. Phys. J. C 76, 225 (2016)

    Article  ADS  Google Scholar 

  5. B.S. De Witt, Phys. Rev. 160, 113 (1967)

    ADS  Google Scholar 

  6. J.M. Lee, T.H. Parker, Bull. (NS) Am. Math. Soc. 17, 37 (1987)

    Article  Google Scholar 

  7. E. Noether, Nachr. Akad. Wiss. G. Math.-Phys. Klasse 2, 235 (1918)

    Google Scholar 

  8. S. Lie, Theorie der Transformationsgruppen (Teubner, Leipzig, 1893)

  9. P. Olver, Application of Lie Groups to Differential Equations (Springer, New York, 1993)

  10. S. Capozziello, R. De Ritis, C. Rubano, P. Scudellaro, Riv. Nuovo Cimento 19, 1 (1996)

    Article  MathSciNet  Google Scholar 

  11. G.W. Bluman, S. Kumei, Symmetries and Differential Equations (Springer-Verlag, New York, 1989)

  12. P.E. Hydon, Symmetry Methods for Differential Equations (Cambridge University Press, Cambridge, 2000)

  13. N.H. Ibragimov (Editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1: Symmetries, Exact Solutions and Conservation Laws (CRC Press, Boca Raton, 1994)

  14. S. Jamal, A.H. Kara, Nonlinear Dyn. 67, 97 (2012)

    Article  Google Scholar 

  15. G. Cicogna, F. Ceccherini, F. Pedorado, SIGMA 2 017, 1 (2006)

    Google Scholar 

  16. N.M. Ivanova, C. Sophocleous, P.G.L. Leach, J. Math. Anal. Appl. 372, 273 (2010)

    Article  MathSciNet  Google Scholar 

  17. M.C. Nucci, G. Sanchini, Symmetry 7, 1613 (2015)

    Article  MathSciNet  Google Scholar 

  18. A. Aslam, M. Jamil, D. Momeni, R. Myrzakulov, M.A. Rashid, M. Raza, Astrophys. Space Sci. 348, 533 (2013)

    Article  ADS  Google Scholar 

  19. S. Capozziello, E. Piedipalumbo, C. Rubano, P. Scudellaro, Phys. Rev. D 80, 104030 (2009)

    Article  ADS  Google Scholar 

  20. S. Basilakos, S. Capozziello, M. De Laurentis, A. Paliathanasis, M. Tsamparlis, Phys. Rev. D 88, 103526 (2013)

    Article  ADS  Google Scholar 

  21. Y. Kucukakca, Eur. Phys. J. C 74, 3086 (2014)

    Article  ADS  Google Scholar 

  22. M. Jamil, S. Ali, D. Momeni, R. Myrzakulov, Eur. Phys. J. C 72, 1998 (2012)

    Article  ADS  Google Scholar 

  23. S. Capozziello, R. de Ritis, P. Scudellaro, Int. J. Mod. Phys. D 2, 465 (1993)

    ADS  Google Scholar 

  24. B. Vakili, Phys. Lett. B 669, 209 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  25. S. Capozziello, A. De Felice, JCAP 08, 016 (2008)

    Article  ADS  Google Scholar 

  26. M.C. Kweyama, K.S. Govinder, S.D. Maharaj, Class. Quantum Grav. 18, 105005 (2011)

    Article  ADS  Google Scholar 

  27. K. Yagdijian, A. Galstian, Commun. Math. Phys. 285, 293 (2009)

    Article  ADS  Google Scholar 

  28. U. Camci, S. Jamal, A.H. Kara, Int J. Theor. Phys. 53, 1483 (2014)

    Article  Google Scholar 

  29. A. Paliathanasis, M. Tsamparlis, Int. J. Geom. Methods Mod. Phys. 11, 14500376 (2014)

    Article  Google Scholar 

  30. A. Paliathanasis, M. Tsamparlis, J. Geom. Phys. 62, 2443 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  31. A.H. Bokhari, A.Y. Al-Dweik, A.H. Kara, M. Karim, F.D. Zaman, J. Math. Phys. 52, 063511 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  32. Y. Bozhkov, I.L. Freire, J. Differ. Equ. 249, 872 (2010)

    Article  ADS  Google Scholar 

  33. K. Yagdijian, A. Galstian, Disc. Cont. Dyn. Syst. S 2, 483 (2009)

    Article  Google Scholar 

  34. S. Jamal, A.H. Kara, A.H. Bokhari, Can. J. Phys. 90, 667 (2012)

    Article  Google Scholar 

  35. S. Jamal, G. Shabbir, Int. J. Geom. Methods Mod. Phys. 13, 1650109 (2016)

    Article  MathSciNet  Google Scholar 

  36. S. Jamal, A. Paliathanasis, Group invariant transformations for the Klein-Gordon equation in three dimensional flat spaces, submitted to J. Geom. Phys. (2016)

  37. A. Paliathanasis, M. Tsamparlis, M.T. Mustafa, Int. J. Geom. Meth. Mod. Phys. 12, 1550033 (2015)

    Article  Google Scholar 

  38. H. Stephani, Differential Equations: their solution using symmetries (Cambridge University Press, Cambridge, 1989)

  39. G.H. Katzin, J. Levine, W.R. Davis, J. Math. Phys. 10, 617 (1969)

    Article  ADS  Google Scholar 

  40. H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselears, E. Herlt, Exact Solutions of Einsteins Field Equations (Cambridge University Press, Cambridge, 2003)

  41. G. Shabbir, A.B. Mehmood, Mod. Phys. Lett. A 22, 807 (2007)

    Article  ADS  Google Scholar 

  42. G. Shabbir, M. Rafiq, Univ. Pol. Bucharest Sci. Bull. Ser. A, Appl. Math. Phys. 72, 121 (2010)

    Google Scholar 

  43. G. Shabbir, A. Ali, Adv. Stud. Theor. Phys. 4, 535 (2010)

    MathSciNet  Google Scholar 

  44. M. Tsamparlis, A. Paliathanasis, Gen. Relativ. Gravit. 43, 1861 (2011)

    Article  ADS  Google Scholar 

  45. S. Jamal, Int. J. Geom. Methods Mod. Phys. 0, 1750070 (2017)

    Article  Google Scholar 

  46. S. Jamal, A.H. Kara, Pramana J. Phys. 77, 1 (2011)

    Article  Google Scholar 

  47. A. Paliathanasis, M. Tsamparlis, S. Basilakos, J.D. Barrow, Phys. Rev. D 91, 123535 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  48. S. Basilakos, J.D. Barrow, Phys. Rev. D 91, 103517 (2015)

    Article  ADS  Google Scholar 

  49. S. Capozziello, M. De Laurentis, S.D. Odintsov, Eur. Phys. J. C 72, 2068 (2012)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits 2001, Johannesburg, South Africa

    Sameerah Jamal

  2. Faculty of Engineering Sciences GIK Institute of Engineering Sciences and Technology, Topi, Swabi, KPK, Pakistan

    Ghulam Shabbir

Authors
  1. Sameerah Jamal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ghulam Shabbir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sameerah Jamal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jamal, S., Shabbir, G. Geometric properties of the Kantowski-Sachs and Bianchi-type Killing algebra in relation to a Klein-Gordon equation. Eur. Phys. J. Plus 132, 70 (2017). https://doi.org/10.1140/epjp/i2017-11375-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2017-11375-2

Search

Navigation