Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2017, Article ID 4384093, 40 pages
https://doi.org/10.1155/2017/4384093
Research Article

Geometrical/Physical Interpretation of the Conserved Quantities Corresponding to Noether Symmetries of Plane Symmetric Space-Times

1School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan
2Departamento de Matemáticas, Facultad de Ciencias, Pontificia Universidad Javeriana, Cra. 7 N. 40-62, Bogotá, Colombia

Correspondence should be addressed to Bismah Jamil; moc.liamg@limaj.hamsib

Received 23 February 2017; Revised 26 April 2017; Accepted 21 May 2017; Published 30 August 2017

Academic Editor: Igor L. Freire

Copyright © 2017 Bismah Jamil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. P. Eisenhart, Continuous Groups of Transformations, Princeton University Press, 1933. View at Scopus
  2. D. Kramer, H. Stephani, E. Herlt, and M. MacCallum, Exact Solutions of Einstein’s Field Equations, Cambridge University Press, Cambridge, UK, 1980. View at MathSciNet
  3. T. Feroze, F. M. Mahomed, and A. Qadir, “The connection between isometries and symmetries of geodesic equations of underlying spaces,” Nonlinear Dynamics, vol. 45, no. 1, pp. 65–74, 2006. View at Google Scholar · View at MathSciNet
  4. Y. Bozhkov and I. L. Freire, “Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation,” Journal of Differential Equations, vol. 249, pp. 872–913, 2010. View at Google Scholar · View at MathSciNet
  5. A. Aslam and A. Qadir, “Noether symmetries of the area-minimizing lagrangian,” Journal of Applied Mathematics, vol. 2012, Article ID 532690, 14 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  6. A. Paliathanasis and M. Tsamparlis, “The geometric origin of Lie point symmetries of the Schrodinger equation and the Klein–Gordon equations,” International Journal of Geometric Methods in Modern Physics, vol. 11, no. 4, Article ID 1450037, 17 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  7. A. H. Bokhari and A. Qadir, “Symmetries of static, spherically symmetric spacetimes,” Journal of Mathematical Physics, vol. 28, no. 5, pp. 1019–1022, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. Ziad, Spherically Symmetric Spacetimes [Ph.D. thesis], Quaid-i-Azam University, Islamabad, 1990.
  9. A. Qadir and M. Ziad, “The classification of spherically symmetric spacetimes,” II Nuovo Cimento B, vol. 110, no. 3, pp. 317–334, 1995. View at Google Scholar · View at MathSciNet
  10. T. B. Farid, A. Qadir, and M. Ziad, “The classification of static plane symmetric spacetimes according to their Ricci collineations,” Journal of Mathematical Physics, vol. 36, no. 10, pp. 5812–5828, 1995. View at Google Scholar · View at MathSciNet
  11. A. H. Bokhari and A. R. Kashif, “Curvature collineations of some static spherically symmetric spacetimes,” Journal of Mathematical Physics, vol. 37, no. 7, pp. 3498–3504, 1996. View at Google Scholar · View at MathSciNet
  12. D. Ahmed and M. Ziad, “Homothetic motions of spherically symmetric spacetimes,” Journal of Mathematical Physics, vol. 38, no. 5, pp. 2547–2552, 1997. View at Google Scholar · View at MathSciNet
  13. T. Feroze, A. Qadir, and M. Ziad, “The classification of plane symmetric spacetimes by isometries,” Journal of Mathematical Physics, vol. 42, no. 10, pp. 4947–4955, 2001. View at Google Scholar · View at MathSciNet
  14. M. Ziad and S. Ehsan, “Classification of plane symmetric lorentzian manifolds according to their homotheties and Classification of plane symmetric lorentzian manifolds according to their homotheties and metrics,” IL Nuovo Cimento, vol. 123, no. 1, pp. 71–83, 2008. View at Google Scholar · View at MathSciNet
  15. A. Qadir and K. Saifullah, “Comments on matter collineations of plane symmetric, cylindrically symmetric and spherically symmetric spacetimes,” Journal of Mathematical Physics, vol. 45, no. 11, pp. 4191-4192, 2004. View at Google Scholar · View at MathSciNet
  16. M. Sharif, “Classification of static plane symmetric spacetimes according to their matter collineations,” Journal of Mathematical Physics, vol. 45, no. 4, pp. 1518–1531, 2004. View at Google Scholar · View at MathSciNet
  17. M. Tsamparlis and A. Paliathanasis, “Lie and Noether symmetries of geodesic equations and collineations,” General Relativity and Gravitation, vol. 42, no. 12, pp. 2957–2980, 2010. View at Google Scholar · View at MathSciNet
  18. M. Tsamparlis, A. Paliathanasis, and L. Karpathopoulos, “Autonomous three-dimensional Newtonian systems which admit Lie and Noether point symmetries,” Journal of Physics. A. Mathematical and Theoretical, vol. 45, no. 27, Article ID 275201, 12 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. H. Azad, A. Y. Al-Dweik, R. Ghanam, and M. T. Mustafa, “Symmetry analysis of wave equation on static spherically symmetric spacetimes with higher symmetries,” Journal of Mathematical Physics, vol. 54, no. 6, Article ID 063509, 23 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. Tsamparlis, “Geometrization of Lie and Noether symmetries with applications in Cosmology,” Journal of Physics: Conference Series, vol. 453, no. 1, Article ID 012020, 2013. View at Publisher · View at Google Scholar · View at Scopus
  21. M. T. Mustafa and A. Y. Al-Dweik, “Noether symmetries and conservation laws of wave equation on static spherically symmetric spacetimes with higher symmetries,” Communications in Nonlinear Science and Numerical Simulation, vol. 23, no. 1-3, pp. 141–152, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  22. M. Tsamparlis, A. Paliathanasis, and A. Qadir, “Noether symmetries and isometries of the minimal surface Lagrangian under constant volume in a Riemannian space,” International Journal of Geometric Methods in Modern Physics, vol. 12, no. 1, Article ID 1550003, 10 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  23. M. Tsamparlis, “Geometrization of Lie and Noether symmetries and applications,” International Journal of Modern Physics: Conference Series, vol. 38, Article ID 1560078, 20 pages, 2015. View at Publisher · View at Google Scholar
  24. E. Buhe, G. Bluman, and A. H. Kara, “Conservation laws for some systems of nonlinear PDEs via the symmetry/adjoint symmetry pair method,” Journal of Mathematical Analysis and Applications, vol. 436, no. 1, pp. 94–103, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  25. H. Stephani and M. MacCallum, Differential Equations—Their Solution Using Symmetries, Cambridge University Press, Cambridge, UK, 1989. View at Publisher · View at Google Scholar
  26. P. E. Hydon, Symmetry Methods for Differential Equations: A Beginner’s Guide, Cambridge University Press, Cambridge, UK, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  27. N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, Chichester, UK, 1999. View at MathSciNet
  28. E. Noether, “Invariante variations probleme,” Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-Phys. Klasse, pp. 235–257, 1918, (see Transport Theory and Statistical Physics, vol. 1, no. 3, pp. 186-207, 1971 for an English translation). View at Google Scholar · View at MathSciNet
  29. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer Science & Business Media, New York, NY, USA, 2000. View at MathSciNet
  30. M. Tsamparlis and A. Paliathanasis, “The geometric nature of Lie and Noether symmetries,” General Relativity and Gravitation, vol. 43, no. 6, pp. 1861–1881, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. G. S. Hall, Symmetries and Curvature Structure in General Relativity, World Scientific, Singapore, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  32. M. Ziad and S. Ehsan, “Classification of plane symmetric lorentzian manifolds according to their homotheties and metrics,” II Nuovo Cimento B, vol. 123, no. 1, pp. 71–83, 2008. View at Google Scholar · View at MathSciNet
  33. F. Ali and T. Feroze, “Classification of plane symmetric static spacetimes according to their noether symmetries,” International Journal of Theoretical Physics, vol. 52, no. 9, pp. 3329–3342, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  34. B. Jamil and T. Feroze, “Classification of a class of plane symmetric static spacetimes according to the noether symmetries,” in Proceedings of the Thirteenth Marcel Grossmann Meeting on General Relativity, vol. 1, pp. 1840–1843, 2015. View at MathSciNet
  35. I. Hussain, F. M. Mahomed, and A. Qadir, “Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy,” General Relativity and Gravitation, vol. 41, no. 10, pp. 2399–2414, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. W.-H. Steeb, Ontinuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, World Scientific Publishing Co Inc, 2007. View at MathSciNet
  37. A. M. Ahmad, A. H. Bokhari, and F. D. Zaman, “Symmetries, conservation laws and wave equation on the Milne metric,” Journal of Applied Mathematics, vol. 2012, Article ID 153817, 7 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  38. J. B. Griffiths and J. Podolský, Exact Spacetimes in Einstein’s General Relativity, Cambridge University Press, Cambridge, UK, 2009. View at MathSciNet
  39. T. Feroze and F. Ali, “Corrigendum to “Noether symmetries and conserved quantities for spaces with a section of zero curvature”,” Journal of Geometry and Physics, vol. 80, pp. 88-89, 2014, Journal of Geometry and Physics, vol. 61, pp. 658-662, 2011. View at Publisher · View at Google Scholar · View at MathSciNet