Vacuum self similar anisotropic cosmologies in F(R)-gravity

  • Pantelis S. Apostolopoulos
Research Article


The implications from the existence of a proper Homothetic Vector Field on the dynamics of vacuum anisotropic models in F(R) gravitational theory are studied. The fact that every Spatially Homogeneous vacuum model is equivalent, formally, with a “flux”-free anisotropic fluid model in standard gravity and the induced power-law form of the functional F(R) due to self-similarity enable us to close the system of equations. We found some new exact anisotropic solutions that arise as fixed points in the associated dynamical system. The non-existence of Kasner-like (Bianchi type I) solutions in proper F(R)-gravity (i.e. \(R\ne 0\)) strengthens the belief that curvature corrections will prevent the shear influence into the past thus permitting an isotropic singularity. We also discuss certain issues regarding the lack of vacuum models of type III, IV, VII\(_{h}\) in comparison with the corresponding results in standard gravity.


F(R)-gravity Self-similarity Dynamical systems Vacuum models Exact solutions 


  1. 1.
    Hall, G.S.: Symmetries and Curvature Structure in General Relativity. World Scientific, Singapore (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Duggal, K.L., Sharma, R.: Symmetries of Spacetimes and Riemannian Manifolds, Mathematics and Its Applications, vol. 487. Springer, US (1999)Google Scholar
  3. 3.
    Wainwright, J., Ellis, G.F.R. (eds.): Dynamical Systems in Cosmology. Cambridge University Press, Cambridge (1997)Google Scholar
  4. 4.
    Ellis, G.F.R., Maartens, R., MacCallum, M.A.H.: Relativistic Cosmology. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Coley, A.A.: Dynamical Systems and Cosmology. Springer, US (2003)Google Scholar
  6. 6.
    Uggla, C., van Elst, H., Wainwright, J., Ellis, G.F.R.: Phys. Rev. D 68, 103502 (2003). arXiv:gr-qc/0304002
  7. 7.
    Sotiriou, T.P., Faraoni, V.: Rev. Mod. Phys. 82, 451 (2010). arXiv:0805.1726 [gr-qc]ADSCrossRefGoogle Scholar
  8. 8.
    De Felice, A., Tsujikawa, S.: Living Rev Rel. 13, 3 (2010). arXiv:1002.4928 [gr-qc]
  9. 9.
    Clifton, T., Ferreira, P.G., Padilla, A., Skordis, C.: Phys. Rept. 513, 1 (2012). arXiv:1106.2476 [astro-ph.CO]ADSCrossRefGoogle Scholar
  10. 10.
    Hawking, S., Israel, W. (eds.): General Relativity: an Einstein Centenary Survey. Cambridge University Press, Cambridge (1979)zbMATHGoogle Scholar
  11. 11.
    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)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ryan Jr., M.P., Shepley, L.C.: Homogeneous Relativistic Cosmologies. Princeton University Press, Princeton (1975)Google Scholar
  13. 13.
    Ellis, G.F.R., van Elst, H.: Cosmological models: cargese lectures 1998. NATO Sci. Ser. C 541, 1 (1999). arXiv:gr-qc/9812046
  14. 14.
    van Elst, H., Uggla, C.: Class. Quantum Gravit. 14, 2673 (1997). arXiv:gr-qc/9603026
  15. 15.
    Apostolopoulos, P.S.: Class. Quantum Gravit. 20, 3371 (2003). arXiv:gr-qc/0306119
  16. 16.
    Apostolopoulos, P.S.: Class. Quantum Gravit. 22, 323 (2005). arXiv:gr-qc/0411102
  17. 17.
    Yano, K.: The Theory of Lie Derivatives and Its Applications. North-Holland, Amsterdam (1955)Google Scholar
  18. 18.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)Google Scholar
  19. 19.
    Leon, G., Saridakis, E.N.: Class. Quantum Gravit. 28, 065008 (2011). arXiv:1007.3956 [gr-qc]
  20. 20.
    Hewitt, C.G., Bridson, R., Wainwright, J.: Gen. Relativ. Gravit. 33, 65 (2001). arXiv:gr-qc/0008037 ADSCrossRefGoogle Scholar
  21. 21.
    Barrow, J.D., Clifton, T.: Class. Quantum Gravit. 23, L1 (2006). arXiv:gr-qc/0509085 ADSCrossRefGoogle Scholar
  22. 22.
    Clifton, T., Barrow, J.D.: Class. Quantum Gravit. 23, 2951 (2006). arXiv:gr-qc/0601118 ADSCrossRefGoogle Scholar
  23. 23.
    Leach, J.A., Carloni, S., Dunsby, P.K.S.: Class. Quantum Gravit. 23, 4915 (2006). arXiv:gr-qc/0603012 ADSCrossRefGoogle Scholar
  24. 24.
    Goheer, N., Leach, J.A., Dunsby, P.K.S.: Class. Quantum Gravit. 24, 5689 (2007). arXiv:0710.0814 [gr-qc]
  25. 25.
    Hervik, S., Barrow, J.D.: Phys. Rev. D 74, 124017 (2006). arXiv:gr-qc/0610013 ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Middleton, J.: Class. Quantum Gravit. 27, 225013 (2010). arXiv:1007.4669 [gr-qc]
  27. 27.
    Leon, G., Roque, A.A.: JCAP 1405, 032 (2014). arXiv:1308.5921 [astro-ph.CO]
  28. 28.
    Leon, G.: Int. J. Mod. Phys. E 20, 19 (2011). arXiv:1403.1984 [gr-qc]ADSCrossRefGoogle Scholar
  29. 29.
    Coley, A., Hervik, S., Papadopoulos, G.O., Pelavas, N.: Class. Quantum Gravit. 26, 105016 (2009). arXiv:0901.0394 [gr-qc]ADSCrossRefGoogle Scholar
  30. 30.
    Alho, A., Carloni, S., and Uggla, C.: JCAP 1608(08), 064 (2016) arXiv:1607.05715 [gr-qc]

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Environmental TechnologyTechnological Educational Institute of Ionian IslandsPanagoulaGreece

Personalised recommendations