Asymptotic Poincare compactification and finite-time singularities

Spiros Cotsakis1

Abstract

We provide an extension of the method of asymptotic decompositions of vector fields with finite-time singularities by applying the central extension technique of Poincare' to the dominant part of the vector field at approach to the singularity. This leads to a bundle of fan-out asymptotic systems whose equilibria at infinity govern the dynamics of the asymptotic solutions of the original system. We show how this method can be useful to describe a single-fluid isotropic universe at the time of maximum expansion, and discuss possible relations of our results to structural stability and non-compact phase spaces.

References

  1. Y. Choquet-Bruhat and S. Cotsakis, J. Geom. Phys. 43, 345 (2002); gr-qc/0201057.
  2. S. Cotsakis, Gen Rel. Grav. 36, 1183 (2004); grqc/ 0309048.
  3. S. Cotsakis and I. Klaoudatou, J. Geom. Phys. 55, 306 (2005); gr-qc/0409022.
  4. S. Cotsakis and I. Klaoudatou, J. Geom. Phys. 57, 1303 (2007); gr-qc/0604029.
  5. S. Cotsakis, Talking about singularities, in Proceedings of MG11 Meeting on General Relativity (World Scientific, 2009); gr-qc/0703084.
  6. J. D. Barrow, Class. Quantum Grav. 21, L79 (2004), gr-qc/0403084; see also J. D. Barrow, Class. Quantum Grav. 21, 5619 (2004); J. D. Barrow, S. Cotsakis, and A. Tsokaros, Class. Quantum Grav. 27, 165017 (2010).
  7. S. Nojiri, S. D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005); hep-th/0501025.
  8. S. Cotsakis and G. Kittou, Phys. Lett. B 712, 16 (2012).
  9. S. Cotsakis and A. Tsokaros, Phys. Lett. B 651, 341 (2007); S. Cotsakis, G. Kolionis, and A. Tsokaros, Phys. Lett. B 721, 1-6 (2013).
  10. I. Antoniadis, S. Cotsakis, and I. Klaoudatou, Class. Quantum Grav. 27, 235018 (2010); arXiv: 1010.6175.
  11. I. Antoniadis, S. Cotsakis, and I. Klaoudatou, Forschr. Phys. 61, 20 (2013); arXiv: 1206.0090.
  12. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, and R. Sundrum, Phys. Lett. B 480, 193 (2000); hepth/ 0001197.
  13. R. Penrose, Conformal treatment of infinity, in Relativity, Groups and Topology, C. DeWitt and B. DeWitt eds. (Gordon and Breach, New York, 1964), pp. 563584.
  14. I. Bendixson, Acta Math. 24, 1 (1901).
  15. S. Cotsakis and J. D. Barrow, The dominant balance at cosmological singularities, J. Phys. Conf. Ser. 68, 012004 (2007), gr-qc/0608137.
  16. S. Lefschetz, Differential Equations: Geometric Theory (2nd Ed., Dover, 1977).
  17. L. Perko, Differential Equations and Dynamical Systems (3rd Ed., Springer, 2001).
  18. F. Dumortier, J. Llibre, and J. C. Arte's, Qualitative Theory of Planar Differential Systems (Springer, 2006).
  19. J. D. Meiss, Differential Dynamical Systems (SIAM, 2007).
  20. J. Wainwright and G. F. R. Ellis, Dynamical Systems in Cosmology (CUP, 1997).
  21. S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Space-Time (CUP, 1973).
  22. Y. Choquet-Bruhat, General Relativity and the Einstein Equations (OUP, 2009).
  23. S. Cotsakis, Structure of infinity in cosmology (to appear in IJMPD), arXiv: 1212.6737.
  24. E. A. Gonza'lez Velasco, Trans. Amer. Math. Soc. 143 201 (1969).
  25. A. Cima, F. Manosas, and J. Villadelprat, Topology 37, 261 (1998).
For more information about this paper please visit Springer's Home Page of this paper.



Back to The Contents Page