Global Geodesic Properties of Gödel-type SpaceTimes

  • Rossella Bartolo
  • Anna Maria Candela
  • José Luis Flores
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 26)

Abstract

The aim of this chapter is to review and complete the study of geodesics on Gödel-type spacetimes from a variational viewpoint in the last decade (say, from [10] to [2]). In particular, we prove some new results on geodesic connectedness and geodesic completeness for these spacetimes.

Keywords

Manifold 

Notes

Acknowledgements

The authors of this chapter acknowledge the partial support of the Spanish Grants with FEDER funds MTM2010-18099 (MICINN). Furthermore, R. Bartolo and A.M. Candela acknowledge also the partial support of M.I.U.R. Research Project PRIN2009 “Metodi Variazionali e Topologici nello Studio di Fenomeni Nonlineari” and of the G.N.A.M.P.A. Research Project 2011 “Analisi Geometrica sulle Varietà di Lorentz ed Applicazioni alla Relatività Generale”; J.L. Flores acknowledges also the partial support of the Regional J. Andalucía Grant P09-FQM-4496, with FEDER funds.

References

  1. 1.
    Bartolo, R., Candela, A.M., Flores, J.L.: Geodesic connectedness of stationary spacetimes with optimal growth. J. Geom. Phys. 56, 2025–2038 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bartolo, R., Candela, A.M., Flores, J.L.: A note on geodesic connectedness in Gödel type spacetimes. Differ. Geom. Appl. 29, 779–786 (2011)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bartolo, R., Candela, A.M., Flores, J.L., Sánchez, M.: Geodesics in static Lorentzian manifolds with critical quadratic behavior. Adv. Nonlinear Stud. 3, 471–494 (2003)MathSciNetMATHGoogle Scholar
  4. 4.
    Benci, V., Fortunato, D.: On the existence of infinitely many geodesics on space–time manifolds. Adv. Math. 105, 1–25 (1994)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Benci, V., Fortunato, D., Giannoni, F.: On the existence of multiple geodesics in static space–times. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 79–102 (1991)MathSciNetMATHGoogle Scholar
  6. 6.
    Calvão, M.O., Damião Soares, I., Tiomno, J.: Geodesics in Gödel–type space–times. Gen. Relat. Gravit. 22, 683–705 (1990)MATHCrossRefGoogle Scholar
  7. 7.
    Candela, A.M., Flores, J.L., Sánchez, M.: On general Plane Fronted Waves. Geodesics. Gen. Relat. Gravit. 35, 631–649 (2003)MATHCrossRefGoogle Scholar
  8. 8.
    Candela, A.M., Flores, J.L., Sánchez, M.: Global hyperboliticity and Palais–Smale condition for action functionals in stationary spacetimes. Adv. Math. 218, 515–536 (2008)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Candela, A.M., Romero, A., Sánchez, M.: Completeness of the trajectories of particles coupled to a general force field. To appear in Arch. Rational Mech. Anal. ArXiv:1202.0523v1.Google Scholar
  10. 10.
    Candela, A.M., Sánchez, M.: Geodesic connectedness in Gödel type space–times. Differ. Geom. Appl. 12, 105–120 (2000)MATHCrossRefGoogle Scholar
  11. 11.
    Candela, A.M., Sánchez, M.: Existence of geodesics in Gödel type space–times. Nonlinear Anal. TMA 47, 1581–1592 (2001)MATHCrossRefGoogle Scholar
  12. 12.
    Candela, A.M., Sánchez, M.: Geodesics in semi–Riemannian manifolds: geometric properties and variational tools. In: Alekseevsky, D.V., Baum, H. (eds.) Recent Developments in Pseudo–Riemannian Geometry, Special Volume in the ESI–Series on Mathematics and Physics, pp. 359–418, EMS Publishing House (2008)Google Scholar
  13. 13.
    Candela, A.M., Salvatore, A., Sánchez, M.: Periodic trajectories in Gödel type space–times. Nonlinear Anal. TMA 51, 607–631 (2002)MATHCrossRefGoogle Scholar
  14. 14.
    Caponio, E., Javaloyes M.A., Sánchez, M.: On the interplay between Lorentzian causality and Finsler metrics of Randers type. Rev. Mat. Iberoam. 27, 919–952 (2011)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Carrion, H.L., Rebouças, M.J., Teixeira, A.F.F.: Gödel–type space–times in induced matter gravity theory. J. Math. Phys. 40, 4011–4027 (1999)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Chandrasekhar, S., Wright, J.P.: The geodesics in Gödel’s universe. Proc. Natl. Acad. Sci. USA 47, 341–347 (1961)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Giannoni, F., Masiello, A.: On the existence of geodesics on stationary Lorentz manifolds with convex boundary. J. Funct. Anal. 101, 340–369 (1991)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Gödel, K.: An example of a new type of cosmological solution of Einstein’s field equations of gravitation. Rev. Mod. Phys. 21, 447–450 (1949)MATHCrossRefGoogle Scholar
  19. 19.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space–Time. Cambridge University Press, London (1973)MATHCrossRefGoogle Scholar
  20. 20.
    Kramer, D., Stephani, H., Herlt, E., MacCallum, M.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (1980)MATHGoogle Scholar
  21. 21.
    Kundt, W.: Trägheitsbahnen in einem von Gödel angegebenen kosmologischen Modell. Z. Phys. 145, 611–620 (1956)CrossRefGoogle Scholar
  22. 22.
    Masiello, A.: Variational Methods in Lorentzian Geometry. Pitman Res. Notes Math. Ser. vol. 309. Longman Sci. Tech., Harlow (1994)Google Scholar
  23. 23.
    Müller, O.: A note on closed isometric embeddings. J. Math. Anal. Appl. 349, 297–298 (2009)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Nash, J.: The imbedding problem for Riemannian manifold. Ann. Math. 63, 20–63 (1956)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    O’Neill, B.: Semi–Riemannian Geometry with Applications to Relativity. Academic, New York (1983)MATHGoogle Scholar
  26. 26.
    Piccione, P., Tausk, D.V.: An index theory for paths that are solutions of a class of strongly indefinite variational problems. Calc. Var. Part. Differ. Equat. 15, 529–551 (2002)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math. vol. 65. Amer. Math. Soc., Providence (1986)Google Scholar
  28. 28.
    Raychaudhuri, A.K., Thakurta, S.N.G.: Homogeneous space-times of the Gödel type. Phys. Rev. D 22, 802–806 (1980)CrossRefGoogle Scholar
  29. 29.
    Rebouças, M.J., Tiomno, J.: Homogeneity of Riemannian space–times of Gödel type. Phys. Rev. D 28, 1251–1264 (1983)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sánchez, M.: Some Remarks on Causality Theory and Variational Methods in Lorentzian Manifolds. Conf. Semin. Mat. Univ. Bari 265 (1997) ArXiv:0712.0600v2Google Scholar
  31. 31.
    Santaló, L.A.: Geodesics in Gödel–Synge spaces. Tensor N.S. 37, 173–178 (1982)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Rossella Bartolo
    • 1
  • Anna Maria Candela
    • 2
  • José Luis Flores
    • 3
  1. 1.Dipartimento di Meccanica Matematica e ManagementPolitecnico di BariBariItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di Bari “Aldo Moro”BariItaly
  3. 3.Departamento de Álgebra, Geometría y Topología, Facultad de CienciasUniversidad de MálagaMálagaSpain

Personalised recommendations