Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Timelike periodic trajectories
in spatially compact Lorentz manifolds

Author: Miguel Sánchez
Journal: Proc. Amer. Math. Soc. 127 (1999), 3057-3066
MSC (1991): Primary 53C50, 53C22
Published electronically: April 23, 1999
MathSciNet review: 1616609
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A result on the existence of timelike periodic trajectories in a general class of Lorentzian manifolds $\mathbb R{}\times M$, with compact $M$, is obtained. The proof is based on arguments concerning closed geodesics and causality theory.

References [Enhancements On Off] (What's this?)

  • [1] V. Benci: Periodic solutions of Lagrangian systems on a compact manifold, J. Diff. Eq. 63 (1986) 135-161. MR 88h:58102
  • [2] J.K. Beem, P.E. Ehrlich: ``Global Lorentzian Geometry", Pure and Applied Mathematics, Marcel Dekker Inc., NY, 1981.
  • [3] V. Benci, D. Fortunato: Periodic trajectories for the Lorentz metric of a static gravitational field, Proc. on ``Variational Methods" (H. Berestycki- J.M. Coron- I. Ekeland, Ed.) Paris (1988) 413-429.
  • [4] V. Benci, D. Fortunato, F. Giannoni: On the existence of multiple geodesics in static space-times, Ann. Inst. Henri Poin. 8 (1991) 79-102. MR 91m:58027
  • [5] V. Benci, D. Fortunato, F. Giannoni: On the existence of periodic trajectories in static Lorentzian manifolds with singular boundary, Nonlinear Analysis, a tribute in honour of Giovanni Prodi, Pisa (1991) 109-133.
  • [6] R. Bartolo, A. Masiello: On the existence of infinitely many trajectories for a class of Lorentzian manifolds like Schwarzschild and Reissner-Nordström spacetimes, J. Math. Anal. Appl. 199 (1996) 14-38. MR 98b:58036
  • [7] A.M. Candela: Lightlike periodic trajectories in space-times, Ann. Mat. Pura Appl., CLXXI (1996) 131-158. MR 98c:58026
  • [8] G.J. Galloway: Closed timelike geodesics, Trans. Amer. Math. Soc. 285 (1984) 379-388. MR 85k:53061
  • [9] G.J. Galloway: Compact Lorentzian manifolds without closed non spacelike geodesics, Proc. Amer. Math. Soc. 98 (1986) 119-123. MR 87i:53094
  • [10] R.P. Geroch: Domain of dependence, J. Math. Phys. 11 (1970) 437-449.
  • [11] F. Giannoni, A. Masiello: On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Func. Anal. 101 No. 2 (1991) 340-369. MR 92k:58053
  • [12] C. Greco: Periodic trajectories in static space-times, Proc. Roy. Soc. Edinb. 113A (1989) 99-103. MR 91e:53043
  • [13] C. Greco: Multiple periodic trajectories for a class of Lorentz metrics of a time-dependent gravitational field, Math. Ann. 287 (1990) 515-521. MR 91d:58045
  • [14] A. Masiello: Timelike periodic trajectories in stationary Lorentz manifolds, Nonlinear Anal. TMA, 19 (1992) 531-545.
  • [15] A. Masiello: On the existence of a timelike trajectory for a Lorentzian metric, Proc. Roy. Soc. Edinb. 125A (1995) 807-815. MR 96j:58037
  • [16] A. Masiello, P. Piccione: Shortening null geodesics in Lorentzian manifolds. Applications to closed light rays, Differential Geom. Appl. 8 (1998) 47-70. CMP 98:07
  • [17] A. Masiello, L. Pisani: Existence of a time-like periodic trajectory for a time-dependent Lorentz metric, Ann. Univ. Ferrara - Sc. Mat. XXXVI (1990) 207-222.
  • [18] B. O'Neill: ``Semi-Riemannian Geometry", Academic Press, San Diego, CA, 1983. MR 85f:53002
  • [19] M. Sánchez: Structure of Lorentzian tori admitting a Killing vector field, Trans. Amer. Math. Soc. 349 No. 3 (1997) 1063-1080. MR 97f:53108
  • [20] M. Sánchez: Geodesics in static spacetimes and t-periodic trajectories, Nonlinear Anal. TMA, 35 (1999) 677-686.
  • [21] M. Sánchez: Some remarks on Causality and Variational Methods in Lorentzian Manifolds, Conf. Sem. Mat. Univ. Bari 265 (1997). CMP 98:09
  • [22] M. Sánchez: Lorentzian manifolds admitting a Killing vector field, Nonlinear Anal. TMA 30 No. 1 (1997) 643-654.
  • [23] R.K. Sachs, H. Wu, ``General Relativity for Mathematicians", Grad. Texts in Math. 48, Springer-Verlag, NY (1977). MR 58:20239a
  • [24] F.J. Tipler: Existence of a closed timelike geodesic in Lorentz spaces, Proc. Amer. Math. Soc. 76 No. 1 (1979) 145-147.MR 80f:83016

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C50, 53C22

Retrieve articles in all journals with MSC (1991): 53C50, 53C22

Additional Information

Miguel Sánchez
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071–Granada, Spain

Keywords: Compact Lorentz manifold, stationary manifold, periodic trajectory, closed geodesic, causality.
Received by editor(s): November 20, 1997
Received by editor(s) in revised form: December 23, 1997
Published electronically: April 23, 1999
Additional Notes: This research was partially supported by a DGICYT Grant No. PB97-0784-C03-01. The author is grateful to Prof. D. Fortunato and Prof. V. Benci for having discussions.
Communicated by: Christopher B. Croke
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society