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General Relativity and Gravitation

, Volume 30, Issue 6, pp 915–932 | Cite as

On the Geometry of Generalized Robertson-Walker Spacetimes: Geodesics

  • Miguel Sánchez
Article

Abstract

The geometry and, especially, the geodesics of a class of spacetimes generalizing Robertson-Walker ones (without any assumption on the fiber) is studied, under a global point of view. Our study covers geodesic connectedness, geodesic completeness and stability of completeness.

GENERALIZED ROBERTSON-WALKER SPACETIME GEODESIC CONNECTEDNESS GEODESIC COMPLETENESS STABILITY OF COMPLETENESS 
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REFERENCES

  1. 1.
    Alías, L. J., Estudillo, F. J. M., Romero, A. (1996). “On the Gaussian curvature of maximal surfaces in n-dimensional generalized Robertson-Walker spacetimes,” Class. Quantum Grav. 13, 3211.Google Scholar
  2. 2.
    Alías, L. J., Romero, A., Sánchez, M. (1995). “Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson-Walker spacetimes,” Gen. Rel. Grav. 27, 71.Google Scholar
  3. 3.
    Alías, L. J., Romero, A., M. Sánchez, M. (1997). “Spacelike hypersurfaces of constant mean curvature and Calabi-Bernstein type problems,” Tohôku Math. J. 49, 337.Google Scholar
  4. 4.
    Avez, A. (1962). “Formule the Gauss-Bonnet-Chern en métrique de signature quelconque,” C. R. Acad. Sci. 255, 2049.Google Scholar
  5. 5.
    Bates, L. (1998). “You Can 't Get There From Here,” J. Diff. Geom. and Applications, to appear.Google Scholar
  6. 6.
    Bedran, M. L., Lesche, B. (1986).“An example of affine collineation in the Robertson-Walker metric,” J. Math. Phys. 27, 2360.Google Scholar
  7. 7.
    Beem, J. K., Ehrlich, P. E. (1987). “Geodesic completeness and stability,” Math. Proc. Camb. Phil. Soc. 102, 319.Google Scholar
  8. 8.
    Beem, J. K., Ehrlich, P. E., Easley, K. L. (1996). Global Lorentzian Geometry (2nd. ed., Pure and Applied Mathematics, vol. 202, Marcel Dekker, New York).Google Scholar
  9. 9.
    Beem, J. K., Parker:, P. E. (1989). “Pseudoconvexity and geodesic connectedness,” Ann. Mat. Pure Appl. 155, 137.Google Scholar
  10. 10.
    Benci, V., Fortunato, D., Giannoni, F. (1991). “On the existence of multiple geodesics in static space-times,” Ann. Inst. Henri Poincaré 8, 79.Google Scholar
  11. 11.
    Benci, V., Fortunato, D., Masiello, A. (1994). “On the geodesic connectedeness of Lorentzian manifolds,” Math. Z. 217, 73.Google Scholar
  12. 12.
    Carot, J., da Costa, J. (1993). “On the geometry of warped spacetimes,” Class. Quantum Grav. 10, 461.Google Scholar
  13. 13.
    Carot, J., Nuñez, L. A., Percoco, U. (1997). “Ricci Collineations for Type B Warped Spacetimes,” Gen. Rel. Grav. 29, 1223.Google Scholar
  14. 14.
    Daftardar, V., Dadhich, N. (1994). “Gradient conformal Killing vectors and exact solutions,” Gen. Rel. Grav. 26, 859.Google Scholar
  15. 15.
    Deszcz, R., Verstraelen, L., Vrancken, L. (1991). “The symmetry of warped product space-times,” Gen. Rel. Grav. 23, 671.Google Scholar
  16. 16.
    Earley, D., Isenberg, J., Marsden, J., Moncrief, V. (1986). “Homothetic and conformal symmetries of solutions to Einstein's equations,” Commun. Math. Phys. 106, 137.Google Scholar
  17. 17.
    Ehlers, J., Geren, P., Sachs, J. K. (1968). “Isotropic solutions of the Einstein-Liouville equations,” J. Math. Phys. 9, 1344.Google Scholar
  18. 18.
    Ferrando, J. J., Morales, J. A., Portilla, M. (1992). “Inhomogeneous space-times admitting isotropic radiation: vorticity-free case,” Phys. Rev. D46, 578.Google Scholar
  19. 19.
    García-Río, E., Kupeli, D. N. (1996). “Singularity versus splitting theorems for stably causal spacetimes,” Annals Glob. Anal. Geom. 14, 301.Google Scholar
  20. 20.
    Geroch, R. (1970). “Domain of dependence,” J. Math. Phys. 11, 437.Google Scholar
  21. 21.
    Giannoni, F., Massiel lo, A. (1995). “Geodesics on product Lorentzian manifolds,” Ann. Inst. Henri Poin caré 8, 27.-60.Google Scholar
  22. 22.
    Haddow, B. M., Carot, J. (1996). “Energy-momentum types of warped spacetimes,” Class. Quantum Grav. 13, 289.Google Scholar
  23. 23.
    Masiello, A. (1994). Variational methods in Lorentzian Geometry (Pitman Research Notes in Mathematics Series, 309, Longman Scientific and Technical, Harlow, Essex).Google Scholar
  24. 24.
    Nomizu, K., Ozeki, H. (1961). “The existence of complete Riemannian metrics,” Proc. Amer. Math. Soc. 12, 889.Google Scholar
  25. 25.
    O'Neill, B. (1983). Semi-Riemannian Geometry with Applications to Relativity (Pure and Applied Ser. vol 103, Academic Press, New York).Google Scholar
  26. 26.
    Ponge, R., Reckziegel, H. (1993). “Twisted products in pseudo-Riemannian geometry,” Geom. Dedicata 48, 15.Google Scholar
  27. 27.
    Romero, A., Sánchez, M. (1994). “On the completeness of certain families of semi-Riemannian manifolds,” Geom. Dedicata 53, 103.Google Scholar
  28. 28.
    Romero, A., Sánchez, M. (1994). “New properties and examples of incomplete Lorentzian tori,” J. Math. Phys. 35, 1992.Google Scholar
  29. 29.
    Romero, A., Sánchez, M. (1994). “Geodesic completeness and conformal Lorentzian moduli space on the torus,” In Proc. III Fall Workshop Differential Geometry and Its Applications, Anales de Fısica, Monografıas 2 189.Google Scholar
  30. 30.
    Romero, A., Sánchez, M. (1993). “On the completeness of geodesics obtained as a limit,” J. Math. Phys. 34, 3768.Google Scholar
  31. 31.
    Romero, A., Sánchez, M. (1998). “Bochner's technique on Lorentzian manifolds and infinitesimal conformal symmetries,” Pac. J. Math., to appear.Google Scholar
  32. 32.
    Sachs, R., Wu, H. (1977). General Relativity for Mathematicians (Graduate Texts in Mathematics 48, Springer, New York).Google Scholar
  33. 33.
    Sánchez, M. (1997). “Geodesic connectedness in Generalized Reissner-Nordström type Lorentz manifolds,” Gen. Rel. Grav. 29, 1023.Google Scholar
  34. 34.
    Sánchez, M. (1997). “Structure of Lorentzian tori with a Killing vector field,” Trans. Amer. Math. Soc. 349, 1063.Google Scholar
  35. 35.
    Sánchez, M. (1997). “Some remarks on Causality Theory and Variational Methods in Lorentzian Manifolds,” Conf. Sem. Mat. Univ. Bari, 256.Google Scholar
  36. 36.
    Sánchez, M. (1997). “Timelike periodic trajectories in spatially compact Lorentzian manifolds.” Preprint.Google Scholar
  37. 37.
    Sánchez, M. (1997). “On the geometry of Generalized Robertson-Walker Spacetimes Curvature and Killing fields.” Preprint.Google Scholar
  38. 38.
    Schmidt, H.-J. (1996). “How should we measure spatial distances?” Gen. Rel. Grav. 28, 899.Google Scholar
  39. 39.
    Schmidt, H.-J. (1996). “Stability and Hamiltonian formulation of higher derivative theories” Phys. Rev. D54, 7906 (Erratum).Google Scholar
  40. 40.
    Seifert, H. J. (1967). “Global connectivity by time-like geodesics,” Zs. für Nature-for schung 22a, 1356.Google Scholar
  41. 41.
    Senovilla, J. M. M. (1998). “Singularity theorems and their consequences,” Gen. Rel. Grav. 30, 701.Google Scholar

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© Plenum Publishing Corporation 1998

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  • Miguel Sánchez

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