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New examples of static spacetimes admitting a unique standard decomposition

  • Alma L. AlbujerEmail author
  • Jónatan Herrera
  • Rafael M. Rubio
Research Article
  • 55 Downloads

Abstract

In this paper we introduce a new general approach for the study of spacetimes admitting a standard static splitting. This approach allows us to give an alternative proof for the uniqueness of splitting in the spatially closed case to the first study made by Sánchez–Senovilla and later by Aledo–Romero–Rubio. However, our technique also allows us to obtain new uniqueness results for standard static models with complete (non necessarily compact) spacelike bases under some mild hypothesis, including some restrictions on the sectional curvature of such bases.

Keywords

Standard static spacetime Standard static splitting Global Killing vector field Maximal hypersurface 

Mathematics Subject Classification

53C50 53C80 83C20 

Notes

References

  1. 1.
    Aledo, J.A., Romero, A., Rubio, R.M.: The existence and uniqueness of standard static splitting. Class. Quantum Gravity 32(10), 105,004 (2015).  https://doi.org/10.1088/0264-9381/32/10/105004 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bartnik, R.: Remarks on cosmological spacetimes and constant mean curvature surfaces. Commun. Math. Phys. 117(4), 615–624 (1988)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry, Pure and Applied Mathematics, vol. 202. Marcel Dekker, New York (1996)zbMATHGoogle Scholar
  4. 4.
    Camacho, C., Neto, A.L.: Geometric Theory of Foliations. Birkhäuser, Basel (1985)CrossRefGoogle Scholar
  5. 5.
    Farkas, H.M., Kra, I.: Riemann Su. Graduate Texts in Mathematics, 2nd edn. Springer, New York (1992)CrossRefGoogle Scholar
  6. 6.
    Farmakis, I., Moskowitz, M.: Fixed Point Theorems and Their Applications. World Scientific Publishing Company, Hackensack (2013)CrossRefGoogle Scholar
  7. 7.
    Galloway, G.J., Vega, C.: Achronal limits, lorentzian spheres, and splitting. Ann. Henri Poincaré 15(11), 2241–2279 (2014).  https://doi.org/10.1007/s00023-013-0305-1 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gromoll, D., Meyer, W.: On complete open manifolds of positive curvature. Ann. Math. 90(1), 75–90 (1969).  https://doi.org/10.2307/1970682 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gutiérrez, M., Olea, B.: Uniqueness of static decompositions. Ann. Glob. Anal. Geom. 39(1), 13–26 (2011).  https://doi.org/10.1007/s10455-010-9222-4 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Harris, S.: Conformally stationary spacetimes. Class. Quantum Gravity 9(7), 1823 (1992)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Javaloyes, M.A., Sánchez, M.: A note on the existence of standard splittings for conformally stationary spacetimes. Class. Quantum Gravity 25(16), 168,001 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kamke, E.: Über die partielle differentialgleichung \(f(x, y)\frac{\partial z}{\partial x} + g(x, y)\frac{\partial z}{\partial x} = h (x, y)\). Math. Z. 41(1), 56–66 (1936).  https://doi.org/10.1007/BF01180405 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kazdan, J.L.: Some applications of partial differential equations to problems in geometry. J.S. Kazdan (1983)Google Scholar
  14. 14.
    Kobayashi, S.: Transformation Groups in Differential Geometry. Springer, Berlin (1995)zbMATHGoogle Scholar
  15. 15.
    O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Pure and Applied Mathematics, vol. 103. Academic Press, New York (1983)zbMATHGoogle Scholar
  16. 16.
    Sachs, R.K., Wu, H.H.: General Relativity for Mathematicians. Springer, New York (1977).  https://doi.org/10.1007/978-1-4612-9903-5
  17. 17.
    Sánchez, M., Senovilla, J.M.M.: A note on the uniqueness of global static decompositions. Class. Quantum Gravity 24(23), 6121 (2007).  https://doi.org/10.1088/0264-9381/24/23/N01 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sharifzadeh, M., Bahrampour, Y.: Some results about the level sets of Lorentzian Busemann function and Bartnik’s conjecture. Commun. Math. Phys. 286(1), 389–398 (2009).  https://doi.org/10.1007/s00220-008-0609-z ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer, New York (1983)zbMATHGoogle Scholar
  20. 20.
    Yau, S.T.: Remarks on the group of isometries of a Riemannian manifold. Topology 16(3), 239–247 (1977).  https://doi.org/10.1016/0040-9383(77)90004-0 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Edificio Albert EinsteinUniversidad de Córdoba, Campus de RabanalesCórdobaSpain

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