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Communications in Mathematical Physics

, Volume 276, Issue 3, pp 611–643 | Cite as

The Causal Boundary of Spacetimes Revisited

  • José L. Flores
Article

Abstract

We present a new development of the causal boundary of spacetimes, originally introduced by Geroch, Kronheimer and Penrose. Given a strongly causal spacetime (or, more generally, a chronological set), we reconsider the GKP ideas to construct a family of completions with a chronology and topology extending the original ones. Many of these completions present undesirable features, like those which appeared in previous approaches by other authors. However, we show that all these deficiencies are due to the attachment of an “excessively big” boundary. In fact, a notion of “completion with minimal boundary” is then introduced in our family such that, when we restrict to these minimal completions, which always exist, all previous objections disappear. The optimal character of our construction is illustrated by a number of satisfactory properties and examples.

Keywords

Ideal Point Causal Structure Timelike Curve Minkowski Plane Conformal Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alaña, V., Flores, J.L.: The causal boundary of product spacetimes. Gen. Relat. Grav., in press, DOI  10.1007/s10714-007-0492-5
  2. 2.
    Berenstein, D., Nastase, H.: On lightcone string field theory from super Yang-Mills and holography. http://arxiv.org/list/hep-th/0205048, 2002
  3. 3.
    Budic R. and Sachs R.K. (1974). Causal boundaries for general relativistic spacetimes. J. Math. Phys. 15: 1302–1309 MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Flores J.L. and Harris S.G. (2007). Topology of causal boundary for Standard Static spacetimes. Class. Quantum Grav. 24: 1211–1260 MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Flores J.L. and Sánchez M. (2003). Causality and conjugate points in general planes waves. Class. Quantum Grav. 20: 2275–2291 MATHCrossRefGoogle Scholar
  6. 6.
    Flores, J.L., Sánchez, M.: The causal boundary of wave-type spacetimes. Preprint (2007)Google Scholar
  7. 7.
    García-Parrado A. and Senovilla J.M. (2003). Causal relationship: A new tool for the causal characterization of Lorentzian manifolds. Class. Quantum Grav. 20: 625–664 MATHCrossRefADSGoogle Scholar
  8. 8.
    García-Parrado A. and Senovilla J.M. (2005). Causal structures and causal boundaries. Class. Quantum Grav. 22: R1–R84 MATHCrossRefADSGoogle Scholar
  9. 9.
    Geroch R.P., Kronheimer E.H. and Penrose R. (1972). Ideal points in spacetime. Proc. Roy. Soc. Lond. A 237: 545–567 ADSMathSciNetGoogle Scholar
  10. 10.
    Harris S.G. (1998). Universality of the future chronological boundary. J. Math. Phys. 39: 5427–5445 MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Harris S.G. (2000). Topology of the future chronological boundary: universality for spacelike boundaries. Class. Quantum Grav. 17: 551–603 MATHCrossRefADSGoogle Scholar
  12. 12.
    Harris S.G. (2001). Causal boundary for Standard Static spacetimes. Nonlinear analysis 47: 2971–2981 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Harris S.G. (2004). Discrete group actions on spacetimes: causality conditions and the causal boundary. Class. Quantum Grav. 21: 1209–1236 MATHCrossRefADSGoogle Scholar
  14. 14.
    Harris S.G. (2004). Boundaries on spacetimes: an outline. Contemp. Math. 359: 65–85 Google Scholar
  15. 15.
    Hubeny V. and Rangamany M. (2002). Causal structures of pp-waves. J. High Energy Phys. 12: 043 CrossRefADSGoogle Scholar
  16. 16.
    Kuang Z.-Q., Li J.-Z. and Liang C.-B. (1986). c-boundary of Taub’s plane-symmetric static vacuum spacetime. Phys. Rev. D 33: 1533–1537 CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Kuang Z.-Q. and Liang C.-B. (1988). On the GKP and BS constructions of the c-boundary. J. Math. Phys. 29: 433–435 MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Kuang Z.-Q. and Liang C.-B. (1992). On the Racz and Szabados constructions of the c-boundary. Phys. Rev. D 46: 4253–4256 CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Marolf D. and Ross S. (2002). Plane Waves: To infinity and beyond!. Class. Quant. Grav. 19: 6289–6302 MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Marolf D. and Ross S.F. (2003). A new recipe for causal completions. Class. Quantum Grav. 20: 4085–4117 MATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    O’Neill, B.: Semi-Riemannian Geometry with applications to Relativity, Series in Pure and Applied Math. 103, N.Y.: Academic Press, 1983Google Scholar
  22. 22.
    Penrose, R.: Conformal treatment of infinity. In: Relativity, Groups and Topology, edited by C.M. de Witt, B. de Witt, New York: Gordon and Breach, 1964; Penrose, R.: Zero rest-mass fields including gravitation: asymptotic behavior. Proc. Roy. Soc. Lond. A 284, 159–203 (1965)Google Scholar
  23. 23.
    Racz, I.: Causal boundary of space-times. Phys. Rev. D 36, 1673–1675 (1987): Racz, I.: Causal boundary for stably causal space-times. Gen. Relat. Grav. 20, 893–904 (1988)Google Scholar
  24. 24.
    Seifert H. (1971). The Causal Boundary of Space-Times. Gen. Rel. Grav. 1: 247–259 MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Szabados L.B. (1988). Causal boundary for strongly causal spaces. Class. Quantum Grav. 5: 121–134 MATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Szabados L.B. (1989). Causal boundary for strongly causal spacetimes: II. Class. Quantum Grav. 6: 77–91 MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Departamento de Álgebra, Geometría y Topología, Facultad de CienciasUniversidad de MálagaMálagaSpain

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