Journal of High Energy Physics

, 2018:123 | Cite as

A novel notion of null infinity for c-boundaries and generalized black holes

  • I. P. Costa e Silva
  • J. L. Flores
  • J. HerreraEmail author
Open Access
Regular Article - Theoretical Physics


We give new definitions of null infinity and black hole in terms of causal boundaries, applicable to any strongly causal spacetime (M, g). These are meant to extend the standard ones given in terms of conformal boundaries, and use the new definitions to prove a classic result in black hole theory for this more general context: if the null infinity is regular (i.e. well behaved in a suitable sense) and (M, g) obeys the null convergence condition, then any closed trapped surface in (M, g) has to be inside the black hole region. As an illustration of this general construction, we apply it to the class of generalized plane waves, where the conformal null infinity is not always well-defined. In particular, it is shown that (generalized) black hole regions do not exist in a large family of these spacetimes.


Black Holes Differential and Algebraic Geometry Spacetime Singularities 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    R. Geroch, E.H. Kronheimer and R. Penrose, Ideal points in space-time, Proc. Roy. Soc. Lond. A 327 (1972) 545.ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett. 10 (1963) 66 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    R. Penrose, Conformal treatment of infinity, in Relativity, Groups and Topology, C.M. de Witt and B. de Witt eds., Gordon and Breach, New York, NY (1964), pp. 566–584 [INSPIRE].
  4. [4]
    S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, Cambridge University Press (1975) [INSPIRE].
  5. [5]
    R. Wald, General Relativity, University of Chicago Press (1984) [INSPIRE].
  6. [6]
    J. Frauendiener, Conformal infinity, Living Rev. Rel. 3 (2000) 1 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  7. [7]
    P.T. Chruściel, Conformal boundary extensions of Lorentzian manifolds, J. Diff. Geom. 84 (2010) 19 [gr-qc/0606101] [INSPIRE].
  8. [8]
    H. Friedrich, On the existence of n-geodesically complete or future complete solutions of einsteins field equations with smooth asymptotic structures, Commun. Math. Phys. 107 (1986) 587.ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    H. Friedrich, On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations, J. Diff. Geom. 34 (1991) 275.MathSciNetCrossRefGoogle Scholar
  10. [10]
    H. Friedrich, Smoothness at null infinity and the structure of initial data, in The Einstein Equations and the Large Scale Behavior of Gravitational Fields, P.T. Chrusciel and H. Frierdrich eds., Birkhauser, Basel (2004), pp. 243–275.Google Scholar
  11. [11]
    P.T. Chruściel and E. Delay, Existence of nontrivial, vacuum, asymptotically simple space-times, Class. Quant. Grav. 19 (2002) L71 [gr-qc/0203053] [INSPIRE].
  12. [12]
    J.L. Flores and M. Sánchez, The Causal boundary of wave-type spacetimes, JHEP 03 (2008) 036 [arXiv:0712.0592] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    L.B. Szabados, Causal Boundary for Strongly Causal Space-time, Class. Quant. Grav. 5 (1988) 121 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    L.B. Szabados, Causal boundary for strongly causal spacetimes. II, Class. Quant. Grav. 6 (1989) 77.Google Scholar
  15. [15]
    D. Marolf and S.F. Ross, A New recipe for causal completions, Class. Quant. Grav. 20 (2003) 4085 [gr-qc/0303025] [INSPIRE].
  16. [16]
    J.L. Flores, The Causal Boundary of spacetimes revisited, Commun. Math. Phys. 276 (2007) 611 [gr-qc/0608063] [INSPIRE].
  17. [17]
    J.L. Flores, J. Herrera and M. Sánchez, On the final definition of the causal boundary and its relation with the conformal boundary, Adv. Theor. Math. Phys. 15 (2011) 991 [arXiv:1001.3270] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  18. [18]
    A. García-Parrado and J.M.M. Senovilla, Causal structures and causal boundaries, Class. Quant. Grav. 22 (2005) R1 [gr-qc/0501069] [INSPIRE].
  19. [19]
    I.P. Costa e Silva, J. Herrera and J.L. Flores, Hausdorff closed limits and the causal boundary of globally hyperbolic spacetimes with timelike boundary, preprint (2018).Google Scholar
  20. [20]
    R. Sachs, Gravitational waves in General Relativity. VI. The outgoing radiation condition, Proc. Roy. Soc. Lond. A 264 (1961) 339.Google Scholar
  21. [21]
    L. Baulieu, A.S. Losev and N.A. Nekrasov, Target space symmetries in topological theories. 1., JHEP 02 (2002) 021 [hep-th/0106042] [INSPIRE].
  22. [22]
    C.P. Herzog and D.T. Son, Schwinger-Keldysh propagators from AdS/CFT correspondence, JHEP 03 (2003) 046 [hep-th/0212072] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    J.K. Beem, P.E. Ehrlich and K.L. Easley, Global Lorentzian Geometry, Pure and Applied Mathematics, vol. 202, Marcel Dekker, New York (1996).Google Scholar
  24. [24]
    B. O’Neill, Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, vol. 103, Academic Press (1983).Google Scholar
  25. [25]
    J.L. Flores, J. Herrera and M. Sánchez, Hausdorff separability of the boundaries for spacetimes and sequential spaces, J. Math. Phys. 57 (2016) 022503.ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    J.L. Flores, J. Herrera and M. Sánchez, Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds, Memoirs Am. Mat. Soc. 226 (2013) 1064 [arXiv:1011.1154] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  27. [27]
    L.A. Aké and J. Herrera, Spacetime coverings and the casual boundary, JHEP 04 (2017) 051 [arXiv:1605.03128] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    E. Minguzzi and M. Sánchez, The Causal hierarchy of spacetimes, in Recent Developments in pseudo-Riemannian geometry, H. Baum and D. Alekseevsky eds., Zürich, EMS Pub. House (2008), pp. 299–358 [DOI:] [gr-qc/0609119] [INSPIRE].
  29. [29]
    R. Penrose, Techniques of Differential Topology in Relativity, SIAM, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics (1972) [DOI:].
  30. [30]
    R. Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14 (1965) 57 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    J.L. Flores and M. Sánchez, Causality and conjugate points in general plane waves, Class. Quant. Grav. 20 (2003) 2275 [gr-qc/0211086] [INSPIRE].
  32. [32]
    A.M. Candela, J.L. Flores and M. Sánchez, On general plane fronted waves: Geodesics, Gen. Rel. Grav. 35 (2003) 631 [gr-qc/0211017] [INSPIRE].
  33. [33]
    P.E. Ehrlich and G.G. Emch, Gravitational waves and causality, Rev. Math. Phys. 04 (1992) 163.MathSciNetCrossRefGoogle Scholar
  34. [34]
    E. Minguzzi, Causality of spacetimes admitting a parallel null vector and weak KAM theory, 2012, arXiv:1211.2685 [INSPIRE].
  35. [35]
    C.J.S. Clarke, On the geodesic completeness of causal space-times, Math. Proc. Cambridge Phil. Soc. 69 (1971) 319.ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • I. P. Costa e Silva
    • 1
  • J. L. Flores
    • 2
  • J. Herrera
    • 3
    Email author
  1. 1.Department of MathematicsUniversidade Federal de Santa CatarinaFlorianópolis-SCBrazil
  2. 2.Departamento de Álgebra, Geometría y Topología, Facultad de CienciasUniversidad de MálagaMálagaSpain
  3. 3.Departamento de Matemáticas, Edificio Albert EinsteinUniversidad de CórdobaCórdobaSpain

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