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Communications in Mathematical Physics

, Volume 322, Issue 2, pp 633–666 | Cite as

Global and Uniqueness Properties of Stationary and Static Spacetimes with Outer Trapped Surfaces

  • Marc MarsEmail author
  • Martin Reiris
Article

Abstract

Global properties of maximal future Cauchy developments of stationary, m-dimensional asymptotically flat initial data with an outer trapped boundary are analyzed. We prove that, whenever the matter model is well posed and satisfies the null energy condition, the future Cauchy development of the data is a black hole spacetime. More specifically, we show that the future Killing development of the exterior of a sufficiently large sphere in the initial data set can be isometrically embedded in the maximal Cauchy development of the data. In the static setting we prove, by working directly on the initial data set, that all Killing prehorizons are embedded whenever the initial data set has an outer trapped boundary and satisfies the null energy condition. By combining both results we prove a uniqueness theorem for static initial data sets with outer trapped boundary.

Keywords

Black Hole Initial Data Killing Vector Smooth Point Cauchy Surface 
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.
    Beig R., Chruściel P.T.: Killing vectors in asymptotically flat space-times: I. Asymptotically translational killing vectors and the rigid positive energy theorem. J. Math. Phys. 37, 1939–1961 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Bunting G., Masoodul Alam A.K.M.: Nonexistence of multiple black holes in asymptotically euclidean static vacuum space-time. Gen. Rel. Grav. 19, 147–154 (1987)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Camacho, C., Neto, A.L.: Geometric theory of foliations. Boston, MA: Birkhauser, Boston Inc., 1985Google Scholar
  4. 4.
    Carrasco, A.: Trapped surfaces in spacetimes with symmetries and applications to uniqueness theorems. Ph.D. Thesis, 2011Google Scholar
  5. 5.
    Carrasco A., Mars M.: On marginally outer trapped surfaces in stationary and static spacetimes. Class Quantum Grav. 25, 055011 (2008)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Carrasco A., Mars M.: Uniqueness theorem for static spacetimes containing marginally outer trapped surfaces. Class Quantum Grav. 28, 175018 (2011)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Chavel, I.: Riemannian geometry, a modern introduction. Cambridge Studies in Advanced Mathematics 98, Cambridge: Cambridge University Press, 2006Google Scholar
  8. 8.
    Chrućiel P.T., Galloway G.J.: Uniqueness of static black-holes without analyticity. Class Quantum Grav. 27, 152001 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    Chruściel, P.T.: The classification of static vacuum space-times containing an asymptotically flat spacelike hypersurface with compact interior. Class Quantum Grav. 16, 661-687 (1999), http://arxiv.org/abs/gr-qc/9809088v2, 2010, Correction to published article
  10. 10.
    Chruściel P.T.: ‘No hair’ theorems - foklore, conjectures, results. Contemporary Math. 170, 23–49 (1994)ADSCrossRefGoogle Scholar
  11. 11.
    Chruściel P.T.: The classification of static vacuum space-times containing an asymptotically flat spacelike hypersurface with compact interior. Class Quantum Grav. 16, 661–687 (1999)ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Chruściel P.T., Lopes Costa J.: On uniqueness of stationary black holes. Astérisque 321, 195–265 (2008)Google Scholar
  13. 13.
    Chruściel P.T., Maerten D.: Killing vectors in Asymptotically flat space–times: Ii. Asymptotically translational killing vectors and the rigid positive energy theorem in higher dimensions. J. Math. Phys. 47, 022502 (2006)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Chruściel P.T., Tod K.P.: The classification of static electro-vacuum spacetimes containing an asymptotically flat spacelike hypersurface with a compact interior. Commun. Math. Phys. 271, 577–589 (2007)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Damour T., Schmidt B.: Reliability of perturbation theory in general relativity. J. Math. Phys. 31, 2441–2453 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press, 1992Google Scholar
  17. 17.
    Galloway G.J.: On the topology of black holes. Commun. Math. Phys. 151, 53–66 (1993)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Galloway G.J.: Maximum principles for null hypersurfaces and null splitting theorems. Ann. Poincaré Phys. Theor. 1, 543–567 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gromoll D., Meyer W.: On differentiable functions with isolated critical points. Topology 8, 361–369 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Shiromizu T., Gibbons G.W., Ida D.: Uniqueness and non-uniqueness of static vacuum black holes in higher dimensions. Prog.Theor.Phys.Suppl. 148, 284–290 (2003)MathSciNetGoogle Scholar
  21. 21.
    Hawking, S.W., Ellis, G.F.R.:The large scale structure of space-time. Cambridge monographs on mathematical physics, Cambridge: Cambridge University Press, 1973Google Scholar
  22. 22.
    Mantegazza C., Mennucci A.C.: Hamilton-jacobi equations and distance functions on riemannian manifolds. Appl. Math. Optim. 47, 1–25 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Miao P.: A remark on boundary effects in static vacuum initial data sets. Class Quantum Grav. 22, L53–L59 (2005)ADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Michor, P.W.: Topics in Differential Geometry. Graduate Texts in Mathematics 93. Providence, VI: American Mathematical Society, 2000Google Scholar
  25. 25.
    Moncrief V.: Spacetime symmetries and linearization stability of the einstein equations. J. Math. Phys. 16, 493–498 (1975)MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Penrose R.: Gravitational collapse – the role of general relativity. Nuovo Cimiento 1, 252–276 (1965)ADSGoogle Scholar
  27. 27.
    Chruściel P.T., Bartnitk R.: Boundary value problems for dirac-type equations, with applications. J. Reine Ange. Math. (Crelle’s Journal) 579, 13–73 (2005)zbMATHGoogle Scholar
  28. 28.
    Rácz I.: On the existence of Killing vector fields. Class. Quantum Grav. 16, 1695–1703 (1999)ADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Rácz I.: Symmetries of spacetime and their relation to initial value problems. Class. Quantum Grav. 18, 5103–5113 (2001)ADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Rácz I., Wald R.M.: Extensions of spacetimes with killing horizons. Class. Quantum Grav. 9, 2643–2656 (1992)ADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Nomizu, K., Kobayashi, S.: Foundations of differential geometry, Vol II. New York: Interscience Publisher, 1969Google Scholar
  32. 32.
    Wald, R.M.: General Relativity. Chicago, IL: The University of Chicago Press, 1984Google Scholar
  33. 33.
    Weyl H.: Zur gravitationstheorie. Ann. Phys. (Berlin) 54, 117–145 (1917)ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad de SalamancaSalamancaSpain
  2. 2.Albert Einstein InstitutMax Planck Institut für GravitationsphysikPotsdamGermany

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