Communications in Mathematical Physics

, Volume 299, Issue 1, pp 89–127 | Cite as

Uniqueness of Smooth Stationary Black Holes in Vacuum: Small Perturbations of the Kerr Spaces

  • S. Alexakis
  • A. D. IonescuEmail author
  • S. Klainerman


The goal of the paper is to prove a perturbative result, concerning the uniqueness of Kerr solutions, a result which we believe will be useful in the proof of their nonlinear stability. Following the program started in Ionescu and Klainerman (Invent. Math. 175:35–102, 2009), we attempt to remove the analyticity assumption in the the well known Hawking-Carter-Robinson uniqueness result for regular stationary vacuum black holes. Unlike (Ionescu and Klainerman in Invent. Math. 175:35–102, 2009), which was based on a tensorial characterization of the Kerr solutions, due to Mars (Class. Quant. Grav. 16:2507–2523, 1999), we rely here on Hawking’s original strategy, which is to reduce the case of general stationary space-times to that of stationary and axi-symmetric spacetimes for which the Carter-Robinson uniqueness result holds. In this reduction Hawking had to appeal to analyticity. Using a variant of the geometric Carleman estimates developed in Ionescu and Klainerman (Invent. Math. 175:35–102, 2009), in this paper we show how to bypass analyticity in the case when the stationary vacuum space-time is a small perturbation of a given Kerr solution. Our perturbation assumption is expressed as a uniform smallness condition on the Mars-Simon tensor. The starting point of our proof is the new local rigidity theorem established in Alexakis et al. (Hawking’s local rigidity theorem without analyticity.[gr-qc], 2009).


Black Hole Stationary Black Hole Carleman Estimate Null Hypersurface Kerr Solution 
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  1. 1.
    Alexakis, S.: Unique continuation for the vacuum Einstein equations. Preprint (2008),[gr-qc], 2009
  2. 2.
    Alexakis, S., Ionescu, A.D., Klainerman, S.: Hawking’s local rigidity theorem without analyticity. Preprint (2009),[gr-qc], 2009
  3. 3.
    Beig R., Simon W.: The stationary gravitational field near spatial infinity. Gen. Rel. Grav. 12, 1003–1013 (1980)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Beig R., Simon W.: On the multipole expansion for stationary space-times. Proc. Roy. Soc. London Ser. A 376, 333–341 (1981)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Bunting, G.L.: Proof of the Uniqueness Conjecture for Black Holes. PhD Thesis, Univ. of New England, Armidale, NSW, 1983Google Scholar
  6. 6.
    Bunting G., Massood-ul-Alam A.K.M.: Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time. Gen. Rel. Grav. 19, 147–154 (1987)zbMATHCrossRefADSGoogle Scholar
  7. 7.
    Carter B.: An axy-symmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971)CrossRefADSGoogle Scholar
  8. 8.
    Carter, B.: Black hole equilibrium states. In: Black holes/Les astres occlus (École d’Été Phys. Théor., Les Houches, 1972), New York: Gordon and Breach, 1973, pp. 57–214Google Scholar
  9. 9.
    Carter, B.: Has the Black Hole Equilibrium Problem Been Solved? In: The Eighth Marcel Grossmann meeting, Part A, B (Jerusalem, 1997), River Edge, NJ: World Sci. Publ., 1999, pp. 136–155Google Scholar
  10. 10.
    Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space, Princeton Math. Series 41, Princeton University Press, 1993Google Scholar
  11. 11.
    Chrusciel P.T.: On completeness of orbits of Killing vector fields. Class. Quant. Grav. 10, 2091–2101 (1993)zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Chrusciel, P.T.: “No Hair” Theorems-Folclore, Conjecture, Results. Diff. Geom. and Math. Phys. (J. Beem and K.L. Duggal) Cont. Math. 170, Providence, RI: AMS, 1994, pp. 23–49Google Scholar
  13. 13.
    Chrusciel P.T.: On the rigidity of analytic black holes. Commun. Math. Phys. 189, 1–7 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Chrusciel P.T., Wald R.M.: Maximal hypersurfaces in stationary asymptotically flat spacetimes. Commun. Math. Phys. 163, 561–604 (1994)zbMATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Chrusciel P.T., Wald R.M.: On the topology of stationary black holes. Class. Quant. Grav. 11, L147–L152 (1993)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Chrusciel P.T., Delay T., Galloway G., Howard R.: Regularity of horizon and the area theorem. Ann. H. Poincaré 2, 109–178 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Chrusciel, P.T., Costa, J.L.: On uniqueness of stationary vacuum black holes. Preprint (2008),[gr-qc], 2008
  18. 18.
    Chrusciel P.T.: On higher dimensional black holes with abelian isometry group. J. Math. Phys. 50, 052501 (2009)CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Friedman J.L., Schleich K., Witt D.M.: Topological censorship. Phys. Rev. Lett. 71, 1846–1849 (1993)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Friedrich H., Rácz I., Wald R.: On the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. Commun. Math. Phys. 204, 691–707 (1999)zbMATHCrossRefADSGoogle Scholar
  21. 21.
    Hawking S.W., Ellis G.F.R.: The large scale structure of space-time. Cambridge Univ. Press, Cambridge (1973)zbMATHCrossRefGoogle Scholar
  22. 22.
    Heusler, M.: Black Hole Uniqueness Theorems, Cambridge Lect. Notes in Phys, Cambridge: Cambridge Univ. Press, 1996Google Scholar
  23. 23.
    Hörmander, L.: The analysis of linear partial differential operators IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 275. Berlin: Springer-Verlag, 1985Google Scholar
  24. 24.
    Ionescu A.D., Klainerman S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175, 35–102 (2009)zbMATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Ionescu A.D., Klainerman S.: Uniqueness results for ill-posed characteristic problems in curved space-times. Commun. Math. Phys. 285, 873–900 (2009)zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Isenberg J., Moncrief V.: Symmetries of Cosmological Cauchy Horizons. Commun. Math. Phys. 89, 387–413 (1983)zbMATHCrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Israel W.: Event horizons in static vacuum space-times. Phys. Rev. Lett. 164, 1776–1779 (1967)ADSGoogle Scholar
  28. 28.
    Klainerman, S., Nicolò, F.: The evolution problem in general relativity. Progress in Mathematical Physics, 25. Boston, MA: Birkhäuser Boston, Inc., 2003Google Scholar
  29. 29.
    Mars M.: A spacetime characterization of the Kerr metric, Class. Quant. Grav. 16, 2507–2523 (1999)zbMATHCrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Mazur P.O.: Proof of Uniqueness for the Kerr-Newman Black Hole Solution. J. Phys. A: Math. Gen. 15, 3173–3180 (1982)zbMATHCrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Robinson D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975)CrossRefADSGoogle Scholar
  32. 32.
    Racz I., Wald R.: Extensions of space-times with Killing horizons. Class. Quant. Gr. 9, 2463–2656 (1992)MathSciNetGoogle Scholar
  33. 33.
    Simon W.: Characterization of the Kerr metric. Gen. Rel. Grav. 16, 465–476 (1984)zbMATHCrossRefADSGoogle Scholar
  34. 34.
    Sudarski D., Wald R.M.: Mass formulas for stationary Einstein Yang-Mills black holes and a simple proof of two staticity theorems. Phys. Rev. D47, 5209–5213 (1993)ADSGoogle Scholar
  35. 35.
    Weinstein G.: On rotating black holes in equilibrium in general relativity. Comm. Pure Appl. Math. 43, 903–948 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.University of Wisconsin – MadisonMadisonUSA
  3. 3.Princeton UniversityPrincetonUSA

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