Skip to main content
SpringerLink
Log in

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

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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. http://arxiv.org/abs/0902.1173v1[gr-qc], 2009).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexakis, S.: Unique continuation for the vacuum Einstein equations. Preprint (2008), http://arxiv.org/abs/0902.1131v2[gr-qc], 2009

  2. Alexakis, S., Ionescu, A.D., Klainerman, S.: Hawking’s local rigidity theorem without analyticity. Preprint (2009), http://arxiv.org/abs/0902.1173v1[gr-qc], 2009

  3. Beig R., Simon W.: The stationary gravitational field near spatial infinity. Gen. Rel. Grav. 12, 1003–1013 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  4. Beig R., Simon W.: On the multipole expansion for stationary space-times. Proc. Roy. Soc. London Ser. A 376, 333–341 (1981)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Bunting, G.L.: Proof of the Uniqueness Conjecture for Black Holes. PhD Thesis, Univ. of New England, Armidale, NSW, 1983

  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)

    Article  MATH  ADS  Google Scholar 

  7. Carter B.: An axy-symmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971)

    Article  ADS  Google Scholar 

  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. 57–214

  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. 136–155

  10. Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space, Princeton Math. Series 41, Princeton University Press, 1993

  11. Chrusciel P.T.: On completeness of orbits of Killing vector fields. Class. Quant. Grav. 10, 2091–2101 (1993)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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. 23–49

  13. Chrusciel P.T.: On the rigidity of analytic black holes. Commun. Math. Phys. 189, 1–7 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Chrusciel P.T., Wald R.M.: Maximal hypersurfaces in stationary asymptotically flat spacetimes. Commun. Math. Phys. 163, 561–604 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Chrusciel P.T., Wald R.M.: On the topology of stationary black holes. Class. Quant. Grav. 11, L147–L152 (1993)

    Article  MathSciNet  Google Scholar 

  16. Chrusciel P.T., Delay T., Galloway G., Howard R.: Regularity of horizon and the area theorem. Ann. H. Poincaré 2, 109–178 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chrusciel, P.T., Costa, J.L.: On uniqueness of stationary vacuum black holes. Preprint (2008), http://arxiv.org/abs/0806.0016v2[gr-qc], 2008

  18. Chrusciel P.T.: On higher dimensional black holes with abelian isometry group. J. Math. Phys. 50, 052501 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  19. Friedman J.L., Schleich K., Witt D.M.: Topological censorship. Phys. Rev. Lett. 71, 1846–1849 (1993)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  Google Scholar 

  21. Hawking S.W., Ellis G.F.R.: The large scale structure of space-time. Cambridge Univ. Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  22. Heusler, M.: Black Hole Uniqueness Theorems, Cambridge Lect. Notes in Phys, Cambridge: Cambridge Univ. Press, 1996

  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, 1985

  24. Ionescu A.D., Klainerman S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175, 35–102 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Ionescu A.D., Klainerman S.: Uniqueness results for ill-posed characteristic problems in curved space-times. Commun. Math. Phys. 285, 873–900 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Isenberg J., Moncrief V.: Symmetries of Cosmological Cauchy Horizons. Commun. Math. Phys. 89, 387–413 (1983)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Israel W.: Event horizons in static vacuum space-times. Phys. Rev. Lett. 164, 1776–1779 (1967)

    ADS  Google Scholar 

  28. Klainerman, S., Nicolò, F.: The evolution problem in general relativity. Progress in Mathematical Physics, 25. Boston, MA: Birkhäuser Boston, Inc., 2003

  29. Mars M.: A spacetime characterization of the Kerr metric, Class. Quant. Grav. 16, 2507–2523 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Mazur P.O.: Proof of Uniqueness for the Kerr-Newman Black Hole Solution. J. Phys. A: Math. Gen. 15, 3173–3180 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  31. Robinson D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975)

    Article  ADS  Google Scholar 

  32. Racz I., Wald R.: Extensions of space-times with Killing horizons. Class. Quant. Gr. 9, 2463–2656 (1992)

    MathSciNet  Google Scholar 

  33. Simon W.: Characterization of the Kerr metric. Gen. Rel. Grav. 16, 465–476 (1984)

    Article  MATH  ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  35. Weinstein G.: On rotating black holes in equilibrium in general relativity. Comm. Pure Appl. Math. 43, 903–948 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    S. Alexakis

  2. University of Wisconsin – Madison, Madison, WI, 53706, USA

    A. D. Ionescu

  3. Princeton University, Princeton, NJ, 08544, USA

    S. Klainerman

Authors
  1. S. Alexakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. D. Ionescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Klainerman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. D. Ionescu.

Additional information

Communicated by P.T. Chruściel

The first author was partially supported by a Clay research fellowship.

The second author was partially supported by a Packard Fellowship.

The third author was partially supported by NSF grant DMS-0070696.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alexakis, S., Ionescu, A.D. & Klainerman, S. Uniqueness of Smooth Stationary Black Holes in Vacuum: Small Perturbations of the Kerr Spaces. Commun. Math. Phys. 299, 89–127 (2010). https://doi.org/10.1007/s00220-010-1072-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-010-1072-1

Keywords

Search

Navigation