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Uniqueness of Kerr–Newman–de Sitter Black Holes with Small Angular Momenta

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Abstract

We show that a stationary solution of the Einstein–Maxwell equations which is close to a non-degenerate Reissner–Nordström–de Sitter solution is in fact equal to a slowly rotating Kerr–Newman–de Sitter solution. The proof uses the nonlinear stability of the Kerr–Newman–de Sitter family of black holes with small angular momenta, recently established by the author, together with an extension argument for Killing vector fields. Our black hole uniqueness result only requires the solution to have high but finite regularity; in particular, we do not make any analyticity assumptions.

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References

  1. 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(1), 89–127 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Alexakis, S., Ionescu, A.D., Klainerman, S.: Rigidity of stationary black holes with small angular momentum on the horizon. Duke Math. J. 163(14), 2603–2615 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carter, B.: Axisymmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26(6), 331 (1971)

    Article  ADS  Google Scholar 

  4. Chrusciel, P.T., Lopes Costa, J., Heusler, M.: Stationary black holes: uniqueness and beyond. Living Rev. Relativ. 15, 7 (2012)

    Article  ADS  MATH  Google Scholar 

  5. Chodosh, O., and Shlapentokh-Rothman, Y.: Time-Periodic Einstein–Klein–Gordon Bifurcations of Kerr. Preprint arXiv:1510.08025 (2015)

  6. Robin Graham, C., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87(2), 186–225 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  8. Hintz, P.: Non-linear stability of the Kerr–Newman–de Sitter family of charged black holes. Preprint arXiv:1612.04489 (2016)

  9. Hintz, P.: Resonance expansions for tensor-valued waves on asymptotically Kerr–de Sitter spaces. J. Spectr. Theory 7, 519–557 (2017). https://doi.org/10.4171/JST/171

  10. Hintz, P., Vasy, A.: Global analysis of quasilinear wave equations on asymptotically Kerr–de Sitter spaces. Int. Math. Res. Not. 2016(17), 5355–5426 (2016)

    Article  MathSciNet  Google Scholar 

  11. Hintz, P., Vasy, A.: The global non-linear stability of the Kerr–de Sitter family of black holes. Preprint arXiv:1606.04014 (2016)

  12. Ionescu, A.D., Klainerman, S.: On the uniqueness of smooth, stationary black holes in vacuum. Inventiones Mathematicae 175(1), 35–102 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Kodama, H., Ishibashi, A.: A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions. Prog. Theor. Phys. 110(4), 701–722 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Kodama, H., Ishibashi, A.: Master equations for perturbations of generalised static black holes with charge in higher dimensions. Prog. Theor. Phys. 111(1), 29–73 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Kodama, H.: Perturbative uniqueness of black holes near the static limit in all dimensions. Prog. Theor. Phys. 112(2), 249–274 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Mazur, P.O.: Proof of uniqueness of the Kerr–Newman black hole solution. J. Phys. A: Math. Gen. 15(10), 3173 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Moncrief, V.: Gauge-invariant perturbations of Reissner–Nordström black holes. Phys. Rev. D 12, 1526–1537 (1975)

    Article  ADS  Google Scholar 

  18. Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34(14), 905 (1975)

    Article  ADS  Google Scholar 

  19. Robinson, D.C.: Four decades of black hole uniqueness theorems. The Kerr spacetime: rotating black holes in general relativity, pp. 115–143 (2009)

  20. Taylor, M.E.: Partial Differential Equations I–III. Springer, Berlin (1996)

    Book  Google Scholar 

  21. Vishveshwara, C.V.: Stability of the Schwarzschild metric. Phys. Rev. D 1, 2870–2879 (1970)

    Article  ADS  Google Scholar 

  22. Wong, W.W.: A space–time characterization of the Kerr–Newman metric. Annales Henri Poincaré 10(3), 453–484 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Wong, W.W.: On the uniqueness of Kerr–Newman black holes. PhD thesis, Princeton University (2009)

  24. Wong, W.W., Yu, P.: Non-existence of multiple-black-hole solutions close to Kerr–Newman. Commun. Math. Phys. 325(3), 965–996 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I am grateful to András Vasy and Maciej Zworski for useful discussions, and to the Miller Institute at the University of California, Berkeley, for support.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA

    Peter Hintz

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  1. Peter Hintz
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Correspondence to Peter Hintz.

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Communicated by James A. Isenberg.

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Hintz, P. Uniqueness of Kerr–Newman–de Sitter Black Holes with Small Angular Momenta. Ann. Henri Poincaré 19, 607–617 (2018). https://doi.org/10.1007/s00023-017-0633-7

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