Abstract
We show the existence of the Hawking vector field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth Einstein–Maxwell space–time without assuming the underlying space–time is analytic. This extends a result of Friedrich et al. (Commun Math Phys 204:691–707, 1971), which holds in the interior of the black hole region. Moreover, we also show, in the presence of an additional Killing vector field T which is tangent to the horizon and not vanishing on the bifurcate sphere, then space–time must be locally axially symmetric without the analyticity assumption. This axial symmetry plays a fundamental role in the classification theory of stationary black holes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alexakis, S.: Unique continuation for the vacuum Einstein equations. gr-qc0902. 1131 (2008, preprint)
Alexakis, S., Ionescu, A.D., Klainerman, S.: Hawking’s local rigidity theorem without analyticity. gr-qc0902.1173 (2009, preprint)
Bunting, G.L.: Proof of the Uniqueness Conjecture for Black Holes, Ph.D. Thesis, University of New England, Armidale (1983)
Carter B.: An axi-symmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971)
Friedrich H., Rácz I., Wald R.: On the rigidity theorem for space-times with a stationary event horizon or a compact Cauchy horizon. Commun. Math. Phys. 204, 691–707 (1999)
Hawking S.W., Ellis G.F.R.: The large scale structure of space–time. Cambridge University Press, London (1973)
Ionescu A.D., Klainerman S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175, 35–102 (2009)
Ionescu A.D., Klainerman S.: Uniqueness results for ill-posed characteristic problems in curved space-times. Commun. Math. Phys. 285, 873–900 (2009)
Israel W.: Event horizons in static electrovac space-times. Commun. Math. Phys. 8, 245–260 (1968)
Rendall A.: Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations. Proc. R. Soc. Lond. A 427, 221–239 (1990)
Robinson D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Piotr T. Chrusciel.
Rights and permissions
About this article
Cite this article
Yu, P. On Hawking’s Local Rigidity Theorem for Charged Black Holes. Ann. Henri Poincaré 11, 1–21 (2010). https://doi.org/10.1007/s00023-010-0033-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-010-0033-8