Abstract
It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner–Nordström data for the Einstein–Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously. Under additional assumptions, it is shown that the Hawking mass blows up identically along this bifurcate null hypersurface, and thus the metric cannot be extended twice differentiably; in fact, it cannot be extended in a weaker sense characterized at the level of the Christoffel symbols. The proof combines estimates obtained in previous work with an elementary Cauchy stability argument. There are no restrictions on the size of the support of the scalar field, and the result applies to both the future and past boundary of spacetime. In particular, it follows that for an open set in the moduli space of solutions around Reissner–Nordström, there is no spacelike component of either the future or the past singularity.
Similar content being viewed by others
References
Aretakis S.: Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations I. Commun. Math. Phys. 307, 17–63 (2011)
Aretakis S.: Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations II. Ann. Henri Poincaré 8, 1491–1538 (2011)
Belinskii V.A., Khalatnikov I.M., Lifshitz E.M.: Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys. 19, 525 (1970)
Bonanno A., Droz S., Israel W., Morsink S.M.: Structure of the charged spherical black hole interior. Proc. R. Soc. Lond. A 450, 553–567 (1995)
Brady P.R., Poisson E.: Cauchy horizon instability for Reissner–Nordstrom black holes in de Sitter space. Class. Quantum Gravity 9, 121–125 (1992)
Brady P.R., Núñez D., Sinha S.: Cauchy horizon singularity without mass inflation. Phys. Rev. D 47, 4239–4243 (1993)
Brady P., Smith J.D.: Black hole singularities: a numerical approach. Phys. Rev. Lett. 75(7), 1256–1259 (1995)
Brady P.R., Moss I.G., Myers R.C.: Cosmic censorship: as strong as ever. Phys. Rev. Lett. 80, 3432–3425 (1998)
Burko L.M.: Structure of the black hole’s Cauchy-horizon singularity. Phys. Rev. Lett. 79(25), 4958–4961 (1997)
Chambers, C.M.: The Cauchy horizon in black hole—de Sitter spacetimes. Ann. Israel Phys. Soc. 13, 33–84 (1997). arXiv:gr-qc/9709025
Choquét-Bruhat Y., Geroch R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–335 (1969)
Christodoulou D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math. 149(1), 183–217 (1999)
Christodoulou D.: On the global initial value problem and the issue of singularities. Class. Quantum Gravity 16, A23–A35 (1999)
Christodoulou, D.: The formation of black holes in general relativity. In: EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2009)
Dafermos M.: Stability and instability of the Cauchy horizon for the spherically-symmetric Einstein–Maxwell-scalar field equations. Ann. Math. 158, 875–928 (2003)
Dafermos M.: The interior of charged black holes and the problem of uniqueness in general relativity Comm. Pure Appl. Math. 58, 445–504 (2005)
Dafermos M.: Spherically symmetric spacetimes with a trapped surface class. Quantum Gravit. 22(11), 2221–2232 (2005)
Dafermos, M., Rodnianski, I.: A note on boundary value problems for black hole evolutions. (2004). arXiv:gr-qc/0403034
Dafermos M., Rodnianski I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Invent. Math. 162, 381–457 (2005)
Dafermos, M., Rendall A.: Strong Cosmic Censorship for Surface-Symmetric Cosmological Spacetimes with Collisionless Matter. (2007). arXiv:gr-qc/0701034
Dafermos, M., Rodnianski, I.: The Black Hole Stability Problem for Linear Scalar Perturbations. (2010). arXiv:1010.5137
Ellis G.F.R., King A.R.: Was the big bang a whimper?. Commun. Math. Phys. 39, 119–156 (1974)
Gnedin M.L., Gnedin N.Y.: Destruction of the Cauchy horizon in the Reissner–Nordström black hole. Class. Quantum Gravit. 10, 1083–1102 (1993)
Gundlach C., Price R., Pullin J.: Late-time behavior of stellar collapse and explosition. II: Nonlinear evolution. Phys. Rev. D 49, 890–899 (1994)
Herman R., Hiscock W.A.: Strength of the mass inflation singularity. Phys. Rev. D 46, 1863–1865 (1992)
Hiscock W.A.: Evolution of the interior of a charged black hole. Phys. Lett. 83A, 110–112 (1981)
Hod S., Piran T.: Mass inflation in dynamic gravitational collapse of a charged scalar field. Phys. Rev. Lett. 81, 1554–1557 (1998)
Israel, W.: Descent into the maelstrom: the black hole interior. In: Teitelboim, C., Zanelli, J. (eds.) The black Hole, 25 Years After. World Scientific, London (1998)
Khan K.A., Penrose R.: Scattering of two impulsive gravitational plane waves. Nature 229, 185–186 (1971)
Klainerman, S., Rodnianski, I., Szeftel, J.: The Bounded L2 Curvature Conjecture. (2012). arXiv:1204.1767
Kommemi J.: The global structure of spherically symmetric charged scalar field spacetimes. Commun. Math. Phys. 323(1), 35–106 (2013)
Kommemi, J.: The Global Structure of Spherically Symmetric Charged Scalar Field Spacetimes. Ph.D. Thesis, University of Cambridge, Cambridge (2013)
Luk, J., Rodnianski, I.: Local Propagation of Impulsive Gravitational Waves. (2012). arXiv:1209.1130
Luk, J., Rodnianski, I.: Nonlinear Interactions of Impulsive Gravitational Waves for the Vacuum Einstein Equations. (2013). arXiv:1301.1072
Ori A.: Inner structure of a charged black hole: an exact mass-inflation solution. Phys. Rev. Lett. 67, 789–792 (1991)
Ori A.: Perturbative approach to the inner structure of a rotating black hole. Gen. Relat. Gravit. 29(7), 881–929 (1997)
Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 57–59 (1965)
Penrose, R.: In: DeWitt, C.M., Wheeler, J.A. (eds.) Battelle Rencontres. W.A. Bejamin, New York, p. 222 (1968)
Poisson E., Israel W.: Inner-horizon instability and mass inflation in black holes. Phys. Rev. Lett. 63(16), 1663–1666 (1989)
Poisson E., Israel W.: Internal structure of black holes. Phys. Rev. D (3) 41(6), 1796–1809 (1990)
Senovilla J.M.M.: On the boundary of the region containing trapped surfaces. AIP Conf. Proc. 1122, 72–87 (2009)
Szekeres P.: Colliding plane gravitational waves. J. Math. Phys. 13, 286–294 (1972)
Tipler F.: Singularities in conformally flat spacetimes. Phys. Lett. 64A, 8–10 (1977)
Williams C.: Asymptotic behavior of spherically symmetric marginally trapped tubes. Ann. Henri Poincaré 9, 1029–1067 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. T. Chruściel
Rights and permissions
About this article
Cite this article
Dafermos, M. Black Holes Without Spacelike Singularities. Commun. Math. Phys. 332, 729–757 (2014). https://doi.org/10.1007/s00220-014-2063-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2063-4