Abstract
A well-known open problem in general relativity, dating back to 1972, has been to prove Price’s law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates. It is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon. In this paper, we prove a well-defined (upper bound) formulation of Price’s law for the collapse of a self-gravitating scalar field with spherically symmetric initial data. We also allow the presence of an additional gravitationally coupled Maxwell field. Our results are obtained by a new mathematical technique for understanding the long-time behavior of large data solutions to the resulting coupled non-linear hyperbolic system of p.d.e.’s in 2 independent variables. The technique is based on the interaction of the conformal geometry, the celebrated red-shift effect, and local energy conservation; we feel it may be relevant for the problem of non-linear stability of the Kerr solution. When combined with previous work of the first author concerning the internal structure of charged black holes, which had assumed the validity of Price’s law, our results can be applied to the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data. Under Christodoulou’s C0-formulation, the conjecture is proven to be false.
Similar content being viewed by others
References
Barack, L.: Late time dynamics of scalar perturbations outside black holes. II. Schwarzschild geometry. Phys. Rev. D 59 (1999)
Barack, L., Ori, A.: Late-time decay of scalar perturbations outside rotating black holes. Phys. Rev. Lett. 82, 4388–4391 (1999)
Beem, J., Ehrlich, P., Easley, K.: Global Lorentzian geometry. Monographs and Textbooks in Pure and Applied Mathematics, vol. 202. New York: Marcel Dekker, Inc. 1996
Bicak, J.: Gravitational collapse with charge and small asymmetries I. Scalar perturbations. Gen. Relativ. Gravitation 3, 331–349 (1972)
Bonanno, A., Droz, S., Israel, W., Morsink, S.M.: Structure of the charged black hole interior. Proc. R. Soc. Lond., Ser. A 450, 553–567 (1995)
Brady, P., Smith, J.: Black hole singularities: a numerical approach. Phys. Rev. Lett. 75, 1256–1259 (1995)
Burko, L., Ori, A.: Late-time evolution of non-linear gravitational collapse. Phys. Rev. D 56, 7828–7832 (1997)
Chae, D.: Global existence of solutions to the coupled Einstein and Maxwell-Higgs system in the spherical symmetry. Ann. Inst. Henri Poincaré 4, 35–62 (2003)
Chandrasekhar, S., Hartle, J.B.: On crossing the Cauchy horizon of a Reissner-Nordström black-hole. Proc. R. Soc. Lond., Ser. A 384, 301–315 (1982)
Ching, E.S.C., Leung, P.T., Suen, W.M., Young, K.: Wave propagation in gravitational systems. Phys. Rev. D 52 (1995)
Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–335 (1969)
Christodoulou, D.: The global initial value problem in general relativity. Proceedings of the Marcel Grossman Meeting, Rome
Christodoulou, D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math. 149, 183–217 (1999)
Christodoulou, D.: A mathematical theory of gravitational collapse. Commun. Math. Phys. 109, 613–647 (1987)
Christodoulou, D.: On the global initial value problem and the issue of singularities. Classical Quantum Gravity 16, A23–A35 (1999)
Christodoulou, D.: Self-gravitating relativistic fluids: a two-phase model. Arch. Ration. Mech. Anal. 130, 343–400 (1995)
Christodoulou, D.: Bounded variation solutions of the spherically symmetric Einstein-scalar field equations. Commun. Pure Appl. Math. 46, 1131–1220 (1992)
Christodoulou, D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math. 44, 339–373 (1991)
Christodoulou, D.: The problem of a self-gravitating scalar field. Commun. Math. Phys. 105, 337–361 (1986)
Christodoulou, D., Tahvildar-Zadeh, A.S.: On the regularity of spherically symmetric wave maps. Commun. Pure Appl. Math 46, 1041–1091 (1993)
Chruściel, P.: On the uniqueness in the large of solutions of the Einstein’s equations (“strong cosmic censorship”). Canberra: Australian National University, Centre for Mathematics and its Applications 1991
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. Commun. Pure Appl. Math. 58, 445–504 (2005)
Dafermos, M.: Stability and instability of the Reissner-Nordström Cauchy horizon and the problem of uniqueness in general relativity. gr-qc/0209052, “Proceedings of the Conference on Non-compact Variational Problems and General Relativity in honor of Haim Brezis and Felix Browder”. Contemp. Math. 350, 99–113 (2004)
Dafermos, M.: Spherically symmetric spacetimes with a trapped surface. Class. Quantum Grav. 22, 2221–2232 (2005)
Dafermos, M.: Price’s law, mass inflation, and strong cosmic censorship. In: Relativity Today. Proceedings of the Seventh Hungarian Relativity Workshop, ed. by I. Rácz. Budapest: Akadémiai Kiadó 2004
Gundlach, C., Price, R.H., Pullin, J.: Late-time behavior of stellar collapse and explosions. I. Linearized perturbations. Phys. Rev. D 49, 883–889 (1994)
Gundlach, C., Price, R.H., Pullin, J.: Late-time behavior of stellar collapse and explosions. II. Nonlinear evolution Phys. Rev. D 49 (1994), 890–899
Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. London, New York: Cambridge University Press 1973
Kay, B., Wald, R.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Classical Quantum Gravity 4, 893–898 (1987)
Machedon, M., Stalker, J.: Decay of solutions to the wave equation on a spherically symmetric background. Preprint
Marsa, R.L., Choptuik, M.W.: Black-hole–scalar-field interactions in spherical symmetry. Phys. Rev. D 54, 4929–4943 (1996)
Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 57–59 (1965)
Poisson, E., Israel, W.: Internal structure of black holes. Phys. Rev. D (3) 41, 1796–1809 (1990)
Price, R.: Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations. Phys. Rev. D (3) 5, 2419–2438 (1972)
Rendall, A.: Local and Global Existence Theorems for the Einstein Equations. Living Rev. Relativ. 3 (2000)
Simpson, M., Penrose, R.: Internal instability in a Reissner-Nordström Black Hole. Int. J. Theor. Phys. 7, 183–197 (1973)
Twainy, F.: The Time Decay of Solutions to the Scalar Wave Equation in Schwarzschild Background. Thesis. San Diego: University of California 1989
Yau, S.T.: Problem Section. In: Seminar on Differential Geometry, ed. by S.T. Yau. Princeton, NJ: Ann. Math. Stud. 1982
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s00222-005-0450-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-005-0450-3