Abstract
This paper contains the second part of a two-part series on the stability and instability of extreme Reissner–Nordström spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation \({\square_{g}\psi=0}\) on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 crossing the future event horizon \({\mathcal{H}^{+}}\). We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon \({\mathcal{H}^{+}}\). A hierarchy of conservations laws on degenerate horizons is also derived.
Article PDF
Similar content being viewed by others
References
Aretakis, S.: The wave equation on extreme Reissner–Nordström black hole spacetimes I: stability and instability results. Commun. Math. Phys. (2011, to appear)
Aretakis, S.: The Price Law for Self-Gravitating Scalar Fields on Extreme Black Hole Spacetimes. In preparation
Blue P., Soffer A.: Phase space analysis on some black hole manifolds. J. Funct. Anal. 256(1), 1–90 (2009)
Christodoulou D., Klainerman S.: The Global Nonlinear Stability of the Minkowski Space. Princeton University Press, Oxford (1994)
Christodoulou D.: On the global initial value problem and the issue of singularities. Class. Quantum Gravity 16(12A), A23–A35 (1999)
Christodoulou D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math. 149(1), 183–217 (1999)
Christodoulou D.: The Action Principle and Partial Differential Equations. Princeton University Press, New Jersey (2000)
Christodoulou D.: The Formation of Black Holes in General Relativity. European Mathematical Society Publishing House, Zurich (2009)
Chruściel P., Nguyen L.: A uniqueness theorem for degenerate Kerr-Newman black holes. Ann. Henri Poincaré 11(4), 585–609 (2010) arXiv:1002.1737
Dafermos M.: Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations. Ann. Math. 158(3), 875–928 (2003)
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., Rodnianski I.: The redshift effect and radiation decay on black hole spacetimes. Commun. Pure Appl. Math. 62, 859–919 (2009)
Dafermos, M., Rodnianski, I.: A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds. Invent. Math. (2011)
Dafermos, M., Rodnianski, I.: Lectures on Black Holes and Linear Waves. arXiv:0811.0354
Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. arXiv:0910.4957
Dafermos, M., Rodnianski, I.: Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: The cases \({|a|\ll M}\) or axisymmetry. arXiv:1010.5132
Dafermos, M., Rodnianski, I.: The black holes stability problem for linear scalar perturbations. arXiv:1010.5137
Wald R.M.: Note on the stability of the Schwarzschild metric. J. Math. Phys. 20, 1056–1058 (1979)
Wald R.M.: General Relativity. The University of Chicago Press, Chicago (1984)
Wald R., Kay B.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Class. Quantum Gravity 4(4), 893–898 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Piotr T. Chrusciel.
Rights and permissions
About this article
Cite this article
Aretakis, S. Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations II. Ann. Henri Poincaré 12, 1491–1538 (2011). https://doi.org/10.1007/s00023-011-0110-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-011-0110-7