Abstract
In this paper, we study the global behavior of solutions to the spherically symmetric coupled Einstein-Klein-Gordon (EKG) system in the presence of a negative cosmological constant. For the Klein-Gordon mass-squared satisfying a ≥ −1 (the Breitenlohner-Freedman bound being a > −9/8), we prove that the Schwarzschild-AdS spacetimes are asymptotically stable: Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching, at an exponential rate, a Schwarzschild-AdS spacetime. The main difficulties in the proof arise from the lack of monotonicity for the Hawking mass and the asymptotically AdS boundary conditions, which render even (part of) the orbital stability intricate. These issues are resolved in a bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained. As a corollary of our estimates on the Klein-Gordon field, one obtains in particular exponential decay in time of spherically-symmetric solutions to the linear Klein-Gordon equation on Schwarzschild-AdS.
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Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime. http://arXiv.org/abs/0908.2265vZ [math.AP], 2000
Blue P., Sterbenz J.: Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space. Commun. Math. Phys. 268(2), 481–504 (2006)
Bony J.-F., Häfner D.: Decay and non-decay of the local energy for the wave equation in the De Sitter-Schwarzschild metric. Commun. Math. Phys. 282, 697–719 (2008)
Breitenlohner P., Freedman D. Z.: Stability in Gauged Extended Supergravity. Ann. Phys. 144, 249 (1982)
Christodoulou D.: The Problem of a Selfgravitating Scalar Field. Commun. Math. Phys. 105, 337–361 (1986)
Christodoulou D.: A mathematical theory of gravitational collapse. Commun. Math. Phys. 109, 613–647 (1987)
Christodoulou D.: Examples of naked singularity formation in the gravitational collapse of a scalar field. Ann Math. 140, 607–653 (1994)
Christodoulou D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math. 149, 183–217 (1999)
Christodoulou, D., Klainerman, S.: The non-linear stability of the Minkowski space. Princeton Mathematical Series, Princeton NJ: Princeton Uniu Press, 1993
Dafermos M.: A note on naked singularities and the collapse of self- gravitating Higgs fields. Adv. Theor. Math. Phys. 9, 575–591 (2005)
Dafermos M.: Spherically symmetric spacetimes with a trapped surface. Class. Quant. Grav. 22, 2221–2232 (2005)
Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In: XVI International Congress on Mathematical Physics, P. Exner (ed.), London: World Scientific, 2009
Dafermos M., Holzegel G.: On the nonlinear stability of higher-dimensional triaxial Bianchi IX black holes. Adv. Theor. Math. Phys. 10, 503–523 (2006)
Dafermos M., Rendall A.: An extension principle for the Einstein-Vlasov system in spherical symmetry. Ann. H. Poincare 6, 1137–1155 (2005)
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 wave equation on Schwarzschild-de Sitter spacetimes. http://arXiv.org/abs/0709.2766v1 [gr-qc], 2008
Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. Institut Mittag-Leffler Report no. 14, 2008/2009 (2008), http://arXiv.org/abs/0811.0354v1 [gr-qc], 2009
Dafermos M., Rodnianski I.: The red-shift effect and radiation decay on black hole spacetimes. Comm. Pure Appl. Math. 62, 859–919 (2009)
Dafermos, M., Rodnianski, I.: Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases \({|a| \ll M}\) or axisymmetry. http://arXiv.org/abs/1010.5132v1 [gr-qc], 2010
Dafermos, M., Rodnianski, I.: The black hole stability problem for linear scalar perturbations. to appear in Proceedings of the 12th Marcel Grossmann Meeting (2010), http://arXiv.org/abs/1010.5137v1 [gr-qc], 2010
Dafermos M., Rodnianski I.: A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds. Invent. Math. 185(3), 467–559 (2011)
Donninger R., Schlag W., Soffer A.: On pointwise decay of linear waves on a Schwarzschild black hole background. Commun. Math. Phys. 309, 51–86 (2012)
Dyatlov S.: Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes. Ann. H. Poincare 13(5), 1101–1166 (2012)
Dyatlov S.: Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole. Commun. Math. Phys. 306, 119–163 (2011)
Friedrich H.: On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587–609 (1986)
Gauntlett J.P., Sonner J., Wiseman T.: Holographic superconductivity in M-Theory. Phys. Rev. Lett. 103, 151601 (2009)
Hartnoll S.A., Herzog C.P., Horowitz G.T.: Building a Holographic Superconductor. Phys. Rev. Lett. 101, 031601 (2008)
Holzegel G.: On the massive wave equation on slowly rotating Kerr-AdS spacetimes. Commun. Math. Phys. 294, 169–197 (2010)
Holzegel, G.: Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem. http://arXiv.org/abs/1010.3216v1 [gr-qc], 2010
Holzegel G.: Stability and decay-rates for the five-dimensional Schwarzschild metric under biaxial perturbations. Adv. Theor. Math. Phys. 14, 1245–1372 (2011)
Holzegel, G.: Well-posedness for the massive wave equation on asymptotically anti-de-Sitter spacetimes. J.Hyperbolic Differ. Equ. 9(2), 239–261 (2012)
Holzegel, G., Smulevici, J.: Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes, to appear in Communications on Pure and Applied Mathematics. http://arXiv.org/abs/1110.6794v1 [gr-qc], 2011
Holzegel, G., Smulevici, J.: Self-gravitating Klein-Gordon fields in asymptotically Anti-de Sitter spacetimes. Ann. Henri Poincaré 13(4), 991–1038 (2012)
Kodama H.: Conserved energy flux for the spherically symmetric system and the back reaction problem in black hole evaporation. Prog. Theor. Phys. 63, 1217 (1980)
Kommemi, J.: The Global structure of spherically symmetric charged scalar field spacetimes. http://arXiv.org/abs/1107.0949v1 [gr-qc], 2011
Luk, J.: A Vector Field Method Approach to Improved Decay for Solutions to the Wave Equation on a Slowly Rotating Kerr Black Hole. http://arXiv.org/abs/1009.0671v2 [gr-qc], 2011
Luk J.: Improved decay for solutions to the linear wave equation on a Schwarzschild black hole. Ann. Henri Poincaré 11(5), 805–880 (2010)
Luk, J.: The Null Condition and Global Existence for Nonlinear Wave Equations on Slowly Rotating Kerr Spacetimes. http://arXiv.org/abs/1009.4109v1 [gr-qc], 2010
Marzuola J., Metcalfe J., Tataru D., Tohaneanu M.: Strichartz Estimates on Schwarzschild Black Hole Backgrounds. Commun. Math. Phys. 293(1), 37–83 (2010)
Melrose, R., Sá Barreto, A., Vasy, A.: Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space. http://arXiv.org/abs/0811.2229v1 [math.Ap], 2008
Metcalfe, J., Tataru, D., Tohaneanu, M.: Price’s Law on Nonstationary Spacetimes. Adv. Math.230(3), 995–1028 (2012)
Ringström H.: Future stability of the Einstein-non-linear scalar field system. Invent. math. 173, 123–208 (2008)
Tataru, D.: Local decay of waves on asymptotically flat stationary space-times, to appear in American Journal of Mathematics, 134(6). http://arXiv.org/abs/0910.5290v2 [math.Ap], 2012
Tataru D., Tohaneanu M.: Local energy estimate on Kerr black hole backgrounds. Int. Math. Res. Not. 2011, 248–292 (2011)
Tohaneanu M.: Strichartz estimates on Kerr black hole backgrounds. Trans AMS 364, 689–702 (2012)
Vasy, A.: The wave equation on asymptotically Anti-de Sitter spaces. Analysis & PDE 5(1), 81–143 (2012)
Vasy, A., Dyatlov, S.: Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces. http://arXiv.org/abs/1012.4391v2 [math.Ap], 2011
Winstanley E.: Dressing a black hole with non-minimally coupled scalar field hair. Class. Quant. Grav. 22, 2233–2248 (2005)
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Communicated by P. T. Chruściel
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Holzegel, G., Smulevici, J. Stability of Schwarzschild-AdS for the Spherically Symmetric Einstein-Klein-Gordon System. Commun. Math. Phys. 317, 205–251 (2013). https://doi.org/10.1007/s00220-012-1572-2
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DOI: https://doi.org/10.1007/s00220-012-1572-2