Abstract
We consider solutions of the scalar wave equation \({\Box_g\phi=0}\), without symmetry, on fixed subextremal Reissner-Nordström backgrounds \({(\mathcal{M}, g)}\) with nonvanishing charge. Previously, it has been shown that for ϕ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon \({\mathcal{H}^+}\). Using this, we show here that ϕ is in fact uniformly bounded, \({|\phi| \leq C}\), in the black hole interior up to and including the bifurcate Cauchy horizon \({\mathcal{C}\mathcal{H}^+}\), to which ϕ in fact extends continuously. The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates. In a forthcoming companion paper we will extend the result to subextremal Kerr backgrounds with nonvanishing rotation.
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Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009). arXiv:0908.2265
Aretakis, S.: Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I. Commun. Math. Phys. 307, 17–16 (2011). arXiv:gr-qc/1110.2007v1
Aretakis, S.: Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations II. Ann. Henri Poincaré 8, 1491–1538 (2011). arXiv:gr-qc/1110.2009v1
Aretakis, S.: Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds. J. Funct. Anal. 263, 2770–2831 (2012). arXiv:gr-qc/1110.2006v1
Aretakis, S.: Horizon instability of extremal black holes (2012). arXiv:gr-qc/1206.6598v2
Blue, P., Soffer, A.: Phase space analysis on some black hole manifolds. J. Funct. Anal. 256, 1–90 (2009). arXiv:math.AP/0511281
Christodoulou D.: The Action Principle and Partial Differential Equations. Princeton University Press, Princeton (2000)
Christodoulou D.: On the global initial value problem and the issue of singularities. Class. Quantum Gravit. 16, A23–A35 (1999)
Costa J.L., Girão P.M., Natário J., Silva J.: On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 1: well posedness and breakdown criterion. Class. Quantum. Gravit. 32, 015017 (2015)
Costa, J.L., Girão, P.M., Natário, J., Silva, J.: On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 2: structure of the solutions and stability of the Cauchy horizon (2014). arXiv:gr-qc/1406.7253
Costa, J.L., Girão, P.M., Natário, J., Silva, J.: On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 3: mass inflation and extendibility of the solutions (2014). arXiv:gr-qc/1406.7261
Dafermos M.: Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-Scalar field equations. Ann. Math. Second Ser. 158(3), 875–928 (2003)
Dafermos M.: The interior of charged black holes and the problem of uniqueness in general relativity. Commun. Pure Appl. Math. LVIII, 0445–0504 (2005)
Dafermos, M.: Black holes without spacelike singularities. Commun. Math. Phys. 332, 729–757 (2014) arXiv:gr-qc/1201.1797v1
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(7), 859–919 (2009). arXiv:gr-qc/0512119
Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In: Exner P (ed.) XVIth International Congress on Mathematical Physics. World Scientific, London, pp. 421–433 (2009) arXiv:0910.4957v1 [math.AP]
Dafermos, M., Rodnianski, I.: Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: the cases |a| ≪ M or axisymmetry (2010). arXiv:1010.5132v1
Dafermos, M., Rodnianski, I.: The black hole stability problem for linear scalar perturbations. In: Damour T et al. (ed.) Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, vol. 2011, pp. 132–189. World Scientific, Singapore (2010). arXiv:1010.5137
Dafermos, M., Rodnianski, I.: A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds. Invent. Math. 185, 467–559 (2011). arXiv:gr-qc/0805.4309v1
Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. In: Clay Mathematics Proceedings on American Mathematical Society 17, 97–205 (2013). arXiv:gr-qc/0811.0354
Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a| < M (2014). arXiv:1402.7034v1
Dyatlov S.: Exponential energy decay for Kerr-de Sitter black holes beyond event horizons. Math. Res. Lett. 18(5), 1023–1035 (2011)
Gajic, D.: Linear waves in the interior of extremal black holes I (2015). arXiv:1509.06568 [gr-qc]
Hawking S.W., Ellis G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1975)
Klainerman, S.: Uniform decay estimates and the lorentz invariance of the classical wave equation. Commun. Pure Appl. Math. 38(3), 321–332 (1985)
Kommemi, J.: The global structure of spherically symmetric charged scalar field spacetimes. PhD thesis (2014)
Lucietti J., Murata K., Reall H.S., Tanahashi N.: On the horizon instability of an extreme Reissner-Nordström black hole. JHEP 1303, 035 (2013)
Lucietti, J., Reall, H.S.: Gravitational instability of an extreme Kerr black hole. Phys. Rev. D 86, 104030 (2012). arXiv:gr-qc/1208.1437
Luk J.: Improved decay for solutions to the linear wave equation on a Schwarzschild black hole. Ann. Henri Poincaré 11, 805–880 (2010)
Luk J.: A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole. Anal. PDE 5(3), 553–625 (2012)
Luk, J.: Weak null singularities in general relativity (2013). arXiv:1311.4970v1
McNamara J.M.: Behaviour of scalar perturbations of a Reissner-Nordström black hole inside the event horizon. Proc. R. Soc. Lond. A. 362, 121–134 (1978)
McNamara J.M.: Instability of black hole inner horizons. Proc. R. Soc. Lond. A. 358, 499–517 (1978)
Metcalfe J., Tataru D., Tohaneanu M.: Price’s law on nonstationary space-times. Adv. Math. 230, 995–1028 (2012)
Murata K., Reall H.S., Tanahashi N.: What happens at the horizon(s) of an extreme black hole?. Class. Quantum. Gravit. 30, 235007 (2013)
Ori A.: Inner structure of a charged black hole: an exact mass-inflation solution. Phys. Rev. Lett. 67, 789–792 (1991)
Penrose, R.: Singularities and time-asymmetry. In: Hawking S.W., IsraelW.I. (eds.) General relativity: an Einstein centenary survey, pp. 581–638, Cambridge University Press, Cambridge (1979)
Poisson E., Israel W.: Internal structure of black holes. Phys. Rev. D 41, 1796–1809 (1990)
Sbierski, J.: Characterisation of the energy of Gaussian beams on Lorentzian manifolds—with applications to black hole spacetimes (2013). arXiv:gr-qc/1311.2477
Schlue V.: Decay of linear waves on higher-dimensional Schwarzschild black holes. Anal. PDE 6(3):515–600 (2010). arXiv:gr-qc/1012.5963
Shlapentokh-Rothman Y.: Quantitative mode stability for the wave equation on the Kerr spacetime. Ann. Henri Poincaré 16(1), 289–345 (2015)
Simpson M., Penrose R.: Internal instability in a Reissner-Nordström black hole. Int. J. Theor. Phys. 17, 183–197 (1973)
Tataru, D., Tohaneanu, M.: A local energy estimate on Kerr black hole backgrounds. IMRN 2011(2), 248–292 (2011)
Taylor M.E.: Partial Differential Equations I, Basic Theory. Springer, Berlin (2010)
Whiting, B.F.: Mode stability of the Kerr black hole. J. Math. Phys. 30(6), 1301–1305 (1989)
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Communicated by P. T. Chruściel
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Franzen, A.T. Boundedness of Massless Scalar Waves on Reissner-Nordström Interior Backgrounds. Commun. Math. Phys. 343, 601–650 (2016). https://doi.org/10.1007/s00220-015-2440-7
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DOI: https://doi.org/10.1007/s00220-015-2440-7