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A proof of Price’s Law on Schwarzschild black hole manifolds for all angular momenta. (English) Zbl 1205.83041

Summary: Price’s Law states that linear perturbations of a Schwarzschild black hole fall off as \(t^{-2\ell-3}\) for \(t\rightarrow \infty \) provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be \(t^{ 2\ell-4}\). We give a proof of \(t^{-2\ell-2}\) decay for general data in the form of weighted \(L^1\) to \(L^\infty\) bounds for solutions of the Regge-Wheeler equation. For initially static perturbations we obtain \(t^{-2\ell-3}\). The proof is based on an integral representation of the solution which follows from self-adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.

MSC:

83C57 Black holes
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
35L05 Wave equation
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[1] Abramowitz, Milton; Stegun, Irene A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nat. Bureau Standards Appl. Math. Ser., vol. 55 (1964), U.S. Government Printing Office: U.S. Government Printing Office Washington, D.C., for sale by the superintendent of documents · Zbl 0171.38503
[2] Aichelburg, Peter C.; Bizoń, Piotr; Zbislaw, Tabor, Bifurcation and fine structure phenomena in critical collapse of a self-gravitating \(σ\)-field, Classical Quantum Gravity, 23, 16, S299-S306 (2006) · Zbl 1191.83001
[3] Andersson, Lars; Blue, Pieter, Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009), Preprint · Zbl 1373.35307
[4] Barack, Leor; Ori, Amos, Late-time decay of scalar perturbations outside rotating black holes, Phys. Rev. Lett., 82, 22, 4388-4391 (1999) · Zbl 0949.83044
[5] Bizoń, Piotr; Chmaj, Tadeusz; Rostworowski, Andrzej, Late-time tails of a Yang-Mills field on Minkowski and Schwarzschild backgrounds, Classical Quantum Gravity, 24, 13, F55-F63 (2007) · Zbl 1120.83028
[6] Bizoń, Piotr; Chmaj, Tadeusz; Rostworowski, Andrzej; Zajac, Stanislaw, Late-time tails of wave maps coupled to gravity (2009), Preprint · Zbl 1181.83057
[7] Blaksley, Carl J.; Burko, Lior M., Late-time tails in the Reissner-Nordström spacetime revisited, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 76, 10, 104035 (2007)
[8] Blue, P.; Soffer, A., Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates, Adv. Differential Equations, 8, 5, 595-614 (2003) · Zbl 1044.58033
[9] Blue, Pieter; Soffer, Avy, A space-time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold, Lett. Math. Phys., 81, 3, 227-238 (2007) · Zbl 1137.58011
[10] Blue, P.; Soffer, A., Phase space analysis on some black hole manifolds, J. Funct. Anal., 256, 1, 1-90 (2009) · Zbl 1158.83007
[11] Blue, Pieter; Sterbenz, Jacob, Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space, Comm. Math. Phys., 268, 2, 481-504 (2006) · Zbl 1123.58018
[12] Burko, Lior M.; Khanna, Gaurav, Late-time Kerr tails revisited, Classical Quantum Gravity, 26, 1, 015014 (2009), 28 · Zbl 1157.83329
[13] Chandrasekhar, S., On the equations governing the perturbations of the Schwarzschild black hole, Proc. R. Soc. Lond. Ser. A, 343, 289-298 (1975)
[14] Chandrasekhar, S., The Mathematical Theory of Black Holes, Oxf. Class. Texts Phys. Sci. (1998), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, reprint of the 1992 edition · Zbl 0912.53053
[15] Christodoulou, Demetrios; Klainerman, Sergiu, The Global Nonlinear Stability of the Minkowski Space, Princeton Math. Ser., vol. 41 (1993), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0827.53055
[16] Cohen, Jeffrey M.; Wald, Robert M., Point charge in the vicinity of a Schwarzschild black hole, J. Math. Phys., 12, 1845-1849 (1971)
[17] Dafermos, Mihalis; Rodnianski, Igor, A proof of Price’s law for the collapse of a self-gravitating scalar field, Invent. Math., 162, 2, 381-457 (2005) · Zbl 1088.83008
[18] Dafermos, Mihalis; Rodnianski, Igor, The red-shift effect and radiation decay on black hole spacetimes (2005), Preprint · Zbl 1169.83008
[19] Dafermos, Mihalis; Rodnianski, Igor, A note on energy currents and decay for the wave equation on a Schwarzschild background (2007), Preprint · Zbl 1211.83019
[20] Dafermos, Mihalis; Rodnianski, Igor, Lectures on black holes and linear waves (2008), Preprint · Zbl 1079.35069
[21] Damour, Thibault; Nagar, Alessandro, Comparing effective-one-body gravitational waveforms to accurate numerical data, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 77, 2, 024043 (2008)
[22] Damour, Thibault; Nagar, Alessandro, Improved analytical description of inspiralling and coalescing black-hole binaries, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 79, 8, 081503 (2009)
[23] Damour, Thibault; Nagar, Alessandro; Dorband, Ernst Nils; Pollney, Denis; Rezzolla, Luciano, Faithful effective-one-body waveforms of equal-mass coalescing black-hole binaries, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 77, 8, 084017 (2008)
[24] Damour, Thibault; Nagar, Alessandro; Hannam, Mark; Husa, Sascha; Brügmann, Bernd, Accurate effective-one-body waveforms of inspiralling and coalescing black-hole binaries, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 78, 4, 044039 (2008)
[25] Deift, P.; Trubowitz, E., Inverse scattering on the line, Comm. Pure Appl. Math., 32, 2, 121-251 (1979) · Zbl 0388.34005
[26] Roland Donninger, Wilhelm Schlag, Decay estimates for the one-dimensional wave equation with an inverse power potential, Int. Math. Res. Not. (2010), in press.; Roland Donninger, Wilhelm Schlag, Decay estimates for the one-dimensional wave equation with an inverse power potential, Int. Math. Res. Not. (2010), in press. · Zbl 1241.35018
[27] Donninger, Roland; Schlag, Wilhelm; Soffer, Avy, On pointwise decay of linear waves on a Schwarzschild black hole background (2009), Preprint · Zbl 1242.83054
[28] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T., Decay of solutions of the wave equation in the Kerr geometry, Comm. Math. Phys., 264, 2, 465-503 (2006) · Zbl 1194.83015
[29] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T., Linear waves in the Kerr geometry: A mathematical voyage to black hole physics, Bull. Amer. Math. Soc., 46, 635-659 (2009) · Zbl 1177.83082
[30] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T., A rigorous treatment of energy extraction from a rotating black hole, Comm. Math. Phys., 287, 3, 829-847 (2009) · Zbl 1200.83068
[31] Gundlach, Carsten; Price, Richard H.; Pullin, Jorge, Late-time behavior of stellar collapse and explosions. I. Linearized perturbations, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 49, 2, 883-889 (1994)
[32] Hod, Shahar, How pure is the tail of gravitational collapse?, Classical Quantum Gravity, 26, 2, 028001 (2009), 4 · Zbl 1158.83315
[33] Hořava, Petr, Quantum gravity at a Lifshitz point, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 79, 8, 084008 (2009)
[34] Kay, Bernard S.; Wald, Robert M., Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere, Classical Quantum Gravity, 4, 4, 893-898 (1987) · Zbl 0647.53065
[35] Kronthaler, Johann, The Cauchy problem for the wave equation in the Schwarzschild geometry, J. Math. Phys., 47, 4, 042501 (2006), 29 · Zbl 1111.83008
[36] Kronthaler, Johann, Decay rates for spherical scalar waves in the Schwarzschild geometry (2007), Preprint · Zbl 1111.83008
[37] Lindblad, Hans; Rodnianski, Igor, Global existence for the Einstein vacuum equations in wave coordinates, Comm. Math. Phys., 256, 1, 43-110 (2005) · Zbl 1081.83003
[38] Luk, Jonathan, Improved decay for solutions to the linear wave equation on a Schwarzschild black hole (2009), Preprint · Zbl 1208.83068
[39] Marzuola, Jeremy; Metcalfe, Jason; Tataru, Daniel; Tohaneanu, Mihai, Strichartz estimates on Schwarzschild black hole backgrounds (2008), Preprint · Zbl 1202.35327
[40] Matsas, G. E.A.; Richartz, M.; Saa, A.; da Silva, A. R.R.; Vanzella, D. A.T., Can quantum mechanics fool the cosmic censor?, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 79, 10, 101502 (2009)
[41] Metcalfe, J.; Tataru, D., Global parametrices and dispersive estimates for variable coefficient wave equations (2007), Preprint
[42] Metcalfe, J.; Tataru, D., Decay estimates for variable coefficient wave equations in exterior domains (2008), Preprint
[43] Poisson, Eric, Radiative falloff of a scalar field in a weakly curved spacetime without symmetries, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 66, 4, 044008 (2002)
[44] Price, Richard H., Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations, Phys. Rev. D (3), 5, 2419-2438 (1972)
[45] Price, Richard H., Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields, Phys. Rev. D (3), 5, 2439-2454 (1972)
[46] Price, Richard H.; Burko, Lior M., Late time tails from momentarily stationary, compact initial data in Schwarzschild spacetimes, Phys. Rev. D, 70, 8, 084039 (Oct. 2004)
[47] Pürrer, Michael; Aichelburg, Peter C., Tails for the Einstein-Yang-Mills system, Classical Quantum Gravity, 26, 3, 035004 (2009), 10 · Zbl 1159.83318
[48] Pürrer, Michael; Husa, Sascha; Aichelburg, Peter C., News from critical collapse: Bondi mass, tails, and quasinormal modes, Phys. Rev. D (Particles, Fields, Gravitation and Cosmology), 71, 10, 104005 (2005)
[49] Regge, Tullio; Wheeler, John A., Stability of a Schwarzschild singularity, Phys. Rev., 108, 4, 1063-1069 (1957) · Zbl 0079.41902
[50] Roman, Steven, The formula of Faà di Bruno, Amer. Math. Monthly, 87, 10, 805-809 (1980) · Zbl 0513.05009
[51] Schlag, W., Dispersive estimates for Schrödinger operators: a survey, (Mathematical Aspects of Nonlinear Dispersive Equations. Mathematical Aspects of Nonlinear Dispersive Equations, Ann. of Math. Stud., vol. 163 (2007), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ), 255-285 · Zbl 1143.35001
[52] Schlag, Wilhelm; Soffer, Avy; Staubach, Wolfgang, Decay for the wave and Schrödinger evolutions on manifolds with conical ends. I, Trans. Amer. Math. Soc., 362, 1, 19-52 (2010) · Zbl 1185.35046
[53] Schlag, Wilhelm; Soffer, Avy; Staubach, Wolfgang, Decay for the wave and Schrödinger evolutions on manifolds with conical ends. II, Trans. Amer. Math. Soc., 362, 1, 289-318 (2010) · Zbl 1187.35032
[54] Tataru, Daniel, Local decay of waves on asymptotically flat stationary space-times (2009), Preprint · Zbl 1266.83033
[55] Tataru, Daniel; Tohaneanu, Mihai, Local energy estimate on Kerr black hole backgrounds (2008), Preprint · Zbl 1209.83028
[56] Teschl, Gerald, Mathematical Methods in Quantum Mechanics, Grad. Stud. Math., vol. 99 (2009), American Mathematical Society: American Mathematical Society Providence, RI, with applications to Schrödinger operators · Zbl 1166.81004
[57] Vishveshwara, C. V., Stability of the Schwarzschild metric, Phys. Rev. D, 1, 10, 2870-2879 (1970)
[58] Wald, Robert M., Note on the stability of the Schwarzschild metric, J. Math. Phys., 20, 6, 1056-1058 (1979)
[59] Zerilli, Frank J., Effective potential for even-parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett., 24, 13, 737-738 (1970)
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