Abstract
We prove sharp pointwise t −3 decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates t −4, and t −6, respectively. We proceed by decomposition into angular momentum ℓ and summation of the decay estimates on the Regge-Wheeler equation for fixed ℓ. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in ℓ is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.
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References
Abramowitz, M., Stegun, I.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 edition. New York: Dover Publications, Inc., 1992
Alexandrova I., Bony J., Ramond T.: Resolvent and scattering matrix at the maximum of the potential. Serdica Math. J 34(1), 267–310 (2008)
Amrein, W., Boutet de Monvel, A., Georgescu, V.: C 0-groups, commutator methods and spectral theory of N-body Hamiltonians. Progress in Mathematics, 135. Basel: Birkhäuser Verlag, 1996
Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime. Preprint http://arXiv.org/abs/0908.2265v2 [math.AP], 2009
Balogh C.B.: Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math 15, 1315–1323 (1967)
Bony, J.-F., Fujiié, S., Ramond, T., Zerzeri, M.: Microlocal solutions of Schrödinger equations at a maximum point of the potential, Preprint 2009
Bony J.-F., Häfner D.: Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric. Commun. Math.Phys. 282(3), 697–719 (2008)
Briet P., Combes J.-M., Duclos P.: On the location of resonances for Schrödinger operators in the semiclassical limit. II. Barrier top resonances. Comm. Par. Diff. Eqs 12(2), 201–222 (1987)
Costin, O., Donninger, R., Schlag, W., Tanveer, S.: Semiclassical low energy scattering for one-dimensional Schrödinger operators with exponentially decaying potentials. To appear in Annales Henri Poincaré. http://arXiv.org/abs/1105.4221v1 [math.SP], 2011
Costin O., Schlag W., Staubach W., Tanveer S.: Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials. J. Funct. Anal. 255(9), 2321–2362 (2008)
Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. Preprint 2008, http://arXiv.org/abs/0811.0354v1 [gr-qc], 2008
Dafermos M., Rodnianski I.: The red-shift effect and radiation decay on black hole spacetimes. Comm. Pure Appl. Math 62(7), 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 185, 467–559 (2011)
Dafermos, M., Rodnianski, I.: Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| < < M or axisymmetry. Preprint http://arXiv.org/abs/1010.5132v1 [gr-qc], 2010
Davies E.B.: Spectral Theory and Differential Operators. Cambridge Univ. Press, Cambridge (1995)
Donninger R., Schlag W.: Decay estimates for the one-dimensional wave equation with an inverse power potential. Int. Math. Res. Not. 2010(22), 4276–4300 (2010)
Donninger R., Schlag W., Soffer A.: A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta. Adv. Math. 226(1), 484–540 (2011)
Finster F., Kamran N., Smoller J., Yau S.-T.: Decay of solutions of the wave equation in the Kerr geometry. Commun. Math. Phys 264(2), 465–503 (2006)
Gérard C., Grigis A.: Precise estimates of tunneling and eigenvalues near a potential barrier. J. Diff. Eqs 72(1), 149–177 (1988)
Graf G.: The Mourre estimate in the semiclassical limit. Lett. Math. Phys. 20(1), 47–54 (1990)
Gustafson S., Sigal I.M.: Mathematical concepts of quantum mechanics. Springer-Verlag, Universitext Berlin (2003)
Hawking S., Ellis G.: The large scale structure of space-time Cambridge Monographs on Mathematical Physics No 1. Cambridge University Press, London-New York (1973)
Helffer, B., Sjöstrand, J.: Semiclassical analysis of Harper’s equation III. Bull. Soc. Math. France, Memoire 39, 1990
Hislop P., Nakamura S.: Semiclassical resolvent estimates. Ann.Inst. H. Poincaré Phys. Théor 51(2), 187–198 (1989)
Hunziker W., Sigal I.M.: Time-dependent scattering theory of N-body quantum systems. Rev. Math. Phys. 12(8), 1033–1084 (2000)
Hunziker W., Sigal I.M., Soffer A.: Minimal escape velocities. Comm. Par. Diff. Eqs 24(11–12), 2279–2295 (1999)
Ivrii V.Ja., Sigal I.M.: Asymptotics of the ground state energies of large Coulomb systems. Ann. of Math 138(2), 243–335 (1993)
Kay B., Wald R.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Class. Quan. Grav 4(4), 893–898 (1987)
Marzuola J., Metcalfe J., Tataru D., Tohaneanu M.: Strichartz estimates on Schwarzschild black hole backgrounds. Commun. Math. Phys 293(1), 37–83 (2010)
Metcalfe, J., Tataru, D., Tohaneanu, M.: Price’s Law on Nonstationary Spacetimes. Preprint http://arXiv.org/abs/1104.5437v2 [math.AP], 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.: A Vector Field Method Approach to Improved Decay for Solutions to the Wave Equation on a Slowly Rotating Kerr Black Hole. Preprint, http://arXiv.org/abs/1009.0671v2 [gr-qc], 2011
Miller, P.D.: Applied asymptotic analysis. Graduate Studies in Mathematics, 75. Providence, RI: Amer. Math. Soc., 2006
Nakamura S.: Semiclassical resolvent estimates for the barrier top energy. Commun. Par. Diff. Eq. 16(4/5), 873–883 (1991)
Olver, F.W.J.: Asymptotics and Special Functions, Wellesley, MA: A K Peters, Ltd. 1997
Mourre E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78(3), 391–408 (1980/81)
Price R.: Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations. Phys. Rev. D 5(3), 2419–2438 (1972)
Price R.: Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields. Phys. Rev. D 5(3), 2439–2454 (1972)
Ramond T.: Semiclassical study of quantum scattering on the line. Commun. Math. Phys. 177(1), 221–254 (1996)
Schlag W., Soffer A., Staubach W.: Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part I. Trans. Amer. Math. Soc. 362(1), 19–52 (2010)
Schlag W., Soffer A., Staubach W.: Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part II. Trans. Amer. Math. Soc. 362(1), 289–318 (2010)
Sigal I.M., Soffer A.: Long-range many-body scattering. Invent. Math 99, 115–143 (1990)
Sigal, I.M., Soffer, A.: Local decay and velocity bounds. Preprint, Princeton University, 1988
Sjöstrand, J.: Semiclassical Resonances Generated by Nondegenerate Critical Points. In: Pseudodifferential Operators (Oberwolfach, 1986), Lecture Notes in Math., Vol. 1256, Berlin: Springer-Verlag, 1987, pp. 402–429
Skibsted E.: Propagation estimates for N-body Schroedinger operators. Commun. Math. Phys 142(1), 67–98 (1991)
Tataru, D.: Local decay of waves on asymptotically flat stationary space-times, Preprint 2009, http://arXiv.org/abs/0910.5290v2 [math.AP], 2010
Tataru D., Tohaneanu M.: A local energy estimate on Kerr black hole backgrounds. Int. Math. Res. Not. IMRN 2011(2), 248–292 (2011)
Tohaneanu, M.: Strichartz estimates on Kerr black hole backgrounds. Preprint http://arXiv.org/abs/0910.1545v1 [math.AP], 2009
Wald R.: General relativity. University of Chicago Press, Chicago, IL (1984)
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Communicated by P. T. Chruściel
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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). https://doi.org/10.1007/s00220-011-1393-8
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DOI: https://doi.org/10.1007/s00220-011-1393-8