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Phase space analysis on some black hole manifolds. (English) Zbl 1158.83007

Summary: The Schwarzschild and Reissner-Nordstrøm solutions to Einstein’s equations describe space-times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space-time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted \(L^{6}\) norm in space decays like \(t^{-1/3}\). This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an \(\epsilon\) loss of angular derivatives.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C57 Black holes
35Q75 PDEs in connection with relativity and gravitational theory
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[1] Bachelot, A., Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric, Ann. Inst. H. Poincaré Phys. Théor., 61, 411-441 (1994) · Zbl 0809.35141
[2] Bachelot, A.; Nicolas, J.-P., Equation non linéaire de Klein-Gordon dans des métriques de type Schwarzschild, C. R. Acad. Sci. Paris Sér. I Math., 316, 1047-1050 (1993) · Zbl 0776.35052
[3] P. Blue, Decay estimates and phase space analysis for wave equations on some black hole metrics, PhD thesis, Rutgers, The State University of New Jersey, New Brunswick, NJ, October 2004; P. Blue, Decay estimates and phase space analysis for wave equations on some black hole metrics, PhD thesis, Rutgers, The State University of New Jersey, New Brunswick, NJ, October 2004
[4] Blue, P.; Soffer, A., Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates, Adv. Differential Equations, 8, 595-614 (2003) · Zbl 1044.58033
[5] Blue, P.; Soffer, A., The wave equation on the Schwarzschild metric. II. Local decay for the spin-2 Regge-Wheeler equation, J. Math. Phys., 4, 012502 (2005), 9 pp · Zbl 1076.58020
[6] Blue, P.; Soffer, A., Errata for “Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds”, “Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates”, and “The wave equation on the Schwarzschild metric. II: Local decay for the spin-2 Regge-Wheeler equation” (2006), preprint · Zbl 1076.58020
[7] Blue, P.; Soffer, A., Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole (2006), preprint
[8] Blue, P.; Soffer, A., 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
[9] Blue, P.; Sterbenz, J., Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space, Comm. Math. Phys., 268, 481-504 (2006) · Zbl 1123.58018
[10] Christodoulou, D.; Klainerman, S., The Global Nonlinear Stability of the Minkowski Space, Princeton Math. Ser., vol. 41 (1993), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0733.35105
[11] Claudel, C.-M.; Virbhadra, K. S.; Ellis, G. F.R., The geometry of photon surfaces, J. Math. Phys., 42, 2, 818-838 (2001) · Zbl 1061.83525
[12] Dafermos, M., The interior of charged black holes and the problem of uniqueness in general relativity, Comm. Pure Appl. Math., 58, 445-504 (2005) · Zbl 1071.83037
[13] Dafermos, M.; Rodnianski, I., 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
[14] Dafermos, M.; Rodnianski, I., Small-amplitude nonlinear waves on a black hole background, J. Math. Pures Appl. (9), 84, 9, 1147-1172 (2005) · Zbl 1079.35069
[15] Dafermos, M.; Rodnianski, I., The red-shift effect and radiation decay on black hole spacetimes (2005), preprint
[16] Dafermos, M.; Rodnianski, I., A note on energy currents and decay for the wave equation on a Schwarzschild background (2007), preprint
[17] DeBièvre, S.; Hislop, P. D.; Sigal, I. M., Scattering theory for the wave equation on noncompact manifolds, Rev. Math. Phys., 4, 575-608 (1992) · Zbl 0778.58064
[18] Dereziński, J.; Gérard, C., Scattering Theory of Classical and Quantum \(N\)-Particle Systems (1997), Springer · Zbl 0899.47007
[19] Dimock, J., Scattering for the wave equation on the Schwarzschild metric, Gen. Relativity Gravitation, 17, 353-369 (1985) · Zbl 0618.35088
[20] Dimock, J.; Kay, B. S., Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric I, Ann. Phys., 175, 366-426 (1987) · Zbl 0628.53080
[21] Ellis, G. F.R.; Hawking, S. F., The Large Scale Structure of Space-Time, Cambridge Monogr. Math. Phys. (1973), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0265.53054
[22] Evans, L., Partial Differential Equations, Grad. Stud. Math., vol. 19 (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[23] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T., Decay rates and probability estimates for massive Dirac particles in the Kerr-Newman black hole geometry, Comm. Math. Phys., 230, 201-244 (2002) · Zbl 1026.83029
[24] 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
[25] Ginibre, J.; Velo, G., Conformal invariance and time decay for nonlinear wave equations. I, Ann. Inst. H. Poincaré, 47, 221-261 (1973) · Zbl 0644.35067
[26] Häfner, D., Complétude asymptotique pour l’équation des ondes dans une classe d’espaces-temps stationnaires et asymptotiquement plats, Ann. Inst. H. Poincaré, 51, 779-833 (2001) · Zbl 0981.35031
[27] Hassell, A.; Tao, T.; Wunsch, J., A Strichartz inequality for the Schrödinger equation on non-trapping asymptotically conic manifolds, Comm. Partial Differential Equations, 30, 157-205 (2005) · Zbl 1068.35119
[28] Hunziker, W.; Sigal, I. M., The quantum \(N\)-body problem, J. Math. Phys., 41, 3448-3510 (2000) · Zbl 0981.81026
[29] Kay, B. S.; Wald, R. M., Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere, Classical Quantum Gravity, 4, 893-898 (1987) · Zbl 0647.53065
[30] Lindblad, H.; Rodnianski, I., Global existence for the Einstein vacuum equations in wave coordinates, Comm. Math. Phys., 256, 1, 43-110 (2005) · Zbl 1081.83003
[31] Machihara, S.; Nakamura, M.; Nakanishi, K.; Ozawa, T., Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal., 219, 1-20 (2005) · Zbl 1060.35025
[32] Misner, C. W.; Thorne, K. S.; Wheeler, J. A., Gravitation (1973), Freeman: Freeman New York
[33] Nicolas, J. P., Nonlinear Klein-Gordon equation on Schwarzschild-like metrics, J. Math. Pures Appl. (9), 74, 35-58 (1995) · Zbl 0853.35123
[34] Regge, T.; Wheeler, J. A., Stability of a Schwarzschild singularity, Phys. Rev., 108, 1063-1069 (1957) · Zbl 0079.41902
[35] Sigal, I. M.; Soffer, A., The \(N\)-particle scattering problem: Asymptotic completeness for short-range systems, Ann. of Math. (2), 126, 35-108 (1987) · Zbl 0646.47009
[36] Soffer, A., On the many body problem in quantum mechanics, Astérisque, 207, 109-152 (1992) · Zbl 0809.47009
[37] Stein, E., Singular Integrals and Differentiability Properties of Functions (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ
[38] Sterbenz, J., Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not., 4, 187-231 (2005) · Zbl 1072.35048
[39] Stalker, J. G.; Tahvildar-Zadeh, A. S., Scalar waves on a naked-singularity background, Classical Quantum Gravity, 21, 2831-2848 (2004) · Zbl 1078.83029
[40] Virbhadra, K. S.; Ellis, G. F.R., Schwarzschild black hole lensing, Phys. Rev. D, 62 (2000), 084003-1-8
[41] Wald, R. M., Note on the stability of the Schwarzschild metric, J. Math. Phys., 20, 1056-1058 (1979)
[42] Whiting, B. F., Mode stability of the Kerr black hole, J. Math. Phys., 30, 1301-1305 (1989) · Zbl 0689.53041
[43] Zerilli, F. J., Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics, Phys. Rev. D, 3, 2141-2160 (1970) · Zbl 1227.83025
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