Abstract
The phenomenon of wave tails has attracted much attention over the years from both physicists and mathematicians. However, our understanding of this fascinating phenomenon is not complete yet. In particular, most former studies of the tail phenomenon have focused on scattering potentials which approach zero asymptotically (x → ∞) faster than x −2. It is well-known that for these (rapidly decaying) scattering potentials the late-time tails are determined by the first Born approximation and are therefore linear in the amplitudes of the scattering potentials (there are, however, some exceptional cases in which the first Born approximation vanishes and one has to consider higher orders of the scattering problem). In the present study we analyze in detail the late-time dynamics of the Klein-Gordon wave equation with a (slowly decaying) Coulomb-like scattering potential: V (x → ∞) = α/x. In particular, we write down an explicit solution (that is, an exact analytic solution which is not based on the first Born approximation) for this scattering problem. It is found that the asymptotic (t → ∞) late-time behavior of the fields depends non-linearly on the amplitude α of the scattering potential. This non-linear dependence on the amplitude of the scattering potential reflects the fact that the late-time dynamics associated with this slowly decaying scattering potential is dominated by multiple scattering from asymptotically far regions.
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References
R.H. Price, Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations, Phys. Rev. D 5 (1972) 2419 [INSPIRE].
J. Bičák, Gravitational collapse with charge and small asymmetries, I: scalar perturbations, Gen. Rel. Grav. 3 (1972) 331.
Y. Sun and R. Price, Excitation of quasinormal ringing of a Schwarzschild black hole, Phys. Rev. D 38 (1988) 1040 [INSPIRE].
C. Gundlach, R.H. Price and J. Pullin, Late time behavior of stellar collapse and explosions: 1. Linearized perturbations, Phys. Rev. D 49 (1994) 883 [gr-qc/9307009] [INSPIRE].
S. Hod and T. Piran, Late time evolution of charged gravitational collapse and decay of charged scalar hair. 1, Phys. Rev. D 58 (1998) 024017 [gr-qc/9712041] [INSPIRE].
S. Hod and T. Piran, Late time evolution of charged gravitational collapse and decay of charged scalar hair. 2, Phys. Rev. D 58 (1998) 024018 [gr-qc/9801001] [INSPIRE].
S. Hod and T. Piran, Late time tails in gravitational collapse of a selfinteracting (massive) scalar field and decay of a selfinteracting scalar hair, Phys. Rev. D 58 (1998) 044018 [gr-qc/9801059] [INSPIRE].
S. Hod and T. Piran, Late time evolution of charged gravitational collapse and decay of charged scalar hair. 3. Nonlinear analysis, Phys. Rev. D 58 (1998) 024019 [gr-qc/9801060] [INSPIRE].
S. Hod, Late time evolution of realistic rotating collapse and the no hair theorem, Phys. Rev. D 58 (1998) 104022 [gr-qc/9811032] [INSPIRE].
P.R. Brady, C.M. Chambers, W.G. Laarakkers and E. Poisson, Radiative falloff in Schwarzschild-de Sitter space-time, Phys. Rev. D 60 (1999) 064003 [gr-qc/9902010] [INSPIRE].
R.-G. Cai and A. Wang, Late time evolution of the Yang-Mills field in the spherically symmetric gravitational collapse, Gen. Rel. Grav. 31 (1999) 1367 [gr-qc/9910067] [INSPIRE].
S. Hod, High order contamination in the tail of gravitational collapse, Phys. Rev. D 60 (1999) 104053 [gr-qc/9907044] [INSPIRE].
S. Hod, Mode coupling in rotating gravitational collapse of a scalar field, Phys. Rev. D 61 (2000) 024033 [gr-qc/9902072] [INSPIRE].
S. Hod, Mode coupling in realistic rotating gravitational collapse, gr-qc/9902073 [INSPIRE].
L. Barack, Late time decay of scalar, electromagnetic and gravitational perturbations outside rotating black holes, Phys. Rev. D 61 (2000) 024026 [gr-qc/9908005] [INSPIRE].
E. Malec, Diffusion of the electromagnetic energy due to the backscattering off Schwarzschild geometry, Phys. Rev. D 62 (2000) 084034 [gr-qc/0005130] [INSPIRE].
S. Hod, The radiative tail of realistic gravitational collapse, Phys. Rev. Lett. 84 (2000) 10 [gr-qc/9907096] [INSPIRE].
W.G. Laarakkers and E. Poisson, Radiative falloff in Einstein-Straus space-time, Phys. Rev. D 64 (2001) 084008 [gr-qc/0105016] [INSPIRE].
S. Hod, How pure is the tail of gravitational collapse?, Class. Quant. Grav. 26 (2009) 028001 [arXiv:0902.0237] [INSPIRE].
J.-l. Jing, Late-time behavior of massive Dirac fields in a Schwarzschild background, Phys. Rev. D 70 (2004) 065004 [gr-qc/0405122] [INSPIRE].
X. He and J. Jing, Late-time evolution of charged massive Dirac fields in the Kerr-Newman background, Nucl. Phys. B 755 (2006) 313 [gr-qc/0611003] [INSPIRE].
B. Wang, C. Molina and E. Abdalla, Evolving of a massless scalar field in Reissner-Nordstrom Anti-de Sitter space-times, Phys. Rev. D 63 (2001) 084001 [hep-th/0005143] [INSPIRE].
R. Konoplya and A. Zhidenko, Quasinormal modes of black holes: from astrophysics to string theory, Rev. Mod. Phys. 83 (2011) 793 [arXiv:1102.4014] [INSPIRE].
J. Gleiser, R.H. Price and J. Pullin, Late time tails in the Kerr spacetime, Class. Quant. Grav. 25 (2008) 072001 [arXiv:0710.4183] [INSPIRE].
M. Tiglio, L.E. Kidder and S.A. Teukolsky, High accuracy simulations of Kerr tails: Coordinate dependence and higher multipoles, Class. Quant. Grav. 25 (2008) 105022 [arXiv:0712.2472] [INSPIRE].
R. Moderski and M. Rogatko, Late time evolution of a selfinteracting scalar field in the space-time of dilaton black hole, Phys. Rev. D 64 (2001) 044024 [gr-qc/0105056] [INSPIRE].
P. Bizon, T. Chmaj and A. Rostworowski, Anomalously small wave tails in higher dimensions, Phys. Rev. D 76 (2007) 124035 [arXiv:0708.1769] [INSPIRE].
R. Moderski and M. Rogatko, Decay of Dirac massive hair in the background of spherical black hole, Phys. Rev. D 77 (2008) 124007 [arXiv:0805.0665] [INSPIRE].
B. Wang, C.-Y. Lin and C. Molina, Quasinormal behavior of massless scalar field perturbation in Reissner-Nordstrom Anti-de Sitter spacetimes, Phys. Rev. D 70 (2004) 064025 [hep-th/0407024] [INSPIRE].
E. Abdalla, B. Cuadros-Melgar, A. Pavan and C. Molina, Stability and thermodynamics of brane black holes, Nucl. Phys. B 752 (2006) 40 [gr-qc/0604033] [INSPIRE].
X. He, B. Wang, S.-F. Wu and C.-Y. Lin, Quasinormal modes of black holes absorbing dark energy, Phys. Lett. B 673 (2009) 156 [arXiv:0901.0034] [INSPIRE].
K.S. Thorne, Nonspherical gravitational collapse — A short review, in Magic without magic: John Archibald Wheeler, J. Klauder ed, W.H. Freeman, San Francisco U.S.A. (1972).
E. Ching, P. Leung, W. Suen and K. Young, Late time tail of wave propagation on curved space-time, Phys. Rev. Lett. 74 (1995) 2414 [gr-qc/9410044] [INSPIRE].
E. Ching, P. Leung, W. Suen and K. Young, Wave propagation in gravitational systems: Late time behavior, Phys. Rev. D 52 (1995) 2118 [gr-qc/9507035] [INSPIRE].
S. Hod, Wave tails in non-trivial backgrounds, Class. Quant. Grav. 18 (2001) 1311 [gr-qc/0008001] [INSPIRE].
S. Hod, Wave tails in time dependent backgrounds, Phys. Rev. D 66 (2002) 024001 [gr-qc/0201017] [INSPIRE].
P.M. Morse and H. Feshbach, Methods of theoretical physics, McGraw-Hill, U.S.A. (1953).
E.W. Leaver, Spectral decomposition of the perturbation response of the Schwarzschild geometry, Phys. Rev. D 34 (1986) 384 [INSPIRE].
N. Andersson, Evolving test fields in a black hole geometry, Phys. Rev. D 55 (1997) 468 [gr-qc/9607064] [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, Dover Publications, New York U.S.A. (1970).
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Hod, S. Scattering by a long-range potential. J. High Energ. Phys. 2013, 56 (2013). https://doi.org/10.1007/JHEP09(2013)056
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DOI: https://doi.org/10.1007/JHEP09(2013)056