Hidden symmetries and decay for the wave equation on the Kerr spacetime

Abstract

Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.

  • [Abramowitz:SpecialFunctionHandbook] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz, M. and Stegun, I. A., Eds., New York: Dover Publications, 1992.
    @book{Abramowitz:SpecialFunctionHandbook, mrkey = {1225604},
      title = {Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables},
      editor = {Abramowitz, Milton and Stegun, Irene A.},
      publisher = {Dover Publications},
      address = {New York},
      year = {1992},
      pages = {xiv+1046},
      isbn = {0-486-61272-4},
      mrclass = {00A20 (00A22 33-00)},
      mrnumber = {1225604},
      }
  • [AlexakisKlainermanIonescu:Uniqueness] Go to document S. Alexakis, A. D. Ionescu, and S. Klainerman, "Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces," Comm. Math. Phys., vol. 299, iss. 1, pp. 89-127, 2010.
    @article{AlexakisKlainermanIonescu:Uniqueness, mrkey = {2672799},
      author = {Alexakis, S. and Ionescu, A. D. and Klainerman, S.},
      title = {Uniqueness of smooth stationary black holes in vacuum: small perturbations of the {K}err spaces},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {299},
      year = {2010},
      number = {1},
      pages = {89--127},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {53C80 (83C57)},
      mrnumber = {2672799},
      doi = {10.1007/s00220-010-1072-1},
      zblnumber = {1200.83059},
      }
  • [andersson:blue:maxwell:2013arXiv1310.2664A] L. Andersson and P. Blue, Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, 2013.
    @misc{andersson:blue:maxwell:2013arXiv1310.2664A,
      author = {Andersson, Lars and Blue, Pieter},
      title = {Uniform energy bound and asymptotics for the {M}axwell field on a slowly rotating {K}err black hole exterior},
      note = {to appear in {\it J. Hyperbolic Differ. Equ.}},
      arxiv = {1310.2664},
      year = {2013},
      }
  • [andersson:blue:nicolas:MR3021792] L. Andersson, P. Blue, and J. Nicolas, "A decay estimate for a wave equation with trapping and a complex potential," Int. Math. Res. Not., vol. 2013, iss. 3, pp. 548-561, 2013.
    @article{andersson:blue:nicolas:MR3021792, mrkey = {3021792},
      author = {Andersson, Lars and Blue, Pieter and Nicolas, Jean-Philippe},
      title = {A decay estimate for a wave equation with trapping and a complex potential},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2013},
      number = {3},
      pages = {548--561},
      issn = {1073-7928},
      mrclass = {35L15 (35B40)},
      mrnumber = {3021792},
      mrreviewer = {Hidetoshi Tahara},
      volume = {2013},
      zblnumber = {06437181},
     }
  • [Bardeen] J. M. Bardeen, "Timelike and null geodesics in the Kerr metric," in Black Holes/Les Astres Occlus, DeWitt, C. and DeWitt, B. S., Eds., New York: Gordon and Breach Science Publishers, 1973, pp. 215-239.
    @incollection{Bardeen, mrkey = {0408678},
      author = {Bardeen, J. M.},
      booktitle = {Black Holes/{L}es Astres Occlus},
      title = {Timelike and null geodesics in the {K}err metric},
      editor = {DeWitt, C. and DeWitt, B. S.},
      publisher = {Gordon and Breach Science Publishers},
      address = {New York},
      year = {1973},
      pages = {215--239},
      mrclass = {83.00},
      mrnumber = {0408678},
      }
  • [Blue:Maxwell] Go to document P. Blue, "Decay of the Maxwell field on the Schwarzschild manifold," J. Hyperbolic Differ. Equ., vol. 5, iss. 4, pp. 807-856, 2008.
    @article{Blue:Maxwell, mrkey = {2475482},
      author = {Blue, Pieter},
      title = {Decay of the {M}axwell field on the {S}chwarzschild manifold},
      journal = {J. Hyperbolic Differ. Equ.},
      fjournal = {Journal of Hyperbolic Differential Equations},
      volume = {5},
      year = {2008},
      number = {4},
      pages = {807--856},
      issn = {0219-8916},
      mrclass = {35Q75 (35B40 35Q60)},
      mrnumber = {2475482},
      mrreviewer = {Xiaobing Henry Feng},
      doi = {10.1142/S0219891608001714},
      zblnumber = {1169.35057},
      }
  • [BlueSoffer:ODE] Go to document P. Blue and A. Soffer, "A space-time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold," Lett. Math. Phys., vol. 81, iss. 3, pp. 227-238, 2007.
    @article{BlueSoffer:ODE, mrkey = {2355489},
      author = {Blue, Pieter and Soffer, Avy},
      KEY={BSo07},
      title = {A space-time integral estimate for a large data semi-linear wave equation on the {S}chwarzschild manifold},
      journal = {Lett. Math. Phys.},
      fjournal = {Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics},
      volume = {81},
      year = {2007},
      number = {3},
      pages = {227--238},
      issn = {0377-9017},
      coden = {LMPHDY},
      mrclass = {58J45 (35L70 58J90)},
      mrnumber = {2355489},
      mrreviewer = {R. Schimming},
      doi = {10.1007/s11005-007-0177-8},
      zblnumber = {1137.58011},
      }
  • [BlueSoffer:LongPaper] Go to document P. Blue and A. Soffer, "Phase space analysis on some black hole manifolds," J. Funct. Anal., vol. 256, iss. 1, pp. 1-90, 2009.
    @article{BlueSoffer:LongPaper, mrkey = {2475417},
      KEY={BSo09},
      author = {Blue, Pieter and Soffer, Avy},
      title = {Phase space analysis on some black hole manifolds},
      journal = {J. Funct. Anal.},
      fjournal = {Journal of Functional Analysis},
      volume = {256},
      year = {2009},
      number = {1},
      pages = {1--90},
      issn = {0022-1236},
      coden = {JFUAAW},
      mrclass = {83C57 (35Q75)},
      mrnumber = {2475417},
      doi = {10.1016/j.jfa.2008.10.004},
      zblnumber = {1158.83007},
      }
  • [BlueSterbenz] Go to document P. Blue and J. Sterbenz, "Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space," Comm. Math. Phys., vol. 268, iss. 2, pp. 481-504, 2006.
    @article{BlueSterbenz, mrkey = {2259204},
      author = {Blue, Pieter and Sterbenz, Jacob},
      KEY={BSt06},
      title = {Uniform decay of local energy and the semi-linear wave equation on {S}chwarzschild space},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {268},
      year = {2006},
      number = {2},
      pages = {481--504},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {58J45 (83C47)},
      mrnumber = {2259204},
      mrreviewer = {Simone Calogero},
      doi = {10.1007/s00220-006-0101-6},
      zblnumber = {1123.58018},
      }
  • [Carter] Go to document B. Carter, "Global structure of the Kerr family of gravitational fields," Phys. Rev., vol. 174, pp. 1559-1571, 1968.
    @article{Carter,
      author = {Carter, B.},
      title = {Global structure of the {K}err family of gravitational fields},
      journal = {Phys. Rev.},
      volume = {174},
      year = {1968},
      pages = {1559--1571},
      zblnumber = {0167.56301},
      doi = {10.1103/PhysRev.174.1559},
      }
  • [Carter:KillingTensor] Go to document B. Carter, "Killing tensor quantum numbers and conserved currents in curved space," Phys. Rev. D, vol. 16, iss. 12, pp. 3395-3414, 1977.
    @article{Carter:KillingTensor, mrkey = {0459413},
      author = {Carter, Brandon},
      title = {Killing tensor quantum numbers and conserved currents in curved space},
      journal = {Phys. Rev. D},
      fjournal = {Physical Review. D. Particles and Fields. Third Series},
      volume = {16},
      year = {1977},
      number = {12},
      pages = {3395--3414},
      issn = {0556-2821},
      mrclass = {81.53},
      mrnumber = {0459413},
      mrreviewer = {A. Staruszkiewicz},
      doi = {10.1103/PhysRevD.16.3395},
      }
  • [Caviglia] Go to document G. Caviglia, "Conformal Killing tensors of order 2 for the Schwarzschild metric," Meccanica, vol. 18, pp. 131-135, 1983.
    @article{Caviglia,
      author = {Caviglia, G.},
      title = {Conformal {K}illing tensors of order 2 for the {S}chwarzschild metric},
      journal = {Meccanica},
      volume = {18},
      year = {1983},
      pages = {131--135},
      zblnumber = {0543.53025},
      doi = {10.1007/BF02128579},
      }
  • [ChanuEtAl] Go to document C. Chanu, L. Degiovanni, and R. G. McLenaghan, "Geometrical classification of Killing tensors on bidimensional flat manifolds," J. Math. Phys., vol. 47, iss. 7, p. 073506, 2006.
    @article{ChanuEtAl, mrkey = {2250306},
      author = {Chanu, C. and Degiovanni, L. and McLenaghan, R. G.},
      title = {Geometrical classification of {K}illing tensors on bidimensional flat manifolds},
      journal = {J. Math. Phys.},
      fjournal = {Journal of Mathematical Physics},
      volume = {47},
      year = {2006},
      number = {7},
      pages = {073506, 20},
      issn = {0022-2488},
      coden = {JMAPAQ},
      mrclass = {53B20 (37J35)},
      mrnumber = {2250306},
      mrreviewer = {Manuel F. Ra{ñ}ada},
      doi = {10.1063/1.2217649},
      zblnumber = {1112.53008},
      }
  • [ChristodoulouKlainerman:MinkowskiStability] D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton, NJ: Princeton Univ. Press, 1993.
    @book{ChristodoulouKlainerman:MinkowskiStability, mrkey = {1316662},
      author = {Christodoulou, Demetrios and Klainerman, Sergiu},
      title = {The Global Nonlinear Stability of the {M}inkowski Space},
      series = {Princeton Math. Series},
      number = {41},
      publisher = {Princeton Univ. Press},
      address = {Princeton, NJ},
      year = {1993},
      pages = {x+514},
      isbn = {0-691-08777-6},
      mrclass = {83C05 (35Q75 58G16 83C35)},
      mrnumber = {1316662},
      mrreviewer = {Alan D. Rendall},
      zblnumber = {0827.53055},
      }
  • [dafermos:rodnianski:Shlapentokh-Rothman:2014arXiv1402.7034] M. Dafermos, I. Rodnianski, and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $|a| < {M}$, 2014.
    @misc{dafermos:rodnianski:Shlapentokh-Rothman:2014arXiv1402.7034,
      author = {Dafermos, M. and Rodnianski, I. and Shlapentokh-Rothman, Y.},
      title = {Decay for solutions of the wave equation on {K}err exterior spacetimes {III: T}he full subextremal case $|a| < {M}$},
      year = {2014},
      arxiv = {1402.7034},
      }
  • [DafermosRodnianski:PriceLaw] Go to document M. Dafermos and I. Rodnianski, "A proof of Price’s law for the collapse of a self-gravitating scalar field," Invent. Math., vol. 162, iss. 2, pp. 381-457, 2005.
    @article{DafermosRodnianski:PriceLaw, mrkey = {2199010},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {A proof of {P}rice's law for the collapse of a self-gravitating scalar field},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {162},
      year = {2005},
      number = {2},
      pages = {381--457},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {83C05 (35Q75 83C22 83C75)},
      mrnumber = {2199010},
      mrreviewer = {Simone Calogero},
      doi = {10.1007/s00222-005-0450-3},
      zblnumber = {1088.83008},
      }
  • [DafermosRodnianski:MorawetzWithoutHarmonics] M. Dafermos and I. Rodnianski, A note on energy currents and decay for the wave equation on a Schwarzschild background, 2007.
    @misc{DafermosRodnianski:MorawetzWithoutHarmonics,
      author = {Dafermos, M. and Rodnianski, I.},
      title = {A note on energy currents and decay for the wave equation on a {S}chwarzschild background},
      year = {2007},
      arxiv = {0710.0171},
      }
  • [DafermosRodnianski:RedShiftSchwarzschild] Go to document M. Dafermos and I. Rodnianski, "The red-shift effect and radiation decay on black hole spacetimes," Comm. Pure Appl. Math., vol. 62, iss. 7, pp. 859-919, 2009.
    @article{DafermosRodnianski:RedShiftSchwarzschild, mrkey = {2527808},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {The red-shift effect and radiation decay on black hole spacetimes},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {62},
      year = {2009},
      number = {7},
      pages = {859--919},
      issn = {0010-3640},
      coden = {CPAMA},
      mrclass = {83C57 (35B40 35L05)},
      mrnumber = {2527808},
      mrreviewer = {Guilherme De Berredo-Peixoto},
      doi = {10.1002/cpa.20281},
      zblnumber = {1169.83008},
      }
  • [Dafermos:Rodnianski:2010maph.conf..421D] Go to document M. Dafermos and I. Rodnianski, "A new physical-space approach to decay for the wave equation with applications to black hole spacetimes," in XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, pp. 421-432.
    @incollection{Dafermos:Rodnianski:2010maph.conf..421D, mrkey = {2730803},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {A new physical-space approach to decay for the wave equation with applications to black hole spacetimes},
      booktitle = {X{VI}th {I}nternational {C}ongress on {M}athematical {P}hysics},
      pages = {421--432},
      publisher = {World Sci. Publ., Hackensack, NJ},
      year = {2010},
      mrclass = {58J45 (35L10 83C05 83C57)},
      mrnumber = {2730803},
      mrreviewer = {Alan D. Rendall},
      doi = {10.1142/9789814304634_0032},
      zblnumber = {1211.83019},
      }
  • [DafermosRodnianski:KerrEnergyBound] Go to document M. Dafermos and I. Rodnianski, "A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds," Invent. Math., vol. 185, iss. 3, pp. 467-559, 2011.
    @article{DafermosRodnianski:KerrEnergyBound, mrkey = {2827094},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {A proof of the uniform boundedness of solutions to the wave equation on slowly rotating {K}err backgrounds},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {185},
      year = {2011},
      number = {3},
      pages = {467--559},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {58J45 (35Q76 35R01 83C05 83C57)},
      mrnumber = {2827094},
      mrreviewer = {Alan D. Rendall},
      doi = {10.1007/s00222-010-0309-0},
      zblnumber = {1226.83029},
      }
  • [DafermosRodnianski:LectureNotes] M. Dafermos and I. Rodnianski, "Lectures on black holes and linear waves," in Evolution Equations, Providence, RI: Amer. Math. Soc., 2013, vol. 17, pp. 97-205.
    @incollection{DafermosRodnianski:LectureNotes, mrkey = {3098640},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {Lectures on black holes and linear waves},
      booktitle = {Evolution Equations},
      series = {Clay Math. Proc.},
      volume = {17},
      pages = {97--205},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2013},
      mrclass = {83C57 (35Q75 35R01 58J99 83C05)},
      mrnumber = {3098640},
      mrreviewer = {Hans-Peter K{ü}nzle},
      }
  • [donninger:schlag:soffer:2009arXiv0911.3179D] Go to document R. Donninger, W. Schlag, and A. Soffer, "On pointwise decay of linear waves on a Schwarzschild black hole background," Comm. Math. Phys., vol. 309, iss. 1, pp. 51-86, 2012.
    @article{donninger:schlag:soffer:2009arXiv0911.3179D, mrkey = {2864787},
      author = {Donninger, Roland and Schlag, Wilhelm and Soffer, Avy},
      title = {On pointwise decay of linear waves on a {S}chwarzschild black hole background},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {309},
      year = {2012},
      number = {1},
      pages = {51--86},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {58J45 (58J90 83C57)},
      mrnumber = {2864787},
      mrreviewer = {Atanas G. Stefanov},
      doi = {10.1007/s00220-011-1393-8},
      zblnumber = {1242.83054},
      }
  • [ErdelyiEtAl] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vols. I, II, New York: McGraw-Hill Book Company, 1953.
    @book{ErdelyiEtAl, mrkey = {0058756},
      author = {Erd{é}lyi, Arthur and Magnus, Wilhelm and Oberhettinger, Fritz and Tricomi, Francesco G.},
      title = {Higher Transcendental Functions. {V}ols. {I},
      {II}},
      publisher = {McGraw-Hill Book Company},
      address = {New York},
      year = {1953},
      pages = {xxvi+302, xvii+396},
      mrclass = {33.0X},
      mrnumber = {0058756},
      mrreviewer = {E. T. Copson},
      zblnumber = {0051.30303},
      zblnumber = {0052.29502},
      }
  • [FinsterKamranSmollerYau:DiracDecay] Go to document F. Finster, N. Kamran, J. Smoller, and S. -T. Yau, "Decay rates and probability estimates for massive Dirac particles in the Kerr-Newman black hole geometry," Comm. Math. Phys., vol. 230, iss. 2, pp. 201-244, 2002.
    @article{FinsterKamranSmollerYau:DiracDecay, mrkey = {1936790},
      author = {Finster, F. and Kamran, N. and Smoller, J. and Yau, S.-T.},
      title = {Decay rates and probability estimates for massive {D}irac particles in the {K}err-{N}ewman black hole geometry},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {230},
      year = {2002},
      number = {2},
      pages = {201--244},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {83C47 (81T20 83C57)},
      mrnumber = {1936790},
      mrreviewer = {V{\'ı}ctor M. Villalba},
      doi = {10.1007/s002200200648},
      zblnumber = {1026.83029},
      }
  • [FinsterKamranSmollerYau:KerrWaveRepresentation] Go to document F. Finster, N. Kamran, J. Smoller, and S. -T. Yau, "An integral spectral representation of the propagator for the wave equation in the Kerr geometry," Comm. Math. Phys., vol. 260, iss. 2, pp. 257-298, 2005.
    @article{FinsterKamranSmollerYau:KerrWaveRepresentation, mrkey = {2177321},
      author = {Finster, F. and Kamran, N. and Smoller, J. and Yau, S.-T.},
      title = {An integral spectral representation of the propagator for the wave equation in the {K}err geometry},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {260},
      year = {2005},
      number = {2},
      pages = {257--298},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {58J45 (34L40 35L15 47F05 83C10)},
      mrnumber = {2177321},
      mrreviewer = {Simone Calogero},
      doi = {10.1007/s00220-005-1390-x},
      zblnumber = {1089.83017},
      }
  • [Friedrich:MinkowskiStability] Go to document H. Friedrich, "Cauchy problems for the conformal vacuum field equations in general relativity," Comm. Math. Phys., vol. 91, iss. 4, pp. 445-472, 1983.
    @article{Friedrich:MinkowskiStability, mrkey = {0727195},
      author = {Friedrich, Helmut},
      title = {Cauchy problems for the conformal vacuum field equations in general relativity},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {91},
      year = {1983},
      number = {4},
      pages = {445--472},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {83C05 (58D25 83C30)},
      mrnumber = {0727195},
      mrreviewer = {Garry Ludwig},
      doi = {10.1007/BF01206015},
      zblnumber = {0555.35116}
    }
  • [FrolovNovikov] Go to document V. P. Frolov and I. D. Novikov, Black Hole Physics, Dordrecht: Kluwer Academic Publishers Group, 1998, vol. 96.
    @book{FrolovNovikov, mrkey = {1668599},
      author = {Frolov, Valeri P. and Novikov, Igor D.},
      title = {Black Hole Physics},
      series = {Fundamental Theories of Physics},
      volume = {96},
      note = {basic concepts and new developments, Chapter 4 and Section 9.9 written jointly with N. Andersson},
      publisher = {Kluwer Academic Publishers Group},
      address = {Dordrecht},
      year = {1998},
      pages = {xxii+770},
      isbn = {0-7923-5145-2},
      mrclass = {83C57 (83-01)},
      mrnumber = {1668599},
      mrreviewer = {Asghar Qadir},
      doi = {10.1007/978-94-011-5139-9},
      zblnumber = {0978.83001},
      }
  • [Hafner:KerrKGScattering] Go to document D. Häfner, "Sur la théorie de la diffusion pour l’équation de Klein-Gordon dans la métrique de Kerr," Dissertationes Math. $($Rozprawy Mat.$)$, vol. 421, p. 102, 2003.
    @article{Hafner:KerrKGScattering, mrkey = {2031494},
      author = {H{ä}fner, Dietrich},
      title = {Sur la théorie de la diffusion pour l'équation de {K}lein-{G}ordon dans la métrique de {K}err},
      journal = {Dissertationes Math. $($Rozprawy Mat.$)$},
      fjournal = {Dissertationes Mathematicae (Rozprawy Matematyczne)},
      volume = {421},
      year = {2003},
      pages = {102},
      issn = {0012-3862},
      coden = {DSMAAH},
      mrclass = {58J45 (35L15 35P25 58J50 81T20)},
      mrnumber = {2031494},
      mrreviewer = {Tohru Ozawa},
      doi = {10.4064/dm421-0-1},
      zblnumber = {1075.35093},
      }
  • [HafnerNicolas:KerrDiracScattering] Go to document D. Häfner and J. Nicolas, "Scattering of massless Dirac fields by a Kerr black hole," Rev. Math. Phys., vol. 16, iss. 1, pp. 29-123, 2004.
    @article{HafnerNicolas:KerrDiracScattering, mrkey = {2047861},
      author = {H{ä}fner, Dietrich and Nicolas, Jean-Philippe},
      title = {Scattering of massless {D}irac fields by a {K}err black hole},
      journal = {Rev. Math. Phys.},
      fjournal = {Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics},
      volume = {16},
      year = {2004},
      number = {1},
      pages = {29--123},
      issn = {0129-055X},
      mrclass = {83C57 (58J90 81T20)},
      mrnumber = {2047861},
      mrreviewer = {Michael Keyl},
      doi = {10.1142/S0129055X04001911},
      zblnumber = {1064.83036},
      }
  • [Holzegel] G. Holzegel, Ultimately Schwarzschildean spacetimes and the black hole stability problem, 2010.
    @misc{Holzegel,
      author = {Holzegel, G.},
      title = {Ultimately {S}chwarzschildean spacetimes and the black hole stability problem},
      year = {2010},
      arxiv = {1010.3216},
      }
  • [Klainerman:LorentzGenerators] Go to document S. Klainerman, "Uniform decay estimates and the Lorentz invariance of the classical wave equation," Comm. Pure Appl. Math., vol. 38, iss. 3, pp. 321-332, 1985.
    @article{Klainerman:LorentzGenerators, mrkey = {0784477},
      author = {Klainerman, Sergiu},
      title = {Uniform decay estimates and the {L}orentz invariance of the classical wave equation},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {38},
      year = {1985},
      number = {3},
      pages = {321--332},
      issn = {0010-3640},
      coden = {CPAMA},
      mrclass = {35L70 (35B40)},
      mrnumber = {0784477},
      mrreviewer = {C. Bardos},
      doi = {10.1002/cpa.3160380305},
      zblnumber = {0635.35059},
      }
  • [Koornwinder] Go to document T. H. Koornwinder, "Book Review: Symmetry and separation of variables," Bull. Amer. Math. Soc., vol. 1, iss. 6, pp. 1014-1019, 1979.
    @article{Koornwinder, mrkey = {1567202},
      author = {Koornwinder, Tom H.},
      title = {Book {R}eview: {S}ymmetry and separation of variables},
      journal = {Bull. Amer. Math. Soc.},
      fjournal = {American Mathematical Society. Bulletin. New Series},
      volume = {1},
      year = {1979},
      number = {6},
      pages = {1014--1019},
      issn = {0273-0979},
      coden = {BAMOAD},
      mrclass = {Contributed Item},
      mrnumber = {1567202},
      doi = {10.1090/S0273-0979-1979-14723-2},
      }
  • [Kronthaler] J. Kronthaler, Decay rates for spherical scalar waves in the Schwarzschild geometry, 2007.
    @misc{Kronthaler,
      author = {Kronthaler, J.},
      title = {Decay rates for spherical scalar waves in the {S}chwarzschild geometry},
      year = {2007},
      arxiv = {0709.3703},
      }
  • [LabaSoffer] I. Łaba and A. Soffer, "Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds," Helv. Phys. Acta, vol. 72, iss. 4, pp. 274-294, 1999.
    @article{LabaSoffer, mrkey = {1732864},
      author = {{\L}aba, I. and Soffer, A.},
      title = {Global existence and scattering for the nonlinear {S}chrödinger equation on {S}chwarzschild manifolds},
      journal = {Helv. Phys. Acta},
      fjournal = {Helvetica Physica Acta. Physica Theoretica. Societatis Physicae Helveticae Commentaria Publica},
      volume = {72},
      year = {1999},
      number = {4},
      pages = {274--294},
      issn = {0018-0238},
      coden = {HPACAK},
      mrclass = {58J50 (35Q75 81Q05)},
      mrnumber = {1732864},
      mrreviewer = {Ricardo Weder},
      zblnumber = {0976.58019},
      }
  • [LindbladRodnianski] Go to document H. Lindblad and I. Rodnianski, "Global existence for the Einstein vacuum equations in wave coordinates," Comm. Math. Phys., vol. 256, iss. 1, pp. 43-110, 2005.
    @article{LindbladRodnianski, mrkey = {2134337},
      author = {Lindblad, Hans and Rodnianski, Igor},
      title = {Global existence for the {E}instein vacuum equations in wave coordinates},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {256},
      year = {2005},
      number = {1},
      pages = {43--110},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {83C05 (35Q75 58J45)},
      mrnumber = {2134337},
      mrreviewer = {Marcel Dossa},
      doi = {10.1007/s00220-004-1281-6},
      zblnumber = {1081.83003},
      }
  • [Luk:Kerr:improveddecay:2010arXiv1009.0671L] Go to document J. Luk, "A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole," Anal. PDE, vol. 5, iss. 3, pp. 553-625, 2012.
    @article{Luk:Kerr:improveddecay:2010arXiv1009.0671L, mrkey = {2994507},
      author = {Luk, Jonathan},
      title = {A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating {K}err black hole},
      journal = {Anal. PDE},
      fjournal = {Analysis \& PDE},
      volume = {5},
      year = {2012},
      number = {3},
      pages = {553--625},
      issn = {2157-5045},
      mrclass = {83C57 (35B40 35Q75 35R01 58J99)},
      mrnumber = {2994507},
      mrreviewer = {Micha{\l}Eckstein},
      doi = {10.2140/apde.2012.5.553},
      zblnumber = {1267.83065},
      }
  • [MarzuolaMetcalfeTataruTohaneanu] Go to document J. Marzuola, J. Metcalfe, D. Tataru, and M. Tohaneanu, "Strichartz estimates on Schwarzschild black hole backgrounds," Comm. Math. Phys., vol. 293, iss. 1, pp. 37-83, 2010.
    @article{MarzuolaMetcalfeTataruTohaneanu, mrkey = {2563798},
      author = {Marzuola, Jeremy and Metcalfe, Jason and Tataru, Daniel and Tohaneanu, Mihai},
      title = {Strichartz estimates on {S}chwarzschild black hole backgrounds},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {293},
      year = {2010},
      number = {1},
      pages = {37--83},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {58J45 (35Q75 83C57)},
      mrnumber = {2563798},
      mrreviewer = {Atanas G. Stefanov},
      doi = {10.1007/s00220-009-0940-z},
      zblnumber = {1202.35327},
      }
  • [miller:MR0460751] W. Miller Jr., Symmetry and Separation of Variables, Reading, MA: Addison-Wesley Publishing Co., 1977, vol. 4.
    @book{miller:MR0460751, mrkey = {0460751},
      author = {Miller, Jr., Willard},
      title = {Symmetry and Separation of Variables},
      note = {with a foreword by Richard Askey},
      series = {Encyclopedia Math. Appl.},
      volume = {4},
      publisher = {Addison-Wesley Publishing Co.},
      address = {Reading, MA},
      year = {1977},
      pages = {xxx+285},
      isbn = {0-201-13503-5},
      mrclass = {33A75},
      mrnumber = {0460751},
      mrreviewer = {Ernest G. Kalnins},
      zblnumber = {0368.35002},
      }
  • [morawetz:1961:MR0132908] Go to document C. S. Morawetz, "The decay of solutions of the exterior initial-boundary value problem for the wave equation," Comm. Pure Appl. Math., vol. 14, pp. 561-568, 1961.
    @article{morawetz:1961:MR0132908, mrkey = {0132908},
      author = {Morawetz, Cathleen S.},
      title = {The decay of solutions of the exterior initial-boundary value problem for the wave equation},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {14},
      year = {1961},
      pages = {561--568},
      issn = {0010-3640},
      mrclass = {35.76},
      mrnumber = {0132908},
      mrreviewer = {H. G. Garnir},
      doi = {10.1002/cpa.3160140327},
      zblnumber = {0101.07701},
      }
  • [MR1240056] P. J. Olver, Applications of Lie Groups to Differential Equations, Second ed., New York: Springer-Verlag, 1993, vol. 107.
    @book{MR1240056, mrkey = {1240056},
      author = {Olver, Peter J.},
      title = {Applications of {L}ie Groups to Differential Equations},
      series = {Graduate Texts in Math.},
      volume = {107},
      edition = {Second},
      publisher = {Springer-Verlag},
      year = {1993},
      pages = {xxviii+513},
      isbn = {0-387-94007-3; 0-387-95000-1},
      mrclass = {58G35 (22E60 35A30 58F05 58F07)},
      mrnumber = {1240056},
      address = {New York},
      zblnumber = {0785.58003},
      }
  • [ONeill] B. O’Neill, The Geometry of Kerr Black Holes, Wellesley, MA: A K Peters Ltd., 1995.
    @book{ONeill, mrkey = {1328643},
      author = {O'Neill, B.},
      title = {The Geometry of {K}err Black Holes},
      publisher = {A K Peters Ltd.},
      address = {Wellesley, MA},
      year = {1995},
      mrnumber = {1328643},
      zblnumber = {0828.53078},
      }
  • [price:1972:I] Go to document R. H. Price, "Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields," Phys. Rev. D, vol. 5, pp. 2439-2454, 1972.
    @article{price:1972:I, mrkey = {0376104},
      author = {Price, Richard H.},
      title = {Nonspherical perturbations of relativistic gravitational collapse. {II}. {I}nteger-spin, zero-rest-mass fields},
      journal = {Phys. Rev. D},
      fjournal = {Physical Review. D. Particles and Fields. Third Series},
      volume = {5},
      year = {1972},
      pages = {2439--2454},
      issn = {0556-2821},
      mrclass = {85.35},
      mrnumber = {0376104},
      doi = {10.1103/PhysRevD.5.2439},
      }
  • [price:1972:II] Go to document R. ~H. Price, "Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields," Phys. Rev. D, vol. 5, pp. 2439-2454, 1972.
    @article{price:1972:II,
      author = {Price, R.~H.},
      title = {{Nonspherical Perturbations of Relativistic Gravitational Collapse. {II. I}nteger-Spin, Zero-Rest-Mass Fields}},
      journal = {Phys. Rev. D},
      year = {1972},
      volume = {5},
      pages = {2439--2454},
      doi = {10.1103/PhysRevD.5.2439},
      }
  • [ralston:MR0254433] Go to document J. V. Ralston, "Solutions of the wave equation with localized energy," Comm. Pure Appl. Math., vol. 22, pp. 807-823, 1969.
    @article{ralston:MR0254433, mrkey = {0254433},
      author = {Ralston, James V.},
      title = {Solutions of the wave equation with localized energy},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {22},
      year = {1969},
      pages = {807--823},
      issn = {0010-3640},
      mrclass = {35.76},
      mrnumber = {0254433},
      mrreviewer = {Teruo Ikebe},
      doi = {10.1002/cpa.3160220605},
      zblnumber = {0209.40402},
      }
  • [sbierski:2013arXiv1311.2477S] J. Sbierski, Characterisation of the energy of Gaussian beams on Lorentzian manifolds – with applications to black hole spacetimes, 2013.
    @misc{sbierski:2013arXiv1311.2477S,
      author = {Sbierski, J.},
      title = {Characterisation of the energy of {G}aussian beams on {L}orentzian manifolds - with applications to black hole spacetimes},
      year = {2013},
      arxiv = {1311.2477},
      }
  • [shlapentokh-rothman:QuantitativeModeStability] Go to document Y. Shlapentokh-Rothman, "Quantitative mode stability for the wave equation on the Kerr spacetime," Ann. Henri Poincaré, vol. 16, iss. 1, pp. 289-345, 2015.
    @article{shlapentokh-rothman:QuantitativeModeStability, mrkey = {3296646},
      author = {Shlapentokh-Rothman, Yakov},
      title = {Quantitative mode stability for the wave equation on the {K}err spacetime},
      journal = {Ann. Henri Poincaré},
      fjournal = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      volume = {16},
      year = {2015},
      number = {1},
      pages = {289--345},
      issn = {1424-0637},
      mrclass = {58J45 (35L10 35R01 83C05)},
      mrnumber = {3296646},
      doi = {10.1007/s00023-014-0315-7},
      zblnumber = {06394165},
     }
  • [Starobinskii:1973JETP…37…28S] Go to document A. ~A. Starobinskiv i, "Amplification of waves during reflection from a rotating “black hole”," Soviet J. Experimental Theor. Phys., vol. 37, p. 28, 1973.
    @article{Starobinskii:1973JETP...37...28S,
      author = {Starobinski{\v i},
      A.~A.},
      title = {Amplification of waves during reflection from a rotating ``black hole''},
      journal = {Soviet J. Experimental Theor. Phys.},
      volume = {37},
      year = {1973},
      pages = {28},
      URL = {http://www.jetp.ac.ru/cgi-bin/e/index/e/37/1/p28?a=list},
     }
  • [Tataru:Price] Go to document D. Tataru, "Local decay of waves on asymptotically flat stationary space-times," Amer. J. Math., vol. 135, iss. 2, pp. 361-401, 2013.
    @article{Tataru:Price, mrkey = {3038715},
      author = {Tataru, Daniel},
      title = {Local decay of waves on asymptotically flat stationary space-times},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {135},
      year = {2013},
      number = {2},
      pages = {361--401},
      issn = {0002-9327},
      mrclass = {35L15 (35B40 35R01 83C10)},
      mrnumber = {3038715},
      mrreviewer = {Calvin Tadmon},
      doi = {10.1353/ajm.2013.0012},
      zblnumber = {1266.83033},
      }
  • [TataruTohaneanu] Go to document D. Tataru and M. Tohaneanu, "A local energy estimate on Kerr black hole backgrounds," Int. Math. Res. Not., vol. 2011, iss. 2, pp. 248-292, 2011.
    @article{TataruTohaneanu, mrkey = {2764864},
      author = {Tataru, Daniel and Tohaneanu, Mihai},
      title = {A local energy estimate on {K}err black hole backgrounds},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2011},
      number = {2},
      pages = {248--292},
      issn = {1073-7928},
      mrclass = {58J45 (35Q75 83C57)},
      mrnumber = {2764864},
      mrreviewer = {Alan D. Rendall},
      doi = {10.1093/imrn/rnq069},
      volume = {2011},
      zblnumber = {1209.83028},
      }
  • [Teo] Go to document E. Teo, "Spherical photon orbits around a Kerr black hole," Gen. Relativity Gravitation, vol. 35, iss. 11, pp. 1909-1926, 2003.
    @article{Teo, mrkey = {2016715},
      author = {Teo, Edward},
      title = {Spherical photon orbits around a {K}err black hole},
      journal = {Gen. Relativity Gravitation},
      fjournal = {General Relativity and Gravitation},
      volume = {35},
      year = {2003},
      number = {11},
      pages = {1909--1926},
      issn = {0001-7701},
      coden = {GRHVA8},
      mrclass = {83C10 (83C15 83C57)},
      mrnumber = {2016715},
      doi = {10.1023/A:1026286607562},
      zblnumber = {1047.83024},
      }
  • [Teukolsky] S. Teukolsky, "Perturbations of a rotating black hole I: fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations," Astrophys. J., vol. 185, pp. 635-647, 1973.
    @article{Teukolsky,
      author = {Teukolsky, S.},
      title = {Perturbations of a rotating black hole {I}: fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations},
      journal = {Astrophys. J.},
      volume = {185},
      year = {1973},
      pages = {635--647},
      }
  • [tohaneanu:strichartz:kerr:MR2846348] Go to document M. Tohaneanu, "Strichartz estimates on Kerr black hole backgrounds," Trans. Amer. Math. Soc., vol. 364, iss. 2, pp. 689-702, 2012.
    @article{tohaneanu:strichartz:kerr:MR2846348, mrkey = {2846348},
      author = {Tohaneanu, Mihai},
      title = {Strichartz estimates on {K}err black hole backgrounds},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {364},
      year = {2012},
      number = {2},
      pages = {689--702},
      issn = {0002-9947},
      coden = {TAMTAM},
      mrclass = {35Q75 (35A02 35B30 35B40 35B45)},
      mrnumber = {2846348},
      doi = {10.1090/S0002-9947-2011-05405-X},
      zblnumber = {1234.35275},
      }
  • [wald:MR757180] Go to document R. M. Wald, General Relativity, Chicago, IL: University of Chicago Press, 1984.
    @book{wald:MR757180, mrkey = {0757180},
      author = {Wald, Robert M.},
      title = {General Relativity},
      publisher = {University of Chicago Press},
      address = {Chicago, IL},
      year = {1984},
      pages = {xiii+491},
      isbn = {0-226-87032-4; 0-226-87033-2},
      mrclass = {83-02 (83-01)},
      mrnumber = {0757180},
      mrreviewer = {Arthur Greenspoon},
      doi = {10.7208/chicago/9780226870373.001.0001},
      zblnumber = {0549.53001},
      }
  • [PenroseWalker:1970] Go to document M. Walker and R. Penrose, "On quadratic first integrals of the geodesic equations for type $\{22\}$ spacetimes," Comm. Math. Phys., vol. 18, pp. 265-274, 1970.
    @article{PenroseWalker:1970, mrkey = {0272351},
      author = {Walker, Martin and Penrose, Roger},
      title = {On quadratic first integrals of the geodesic equations for type {$\{22\}$} spacetimes},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {18},
      year = {1970},
      pages = {265--274},
      issn = {0010-3616},
      mrclass = {83.53},
      mrnumber = {0272351},
      mrreviewer = {J. R. Vanstone},
      doi = {10.1007/BF01649445},
      zblnumber = {0197.26404},
      }
  • [Whiting] Go to document B. F. Whiting, "Mode stability of the Kerr black hole," J. Math. Phys., vol. 30, iss. 6, pp. 1301-1305, 1989.
    @article{Whiting, mrkey = {0995773},
      author = {Whiting, Bernard F.},
      title = {Mode stability of the {K}err black hole},
      journal = {J. Math. Phys.},
      fjournal = {Journal of Mathematical Physics},
      volume = {30},
      year = {1989},
      number = {6},
      pages = {1301--1305},
      issn = {0022-2488},
      coden = {JMAPAQ},
      mrclass = {83C25 (83C15)},
      mrnumber = {0995773},
      doi = {10.1063/1.528308},
      zblnumber = {0689.53041},
      }

Authors

Lars Andersson

Albert Einstein Institute, Max Planck Institute for Gravitational Physics, Potsdam, Germany Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden

Pieter Blue

The School of Mathematics and the Maxwell Institute, University of Edinburgh, Edinburgh, Scotland, UK