Some aspects of spectral and microlocal methods in Mathematical General Relativity

Dietrich Häfner

doi:10.5802/crmeca.273

Article de synthèse

Some aspects of spectral and microlocal methods in Mathematical General Relativity
[Quelques aspects des méthodes spectrales et microlocales en relativité générale mathématique]

Dietrich Häfner ¹

¹ Université Grenoble Alpes, Institut Fourier, 100 rue des maths, 38610 Gières, France

Comptes Rendus. Mécanique, Volume 353 (2025), pp. 1-28.

Résumé
Abstract

Nous passons en revue quelques-unes des idées importantes de l’analyse spectrale et microlocale qui ont été appliquées à des problèmes de relativité générale mathématique au cours des dernières décennies.

We review some of the important ideas of spectral and microlocal analysis which have been applied to problems in Mathematical General Relativity over the last decades.

Reçu le : 2024-10-18
Accepté le : 2024-11-04
Publié le : 2025-01-08

DOI : 10.5802/crmeca.273

Keywords: Spectral theory, Microlocal analysis, General Relativity
Mots-clés : Théorie spectrale, Analyse microlocale, Relativité générale

Affiliations des auteurs :

Dietrich Häfner ¹

¹ Université Grenoble Alpes, Institut Fourier, 100 rue des maths, 38610 Gières, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Dietrich Häfner. Some aspects of spectral and microlocal methods in Mathematical General Relativity. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 1-28. doi : 10.5802/crmeca.273. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.273/

Bibliographie
Cité par

[1] J. J. Duistermaat; L. Hörmander Fourier integral operators. II, Acta Math., Volume 128 (1972), pp. 183-269 | DOI

[2] M. J. Radzikowski Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys., Volume 179 (1996) no. 3, pp. 529-553 | DOI

[3] C. Gérard; M. Wrochna Construction of Hadamard states by pseudo-differential calculus, Commun. Math. Phys., Volume 325 (2014) no. 2, pp. 713-755 | DOI

[4] J. Dimock; B. S. Kay Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric. I, Ann. Phys., Volume 175 (1987), pp. 366-426 | DOI

[5] A. Bachelot Asymptotic completeness for the Klein–Gordon equation on the Schwarzschild metric, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 61 (1994) no. 4, pp. 411-441

[6] A. Bachelot The Hawking effect, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 70 (1999) no. 1, pp. 41-99

[7] C. Dappiaggi; V. Moretti; N. Pinamonti Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime, Adv. Theor. Math. Phys., Volume 15 (2011) no. 2, pp. 355-447 | DOI

[8] D. Häfner Creation of Fermions by Rotating Charged Black Holes, Mém. Soc. Math. Fr., Nouv. Sér., 117, Société Mathématique de France (SMF), Paris, 2009 | DOI

[9] C. Gérard; D. Häfner; M. Wrochna The Unruh state for massless fermions on Kerr spacetime and its Hadamard property, Ann. Sci. Éc. Norm. Supér. (4), Volume 56 (2023) no. 1, pp. 127-196 | DOI

[10] D. Häfner; J.-P. Nicolas Scattering of massless Dirac fields by a Kerr black hole, Rev. Math. Phys., Volume 16 (2004) no. 1, pp. 29-123 | DOI

[11] C. K. M. Klein Construction of the Unruh state for a real scalar field on the Kerr–de Sitter spacetime, Ann. Henri Poincaré, Volume 24 (2023) no. 7, pp. 2401-2442 | DOI

[12] D. Häfner On the scattering theory for a Klein–Gordon equation in the Kerr metric, Diss. Math., Volume 421 (2003), pp. 1-102 (French) | DOI

[13] M. Dafermos; I. Rodnianski; Y. Shlapentokh-Rothman A scattering theory for the wave equation on Kerr black hole exteriors, Ann. Sci. Éc. Norm. Supér. (4), Volume 51 (2018) no. 2, pp. 371-486

[14] V. Georgescu; C. Gérard; D. Häfner Asymptotic completeness for superradiant Klein–Gordon equations and applications to the de Sitter–Kerr metric, J. Eur. Math. Soc. (JEMS), Volume 19 (2017) no. 8, pp. 2371-2444 | DOI

[15] P. Hintz; A. Vasy Analysis of linear waves near the Cauchy horizon of cosmological black holes, J. Math. Phys., Volume 58 (2017) no. 8, 081509 | DOI

[16] D. Christodoulou; S. Klainerman The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series, 41, Princeton University Press, Princeton, NJ, 1993

[17] A. Bachelot; A. Motet-Bachelot Resonances of a Schwarzschild black hole, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 59 (1993) no. 1, pp. 3-68 (French)

[18] A. S. Barreto; M. Zworski Distribution of resonances for spherical black holes, Math. Res. Lett., Volume 4 (1997) no. 1, pp. 103-121 | DOI

[19] J.-F. Bony; D. Häfner Decay and non-decay of the local energy for the wave equation on the De Sitter–Schwarzschild metric, Commun. Math. Phys., Volume 282 (2008) no. 3, pp. 697-719 | DOI

[20] R. Melrose; A. S. Barreto; A. Vasy Asymptotics of solutions of the wave equation on de Sitter–Schwarzschild space, Commun. Partial Differ. Equ., Volume 39 (2014) no. 3, pp. 512-529 | DOI

[21] S. Dyatlov Quasi-normal modes and exponential energy decay for the Kerr–de Sitter black hole, Commun. Math. Phys., Volume 306 (2011) no. 1, pp. 119-163 | DOI

[22] S. Dyatlov Exponential energy decay for Kerr–de Sitter black holes beyond event horizons, Math. Res. Lett., Volume 18 (2011) no. 5, pp. 1023-1035 | DOI

[23] A. Vasy Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math., Volume 194 (2013) no. 2, pp. 381-513 | DOI

[24] A. Vasy; M. Wrochna Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes, Ann. Henri Poincaré, Volume 19 (2018) no. 5, pp. 1529-1586 | DOI

[25] J. Wunsch; M. Zworski Resolvent estimates for normally hyperbolic trapped sets, Ann. Henri Poincaré, Volume 12 (2011) no. 7, pp. 1349-1385 | DOI

[26] S. Dyatlov Spectral gaps for normally hyperbolic trapping, Ann. Inst. Fourier, Volume 66 (2016) no. 1, pp. 55-82 | DOI

[27] P. Hintz Normally hyperbolic trapping on asymptotically stationary spacetimes, Probab. Math. Phys., Volume 2 (2021) no. 1, pp. 71-126 | DOI

[28] P. Hintz; A. Vasy The global non-linear stability of the Kerr–de Sitter family of black holes, Acta Math., Volume 220 (2018) no. 1, pp. 1-206 | DOI

[29] O. Petersen; A. Vasy Stationarity and Fredholm theory in subextremal Kerr–de Sitter spacetimes, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 20 (2024), 052

[30] P. Hintz Mode stability and shallow quasinormal modes of Kerr–de Sitter black holes away from extremality, J. Eur. Math. Soc. (2024) (Online first) | DOI

[31] J.-F. Bony; D. Häfner Low frequency resolvent estimates for long range perturbations of the Euclide Laplace, Math. Res. Lett., Volume 17 (2010) no. 2, pp. 301-306 | DOI

[32] J.-F. Bony; D. Häfner Local energy decay for several evolution equations on asymptotically Euclidean manifolds, Ann. Sci. Éc. Norm. Supér. (4), Volume 45 (2012) no. 2, pp. 311-335

[33] A. Vasy Limiting absorption principle on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach, Commun. Partial Differ. Equ., Volume 46 (2021) no. 5, pp. 780-822 | DOI

[34] A. Vasy Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach, Commun. Partial Differ. Equ., Volume 46 (2021) no. 5, pp. 823-863 | DOI

[35] D. Baskin; A. Vasy; J. Wunsch Asymptotics of radiation fields in asymptotically Minkowski space, Am. J. Math., Volume 137 (2015) no. 5, pp. 1293-1364 | DOI

[36] D. Baskin; A. Vasy; J. Wunsch Asymptotics of scalar waves on long-range asymptotically Minkowski spaces, Adv. Math., Volume 328 (2018), pp. 160-216 | DOI

[37] D. Gajic; C. M. Warnick Quasinormal modes on Kerr spacetimes, preprint, 2024 | arXiv

[38] T. Stucker Quasinormal modes for the Kerr black hole, preprint, 2024 | arXiv

[39] D. Häfner; P. Hintz; A. Vasy Linear stability of slowly rotating Kerr black holes, Invent. Math., Volume 223 (2021) no. 3, pp. 1227-1406 | DOI

[40] K. S. Thorne Disk accretion onto a black hole. 2. Evolution of the hole, Astrophys. J., Volume 191 (1974), pp. 507-520

[41] S. Dyatlov Asymptotics of linear waves and resonances with applications to black holes, Commun. Math. Phys., Volume 335 (2015) no. 3, pp. 1445-1485 | DOI

[42] B. F. Whiting Mode stability of the Kerr black hole, J. Math. Phys., Volume 30 (1989) no. 6, pp. 1301-1305 | DOI

[43] L. Andersson; D. Häfner; B. F. Whiting Mode analysis for the linearized Einstein equations on the Kerr metric : the large $𝔞$ case, J. Eur. Math. Soc. (2024) (Online first) | DOI

[44] P. Millet Optimal decay for solutions of the Teukolsky equation on the Kerr metric for the full subextremal range |a| $<$ M, preprint, 2023 | arXiv

[45] P. Hintz A sharp version of Price’s law for wave decay on asymptotically flat spacetimes, Commun. Math. Phys., Volume 389 (2022) no. 1, pp. 491-542 | DOI

[46] S. Klainerman; J. Szeftel Kerr stability for small angular momentum, Pure Appl. Math. Q., Volume 19 (2023) no. 3, pp. 791-1678 | DOI

[47] S. Klainerman; J. Szeftel Construction of GCM spheres in perturbations of Kerr, Ann. PDE, Volume 8 (2022) no. 2, 17 | DOI

[48] S. Klainerman; J. Szeftel Effective results on uniformization and intrinsic GCM spheres in perturbations of Kerr, Ann. PDE, Volume 8 (2022) no. 2, 18 | DOI

[49] E. Giorgi; S. Klainerman; J. Szeftel Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes, preprint, 2022 | arXiv

[50] D. Shen Construction of GCM hypersurfaces in perturbations of Kerr, Ann. PDE, Volume 9 (2023) no. 1, 11 | DOI

[51] M. Dafermos; G. Holzegel; I. Rodnianski; M. Taylor The non-linear stability of the Schwarzschild family of black holes, preprint, 2021 | arXiv

[52] O. Gannot Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis, Commun. Math. Phys., Volume 330 (2014) no. 2, pp. 771-799 | DOI

[53] G. Holzegel; J. Smulevici Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes, Anal. PDE, Volume 7 (2014) no. 5, pp. 1057-1090 | DOI

[54] M. Dafermos The interior of charged black holes and the problem of uniqueness in general relativity, Commun. Pure Appl. Math., Volume 58 (2005) no. 4, pp. 445-504 | DOI

[55] J. Luk; S.-J. Oh Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data. I: The interior of the black hole region, Ann. Math. (2), Volume 190 (2019) no. 1, pp. 1-111 | DOI

[56] J. Luk; S.-J. Oh Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data II: the exterior of the Black Hole region, Ann. PDE, Volume 5 (2019) no. 1, 6 | DOI

[57] V. Cardoso; J. L. Costa; K. Destounis; P. Hintz; A. Jansen Quasinormal modes and strong cosmic censorship, preprint, 2017 | arXiv

[58] S. Hollands; R. M. Wald; J. Zahn Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime, Class. Quantum Gravity, Volume 37 (2020) no. 11, 115009 | DOI

[59] C. Kehle; Y. Shlapentokh-Rothman A scattering theory for linear waves on the interior of Reissner–Nordström black holes, Ann. Henri Poincaré, Volume 20 (2019) no. 5, pp. 1583-1650 | DOI

[60] S. Chandrasekhar; J. B. Hartle On crossing the Cauchy horizon of a Reissner–Nordström black-hole, Proc. R. Soc. Lond., Ser. A, Volume 384 (1982) no. 1787, pp. 301-315 | DOI

[61] D. Häfner; M. Mokdad; J.-P. Nicolas Scattering theory for Dirac fields inside a Reissner–Nordström-type black hole, J. Math. Phys., Volume 62 (2021) no. 8, 081503 | DOI

[62] M. Mokdad; M. Provci Scattering of Dirac fields in the interior of Kerr–Newman(-Anti)–de Sitter black holes, preprint, 2023 | arXiv

[63] P. Hintz Linear waves on asymptotically flat spacetimes. I, preprint, 2023 | arXiv

[64] P. Hintz; A. Vasy Microlocal analysis near null infinity in asymptotically flat spacetimes, preprint, 2023 | arXiv

[65] N. Fenichel Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., Volume 21 (1971), pp. 193-226 | DOI

[66] M. W. Hirsch; C. C. Pugh; M. Shub Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer, Cham, 1977

[67] Y. Fourès-Bruhat Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math., Volume 88 (1952), pp. 141-225 (French) | DOI

[68] D. Häfner; P. Hintz; A. Vasy Correction to: “Linear stability of slowly rotating Kerr black holes”, Invent. Math., Volume 236 (2024) no. 1, pp. 477-481 | DOI

[69] L. Andersson; T. Bäckdahl; P. Blue; S. Ma Stability for linearized gravity on the Kerr spacetime, preprint, 2019 | arXiv

[70] M. Dafermos; G. Holzegel; I. Rodnianski The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Math., Volume 222 (2019) no. 1, pp. 1-214 | DOI

[71] P.-K. Hung; J. Keller; M.-T. Wang Linear stability of Schwarzschild spacetime: decay of metric coefficients, J. Differ. Geom., Volume 116 (2020) no. 3, pp. 481-541 | DOI

[72] P. Millet Geometric background for the Teukolsky equation revisited, Rev. Math. Phys., Volume 36 (2024) no. 3, 2430003 | DOI

[73] Y. Shlapentokh-Rothman; R. T. da Costa Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range |a| $<$ M: frequency space analysis, preprint, 2023 | arXiv

[74] Y. Shlapentokh-Rothman; R. T. da Costa Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range |a| $<$ M: physical space analysis, preprint, 2023 | arXiv

[75] S. Ma; L. Zhang Sharp decay for Teukolsky equation in Kerr spacetimes, Commun. Math. Phys., Volume 401 (2023) no. 1, pp. 333-434 | DOI

[76] L. Hörmander The Analysis of Linear Partial Differential Operators. III: Pseudo-Differential Operators, Grundlehren der mathematischen Wissenschaften, 274, Springer, Cham, 1985

[77] M. Zworski Semiclassical Analysis, Graduate Studies in Mathematics, 138, American Mathematical Society (AMS), Providence, RI, 2012

[78] J. Wunsch Microlocal analysis and evolution equations: lecture notes from 2008 CMI/ETH summer school April 25, 2013, Evolution Equations. Proceedings of the Clay Mathematics Institute Summer School, ETH, Zürich, Switzerland, June 23–July 18, 2008 (D. Ellwood et al., eds.) (Clay Mathematics Proceedings), Volume 17, American Mathematical Society (AMS), Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2013, pp. 1-72

[79] P. Hintz Introduction to microlocal analysis https://people.math.ethz.ch/~hintzp/notes/micro.pdf (Lecture Notes 2023)

[80] P. Hintz Global analysis and nonlinear wave equations on cosmological spacetimes, PhD thesis, Stanford university (2015) https://people.math.ethz.ch/~hintzp/thesis-augmented.pdf

[81] R. B. Melrose Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces, in Spectral and scattering theory, Proceedings of the 30th Taniguchi international workshop, held at Sanda, Hyogo, Japan, Marcel Dekker, Basel, 1994, pp. 85-130

[82] M. Zworski Resonances for asymptotically hyperbolic manifolds: Vasy’s method revisited, J. Spectr. Theory, Volume 6 (2016) no. 4, pp. 1087-1114 | DOI

[83] S. Dyatlov; M. Zworski Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér. (4), Volume 49 (2016) no. 3, pp. 543-577 http://hdl.handle.net/1721.1/115500

[84] Y. Shlapentokh-Rothman Exponentially growing finite energy solutions for the Klein–Gordon equation on sub-extremal Kerr spacetimes, Commun. Math. Phys., Volume 329 (2014) no. 3, pp. 859-891 | DOI

[85] N. Besset; D. Häfner Existence of exponentially growing finite energy solutions for the charged Klein–Gordon equation on the de Sitter–Kerr–Newman metric, J. Hyperbolic Differ. Equ., Volume 18 (2021) no. 2, pp. 293-310 | DOI

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