On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications

On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications
[Sur les propriétés de l’opérateur de contraintes relativistes dans des espaces à poids, et applications]

Chru, Piotr T. ; Delay, Erwann

Mémoires de la Société Mathématique de France, no. 94 (2003) , 109 p.

Résumé
Abstract

Nous étudions les propriétés de surjectivité de l’application de contraintes en relativité générale dans une large classe d’espaces fonctionnels à poids, généralisant ainsi une analyse de Corvino et Schoen. Comme corollaire on obtient plusieurs résultats de perturbation, de recollement, ou d’extension. Ainsi, nous démontrons l’existence d’espaces-temps non triviaux, sans singularités, solutions d’équations d’Einstein du vide, qui sont stationnaires dans un voisinage de $i^{0}$ . Pour des données initiales proches de celles de Minkowski ceci conduit, sous une condition de parité approximative, à des espaces-temps avec un infini isotrope $ℐ$ global et lisse. Nous prouvons l’existence de données initiales pour des trous noirs multiples qui sont exactement kerriennes, ou exactement schwarzschildiennes, dans une région asymptotique, mais aussi près de chaque composante connexe de l’horizon apparent. Nous montrons que pour des métriques génériques les perturbations des données initiales introduites par les recollements du type Isenberg-Mazzeo-Pollack peuvent être localisées, de sorte que les données initiales sur la variété obtenue en prenant la somme connexe coincident avec les données initiales originelles, sauf dans un petit voisinage de la zone de recollement. Nous prouvons l’existence de solutions asymptotiquement plates qui sont statiques ou stationnaires modulo des termes en $r^{- m}$ , avec $m$ arbitrairement prescrit, et avec des moments multipolaires qu’on peut spécifier librement dans certains ouverts.

Generalizing an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted function spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularity-free, vacuum space-times which are stationary in a neighborhood of $i^{0}$ ; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global $ℐ$ ; we prove existence of initial data for many black holes which are exactly Kerr — or exactly Schwarzschild — both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localized, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to $r^{- m}$ terms, for any fixed $m$ , and with multipole moments freely prescribable within certain ranges.

MR 2031583 | Zbl 1058.83007 | 3 citations dans Numdam

DOI : https://doi.org/10.24033/msmf.407
Classification : 83C05
Mots clés : Données initiales en relativité générale, trous noirs non connexes, espaces-temps asymptotiquement simples, recollements de données initiales

BibTeX
RIS

@book{MSMF_2003_2_94__1_0,
     author = {Chru, Piotr T. and Delay, Erwann},
     title = {On mapping properties of the~general relativistic constraints operator in~weighted function spaces, with~applications},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {94},
     year = {2003},
     doi = {10.24033/msmf.407},
     zbl = {1058.83007},
     mrnumber = {2031583},
     language = {en},
     url = {http://archive.numdam.org/item/MSMF_2003_2_94__1_0/}
}

TY  - BOOK
AU  - Chru, Piotr T.
AU  - Delay, Erwann
TI  - On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications
T3  - Mémoires de la Société Mathématique de France
PY  - 2003
DA  - 2003///
IS  - 94
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/MSMF_2003_2_94__1_0/
UR  - https://zbmath.org/?q=an%3A1058.83007
UR  - https://www.ams.org/mathscinet-getitem?mr=2031583
UR  - https://doi.org/10.24033/msmf.407
DO  - 10.24033/msmf.407
LA  - en
ID  - MSMF_2003_2_94__1_0
ER  -

Chru, Piotr T.; Delay, Erwann. On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mémoires de la Société Mathématique de France, Série 2, , no. 94 (2003), 109 p. doi : 10.24033/msmf.407. http://numdam.org/item/MSMF_2003_2_94__1_0/

Bibliographie
Cité par

[1] L. Andersson – « Elliptic systems on manifolds with asymptotically negative curvature », Indiana Univ. Math. Jour. 42 (1993), p. 1359–1388. | MR 1266098 | Zbl 0792.58042

[2] L. Andersson et P. Chruściel – « On “hyperboloidal" Cauchy data for vacuum Einstein equations and obstructions to smoothness of Scri », Commun. Math. Phys. 161 (1994), p. 533–568. | MR 1269390 | Zbl 0793.53084

[3] —, « On asymptotic behavior of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions” », Dissert. Math. 355 (1996), p. 1–100. | MR 1405962

[4] L. Andersson, P. Chruściel et H. Friedrich – « On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einsteins field equations », Commun. Math. Phys. 149 (1992), p. 587–612. | MR 1186044 | Zbl 0764.53027

[5] T. Aubin – « Espaces de Sobolev sur les variétés Riemanniennes », Bull. Sci. Math., II. Ser. 100 (1976), p. 149–173. | MR 488125 | Zbl 0328.46030

[6] —, Nonlinear analysis on manifolds. Monge-Ampère equations, Springer, New York, Heidelberg, Berlin, 1982.

[7] R. Bartnik – « The mass of an asymptotically flat manifold », Comm. Pure Appl. Math. 39 (1986), p. 661–693. | MR 849427 | Zbl 0598.53045

[8] R. Beig et P. Chruściel – « Killing Initial Data », Class. Quantum. Grav. 14 (1996), p. A83–A92, A special issue in honour of Andrzej Trautman on the occasion of his 64th Birthday, J.Tafel, editor.

[9] —, « Killing vectors in asymptotically flat space-times: I. Asymptotically translational Killing vectors and the rigid positive energy theorem », Jour. Math. Phys. 37 (1996), p. 1939–1961, gr-qc/9510015. | MR 1380882 | Zbl 0864.53063

[10] —, « The isometry groups of asymptotically flat, asymptotically empty space-times with timelike ADM four-momentum », Commun. Math. Phys. 188 (1997), p. 585–597, gr-qc/9610034. | MR 1473313 | Zbl 0890.53063

[11] R. Beig et N. Ó. Murchadha – « The Poincaré group as the symmetry group of canonical general relativity », Ann. Phys. 174 (1987), p. 463–498. | MR 881482 | Zbl 0617.70021

[12] R. Beig et W. Simon – « On the multipole expansion for stationary space-times », Proc. Roy. Soc. London A 376 (1981), p. 333–341. | MR 618449 | Zbl 0456.53037

[13] H. Bondi, M. Van Der Burg et A. Metzner – « Gravitational waves in general relativity VII: Waves from axi-symmetric isolated systems », Proc. Roy. Soc. London A 269 (1962), p. 21–52. | MR 147276 | Zbl 0106.41903

[14] G. Bunting et A. Masood-ul-Alam – « Nonexistence of multiple black holes in asymptotically euclidean static vacuum space-time », Gen. Rel. Grav. 19 (1987), p. 147–154. | MR 876598

[15] D. Christodoulou – « The boost problem for weakly coupled quasilinear hyperbolic systems of the second order », Jour. Math. Pures et Appl. 60 (1981), p. 99–130. | MR 616009 | Zbl 0474.35062

[16] D. Christodoulou et Y. Choquet-Bruhat – « Elliptic systems in $H_{s, δ}$ spaces on manifolds which are Euclidean at infinity », Acta. Math. 146 (1981), p. 129–150. | MR 594629 | Zbl 0484.58028

[17] D. Christodoulou et N. Ó. Murchadha – « The boost problem in general relativity », Comm. Math. Phys. 80 (1980), p. 271–300. | MR 623161 | Zbl 0477.35081

[18] P. Chruściel – « On the relation between the Einstein and the Komar expressions for the energy of the gravitational field », Ann. Inst. H. Poincaré 42 (1985), p. 267–282. | EuDML 76282 | Numdam | MR 797276 | Zbl 0645.53063

[19] —, « Boundary conditions at spatial infinity from a Hamiltonian point of view », in Topological Properties and Global Structure of Space-Time (P. Bergmann et V. de Sabbata, éds.), Plenum Press, New York, 1986, pp. 49-59, URL http://www.phys.univ-tours.fr/~piotr/scans.

[20] —, « On angular momentum at spatial infinity », Class. Quantum Grav. 4 (1987), p. L205–L210, erratum p. 1049. | Zbl 0624.53050

[21] —, « Quelques inégalités dans les espaces de Sobolev à poids », Tours preprint, unpublished, http://www.phys.univ-tours.fr/~piotr/papers/wpi, 1987.

[22] —, « On the invariant mass conjecture in general relativity », Commun. Math. Phys. 120 (1988), p. 233–248. | MR 973533 | Zbl 0661.53060

[23] P. Chruściel et E. Delay – « Existence of non-trivial asymptotically simple vacuum space-times », Class. Quantum Grav. 19 (2002), p. L71–L79, gr-qc/0203053, erratum Class. Quantum Grav. 19 (2002), 3389. | MR 1920322 | Zbl 1005.83009

[24] —, « Manifold structures for sets of solutions of the general relativistic constraint equations », gr-qc/0309001, 2003.

[25] P. Chruściel et M. Herzlich – « The mass of asymptotically hyperbolic Riemannian manifolds », Pacific Jour. Math. (2001), in press, dg-ga/0110035. | MR 2038048

[26] P. Chruściel et O. Lengard – « Solutions of wave equations in the radiating regime », Bull. Soc. Math. de France (2003), in press, math.AP/0202015.

[27] P. Chruściel et R. Mazzeo – « On “many-black-hole” vacuum spacetimes », Class. Quantum Grav. 20 (2003), p. 729–754, gr-qc/0210103. | MR 1959399 | Zbl 1033.83021

[28] P. Chruściel et G. Nagy – « The Hamiltonian mass of asymptotically anti-de Sitter space-times », Class. Quantum Grav. 18 (2001), p. L61–L68, hep-th/0011270. | MR 1834138 | Zbl 0978.83005

[29] P. Chruściel et W. Simon – « Towards the classification of static vacuum spacetimes with negative cosmological constant », Jour. Math. Phys. 42 (2001), p. 1779–1817, gr-qc/0004032. | MR 1820431 | Zbl 1009.83009

[30] J. Corvino – « Scalar curvature deformation and a gluing construction for the Einstein constraint equations », Commun. Math. Phys. 214 (2000), p. 137–189. | MR 1794269 | Zbl 1031.53064

[31] —, lecture in Oberwolfach, July 2000.

[32] J. Corvino et R. Schoen – « On the asymptotics for the vacuum Einstein constraint equations », gr-qc/0301071, 2003.

[33] —, « Vacuum spacetimes which are identically Schwarzschild near spatial infinity », talk given at the Santa Barbara Conference on Strong Gravitational Fields, June 22-26, 1999, http://doug-pc.itp.ucsb.edu/online/gravity_c99/schoen/.

[34] S. Dain – « Initial data for stationary space-times near space-like infinity », Class. Quantum Grav. 18 (2001), p. 4329–4338, gr-qc/0107018. | MR 1866795 | Zbl 1005.83007

[35] T. Damour et B. Schmidt – « Reliability of perturbation theory in general relativity », Jour. Math. Phys. 31 (1990), p. 2441–2453. | MR 1072957 | Zbl 0723.53050

[36] I. Ekeland et R. Temam – Convex analysis and variational problems, Studies in Math. and its Appl., vol. 1, North Holland, Amsterdam, 1976. | MR 463994

[37] H. Friedrich – « On static and radiative spacetimes », Commun. Math. Phys. 119 (1988), p. 51–73. | MR 968480 | Zbl 0658.53074

[38] —, « Einstein equations and conformal structure: Existence of anti-de-Sitter-type space-times », Jour. Geom. and Phys. 17 (1995), p. 125–184. | MR 1356135 | Zbl 0840.53055

[39] —, « Einstein’s equation and geometric asymptotics », in Gravitation and Relativity: At the turn of the Millennium, Proceedings of GR15 (N. Dahdich et J. Narlikar, éds.), IUCAA, Pune, 1998, p. 153–176.

[40] —, « Conformal Einstein evolution », in Proceedings of the Tübingen Workshop on the Conformal Structure of Space-times (H. Friedrich et J. Frauendiener, éds.), Lecture Notes in Physics, vol. 604, Springer, 2002, gr-qc/0209018, p. 1–50. | Zbl 1054.83006

[41] C. Graham et J. Lee – « Einstein metrics with prescribed conformal infinity on the ball », Adv. Math. 87 (1991), p. 186–225. | MR 1112625 | Zbl 0765.53034

[42] V. Guillemin et A. Pollack – Differential topology, Prentice-Hall, Englewood Cliffs, N.J, 1974. | MR 348781 | Zbl 0361.57001

[43] R. Hansen – « Multipole moments of stationary space-times », Jour. Math. Phys. 15 (1974), p. 46–52. | MR 337249 | Zbl 1107.83304

[44] E. Hebey – Sobolev spaces on Riemannian manifolds, Lect. Notes in Math., vol. 1635, Springer, Berlin, 1996. | MR 1481970 | Zbl 0866.58068

[45] J. Isenberg, R. Mazzeo et D. Pollack – « Gluing and wormholes for the Einstein constraint equations », Commun. Math. Phys. 231 (2002), p. 529–568, gr-qc/0109045. | MR 1946448 | Zbl 1013.83008

[46] —, « On the topology of vacuum spacetimes », Annales Henri Poincaré (2003), in press, gr-qc/0206034. | Zbl 1026.83008

[47] D. Joyce – « Constant scalar curvature metrics on connected sums », Thèse, Lincoln College, Oxford, 2001, math.DG/0108022.

[48] J. Kánnár – « Hyperboloidal initial data for the vacuum Einstein equations with cosmological constant », Class. Quantum Grav. 13 (1996), p. 3075–3084. | MR 1419635 | Zbl 0869.53057

[49] J. Klenk – « Existence of stationary vacuum solutions of Einstein’s equations in an exterior domain », Jour. Aust. Math. Soc., Ser. B 41 (1999), p. 231–247 (English). | MR 1723658 | Zbl 0952.83017

[50] D. Kramer, H. Stephani, M. Maccallum et E. Herlt – « Exact solutions of Einstein’s field equations », (E. Schmutzer, éd.), Cambridge University Press, Cambridge, 1980. | MR 614593 | Zbl 0449.53018

[51] O. Ladyzhenskaya, V. Solonnikov et N. Ural’Tseva – Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith, Translations of Mathematical Monographs. 23. Providence, RI: American Mathematical Society. XI, 648 p. , 1968. | MR 241822

[52] J. Lee – « Fredholm operators and Einstein metrics on conformally compact manifolds », math.DG/0105046, 2001.

[53] A. Lightman, W. Press, R. Price et S. Teukolsky – Problem book in relativity and gravitation, Princeton University Press, Princeton, N.J., 1975. | MR 418811 | Zbl 0339.53002

[54] C. Morrey – Multiple integrals in the calculus of variation, Springer Verlag, Berlin, Heidelberg, New York, 1966. | MR 202511

[55] R. Myers et M. Perry – « Black holes in higher dimensional space-times », Ann. Phys. 172 (1986), p. 304–347. | MR 868295 | Zbl 0601.53081

[56] T. Regge et C. Teitelboim – « Role of surface integrals in the Hamiltonian formulation of general relativity », Ann. Phys. 88 (1974), p. 286–318. | MR 359663 | Zbl 0328.70016

[57] O. Reula – « On existence and behaviour of asymptotically flat solutions to the stationary Einstein equations », Commun. Math. Phys. 122 (1989), p. 615–624. | MR 1002835 | Zbl 0667.53056

[58] R. Sachs – « Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time », Proc. Roy. Soc. London A 270 (1962), p. 103–126. | MR 149908 | Zbl 0101.43605

[59] U. Schaudt – « On the Dirichlet problem for stationary and axisymmetric Einstein equations », Commun. Math. Phys. 190 (1998), p. 509–540. | MR 1600303 | Zbl 0917.35033

[60] W. Simon et R. Beig – « The multipole structure of stationary space-times », Jour. Math. Phys. 24 (1983), p. 1163–1171. | MR 702096 | Zbl 0515.53025

[61] B. Smith et G. Weinstein – « On the connectedness of the space of initial data for the Einstein equations », Electron. Res. Announc. Am. Math. Soc. 6 (2000), p. 52–63. | EuDML 232919 | MR 1777856 | Zbl 0969.83006

[62] V. Solovyev – « Generator algebra of the asymptotic Poincaré group in the general theory of relativity », Teor. i Mat. Fiz. 65 (1985), p. 400–414, in Russian; english translation avail. in Theor. Math. Phys. 1986, p. 1240.

[63] H. Triebel – Interpolation theory, functions spaces and differential operators, North Holland, Amsterdam, 1978. | MR 503903

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