[Géométrie globale des espaces–temps -symétriques de faible régularité]
Nous définissons la classe des espaces–temps à symétrie de faible régularité, et nous étudions leur géométrie globale. Nous formulons le problème de données initiales pour les équations d'Einstein sous une faible régularité. Nous établissons l'existence d'un feuilletage global par les surfaces de niveau de la fonction d'aire R des surfaces de symétrie, de sorte que chaque feuille induit une hypersurface initiale. A l'exception des espaces–temps plats de Kasner (connus explicitement), la fonction R prend toutes valeurs positives. Nos hypothèses imposent seulement que le gradient de R est continu et que les coefficients métriques sont dans l'espace de Sobolev (ou sont moins réguliers).
We define the class of weakly regular spacetimes with -symmetry, and investigate their global geometrical structure. We formulate the initial value problem for the Einstein vacuum equations with weak regularity, and establish the existence of a global foliation by the level sets of the area R of the orbits of symmetry, so that each leaf can be regarded as an initial hypersurface. Except for the flat Kasner spacetimes which are known explicitly, R takes all positive values. Our weak regularity assumptions only require that the gradient of R is continuous while the metric coefficients belong to the Sobolev space (or have even less regularity).
Accepté le :
Publié le :
Philippe G. LeFloch 1 ; Jacques Smulevici 2
@article{CRMATH_2010__348_21-22_1231_0, author = {Philippe G. LeFloch and Jacques Smulevici}, title = {Global geometry of $ {T}^{2}$-symmetric spacetimes with weak regularity}, journal = {Comptes Rendus. Math\'ematique}, pages = {1231--1233}, publisher = {Elsevier}, volume = {348}, number = {21-22}, year = {2010}, doi = {10.1016/j.crma.2010.09.009}, language = {en}, }
TY - JOUR AU - Philippe G. LeFloch AU - Jacques Smulevici TI - Global geometry of $ {T}^{2}$-symmetric spacetimes with weak regularity JO - Comptes Rendus. Mathématique PY - 2010 SP - 1231 EP - 1233 VL - 348 IS - 21-22 PB - Elsevier DO - 10.1016/j.crma.2010.09.009 LA - en ID - CRMATH_2010__348_21-22_1231_0 ER -
Philippe G. LeFloch; Jacques Smulevici. Global geometry of $ {T}^{2}$-symmetric spacetimes with weak regularity. Comptes Rendus. Mathématique, Volume 348 (2010) no. 21-22, pp. 1231-1233. doi : 10.1016/j.crma.2010.09.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.009/
[1] Global foliations of vacuum spacetimes with isometry, Ann. Phys., Volume 260 (1997), pp. 117-148
[2] General Relativity and the Einstein Equations, Oxford Math. Monographs, Oxford Univ. Press, 2009
[3] Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys., Volume 14 (1969), pp. 329-335
[4] On spacetimes with symmetric compact Cauchy surfaces, Ann. Phys., Volume 202 (1990), pp. 100-150
[5] Théorèmes d'existence pour certains systèmes d'équations aux dérivées partielles non-linéaires, Acta Math., Volume 88 (1952), pp. 141-225
[6] On the area of the symmetry orbits in -symmetric spacetimes, Classical Quantum Gravity, Volume 20 (2003), pp. 3783-3796
[7] Rough solutions of the Einstein vacuum equations, Ann. of Math., Volume 161 (2005), pp. 1143-1193
[8] Definition and weak stability of spacetimes with distributional curvature, Port. Math., Volume 64 (2007), pp. 535-573
[9] A global foliation of Einstein–Euler spacetimes with Gowdy symmetry on (preprint) | arXiv
[10] P.G. LeFloch, J. Smulevici, Global geometry of future expanding -symmetric spacetimes with weak regularity, in preparation.
[11] Shock waves and gravitational waves in matter spacetimes with Gowdy symmetry, Port. Math., Volume 62 (2005), pp. 349-370
[12] The characteristic initial value problem for plane symmetric spacetimes with weak regularity (preprint) | arXiv
[13] Global properties of Gowdy spacetimes with topology, Ann. Phys., Volume 132 (1981), pp. 87-107
[14] Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry, Comm. Math. Phys., Volume 189 (1997), pp. 145-164
[15] On a wave map equation arising in general relativity, Comm. Pure Appl. Math., Volume 57 (2004), pp. 657-703
[16] J. Smulevici, On the area of the symmetry orbits in spacetimes with toroidal or hyperbolic symmetry, Analysis and PDE, submitted for publication.
Cité par Sources :
Commentaires - Politique