Global geometry of $ {T}^{2}$-symmetric spacetimes with weak regularity

Philippe G. LeFloch; Jacques Smulevici

doi:10.1016/j.crma.2010.09.009

Mathematical Physics

Global geometry of

T^{2}

-symmetric spacetimes with weak regularity
[Géométrie globale des espaces–temps

T^{2}

-symétriques de faible régularité]

Présenté par : Yvonne Choquet-Bruhat

Philippe G. LeFloch ¹ ; Jacques Smulevici ²

¹ Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4, place Jussieu, 75252 Paris cedex 05, France
² Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germany

Comptes Rendus. Mathématique, Volume 348 (2010) no. 21-22, pp. 1231-1233.

Résumé
Abstract

Nous définissons la classe des espaces–temps à symétrie $T^{2}$ 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 $H^{1}$ (ou sont moins réguliers).

We define the class of weakly regular spacetimes with $T^{2}$ -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 $H^{1}$ (or have even less regularity).

Reçu le : 2010-06-18
Accepté le : 2010-09-09
Publié le : 2010-09-23

DOI : 10.1016/j.crma.2010.09.009

Affiliations des auteurs :

Philippe G. LeFloch ¹ ; Jacques Smulevici ²

¹ Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4, place Jussieu, 75252 Paris cedex 05, France
² Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germany

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     title = {Global geometry of $ {T}^{2}$-symmetric spacetimes with weak regularity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1231--1233},
     publisher = {Elsevier},
     volume = {348},
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     year = {2010},
     doi = {10.1016/j.crma.2010.09.009},
     language = {en},
}

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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/

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