Global geometry of T2-symmetric spacetimes with weak regularity

LeFloch, Philippe G.; Smulevici, Jacques

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

Nous définissons la classe des espaces–temps à symétrie T2 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 H1 (ou sont moins réguliers).

We define the class of weakly regular spacetimes with T2-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 H1 (or have even less regularity).

[1] Berger, B.K.; Chruściel, P.; Isenberg, J.; Moncrief, V. Global foliations of vacuum spacetimes with T2 isometry, Ann. Phys., Volume 260 (1997), pp. 117-148

[2] Choquet-Bruhat, Y. General Relativity and the Einstein Equations, Oxford Math. Monographs, Oxford Univ. Press, 2009

[3] Choquet-Bruhat, Y.; Geroch, R. Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys., Volume 14 (1969), pp. 329-335

[4] Chruściel, P. On spacetimes with U(1)×U(1) symmetric compact Cauchy surfaces, Ann. Phys., Volume 202 (1990), pp. 100-150

[5] Foures-Bruhat, Y. Théorèmes d'existence pour certains systèmes d'équations aux dérivées partielles non-linéaires, Acta Math., Volume 88 (1952), pp. 141-225

[6] Isenberg, J.; Weaver, M. On the area of the symmetry orbits in T2-symmetric spacetimes, Classical Quantum Gravity, Volume 20 (2003), pp. 3783-3796

[7] Klainerman, S.; Rodnianski, I. Rough solutions of the Einstein vacuum equations, Ann. of Math., Volume 161 (2005), pp. 1143-1193

[8] LeFloch, P.G.; Mardare, C. Definition and weak stability of spacetimes with distributional curvature, Port. Math., Volume 64 (2007), pp. 535-573

[9] LeFloch, P.G.; Rendall, A. A global foliation of Einstein–Euler spacetimes with Gowdy symmetry on T3 (preprint) | arXiv

[10] P.G. LeFloch, J. Smulevici, Global geometry of future expanding T2-symmetric spacetimes with weak regularity, in preparation.

[11] LeFloch, P.G.; Stewart, J.M. Shock waves and gravitational waves in matter spacetimes with Gowdy symmetry, Port. Math., Volume 62 (2005), pp. 349-370

[12] LeFloch, P.G.; Stewart, J.M. The characteristic initial value problem for plane symmetric spacetimes with weak regularity (preprint) | arXiv

[13] Moncrief, V. Global properties of Gowdy spacetimes with T3×R topology, Ann. Phys., Volume 132 (1981), pp. 87-107

[14] Rendall, A.D. Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry, Comm. Math. Phys., Volume 189 (1997), pp. 145-164

[15] Ringström, H. 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.