JournalsjemsVol. 17 , No. 5pp. 1229–1292

Weakly regular T2T^2-symmetric spacetimes. The global geometry of future Cauchy developments

  • Philippe G. LeFloch

    Université Pierre et Marie Curie - Paris 6, France
  • Jacques Smulevici

    Université Paris-Sud, Orsay, France
Weakly regular $T^2$-symmetric spacetimes. The global geometry of future Cauchy developments cover
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Abstract

We provide a geometric well-posedness theory for the Einstein equations within the class of weakly regular vacuum spacetimes with T2T^2-symmetry, as defined in the present paper, and we investigate their global causal structure. Our assumptions allow us to give a meaning to the Einstein equations under weak regularity as well as to solve the initial value problem under the assumed symmetry. First, introducing a frame adapted to the symmetry and identifying certain cancellation properties taking place in the standard expressions of the connection and the curvature, we formulate the initial value problem for the Einstein field equations under the proposed weak regularity assumptions. Second, considering the Cauchy development of any weakly regular initial data set and denoting by RR the area of the orbits of symmetry, we establish the existence of a global foliation by the level sets of RR such that RR grows to infinity in the future direction. Our weak regularity assumptions only require that RR is Lipschitz continuous while the metric coefficients describing the initial geometry of the symmetry orbits are in the Sobolev space H1H^1 and the remaining coefficients have even weaker regularity.

Cite this article

Philippe G. LeFloch, Jacques Smulevici, Weakly regular T2T^2-symmetric spacetimes. The global geometry of future Cauchy developments. J. Eur. Math. Soc. 17 (2015), no. 5 pp. 1229–1292

DOI 10.4171/JEMS/530