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A Global Foliation of Einstein–Euler Spacetimes with Gowdy-Symmetry on T 3

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Abstract

We investigate the initial value problem for the Einstein–Euler equations of general relativity under the assumption of Gowdy symmetry on T 3, and we construct matter spacetimes with low regularity. These spacetimes admit both impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularities propagating at light speed) and shock waves in the fluid (that is, discontinuities propagating at about the sound speed). Given an initial data set, we establish the existence of a future development, and we provide a global foliation in terms of a globally and geometrically defined time-function, closely related to the area of the orbits of the symmetry group. The main difficulty lies in the low regularity assumed on the initial data set which requires a distributional formulation of the Einstein–Euler equations.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252, Paris, France

    Philippe G. LeFloch

  2. Max-Planck-Institut für Gravitationsphysik, Albert-Einstein Institut, Am Mühlenberg 1, 14476, Potsdam, Germany

    Alan D. Rendall

Authors
  1. Philippe G. LeFloch
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  2. Alan D. Rendall
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Correspondence to Philippe G. LeFloch.

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Communicated by C. M. Dafermos

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LeFloch, P.G., Rendall, A.D. A Global Foliation of Einstein–Euler Spacetimes with Gowdy-Symmetry on T 3 . Arch Rational Mech Anal 201, 841–870 (2011). https://doi.org/10.1007/s00205-011-0425-z

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  • DOI: https://doi.org/10.1007/s00205-011-0425-z

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