Einstein-Euler equations for matter spacetimes with Gowdy symmetry

LeFloch, Philippe G.

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 23, 15 p.

We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on $T^3$. Given an arbitrary initial data set, we establish the existence of a globally hyperbolic future development and we provide a global foliation of this spacetime in terms of a geometrically defined time-function coinciding with the area of the orbits of the symmetry group. This allows us to construct matter spacetimes with weak regularity which admit, both, impulsive gravitational waves and shock waves. The cosmic censorhip conjecture is established in the polarized case.

[1] Andréasson H., Global foliations of matter spacetimes with Gowdy symmetry, Commun. Math. Phys. 206 (1999), 337–366. | MR | Zbl

[2] Andréasson H., Rendall A.D., and Weaver M., Existence of CMC and constant areal time foliations in $T^2$ symmetric spacetimes with Vlasov matter, Comm. Partial Differential Equations 29 (2004), 237–262. | MR | Zbl

[3] Barnes A.P., LeFloch P.G., Schmidt B.G., and Stewart J.M., The Glimm scheme for perfect fluids on plane-symmetric Gowdy spacetimes, Class. Quantum Grav. 21 (2004), 5043–5074. | MR | Zbl

[4] Berger B.K., Chruściel P., Isenberg J., and Moncrief V., Global foliations of vacuum spacetimes with $T^2$ isometry, Ann. Phys. 260 (1997), 117–148. | MR | Zbl

[5] Berger B.K., Chruściel P., and Moncrief V., On asymptotically flat spacetimes with $G_2$-invariant Cauchy surfaces, Ann. Phys. 237 (1995), 322–354. | MR | Zbl

[6] Choquet-Bruhat Y., Théorème d’existence pour certains systèmes d’équations aux dérivées partielles nonlinéaires, Acta Math. 88 (1952), 141Ð-225. | MR | Zbl

[7] Choquet-Bruhat Y., General relativity and the Einstein equations, Oxford Univ. Press, 2009. | MR | Zbl

[8] Choquet-Bruhat Y. and Geroch R., Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys. 14 (1969), 329Ð-335. | MR | Zbl

[9] Christodoulou D. and Klainerman S., The global nonlinear stability of the Minkowski space, Princeton Univ. Press, 1993. | MR | Zbl

[10] Chruściel P., On spacetimes with $U\left(1\right)\times U\left(1\right)$ symmetric compact Cauchy surfaces, Ann. Phys. 202 (1990), 100–150. | MR | Zbl

[11] Chruściel P., Isenberg J., and Moncrief V., Strong cosmic censorship in polarized Gowdy spacetimes, Class. Quantum Grav. 7 (1990), 1671–1680. | MR | Zbl

[12] Dafermos M. and Rendall A.D., Strong cosmic censorship for $T^2$-symmetric cosmological spacetimes with collisionless matter, Preprint gr-qc/0610075.

[13] Eardley D. and Moncrief V., The global existence problem and cosmic censorship in general relativity, Gen. Relat. Grav. 13 (1981), 887–892. | MR

[14] Gowdy R., Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: topologies and boundary conditions, Ann. Phys. 83 (1974), 203–241. | MR | Zbl

[15] Hawking S.W. and Penrose R., The singularities of gravitational collapse and cosmology, Proc. Roy. Soc. A314 (1970), 529–548. | MR | Zbl

[16] Isenberg J. and Moncrief V., Asymptotic behavior of the gravitational field and the nature of singularities in Gowdy spacetimes, Ann. Phys. 99 (1990), 84–122. | MR | Zbl

[17] Isenberg J. and Weaver M., On the area of the symmetry orbits in $T^2$ symmetric spacetimes, Class. Quantum Grav. 20 (2003), 3783–3796. | MR | Zbl

[18] LeFloch P.G. and Mardare C., Definition and weak stability of spacetimes with distributional curvature, Port. Math. 64 (2007), 535–573. | MR | Zbl

[19] LeFloch P.G. and Rendall A., Stability and global foliation of matter spacetimes on $T^3$ with gravitational waves and shock waves, in preparation.

[20] LeFloch P.G. and Stewart J.M., Shock waves and gravitational waves in matter spacetimes with Gowdy symmetry, Port. Math. 62 (2005), 349–370. | MR

[21] LeFloch P.G. and Stewart J.M., Causal structure of plane symmetric spacetimes with self-gravitating relativistic fluids, in preparation.

[22] Moncrief V., Global properties of Gowdy spacetimes with $T^3\times R$ topology, Ann. Phys. 132 (1981), 87–107. | MR

[23] Penrose R., Gravitational collapse and spacetime singularities, Phys. Rev. Lett. 14 (1965), 57–59. | MR | Zbl

[24] Rendall A.D., Cosmic censorship and the Vlasov equation, Class. Quantum Grav. 9 (1992), 99–104. | MR

[25] Rendall A.D., Crushing singularities in spacetimes with spherical, plane, and hyperbolic symmetry, Class. Quantum Grav. 12 (1995), 1517–1533. | MR | Zbl

[26] Ringström H., Curvature blow-up on a dense subset of the singularity in $T^3$-Gowdy, J. Hyperbolic Differ. Equ. 2 (2005), 547–564. | MR

[27] Ringström H., Strong cosmic censorship in $T^3$-Gowdy spacetimes, Ann. Math., to appear.

[28] Tegankong D., Noutchegueme N., and Rendall A.D., Local existence and continuation criteria for solutions of the Einstein-Vlasov-scalar field system with surface symmetry, J. Hyperbolic Differ. Equ. 1 (2004), 691–724. | MR | Zbl