Vol. 9, No. 2, 2016

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
Future asymptotics and geodesic completeness of polarized $T^2$-symmetric spacetimes

Philippe G. LeFloch and Jacques Smulevici

Vol. 9 (2016), No. 2, 363–395

We investigate the late-time asymptotics of future-expanding, polarized vacuum Einstein spacetimes with T2-symmetry on T3, which, by definition, admit two spacelike Killing fields. Our main result is the existence of a stable asymptotic regime within this class; that is, we provide here a full description of the late-time asymptotics of the solutions to the Einstein equations when the initial data set is close to the asymptotic regime. Our proof is based on several energy functionals with lower-order corrections (as is standard for such problems) and the derivation of a simplified model that we exhibit here. Roughly speaking, the Einstein equations in the symmetry class under consideration consist of a system of wave equations coupled to constraint equations plus a system of ordinary differential equations. The unknowns involved in the system of ordinary equations are blowing up in the future timelike directions. One of our main contributions is the derivation of novel effective equations for suitably renormalized unknowns. Interestingly, this renormalization is not performed with respect to a fixed background, but does involve the energy of the coupled system of wave equations. In addition, we construct an open set of initial data that are arbitrarily close to the expected asymptotic behavior. We emphasize that, in comparison, the class of Gowdy spacetimes exhibits a very different dynamical behavior to the one we uncover in the present work for general polarized T2-symmetric spacetimes. Furthermore, all the conclusions of this paper are valid within the framework of weakly T2-symmetric spacetimes previously introduced by the authors.

Einstein equations, $T^2$-symmetry, future expanding spacetime, late-time asymptotics, geodesic completeness
Mathematical Subject Classification 2010
Primary: 83C05, 83C20, 35Q76
Received: 9 August 2014
Revised: 14 October 2015
Accepted: 6 December 2015
Published: 24 March 2016
Philippe G. LeFloch
Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique
Université Pierre et Marie Curie (Paris 6)
4 Place Jussieu
75252 Paris
Jacques Smulevici
Laboratoire de Mathématiques
Université Paris-Sud 11
Bât. 425
91405 Orsay