The Einstein-Vlasov-Scalar Field System with Gowdy or T 2 Symmetry in Contracting Direction
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The Einstein-Vlasov-Scalar Field System with Gowdy or T 2 Symmetry in Contracting Direction Alex Lassiye Tchuani1 · David Tegankong2 · Norbert Noutchegueme1
Received: 29 August 2017 / Revised: 4 January 2018 / Accepted: 5 January 2018 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018
Abstract We prove in the case of cosmological models for the Einstein-Vlasov-scalar field system with Gowdy symmetry, that the solutions exist globally in the past. The sources of the equations are generated by a distribution function and a scalar field, subject to the Vlasov and the wave equations respectively. The result is generalized for the case of T 2 symmetry. Using previous results, we deduce geodesic completeness. Keywords Einstein · Vlasov · Scalar field · Gowdy symmetry · T 2 symmetry · Hyperbolic differential equations · Global existence · Geodesic completeness Mathematics Subject Classification (2010) 83C20 · 83C22 · 34B05
1 Introduction The question of global existence solutions of Einstein equations with matter or not is very important in general relativity. Here, for long, the practice has been to study existence of solutions under symmetry assumptions. Spacetimes with Gowdy or T 2 symmetry have received much attention by different authors for many years, see [1, 3, 8, 11, 15] and the references therein.
David Tegankong
[email protected] Alex Lassiye Tchuani [email protected] Norbert Noutchegueme [email protected] 1
Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon
2
Department of Mathematics, ENS, University of Yaounde 1, P.O. Box 47, Yaounde, Cameroon
A. L. Tchuani et al.
In this paper, we prove past global existence, with respect to a geometrically defined time, for Einstein equations coupled to the Vlasov and non-linear wave equations. In the surface symmetry case with the Vlasov and linear scalar field, we have shown in [12] (without any restriction on the data) and [13] global existence in the past time direction. The symmetry assumption here is the Gowdy and T 2 one, and thereby we extend respectively Andr´easson’s result in [1] and Smulevici’s one in [11] for the Vlasov case. Andr´easson’s result was the first result to provide a global foliation in the contracting and expanding direction of a spacetime containing both matter and gravitational waves. In that paper, Andr´easson generalized in the two time directions (contracting and expanding) the Moncrief result for the vacuum case. His method of proof is inspired by a result in [3] for vacuum spacetimes admitting a T 2 isometry group acting on T 3 space-like surfaces. Gowdy spacetimes are a special case of these spacetimes. Andr´easson proves that T 3 × R spacetimes with Gowdy symmetry admit global foliations by conformal coordinates in the contracting direction. We extend the result in [1] by introducing a scalar field. It can be seen as a step towards certain questions of physical interest. In recent years, cosmologi
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