On surface-symmetric spacetimes with collisionless and charged matter

Some future global properties of cosmological solutions for the Einstein-Vlasov-Maxwell system with surface symmetry are presented. Global existence is proved, the homogeneous spacetimes are future complete for causal trajectories, and the same is true for inhomogeneous plane-symmetric solutions with small initial data. In the latter case some decay properties are also obtained at late times. Similar but slightly weaker results hold for hyperbolic symmetry.Comment: 34 pages, version to be published in AH

On surface-symmetric spacetimes with collisionless and charged matter

In this talk we present some global properties of cosmological solutions of the surface-symmetric Einstein equations coupled to collisionless and charged matter described by the Vlasov and Maxwell equations. This involves two issues which are the existence of a global time coordinate t and the asymptotic behaviour of the solution at late times. For details concerning the proof of results stated in the following we refer to [1] and references therein.In this talk we present some global properties of cosmological solutions of the surface-symmetric Einstein equations coupled to collisionless and charged matter described by the Vlasov and Maxwell equations. This involves two issues which are the existence of a global time coordinate t and the asymptotic behaviour of the solution at late times. For details concerning the proof of results stated in the following we refer to [1] andreferences therein

On the spherically symmetric Einstein-Yang-Mills-Higgs equations in Bondi coordinates

We revisit and generalize, to the Einstein-Yang-Mills-Higgs system, previous results of D. Christodoulou and D. Chae concerning global solutions for the Einstein-scalar field and the Einstein-Maxwell-Higgs equations. The novelty of the present work is twofold. For one thing the assumption on the self-interaction potential is improved. For another thing explanation is furnished why the solutions obtained here and those proved by Chae for the Einstein-Maxwell-Higgs decay more slowly than those established by Christodoulou in the case of self-gravitating scalar fields. Actually this latter phenomenon stems from the non-vanishing local charge in Einstein-Maxwell-Higgs and Einstein-Yang-Mills-Higgs models.Comment: 25 page

On the Einstein-Vlasov system with cosmological constant

Das Einstein-Vlasov-System beschreibt die Zeitentwicklung stossfreier Materie im Rahmen der Allgemeinen Relativitätstheorie. Ziel dieser Arbeit ist es, möglichst viele Informationen zu bekommen über globale Lösungen des Anfangswertproblems für das Einstein-Vlasov-System mit kosmologischer Konstante Lambda und sphärischer, ebener oder hyperbolischer Symmetrie, geschrieben in Flächenkoordinaten. Die vorliegende Untersuchung befasst sich mit Raumzeiten, die eine kompakte Cauchy-Hyperfläche besitzen, und in diesem Fall werden Daten auf einer kompakten dreidimensionalen Mannigfaltigkeit gegeben. Die Ergebnisse von G. Rein über lokale Existenz und Fortsetzungskriterien für Raumzeiten mit Lambda=0 werden auf den Fall mit Lambda ungleich Null erweitert. Es wird ausserdem die Lösbarkeit der Zwangsbedingungen bewiesen, die durch die Anfangsdaten erfüllt werden müssen. Es wird gezeigt dass im Fall Lambda kleiner Null keine in der Zukunft globale Lösung existieren kann, so dass die Untersuchung in der expandierenden Richtung sich auf den Fall Lambda grösser Null beschränkt. Mit der Annahme ebener (k=0) oder hyperbolischer (k=-1) Symmetrie und Lambda grösser Null, wird gezeigt, dass der Flächenradius in der Zukunft gegen unendlich strebt so dass globale Existenz in der Zukunft gilt, dass die Raumzeiten in der Zukunft geodätisch vollständig sind, und dass die Expansion zu späten Zeiten isotrop und exponentiell wird. Dadurch wird in dieser Klasse von Raumzeiten eine Form einer Aussage bewiesen, die als "cosmic no hair theorem" bezeichnet wird. Entsprechende Ergebnisse werden auch im sphärisch symmetrischen Fall (k=1) bewiesen, vorausgesetzt dass die Anfangszeit hinreichend gross ist. Ausserdem wird das Verhalten des Energie-Impuls-Tensors zu späten Zeiten analysiert. Zusätzlich wird globale Existenz in der Vergangenheit bewiesen für generische Anfangsdaten wenn Lambda kleiner gleich Null und k grösser gleich Null. Ausserdem werden einige bekannte Ergebnisse verallgemeinert in dem die Existenz bis t=0 für kleine Daten wenn Lambda kleiner Null und k=-1 oder wenn Lambda grösser Null bewiesen wird. In diesem Fall wird bewiesen dass eine Krümmungsinvariante, der Kretschmann-Skalar, für t gegen Null explodiert, so dass es eine Singularität bei t=0 gibt. Anschliessend wird die Natur dieser Anfangssingularität analysiert und es wird gezeigt, dass das asymptotische Verhalten dem einer Kasner-Lösung ähnelt.The Einstein-Vlasov system governs the time evolution of a self-gravitating collisionless gas in the context of general relativity. The aim of this thesis is to obtain as much information as possible about global solutions of the initial value problem for the Einstein-Vlasov system with cosmological constant and spherical, plane or hyperbolic symmetry, written in areal coordinates. Our investigation is concerned with the spacetimes possessing a compact Cauchy hypersurface, in this case the data are given on a compact 3-manifold. The results on the local existence and continuation criteria obtained by G. Rein for the Einstein-Vlasov system with vanishing cosmological constant are extended to the case with a non-zero cosmological constant. We also prove the solvability of the constraint problem on the initial data. We show that there is no global solution in the future when the cosmological constant is negative so that the study in the expanding direction deals only with the positive cosmological constant case. Under the assumption of plane (k=0) or hyperbolic (k=-1) symmetry and that the cosmological constant Lambda is positive we prove that the area radius goes to infinity and so global existence in the future time direction is shown, the spacetimes are future geodesically complete, and the expansion becomes isotropic and exponential at late times. This proves a form of the so-called cosmic no-hair theorem in this class of spacetimes. These results are also proved in the spherically symmetric case (k=1) provided that the initial time is sufficiently large. Furthermore we analyze the behaviour of the energy-momentum tensor at late times. In addition, in the past time direction we prove global existence for generic data if Lambda is non-positive and k is non-negative. Besides this we generalize some known results in the literature by proving existence up to t=0 for small data in the cases Lambda negative, k=-1 and Lambda positive, by proving that the curvature invariant called Kretschmann scalar blows up as t tends to zero so that there is a singularity at t=0. Furthermore we analyze the nature of this initial singularity and also show that the asymptotics is Kasner-like at early times

Global existence and asymptotic behaviour in the future for the Einstein-Vlasov system with positive cosmological constant

The behaviour of expanding cosmological models with collisionless matter and a positive cosmological constant is analysed. It is shown that under the assumption of plane or hyperbolic symmetry the area radius goes to infinity, the spacetimes are future geodesically complete, and the expansion becomes isotropic and exponential at late times. This proves a form of the cosmic no hair theorem in this class of spacetimes