Cosmological solutions of the Einstein-Vlasov-scalar field system
vorgelegt von
Diplom-Mathematiker
David Tegankong
von der Fakult¨at II-Mathematik und Naturwissenschaften
der Technische Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
- Dr. rer. nat. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. R. Nabben
Berichter: Priv.-Doz. Dr. A. D. Rendall
Berichter: Prof. Dr. A. Unterreiter
Tag der wissenschaftlichen Aussprache: 19. Mai 2005
Berlin 2005
D 83
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To my wife Blandine Pulcherie Tamatcho and our child Marl`ene, Audrey and
Brice Tegankong.
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Acknowledgements
I am grateful to my supervisors Pr. Alan D. Rendall and Pr. Norbert Noutchegueme
for helpful suggestions that both gave me during this work. A major part of
this work was carried out during two three-months stays at the Albert Einstein
Institute where I received pleasure of collaboration of Pr. Alan D. Rendall. So
I would like to give him many thanks.
I gratefully acknowledge the financial support of the VolkswagenStiftung,
Federal Republic of Germany, that allows me to do this work.
Besides, I thank Frau Hauschild and Frau Lampe for a warm welcome they
brought me during each of my visit at the Albert Einstein Institute in G¨olm.
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Contents
Abstract 1
Introduction 2
1 Equations and preliminary results 5
1.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Auxiliary system and preliminary results . . . . . . . . . . . . . . 10
1.3 The reduced system . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Local existence and continuation of solutions 26
2.1 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Local existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Continuation criteria . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Global existence and asymptotic behaviour of solutions in the
past 50
3.1 Global existence in the past . . . . . . . . . . . . . . . . . . . . . 50
3.2 On past asymptotic behaviour . . . . . . . . . . . . . . . . . . . . 54
4 Global existence and asymptotic behaviour of solutions in the
future 62
4.1 Global existence in the future . . . . . . . . . . . . . . . . . . . . 62
4.2 The future asymptotic behaviour . . . . . . . . . . . . . . . . . . 68
4.2.1 Integration of equations . . . . . . . . . . . . . . . . . . . 69
4.2.2 Geodesic completeness . . . . . . . . . . . . . . . . . . . . 70
Conclusion 75
Appendices 76
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Abstract
The aim of this thesis is to obtain as much information as possible, about global
solutions of the Cauchy problem for the Einstein-Vlasov-scalar field system with
spherical, plane and hyberbolic symmetries written in areal coordinates. The
sources of this system are generated by both a distribution function and a linear
scalar field subject to the Vlasov and wave equations respectively. This system
describes the evolution of self-gravitating collisionless matter and scalar waves
within the context of general relativity. We consider the cosmological case. That
is spacetimes possess a compact Cauchy hypersurface and then, data are given
on a compact 3-manifold.
We extend the local-in-time results obtained by G. Rein for the Einstein-
Vlasov system with collisionless matter alone. This extension concerns pointwise
estimates for hyperbolic equations by the method of characteristics. This means
that the system is transformed to a system of ordinary differential equations
which are integrated along characteristics. The constraint equation on the initial
data reduced to an ordinary differential equation of first order and is solved. In
the past direction, we show global existence results for general data. The proof
is based on a change of variables inspired by the work of M. Weaver. The nature
of singularity is analyzed. The curvature invariant called Kretschmann scalar
blows up as ttends to 0 so that there is a singularity at tequal zero.
We prove that there is no global solution in the future in the spherical
symmetry case. In the plane and hyperbolic symmetries, the area radius goes
to infinity and so we obtain global solutions in the expanding direction. In the
special case of plane symmetry without Vlasov contribution, we show that the
asymptotics are Kasner-like at early time. Moreover the spacetime obtained in
this case is future geodesically complete.
We conclude the work by showing that the spatially homogeneous solutions
of the plane and hyperbolic symmetric Einstein-Vlasov-scalar field system exist
globally in the future and the corresponding spacetimes are geodesically com-
plete. Future asymptotics are Kasner-like in the plane symmetric case.
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