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General Relativity
Initial Value Problem
Phase Space
Related Publications
(9)
Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry
On the area of the symmetry orbits in T2 symmetric spacetimes with Vlasov matter
On the EinsteinVlasov system with hyperbolic symmetry
Constant mean curvature foliations in cosmological spacetimes
Analytic description of singularities in Gowdy spacetimes
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Cosmological solutions of the VlasovEinstein system with spherical, plane, and hyperbolic symmetry
Cosmological solutions of the VlasovEinstein system with spherical, plane, and hyperbolic symmetry,10.1017/S0305004100074569,Mathematical Proceedings
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Cosmological solutions of the VlasovEinstein system with spherical, plane, and hyperbolic symmetry
(
Citations: 33
)
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Gerhard Rein
The VlasovEinstein system describes a selfgravitating, collisionless gas within the framework of general relativity. The author investigates the
initial value problem
in a cosmological setting with spherical, plane, or hyperbolic symmetry and proves that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phasespace distribution of the matter remain bounded.
Journal:
Mathematical Proceedings of The Cambridge Philosophical Society  MATH PROC CAMBRIDGE PHIL SOC
, vol. 119, no. 04, 1996
DOI:
10.1017/S0305004100074569
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Citation Context
(20)
...contexts: Gowdy spacetimes [17], plane symmetric spacetimes with a massless scalar field [18], polarized and halfpolarized U(1) symmetric vacuum spacetimes [19], spacetimes with collisionless matter and spherical, plane or hyperbolic symmetry [
20
], and a particular subset of general Gowdy spacetimes [21]...
Thibault Damour
,
et al.
Describing general cosmological singularities in Iwasawa variables
...The method, which has been used for instance in [
8
] for the EinsteinVlasov system in plane symmetry, consists on constructing an iteration and proving its convergence...
S. B. TCHAPNDA
,
et al.
THE PLANE SYMMETRIC EINSTEINDUST SYSTEM WITH POSITIVE COSMOLOGICAL CO...
...The study of the surfacesymmetric cosmological solutions of the EinsteinVlasov system considered here was initiated in [32,
29
]...
...Past inextendibility for small data under various assumptions on k and � has been shown in [
29
, 37]...
Mihalis Dafermos
,
et al.
Strong cosmic censorship for surfacesymmetric cosmological spacetimes...
...To this end we maintain the notation in [
7
, 12, 13], and follow their work wherever possible...
...The first step consists on generalizing the local existence result in [
7, theorem 3.1
] to the case of charged particles under study:...
...The regularity properties required for a solution are as in [
7
]...
...Instead of considering the subsystem (1.12)(1.15), (1.19), an idea used in [
7
], that we follow here, is to consider an auxiliary system consisting of...
...As in [
7
], the solution of the auxiliary system above is used to construct a sequence of iterative solutions...
Sophonie Blaise Tchapnda
.
On SurfaceSymmetric Spacetimes with Collisionless and Charged Matter
...In the case where the matter is given by the Vlasov equation alone certain results on the initial singularity have been obtained by Rein [
7
]...
...Using results of [5] it was shown that the initial singularity is a curvature singularity (the Kretschmann scalar blows up uniformly there) and a crushing singularity (the mean curvature of the hypersurfaces of constant t blows up uniformly as t → 0). For matter described by the Vlasov equation alone the results already mentioned can be combined with theorems in [
7
] to show that t = 0 is a crushing singularity where the Kretschmann scalar ...
...coordinates. The functions � and µ are periodic in r with period 1. It has been shown in [
7
] that due to the symmetry, f can be written as a function of...
...It seems that if the estimates could be improved slightly they would allow a bootstrap argument on the bound for Q similar to that used in [
7
]...
...Proof We can use the following expression for the Kretschman scalar from [
7
]...
...Proof We use the same argument as in [
7
] and obtain the following : K(t,r) = −(ú � + 2t)e −µ ; ú � = e2µ � 4�t� − k+e...
...are the generalized Kasner exponents and a(r) a continuous function of r. Proof We have as in [
7
]...
...For the EinsteinVlasov system this has been done in [
7
] for small data...
...In Proposition 3.2 an analogue of some parts of the proof of the small data theorem in [
7
] is obtained but this is not sufficient in order to determine the asymptotic behaviour...
David Tegankong
,
et al.
On the nature of initial singularities for solutions of the EinsteinV...
References
(9)
Static solutions of the spherically symmetric VlasovEinstein system
(
Citations: 31
)
Gerhard Rein
Journal:
Mathematical Proceedings of The Cambridge Philosophical Society  MATH PROC CAMBRIDGE PHIL SOC
, vol. 115, no. 03, 1994
Global existence of classical solutions to the vlasovpoisson system in a threedimensional, cosmological setting
(
Citations: 23
)
Gerhard Rein
,
Alan D. Rendall
Journal:
Archive for Rational Mechanics and Analysis  ARCH RATION MECH ANAL
, vol. 126, no. 2, pp. 183201, 1994
Global properties of locally spatially homogeneous cosmological models with matter
(
Citations: 18
)
Alan D. Rendall
Journal:
Mathematical Proceedings of The Cambridge Philosophical Society  MATH PROC CAMBRIDGE PHIL SOC
, vol. 118, no. 03, 1995
Global classical solutions of the VlasovPoisson system in three dimensions for general initial data
(
Citations: 174
)
K PFAFFELMOSER
Journal:
Journal of Differential Equations  J DIFFERENTIAL EQUATIONS
, vol. 95, no. 2, pp. 281303, 1992
Violation of cosmic censorship in the gravitational collapse of a dust cloud
(
Citations: 42
)
Demetrios Christodoulou
Journal:
Communications in Mathematical Physics  COMMUN MATH PHYS
, vol. 93, no. 2, pp. 171195, 1984
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Citations
(33)
Planesymmetric spacetimes with positive cosmological constant. The case of stiff fluids
Philippe G. LeFloch
,
Sophonie B. Tchapnda
Published in 2010.
Classical solutions to the Vlasov–Poisson system in an accelerating cosmological setting
Ho Lee
Journal:
Journal of Differential Equations  J DIFFERENTIAL EQUATIONS
, vol. 249, no. 5, pp. 11111130, 2010
On the area of the symmetry orbits of cosmological spacetimes with toroidal or hyperbolic symmetry
Jacques Smulevici
Published in 2009.
Describing general cosmological singularities in Iwasawa variables
(
Citations: 7
)
Thibault Damour
,
Sophie de Buyl
Journal:
Physical Review D  PHYS REV D
, vol. 77, no. 4, 2008
THE PLANE SYMMETRIC EINSTEINDUST SYSTEM WITH POSITIVE COSMOLOGICAL CONSTANT
(
Citations: 1
)
S. B. TCHAPNDA
,
Max Planck
,
Am Muhlenberg
Journal:
Journal of Hyperbolic Differential Equations  J HYPERBOLIC DIFFER EQU
, vol. 05, no. 03, 2008