One of the first studies on the Einstein–Vlasov system for spatially-homogeneous spacetimes is the work [152] by Rendall. He chooses a Gaussian time coordinate and investigates the maximal range of this time coordinate for solutions evolving from homogeneous data. For Bianchi IX and for Kantowski–Sachs spacetimes he finds that the range is finite and that there is a curvature singularity in both the past and the future time directions. For the other Bianchi types there is a curvature singularity in the past, and to the future spacetime is causally geodesically complete. In particular, strong cosmic censorship holds in these cases.
Although the questions on curvature singularities and geodesic completeness are very important, it is also desirable to have more detailed information on the asymptotic behavior of the solutions, and, in particular, to understand in which situations the choice of matter model is essential for the asymptotics.
In recent years several studies on the Einstein–Vlasov system for spatially locally homogeneous spacetimes have been carried out with the goal to obtain a deeper understanding of the asymptotic structure of the solutions. Roughly, these investigations can be divided into two cases: (i) studies on non-locally rotationally symmetric (non-LRS) Bianchi I models and (ii) studies of LRS Bianchi models.
In case (i) Rendall shows in [153] that solutions converge to dust solutions for late times. Under the
additional assumption of small initial data this result is extended by Nungesser [127], who gives the rate of
convergence of the involved quantities. In [153] Rendall also raises the question of the existance of solutions
with complicated oscillatory behavior towards the initial singularity may exist for Vlasov matter, in contrast
to perfect fluid matter. Note that for a perfect fluid the pressure is isotropic, whereas for Vlasov
matter the pressure may be anisotropic, and this fact could be sufficient to drastically change the
dynamics. This question is answered in [93
], where the existence of a heteroclinic network is
established as a possible asymptotic state. This implies a complicated oscillating behavior, which
differs from the dynamics of perfect fluid solutions. The results in [93] were then put in a more
general context by Calogero and Heinzle [46], where quite general anisotropic matter models are
considered.
In case (ii) the asymptotic behaviour of solutions has been analyzed in [159, 160
, 48
, 47
]. In [159], the
case of massless particles is considered, whereas the massive case is studied in [160
]. Both the nature of the
initial singularity and the phase of unlimited expansion are analyzed. The main concern in these two works
is the behavior of Bianchi models I, II, and III. The authors compare their solutions with the solutions to
the corresponding perfect fluid models. A general conclusion is that the choice of matter model is very
important since, for all symmetry classes studied, there are differences between the collision-less model and
a perfect fluid model, both regarding the initial singularity and the expanding phase. The most striking
example is for the Bianchi II models, where they find persistent oscillatory behavior near the
singularity, which is quite different from the known behavior of Bianchi type II perfect fluid models.
In [160] it is also shown that solutions for massive particles are asymptotic to solutions with
massless particles near the initial singularity. For Bianchi I and II, it is also proven that solutions
with massive particles are asymptotic to dust solutions at late times. It is conjectured that the
same also holds true for Bianchi III. This problem is then settled by Rendall in [157]. The
investigation [48] concerns a large class of anisotropic matter models, and, in particular, it is
shown that solutions of the Einstein–Vlasov system with massless particles oscillate in the limit
towards the past singularity for Bianchi IX models. This result is extended to the massive case
in [47].
Before finishing this section we mention two other investigations on homogeneous models with Vlasov
matter. In [106] Lee considers the homogeneous spacetimes with a cosmological constant for all Bianchi
models except Bianchi type IX. She shows global existence as well as future causal geodesic completeness.
She also obtains the time decay of the components of the energy momentum tensor as ,
and she shows that spacetime is asymptotically dust-like. Anguige [28] studies the conformal
Einstein–Vlasov system for massless particles, which admit an isotropic singularity. He shows that
the Cauchy problem is well posed with data consisting of the limiting density function at the
singularity.
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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