Isotropization of non-diagonal Bianchi I spacetimes with collisionless matter at late times assuming small data

Abstract

Assuming that the space-time is close to isotropic in the sense that the shear parameter is small and that the maximal velocity of the particles is bounded, we have been able to show that for non-diagonal Bianchi I-symmetric spacetimes with collisionless matter the asymptotic behaviour at late times is close to the special case of dust. We also have been able to show that all the Kasner exponents converge to 1 3 and an asymptotic expression for the induced metric has been obtained. The key was a bootstrap argument.The sign conventions of [18] are used. In particular, we use metric signature -+ + + and geometrized units, i.e. the gravitational constant G and the speed of light c are set equal to one. Also the Einstein summation convention that repeated indices are to be summed over is used. Latin indices run from one to three. C will be an arbitrary constant and ǫ will denote a small and strictly positive constant. They both may appear several times in different equations or inequalities without being the same constant. A dot above a letter will denote a derivative with respect to the cosmological (Gaussian) time t.We will use the 3+1 decomposition of the Einstein equations as made in [18]. We use Gauss coordinates, which implies that the lapse function is the identity and the shift vector vanishes, so comparing our metric with (2.28) of [18] we have that α = 1 and β a = 0. The

Self-Similarity Breaking of Cosmological Solutions with Collisionless Matter

“…Nevertheless the shear tends to zero. This means that the methods which have been applied in [7] for solutions close to a self-similar solution, or what one might call an "exact" solution have also been applicable to a case where solutions are close to a non self-similar solution. The key to treat this case was to introduce the compactification of one of the variables as was done in [13].…”

“…The other curvature variable N − and the shear variables Σ + and Σ − are small initially as well. What is obtained is an analogue to the result of [13] using the methods of [7].…”

The Einstein-Vlasov System/Kinetic Theory

“…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.…”

Isotropization of slowly expanding spacetimes

“…Hence, there has been an interest in adopting other techniques as well. Nungesser [19,20,21] and Nungesser et. al.…”