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

Abstract

Assuming that the spacetime 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 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 a metric signature − + + + and geometrized units, i.e. the gravitational constant G and the speed of light c are set equal to 1. Also the Einstein summation convention that repeated indices are to be summed over is used. Latin indices run from 1 to 3. C will be an arbitrary constant and ϵ 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.