Symmetries of Bianchi I space-times (Q2738271)

A space-time \((M,g)\) with a unit timelike vector field \(U\) (in particular, a perfect fluid space-time with 4-velocity \(U\)) is said to admit a kinematic self-similarity if it admits a vector field \(X\) which satisfies \({\mathcal L}_XU=\alpha U\) and \({\mathcal L}_Xh=2\delta h\), where \({\mathcal L}_X\) is the Lie derivative with respect to \(X\), \(\alpha, \beta \in R\), \(h=g+U^\flat \otimes U^\flat\) is the projection tensor field orthogonal to \(U\) and \(U^\flat\) is the 1-form \(g\)-equivalent to \(U\). Note that \(X\) reduces to a homothetic vector field if \(\alpha = \beta\), which is Killing if \(\alpha=\beta=0\), in particular. A kinematic self-similarity is said to be proper if \(\alpha \neq\beta\). A diagonal Bianchi I space-time is a (4-dimensional) spatially homogeneous space-time which admits an Abelian group of isometries \(G_3\), acting on spacelike hypersurfaces, generated by three independent spacelike Killing vector fields. In synchronous coordinates, the Lorentzian metric is \(g=-dt^2 + \sum_{\mu =1}^3 A_{\mu}(t)^2(dx^{\mu})^2\), where \(A_{\mu}(t)\) are functions of the time coordinate only.NEWLINENEWLINENEWLINEIn this paper, the authors study proper (i.e. all the functions \(A_{\mu}(t)\) are assumed to be different) diagonal Bianchi I space-times (which are simply called Bianchi I space-times) and determine which of them admit a conformal vector field, in particular, a homothetic vector field. Moreover, they obtain all Bianchi I space-times which admit a kinematic self-similarity; so, they characterize the Kasner-type space-times. Finally, a new Bianchi I viscous fluid space-time is found, with nonzero bulk viscous stress. The model begins with a big-bang and isotropizes at late time tending to a De Sitter universe.