Abstract
In this paper we use Noether symmetries of the geodesic Lagrangian in Bianchi V spacetimes to study various cosmological solutions of Einstein's field equations. Our first result is the identification of the subalgebras of Noether symmetries of the equations of motions in such spacetimes with dimension 4, 5, 6, 7, 9 or 10 of the maximal algebra of Lie point symmetries of dimension 13. Secondly we give physical interpretation of new cosmological solutions which satisfy positive energy condition and yield critical bounds on the expansion coefficient $\alpha$, in which the underlying non-flat spacetimes carry interesting physical properties. Specifically the energy density behaves in one of the following ways. (i) It is positive and constant for all time. (ii) It varies with time and attains a global maximum after some time and then asymptotically converges to zero. (iii) It increases for all time and attains a maximum value at the asymptotic limit $t\rightarrow \infty$. In particular a non-flat spacetime is obtained that mimics the expansion in a flat FRW universe dominated by vacuum energy such that the expansion factor has the same form in both. However, the energy density is dynamical in the former.
Highlights
Suppose M is a 4-dimensional smooth manifold equipped with a non-degenerate metric g of Lorentzian signature (+, −, −, −)
In [7], the authors have classified Bianchi V spacetimes using the symmetries of the curvature and Weyl tensors known as curvature collineations (CC) and Weyl collineations (WC), respectively
It is well known that Noether algebra is a subalgebra of the Lie algebra of the point symmetries of the differential equations [8], our study yields a classification of the Noether algebras of the equations of motion for the geodesic Lagrangian in Bianchi V spacetimes
Summary
Suppose M is a 4-dimensional smooth manifold equipped with a non-degenerate metric g of Lorentzian signature (+, −, −, −). Homogeneous spacetimes of dimension 1 + 3 are characterized using the Bianchi classification of real Lie algebras of dimension 3 which resulted into nine classes I, II, ..., IX [5]. The famous cosmological solutions of EFEs known as Friedmann–Lemaître–Robertson– Walker (FRW) metrics are both isotropic and homogeneous and are particular cases of Bianchi types I, V, VII, and IX [1]. It is important to investigate other Bianchi homogeneous cosmologies with regard to their invariant geometrical properties. The authors of [6] have done the symmetry analysis of all Bianchi spacetimes in the presence of a dynamical field. In [7], the authors have classified Bianchi V spacetimes using the symmetries of the curvature and Weyl tensors known as curvature collineations (CC) and Weyl collineations (WC), respectively
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