Ricci collineations of the Bianchi type II, VIII, and IX space-times

Abstract

Ricci and contracted Ricci collineations of the Bianchi type II, VIII, and IX space-times, associated with the vector fields of the form (i) one component of ξ a (x b ) is different from zero and (ii) two components of ξ a (x b ) are different from zero, for a, b = 1, 2, 3, 4, are presented. In subcase (i.b), which is ξ a = (0, ξ 2 (x a ), 0, 0), some known solutions are found, and in subcase (i.d), which is ξ a = (0, 0, 0, ξ 4 (x a )), choosing S(t) = const.×R(t), the Bianchi type II, VIII, and IX spacetimes is reduced to the Robertson-Walker metric.

Ricci Collineations of the Bianchi Types I and Iii, and Kantowski–sachs Spacetimes

“…In this case, from eqs. (8), (9), (14), and (17), it follows that ξ 1 = ξ 1 (θ, φ), ξ 2 = ξ 2 (r, θ), and ξ 3 = ξ 3 (r, θ, φ). Then, from eq.…”

“…(13) and (17), ξ 3 = ξ 3 (θ, φ). When R 22 = 0, after a systematic integration of the equations (10) and (14) it is shown that RCV field is…”

“…In recently , we have discussed that RCs and family of CRCs of Bianchi type-II, VIII, and IX spacetimes by considering some RCVs [14]. Now, we consider that the metric for Bianchi types I(δ = 0) and III(δ = −1), and Kantowski-Sachs (δ = +1) cosmological models is in the form [15]- [17],…”

Bianchi type-II universe with linear equation of state

“…We consider a spatially homogeneous and anisotropic (LRS) Bianchi type-II metric is in the form [44][45][46][47] …”

Classification of Kantowski–Sachs metric via conformal Ricci collineations

“…An entire and significant work on CCs was published by Hall [4] in his book "Symmetries and Curvature Structure in General Relativity", where a classical discussion about other symmetries is also given. As the Ricci tensor is the trace of the Riemann tensor, the RCs have a natural geometric significance [7][8][9]. The physical importance of RCs is also investigated in the literature.…”