Noether symmetries of Bianchi type I spacetimes via Rif tree approach

Ashfaque H. Bokhari ◽  
Muhammad Farhan ◽  
Tahir Hussain

In this paper, we have studied Noether symmetries of the general Bianchi type I spacetimes. The Lagrangian associated with the most general Bianchi type I metric is used to find the set of Noether symmetry equations. These equations are analyzed using an algorithm, developed in Maple, to get all possible Bianchi type I metrics admitting different Noether symmetries. The set of Noether symmetry equations is then solved for each metric to obtain the Noether symmetry algebras of dimensions 4, 5, 6 and 9.

2020 ◽  
Vol 17 (11) ◽  
pp. 2050163
Adnan Malik ◽  
M. Farasat Shamir ◽  
Ibrar Hussain

This paper investigates the Noether symmetry approach in modified [Formula: see text] theory of gravity, where [Formula: see text] is the Ricci scalar, [Formula: see text] is a scalar field and [Formula: see text] is the kinetic energy term. For this purpose, we consider Locally Rotationally Symmetric (LRS) Bianchi Type-I spacetimes. The cosmological solutions are developed through Noether symmetry approach. The determining equations are computed in the context of a point-like Lagrangian for [Formula: see text] gravity. In particular, three different cases are taken into account for the LRS Bianchi Type-I spacetimes. Some important conserved quantities for the Lagrangian in this modified theory of gravity are worked out through the determining equations. In this scenario, the graphical behavior of energy density, pressure component and equation of state parameter are reported and analyzed with the help of first integrals of motion. The negative trends of the strong energy conditions actually suggest that the said theory supports expanding Universe with dark energy.

2021 ◽  
pp. 104299
Ashfaque H. Bokhari ◽  
Tahir Hussain ◽  
Jamshed Khan ◽  
Uzma Nasib

2018 ◽  
Vol 96 (7) ◽  
pp. 677-680 ◽  
I. Basaran Oz ◽  
Y. Kucukakca ◽  
N. Unal

In this study, we consider a phantom cosmology in which a scalar field is minimally coupled to gravity. For anisotropic locally rotational symmetric (LRS) Bianchi type I space–time, we use the Noether symmetry approach to determine the potential of such a theory. It is shown that the potential must be in the trigonometric form as a function of the scalar field. We solved the field equations of the theory using the result obtained from the Noether symmetry. Our solution shows that the universe has an accelerating expanding phase.

2021 ◽  
pp. 2150208
Ashfaque H. Bokhari ◽  
Tahir Hussain ◽  
Wajid Hussain ◽  
Fawad Khan

In this paper, we have adopted a new approach to study the Killing vector fields of locally rotationally symmetric and general Bianchi type I spacetimes. Instead of directly integrating the set of Killing’s equations, an algorithm is developed in Maple which converts these equations to the reduced involutive form (Rif) and consequently it imposes some restrictions on the metric functions in the form of a tree, known as Rif tree. The set of Killing’s equations is then solved for each branch of the Rif tree, giving the explicit form of the Killing vector fields. The structure of Lie algebra is presented for each set of the obtained Killing vector fields and some physical implications of the obtained metrics are discussed.

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