Conformal vector fields of Bianchi type-I spacetimes

2021 ◽  
Author(s):  
Suhail Khan ◽  
Maria Bukhari ◽  
Ali H. Alkhaldi ◽  
Akram Ali
Keyword(s):  
Vector Fields ◽  
Bianchi Type ◽  
Conformal Killing ◽  
Type I ◽  
Bianchi Type I ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Integration Techniques ◽  
Metric Functions ◽  
Killing Equations

This paper aims to investigate Conformal Vector Fields (CVFs) of Bianchi type-I spacetimes. A set of 10-coupled Partial Differential Equations (PDEs) is obtained from the conformal Killing equations. These equations are solved by using direct integration techniques to explore the components of CVFs. Utilizing these components, we get a system of three integrability conditions. Finally, we achieve CVFs along with conformal factors for unique possibilities of unknown metric functions from the solution of these conditions. From our results, it is examined that Bianchi type-I spacetimes admit five or fifteen CVFs for specific choices of metric functions.

Existence of conformal vector fields of Bianchi type I space-times in f(R) gravity

2020 ◽  
Vol 17 (08) ◽  
pp. 2050113 ◽  
Author(s):  
Ghulam Shabbir ◽  
Fiaz Hussain ◽  
S. Jamal ◽  
Muhammad Ramzan
Keyword(s):  
Vector Fields ◽  
Bianchi Type ◽  
Flat Space ◽  
Type I ◽  
Direct Integration ◽  
Conformally Flat ◽  
Bianchi Type I ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Integration Techniques

In this paper, Bianchi type I space-times in the [Formula: see text] theory of gravity are classified via conformal vector fields using algebraic and direct integration techniques. In this classification, we show that the conformal vector fields are of dimension four, five, six or fifteen. Additionally, we found that non-conformally flat Bianchi type I space-times admit conformal vector fields of dimension four, five or six. In the case of conformally flat or flat space-times, the dimension of the conformal vector fields is fifteen.

Concircular vector fields on Lorentzian manifold of Bianchi type-I spacetimes

2018 ◽  
Vol 33 (12) ◽  
pp. 1850063
Author(s):  
Amjad Mahmood ◽  
Ahmad T. Ali ◽  
Suhail Khan
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Bianchi Type ◽  
Killing Vector ◽  
Lorentzian Manifold ◽  
Conformal Killing ◽  
Type I ◽  
Bianchi Type I ◽  
Conformal Killing Vector Fields ◽  
Metric Functions

Our aim in this paper is to obtain concircular vector fields (CVFs) on the Lorentzian manifold of Bianchi type-I spacetimes. For this purpose, two different sets of coupled partial differential equations comprising ten equations each are obtained. The first ten equations, known as conformal Killing equations are solved completely and components of conformal Killing vector fields (CKVFs) are obtained in different possible cases. These CKVFs are then substituted into second set of ten differential equations to obtain CVFs. It comes out that Bianchi type-I spacetimes admit four-, five-, six-, seven- or 15-dimensional CVFs for particular choices of the metric functions. In many cases, the CKVFs of a particular metric are same as CVFs while there exists few cases where proper CKVFs are not CVFs.

Killing vector fields of Bianchi type I spacetimes via Rif tree approach

2021 ◽  
pp. 2150208
Author(s):  
Ashfaque H. Bokhari ◽  
Tahir Hussain ◽  
Wajid Hussain ◽  
Fawad Khan
Keyword(s):  
Vector Fields ◽  
Bianchi Type ◽  
Killing Vector ◽  
Type I ◽  
Killing Vector Fields ◽  
New Approach ◽  
Bianchi Type I ◽  
Rotationally Symmetric ◽  
Metric Functions ◽  
Tree Approach

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.

Homothetic vector fields in a specially homogenous Bianchi type-I cosmological model in Lyra geometry

2015 ◽  
Vol 93 (11) ◽  
pp. 1397-1401 ◽  
Author(s):  
A.S. Alofi ◽  
Ragab M. Gad
Keyword(s):  
Cosmological Model ◽  
Riemannian Geometry ◽  
Vector Fields ◽  
Bianchi Type ◽  
Displacement Vector ◽  
Lyra Geometry ◽  
Type I ◽  
Bianchi Type I ◽  
Homothetic Vector ◽  
Vector Formula

In this paper, homothetic vector fields of a spatially homogenous Bianchi type-I cosmological model have been evaluated based on Lyra geometry. Further, we investigate the equation of state in cases when a displacement vector [Formula: see text] is a function of t and when it is constant. We give a comparison between the obtained results, using Lyra geometry, and those obtained previously in the context of general relativity, based on Riemannian geometry.

CLASSIFICATION OF BIANCHI TYPE I SPACETIMES ACCORDING TO THEIR PROPER TELEPARALLEL HOMOTHETIC VECTOR FIELDS IN THE TELEPARALLEL THEORY OF GRAVITATION

2010 ◽  
Vol 25 (25) ◽  
pp. 2145-2153 ◽  
Author(s):  
GHULAM SHABBIR ◽  
SUHAIL KHAN
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Bianchi Type ◽  
Special Choice ◽  
Type I ◽  
Bianchi Type I ◽  
Homothetic Vector ◽  
Teleparallel Theory ◽  
Theory Of Gravitation

In this paper we explored teleparallel homothetic vector fields in Bianchi type I spacetimes in the teleparallel theory of gravitation using direct integration technique. It turns out that the dimensions of the teleparallel homothetic vector fields are 4, 5, 7 or 11 which are same in numbers as in general relativity. In the cases of 4, 5 or 7 proper teleparallel homothetic vector fields exist for the special choice of the spacetimes. In the case of 11 teleparallel homothetic vector fields all the torsion components are zero. The homothetic vector fields of general relativity are recovered in this case and the spacetime become Minkowski.

Classification of vacuum classes of plane fronted gravitational waves via proper conformal vector fields in f(R) gravity

2019 ◽  
Vol 16 (10) ◽  
pp. 1950151 ◽  
Author(s):  
Fiaz Hussain ◽  
Ghulam Shabbir ◽  
Muhammad Ramzan ◽  
Shabeela Malik
Keyword(s):  
Gravitational Waves ◽  
Vector Fields ◽  
Homothetic Vector ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Integration Techniques ◽  
Pp Wave ◽  
Wave Space ◽  
Theory Of Gravity

In this paper, we have studied proper conformal vector fields of pp-wave space-times in the [Formula: see text] theory of gravity using algebraic and direct integration techniques. From this study, we found that a very special class of pp-waves known as plane fronted gravitational waves (GWs) is a solution in the [Formula: see text] theory of gravity. In order to find proper conformal vector fields, plane GWs are further classified in ten cases. Studying each case in detail it turns out that in two cases proper conformal vector fields exist while in the rest of eight cases, conformal vector fields become homothetic vector fields.

scholarly journals A note on some perfect fluid Kantowski–Sachs and Bianchi type III spacetimes and their conformal vector fields in f(R) theory of gravity

2019 ◽  
Vol 34 (11) ◽  
pp. 1950079 ◽  
Author(s):  
Ghulam Shabbir ◽  
Fiaz Hussain ◽  
A. H. Kara ◽  
Muhammad Ramzan
Keyword(s):  
Perfect Fluid ◽  
Vector Fields ◽  
Bianchi Type ◽  
Killing Vector ◽  
Integration Technique ◽  
Killing Vector Fields ◽  
Type Iii ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Theory Of Gravity

The purpose of this paper is to find conformal vector fields of some perfect fluid Kantowski–Sachs and Bianchi type III spacetimes in the [Formula: see text] theory of gravity using direct integration technique. In this study, there exist only eight cases. Studying each case in detail, we found that in two cases proper conformal vector fields exist while in the rest of the cases, conformal vector fields become Killing vector fields. The dimension of conformal vector fields is either 4 or 6.

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