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.

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.

A NOTE ON CLASSIFICATION OF BIANCHI TYPE I SPACETIMES ACCORDING TO THEIR TELEPARALLEL KILLING VECTOR FIELDS

2010 ◽  
Vol 25 (01) ◽  
pp. 55-61 ◽  
Author(s):  
GHULAM SHABBIR ◽  
SUHAIL KHAN
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Bianchi Type ◽  
Killing Vector ◽  
Type I ◽  
Direct Integration ◽  
Integration Technique ◽  
Killing Vector Fields ◽  
Bianchi Type I

In this paper we classify Bianchi type I spacetimes according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the teleparallel Killing vector fields is 3, 4, 6 or 10 which are the same in numbers as in general relativity. In case of 3, 4 or 6 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of t. In the case of 10 Killing vector fields, the spacetime becomes Minkowski and all the torsion components are zero. The Killing vector fields in this case are exactly the same as in the general relativity.

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.

Proper projective symmetry in LRS Bianchi type V spacetimes

2018 ◽  
Vol 33 (13) ◽  
pp. 1850073 ◽  
Author(s):  
Ghulam Shabbir ◽  
K. S. Mahomed ◽  
F. M. Mahomed ◽  
R. J. Moitsheki
Keyword(s):  
Vector Fields ◽  
Bianchi Type ◽  
Killing Vector ◽  
Riemann Tensor ◽  
Direct Integration ◽  
Killing Vector Fields ◽  
Rotationally Symmetric ◽  
Projective Vector ◽  
Type V ◽  
Bianchi Type V

In this paper, we investigate proper projective vector fields of locally rotationally symmetric (LRS) Bianchi type V spacetimes using direct integration and algebraic techniques. Despite the non-degeneracy in the Riemann tensor eigenvalues, we classify proper Bianchi type V spacetimes and show that the above spacetimes do not admit proper projective vector fields. Here, in all the cases projective vector fields are Killing vector fields.

scholarly journals Proper Homothetic Vector Fields of Bianchi Type I Spacetimes via Rif Tree Approach

2021 ◽  
pp. 104299
Author(s):  
Ashfaque H. Bokhari ◽  
Tahir Hussain ◽  
Jamshed Khan ◽  
Uzma Nasib
Keyword(s):  
Vector Fields ◽  
Bianchi Type ◽  
Type I ◽  
Bianchi Type I ◽  
Homothetic Vector ◽  
Tree Approach

scholarly journals Concircular vector fields for Kantowski–Sachs and Bianchi type-III spacetimes

2018 ◽  
Vol 15 (08) ◽  
pp. 1850126 ◽  
Author(s):  
Suhail Khan ◽  
Amjad Mahmood ◽  
Ahmad T. Ali
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Bianchi Type ◽  
Killing Vector ◽  
Conformal Killing ◽  
Conformal Killing Vector ◽  
Killing Vector Fields ◽  
Type Iii ◽  
Conformal Killing Vector Fields ◽  
Metric Functions

This paper intends to obtain concircular vector fields (CVFs) of Kantowski–Sachs and Bianch type-III spacetimes. For this purpose, ten conformal Killing equations and their general solution in the form of conformal Killing vector fields (CKVFs) are derived along with their conformal factors. The obtained conformal Killing vector fields are then placed in Hessian equations to obtain the final form of concircular vector fields. The existence of concircular symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. It is shown that Kantowski–Sachs and Bianchi type-III spacetimes admit four-, six-, or fifteen-dimensional concircular vector fields. It is established that for Einstein spaces, every conformal Killing vector field is a concircular vector field. Moreover, it is explored that every concircular vector field obtained here is also a conformal Ricci collineation.

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