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.

A NOTE ON KILLING VECTOR FIELDS OF BIANCHI TYPE II SPACETIMES IN TELEPARALLEL THEORY OF GRAVITATION

2010 ◽  
Vol 25 (20) ◽  
pp. 1733-1740 ◽  
Author(s):  
GHULAM SHABBIR ◽  
SUHAIL KHAN
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Bianchi Type ◽  
Killing Vector ◽  
Type Ii ◽  
Direct Integration ◽  
Integration Technique ◽  
Killing Vector Fields ◽  
Teleparallel Theory ◽  
Theory Of Gravitation

The aim of this paper is to classify Bianchi type II spacetimes according to their teleparallel Killing vector fields using the direct integration technique. Studying teleparallel Killing vector fields in the above spacetimes, it turns out that the dimensions of the teleparallel Killing vector fields are 4, 5 or 7. A brief comparison between teleparallel and general relativity Killing vector fields are given. It is shown that for the above spacetimes in the presence of torsion we get more conservation laws which are different from the theory of general relativity.

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.

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.

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.

CLASSIFICATION OF CYLINDRICALLY SYMMETRIC STATIC SPACETIMES ACCORDING TO THEIR KILLING VECTOR FIELDS IN TELEPARALLEL THEORY OF GRAVITATION

2010 ◽  
Vol 25 (07) ◽  
pp. 525-533 ◽  
Author(s):  
GHULAM SHABBIR ◽  
SUHAIL KHAN
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Minkowski Spacetime ◽  
Killing Vector Fields ◽  
Plane Symmetric ◽  
Cylindrically Symmetric ◽  
Teleparallel Theory ◽  
Theory Of Gravitation

In this paper we classify cylindrically symmetric static spacetimes according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the teleparallel Killing vector fields are 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 r. In the case of 10 Killing vector fields the spacetime becomes Minkowski spacetime and all the torsion components are zero. The Killing vector fields in this case are exactly the same as in general relativity. Here we also discuss the Lie algebra in each case. It is important to note that this classification also covers the plane symmetric static spacetimes.

A note on proper homothetic vector fields in plane symmetric perfect fluid static spacetimes in f(R, T) theory of gravity

2019 ◽  
Vol 34 (24) ◽  
pp. 1950189 ◽  
Author(s):  
M. Jamil Khan ◽  
Ghulam Shabbir ◽  
M. Ramzan
Keyword(s):  
Perfect Fluid ◽  
Vector Fields ◽  
Killing Vector ◽  
The Other ◽  
Direct Integration ◽  
Integration Technique ◽  
Killing Vector Fields ◽  
Plane Symmetric ◽  
Homothetic Vector ◽  
Theory Of Gravity

The purpose of this paper is to find proper homothetic vector fields in plane symmetric perfect fluid static spacetimes in the [Formula: see text] theory of gravity by using direct integration technique. In this study, there exist six cases. Studying each case in detail, we found that in four cases proper homothetic vector fields exist while in the other two cases homothetic vector fields become Killing 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.

Classification of proper teleparallel conformal symmetry of spherically symmetric static spacetimes using diagonal tetrads

2020 ◽  
Vol 35 (28) ◽  
pp. 2050232
Author(s):  
Muhammad Amer Qureshi ◽  
Ghulam Shabbir ◽  
K. S. Mahomed ◽  
Taha Aziz
Keyword(s):  
Vector Fields ◽  
Conformal Symmetry ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Conformal Vector Field ◽  
Integration Technique ◽  
Killing Vector Fields ◽  
Conformal Vector ◽  
Conformal Vector Fields

We study proper teleparallel conformal vector fields in spherically symmetric static spacetimes. The main objective of this paper is to present the classification for the above-mentioned spacetimes. The problem has been examined by two methods: direct integration technique and diagonal tetrads. We show that the spherically symmetric static spacetimes do not admit proper teleparallel conformal vector field, so are actually the teleparallel killing vector fields.

Sign in / Sign up

Export Citation Format

Share Document