Conformal and Disformal Structure of 3D Circularly Symmetric Static Metric in f(R) Theory of Gravity

Muhammad Ramzan,Fiaz Hussain,Murtaza Ali

doi:10.22581/muet1982.2001.11

Abstract

Conformal vector fields are treated as generalization of homothetic vector fields while disformal vector fields are defined through disformal transformations which are generalization of conformal transformations, therefore it is important to study conformal and disformal vector fields. In this paper, conformal and disformal structure of 3D (Three Dimensional) circularly symmetric static metric is discussed in the framework of f(R) theory of gravity. The purpose of this paper is twofold. Firstly, we have found some dust matter solutions of EFEs (Einstein Field Equations) by considering 3D circularly symmetric static metric in the f(R) theory of gravity. Secondly, we have found CKVFs (Conformal Killing Vector Fields) and DKVFs (Disformal Killing Vector Fields) of the obtained solutions by means of some algebraic and direct integration techniques. A metric version of f(R) theory of gravity is used to explore the solutions and dust matter as a source of energy momentum tensor. This study reveals that no proper DVFs exists. Here, DVFs for the solutions under consideration are either HVFs (Homothetic Vector Fields) or KVFs (Killing Vector Fields) in the f(R) theory of gravity. In this study, two cases have been discussed. In the first case, both CKVFs and DKVFs become HVFs with dimension three. In the second case, there exists two subcases. In the first subcase, DKVFs become HVFs with dimension seven. In the second subcase, CKVFs and DKVFs become KVFs having dimension four.

Highlights

Conformal vector fields are treated as generalization of homothetic vector fields while disformal vector fields are defined through disformal transformations which are generalization of conformal transformations, it is important to study conformal and disformal vector fields
DKVFs are defined by means of disformal transformations which have many applications in TeVes theories for Mond [14], biometric theories of gravity [15], scalar tensor theories [16,17,18,19], disformal inflation [20], rainbow gravity [21] and many others
We have found some dust matter solutions of EFEs in the f(R) theory of gravity and we have extended our analysis by finding CKVFs and DKVFs of the resulting solutions

Summary

Introduction

Conformal vector fields are treated as generalization of homothetic vector fields while disformal vector fields are defined through disformal transformations which are generalization of conformal transformations, it is important to study conformal and disformal vector fields. Conformal and disformal structure of 3D (Three Dimensional) circularly symmetric static metric is discussed in the framework of f(R) theory of gravity. We have found some dust matter solutions of EFEs (Einstein Field Equations) by considering 3D circularly symmetric static metric in the f(R) theory of gravity. A metric version of f(R) theory of gravity is used to explore the solutions and dust matter as a source of energy momentum tensor. DVFs for the solutions under consideration are either HVFs (Homothetic Vector Fields) or KVFs (Killing Vector Fields) in the f(R) theory of gravity. To classify the solutions of EFEs symmetries play an important role as with the help of symmetries one can explain geometry of spacetime [1]. Ali et al [11] investigated proper homothetic vector fields of 3D circularly symmetric static metric. To solve Equation (1), we consider 3D circularly symmetric static metric in the usual coordinates (t,r, )given by (x0, x1, x2) respectively [5]

Objectives

Conclusion