Conharmonic Curvature Inheritance in Spacetime of General Relativity

Musavvir Ali,Mohd Bilal,Mohammad Salman

doi:10.3390/universe7120505

Musavvir Ali, Mohd Bilal + Show 1 more

Open Access

https://doi.org/10.3390/universe7120505

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Journal: Universe	Publication Date: Dec 17, 2021
Citations: 3	License type: CC BY 4.0

Affiliation: Umm al-Qura University, Aligarh Muslim University

Abstract

The motive of the current article is to study and characterize the geometrical and physical competency of the conharmonic curvature inheritance (Conh CI) symmetry in spacetime. We have established the condition for its relationship with both conformal motion and conharmonic motion in general and Einstein spacetime. From the investigation of the kinematical and dynamical properties of the conformal Killing vector (CKV) with the Conh CI vector admitted by spacetime, it is found that they are quite physically applicable in the theory of general relativity. We obtain results on the symmetry inheritance for physical quantities (μ,p,ui,σij,η,qi) of the stress-energy tensor in imperfect fluid, perfect fluid and anisotropic fluid spacetimes. Finally, we prove that the conharmonic curvature tensor of a perfect fluid spacetime will be divergence-free when a Conh CI vector is also a CKV.

Highlights

IntroductionLet (V4 , g) be a spacetime, where V4 is a four-dimensional connected smooth Hausdorff manifold and g is a smooth Lorentz metric of signature (−, +, +, +)
Inheritance in Spacetime of GeneralLet (V4, g) be a spacetime, where V4 is a four-dimensional connected smooth Hausdorff manifold and g is a smooth Lorentz metric of signature (−, +, +, +)
We introduce the notion of conharmonic curvature inheritance symmetry as follows

Summary

Introduction

Let (V4 , g) be a spacetime, where V4 is a four-dimensional connected smooth Hausdorff manifold and g is a smooth Lorentz metric of signature (−, +, +, +). The gravitational field consists of two parts viz., the free gravitational part and the matter part, which is described by the Riemannian curvature tensor in the general theory of relativity The connection between these two parts is explained through Bianchi’s identities [9]. In the present research paper, we raise the following fundamental problem: how are the geometrical symmetries of the spacetime (V4 , g) associated with the conharmonic curvature symmetry vector field, under the condition that this vector is inherited by some of the source terms of the energy-momentum tensor in the field equations? L. Duggal introduced the concept of inheritance symmetry for the curvature tensor of Riemannian spaces with physical applications to the fluid spacetime of general relativity ([2,13]). In an attempt to support our study, which is related to the solution of EFE and conservation law of generators, we have constructed some non-trivial examples that are embedded in the Appendix A after the conclusion

Preliminaries

Relationship of Conh CI with Other Symmetries of Spacetime

Conformal Motion

Conharmonic Motion

Physical Interpretation to Fluid Spacetimes of General Relativity

Conclusions