[Paper Review] Geometrical Properties of a Point-like Global Monopole Spacetime

Absos Ali Shaikh, Faizuddin Ahmed+1 more|arXiv (Cornell University)|Jan 12, 2023

Advanced Differential Geometry Research4 citations

TL;DR

This paper investigates the geometric and symmetry properties of a point-like global monopole (PGM) spacetime, a static, spherically symmetric solution to Einstein's field equations. It demonstrates that the PGM spacetime admits multiple pseudosymmetry structures—such as pseudosymmetry due to Weyl, concircular, and conharmonic curvature tensors—and is 2-quasi-Einstein, generalized quasi-Einstein, and Einstein of degree 2. Crucially, it reveals a fundamental distinction between curvature inheritance notions defined on (1,3)- and (0,4)-type curvature tensors, showing they are not equivalent in this spacetime.

ABSTRACT

The aim of this paper is to study the geometric properties of the point-like global monopole (briefly, PGM) spacetime, which is a static and spherically symmetric solution of the Einstein's field equations. It has shown that PGM spacetime admits various types of pseudosymmetry structures, such as pseudosymmetry due to Weyl conformal curvature tensor, pseudosymmetry due to concircular curvature tensor, pseudosymmetry due to conharmonic curvature tensor, Ricci generalized conformal pseudo-symmetric due to projective curvature tensor, Ricci generalized projective pseudo-symmetric. Moreover, it has proved that PGM spacetime is $2$-quasi Einstein, generalized quasi-Einstein, Einstein manifold of degree $2$, and its Weyl conformal curvature $2$-forms are recurrent. The energy-momentum tensor of the PGM spacetime realizes several types of pseudosymmetry, and its Ricci tensor is compatible with Riemann curvature, Weyl conformal curvature, projective curvature, and conharmonic curvature and concircular curvature. Further, it has shown that PGM spacetime admits motion, curvature collineation, and Ricci collineation. Also, the notion of curvature inheritance (resp., curvature collineation) for the (1,3)-type curvature tensor is not equivalent to the notion of curvature inheritance (resp., curvature collineation) for the (0,4)-type curvature tensor as it has shown that such distinctive properties were possessed by PGM spacetime. Hence the notions of curvature inheritance defined by Duggal \cite{Duggal1992} and Shaikh and Datta \cite{ShaikhDatta2022} are not equivalent.

Motivation & Objective

• To analyze the geometric structures of the point-like global monopole (PGM) spacetime using curvature tensors and their symmetries.
• To determine whether curvature inheritance and curvature collineation notions for (1,3)- and (0,4)-type curvature tensors are equivalent in PGM spacetime.
• To investigate the role of non-Killing vector fields in preserving geometric structures such as curvature and Ricci tensors.
• To classify the PGM spacetime in terms of generalized Einstein and pseudosymmetric conditions, including Ricci generalized pseudosymmetry.

Proposed method

• The study employs the Levi-Civita connection and curvature tensors (Riemann, Weyl, conformal, projective, conharmonic, and concircular) to analyze geometric properties of the PGM spacetime.
• It uses Lie derivatives of curvature and Ricci tensors along vector fields to test for curvature collineation, Ricci collineation, and curvature inheritance.
• The analysis is based on the exact solution of the Einstein field equations for a point-like global monopole, with metric components derived from spherical symmetry and global monopole energy-momentum tensor.
• Algebraic computations are performed using a custom Wolfram Mathematica program to verify tensorial identities and symmetry conditions.
• The paper applies definitions of pseudosymmetry, generalized symmetry, and recurrence conditions to classify the spacetime's curvature behavior.
• It compares two distinct notions of curvature inheritance: one for (1,3)-type tensors (Duggal) and one for (0,4)-type tensors (Shaikh and Datta), using explicit vector field examples.

Experimental results

Research questions

• RQ1 Does the PGM spacetime admit pseudosymmetry with respect to the Weyl conformal curvature tensor, and if so, what is the nature of this pseudosymmetry?
• RQ2 Are the notions of curvature inheritance for (1,3)-type and (0,4)-type curvature tensors equivalent in the PGM spacetime?
• RQ3 Does the PGM spacetime support curvature collineation and Ricci collineation for non-Killing vector fields, and if so, which ones?
• RQ4 What is the classification of the PGM spacetime in terms of Einstein and quasi-Einstein manifolds, and how does this relate to its curvature forms?
• RQ5 How do the energy-momentum tensor and Ricci tensor interact with various curvature tensors in the PGM spacetime?

Key findings

• The PGM spacetime is 2-quasi-Einstein, generalized quasi-Einstein, and Einstein of degree 2, as proven in Theorem 3.1.
• It admits pseudosymmetry due to the Weyl conformal curvature tensor, the concircular curvature tensor, and the conharmonic curvature tensor.
• The spacetime is Ricci generalized pseudosymmetric with respect to the projective curvature tensor, and its energy-momentum tensor satisfies multiple pseudosymmetric-type curvature conditions.
• The Weyl conformal curvature 2-forms of the PGM spacetime are recurrent, indicating a form of generalized symmetry.
• The PGM spacetime admits curvature collineation and Ricci collineation for the (1,3)-type curvature tensor with respect to the non-Killing vector field ∂/∂r, but not curvature collineation for the (0,4)-type tensor R.
• The notions of curvature inheritance for (1,3)- and (0,4)-type curvature tensors are not equivalent in the PGM spacetime, as demonstrated by distinct behaviors under Lie derivatives along ∂/∂r and ∂/∂θ.