[Paper Review] Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes

Absos Ali Shaikh, Biswa Ranjan Datta|arXiv (Cornell University)|Sep 2, 2022

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper investigates Ricci solitons and generalized curvature inheritance in Robinson-Trautman (RT) spacetimes using algebraic computations in Wolfram Mathematica. It establishes that RT spacetimes admit almost Ricci solitons, almost η-Ricci solitons, and generalized curvature inheritance for Ricci, Weyl conformal, concircular, conharmonic, and Weyl projective tensors, with explicit conditions derived for curvature and Ricci collineations.

ABSTRACT

The purpose of the article is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson-Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT spacetime admits almost Ricci soliton, almost $η$-Ricci soliton, almost gradient $η$-Ricci soliton. As a generalization of curvature inheritance \cite{Duggal1992} and curvature collineation \cite{KLD1969}, in this paper, we introduce the notion of extit{generalized curvature inheritance} and examine if RT spacetime admits such a notion. It is shown that RT spacetime also realizes the generalized curvature (resp. Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) inheritance. Finally, several conditions are obtained, under which RT spacetime possesses curvature (resp. Ricci, conharmonic, Weyl projective) inheritance as well as curvature (resp. Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) collineation, and we have also introduced the concept of generalized Lie inheritance and showed that RT spacetime realizes such a notion.

Motivation & Objective

• To investigate the existence of Ricci solitons and curvature inheritance structures in Robinson-Trautman spacetimes.
• To generalize the concepts of curvature inheritance and collineation by introducing generalized curvature inheritance.
• To determine conditions under which RT spacetimes admit curvature, Ricci, and other tensor collineations.
• To analyze the geometric significance of Lie derivative conditions on curvature and Ricci tensors in RT spacetimes.
• To establish the realization of generalized Lie inheritance in RT spacetimes through algebraic tensor analysis.

Proposed method

• Employed algebraic tensor computations using Wolfram Mathematica to compute the Riemann curvature, Ricci tensor, and scalar curvature of RT spacetime.
• Defined and computed Kulkarni-Nijenbhuys-type product tensors: $g\wedge g$, $g\wedge S$, and $S\wedge S$.
• Implemented Lie derivative operators $\mathscr{L}_\xi$ for (0,2) and (0,4)-type tensors to analyze symmetry conditions.
• Used symbolic solving via `Solve` to determine coefficients in Lie derivative equations, identifying generalized curvature inheritance structures.
• Formulated and solved tensor equations of the form $\mathscr{L}_\xi R = xR + yg\wedge g + z g\wedge S + w S\wedge S$ to detect generalized curvature inheritance.
• Applied the same framework to Ricci soliton equations of the form $\mathscr{L}_\xi g + 2S - \mu g = 0$ with variable $\mu$, identifying almost Ricci soliton structures.

Experimental results

Research questions

• RQ1 Under what conditions does a Robinson-Trautman spacetime admit an almost Ricci soliton?
• RQ2 Does the RT spacetime support generalized curvature inheritance for Ricci, Weyl conformal, concircular, conharmonic, and Weyl projective tensors?
• RQ3 What are the necessary and sufficient conditions for curvature, Ricci, and other tensor collineations in RT spacetimes?
• RQ4 Can generalized Lie inheritance be realized in RT spacetimes, and how is it characterized algebraically?
• RQ5 How do the geometric symmetries of RT spacetimes relate to the existence of $\eta$-Ricci solitons and their generalizations?

Key findings

• Robinson-Trautman spacetimes admit almost Ricci solitons under specific conditions derived from Lie derivative analysis.
• The spacetime supports almost $\eta$-Ricci solitons when the soliton constants $\mu$ and $\lambda$ are smooth functions.
• Generalized curvature inheritance is realized for Ricci, Weyl conformal, concircular, conharmonic, and Weyl projective tensors in RT spacetimes.
• Explicit conditions are obtained under which RT spacetimes possess curvature, Ricci, and Weyl projective collineations.
• The spacetime realizes generalized Lie inheritance, as confirmed by solving Lie derivative equations involving curvature and Ricci tensors.
• Algebraic computations in Wolfram Mathematica confirm the existence of these structures through symbolic solving of tensor equations with multiple coefficients.