Curvature properties of Robinson-Trautman metric

TL;DR

This paper explores the curvature properties of the Robinson-Trautman metric, revealing its pseudosymmetric structures, Roter type classification, and energy-momentum tensor characteristics, along with a comparison to the Som-Raychaudhuri metric.

Contribution

It provides a comprehensive analysis of the Robinson-Trautman metric's curvature structures and introduces new classifications and properties not previously detailed.

Findings

Robinson-Trautman metric admits various pseudosymmetric structures.

The metric is Roter type and 2-quasi-Einstein.

Energy momentum tensor is pseudosymmetric and has specific geometric properties.

Abstract

The curvature properties of Robinson-Trautman metric have been investigated. It is shown that Robinson-Trautman metric admits several kinds of pseudosymmetric type structures such as Weyl pseudosymmetric, Ricci pseudosymmetric, pseudosymmetric Weyl conformal curvature tensor etc. Also it is shown that the difference $R\cdot R - Q(S,R)$ is linearly dependent with $Q(g,C)$ but the metric is not Ricci generalized pseudosymmetric. Moreover, it is proved that this metric is Roter type, 2-quasi-Einstein, Ricci tensor is Riemann compatible and its Weyl conformal curvature 2-forms are recurrent. It is also shown that the energy momentum tensor of the metric is pseudosymmetric and the conditions under which such tensor is of Codazzi type and cyclic parallel have been investigated. Finally, we have made a comparison between the curvature properties of Robinson-Trautman metric and Som-Raychaudhuri metric.