Homotheties of Cylindrically Symmetric Static Manifolds and their Global Extension

TL;DR

This paper classifies cylindrically symmetric static manifolds based on their homothety groups, explicitly derives their metrics, and examines their global extensions and physical properties via Einstein's equations.

Contribution

It provides a detailed classification of these manifolds' homothety groups and explores the global extendability and physical interpretation of the resulting spaces.

Findings

Metrics admit homothety groups of orders 4, 5, 7, 11

Certain local homotheties are globally prohibited by topology

Physical nature of spaces analyzed using Einstein's equations

Abstract

Cylindrically symmetric static manifolds are classified according to their homotheties and metrics. In each case the homothety vector fields and the corresponding metrics are obtained explicitly by solving the homothety equations. It turns out that these metrics admit homothety groups $H_m$, where $m=4,5,7,11$. This classification is then used to identify the cylindrically symmetric static spaces admitting the local homotheties, which are globally prohibited due to their topological construction. Einstein's field equations are then used to identify the physical nature of the spaces thus obtained.