On the definition of cylindrical symmetry

TL;DR

This paper reviews and proposes a more general definition of cylindrical symmetry in General Relativity by dropping the orthogonal transitivity requirement, leading to new insights into the structure of cylindrically symmetric spacetimes.

Contribution

It introduces a broader definition of cylindrical symmetry that does not require orthogonal transitivity, and analyzes stationarity and staticity within this framework.

Findings

Proposes a new, more general definition of cylindrical symmetry.

Establishes new results on the isometry group's structure.

Analyzes stationarity and staticity without orthogonal transitivity.

Abstract

The standard definition of cylindrical symmetry in General Relativity is reviewed. Taking the view that axial symmetry is an essential pre-requisite for cylindrical symmetry, it is argued that the requirement of orthogonal transitivity of the isometry group should be dropped, this leading to a new, more general definition of cylindrical symmetry. Stationarity and staticity in cylindrically symmetric spacetimes are then defined, and these issues are analysed in connection with orthogonal transitivity, thus proving some new results on the structure of the isometry group for this class of spacetimes.