Axial symmetry and conformal Killing vectors

TL;DR

This paper investigates the relationship between axial symmetry and conformal Killing vectors in spacetimes, proving that under certain conditions these symmetries must commute, which constrains the structure of such spacetimes.

Contribution

It establishes general conditions under which axial and conformal symmetries commute in axisymmetric spacetimes, independent of Einstein's equations or matter content.

Findings

Axial and conformal Killing vectors must commute if no additional conformal symmetry exists.

In conformally stationary, axisymmetric spacetimes, axial and conformal timelike symmetries commute without restrictions.

The commutator of axial and other symmetries must vanish or imply larger symmetry groups.

Abstract

Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, we prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vector with the two others mush vanish or else the symmetry is larger than that originally considered. The results are completely general and do not depend on Einstein's equations or any particular matter content.