Comment on "Conformally flat stationary axisymmetric metrics"

TL;DR

This paper critiques recent claims of new conformally flat stationary axisymmetric spacetimes, reaffirming Collinson's theorem under certain conditions and providing counterexamples when those conditions are relaxed.

Contribution

It clarifies the conditions under which Collinson's theorem holds and introduces explicit counterexamples when orthogonal transitivity is not assumed.

Findings

Collinson's theorem remains valid with an axis of symmetry and orthogonal transitivity.

Counterexamples exist when orthogonal transitivity is dropped.

The paper refines understanding of conformally flat stationary axisymmetric metrics.

Abstract

Garcia and Campuzano claim to have found a previously overlooked family of stationary and axisymmetric conformally flat spacetimes, contradicting an old theorem of Collinson. In both these papers it is tacitly assumed that the isometry group is orthogonally transitive. Under the same assumption, we point out here that Collinson's result still holds if one demands the existence of an axis of symmetry on which the axial Killing vector vanishes. On the other hand if the assumption of orthogonal transitivity is dropped, a wider class of metrics is allowed and it is possible to find explicit counterexamples to Collinson's result.