Static axisymmetric rings in general relativity: How diverse they are

TL;DR

This paper compares three different static, axially symmetric ring solutions in general relativity, revealing significant geometric differences and contrasting them with the more complex but simpler Kerr ring, highlighting the diversity of static ring geometries.

Contribution

It provides a detailed comparison of three static ring solutions in GR, analyzing their geometric and invariant properties, and contrasting them with the Kerr ring to highlight their differences.

Findings

Static rings differ significantly in geometry and invariants.

The Kerr ring, despite being non-static, appears geometrically simpler.

The Bach-Weyl ring shows unusual deformation, suggesting coordinate issues.

Abstract

Three static and axially symmetric (Weyl-type) ring singularities -- the Majumdar-Papapetrou--type (extremally charged) ring, the Bach-Weyl ring and the Appell ring -- are studied in general relativity in order to show how remarkably the geometries in their vicinity differ from each other. This is demonstrated on basic measures of the rings and on invariant characteristics given by the metric and by its first and second derivatives (lapse, gravitational acceleration and curvature), and also on geodesic motion. The results are also compared against the Kerr space-time which possesses a ring singularity too. The Kerr solution is only stationary, not static, but in spite of the consequent complication by dragging, its ring appears to be simpler than the static rings. We show that this mainly applies to the Bach-Weyl ring, although this straightforward counter-part of the Newtonian homogeneous circular ring is by default being taken as the simplest ring solution, and although the other two static ring sources may seem more "artificial". The weird, directional deformation around the Bach-Weyl ring probably indicates that a more adequate coordinate representation and interpretation of this source should exists.