Static axisymmetric rings in general relativity: How diverse they are
Semerák, O.
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 counterpart 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 exist.