Comment on “Relativistic static thin dust disks with an inner edge: An infinite family of new exact solutions”

Abstract

We consider the family of static, axially symmetric, exact solutions of the vacuum Einstein equations, presented in a recent article by G. A. Gonz\'alez, et al. (Phys. Rev. D 79, 124048 (2009)) (I). These solutions are singular on an infinite disk with a central inner edge. This singularity was interpreted in (I) as corresponding to the presence of dust with positive energy density everywhere on the disk. It was further asserted in (I) that the Riemann tensor is regular everywhere. Unfortunately, as we show in this Comment, neither the physical interpretation of the source nor the assertions about the Riemann tensor are correct. We provide an extended analysis of the geometric properties of the solutions that indicates that the Riemann tensor is, in fact, singular on the edge of the disk, and that the disk itself contains a singularity on this edge that acts as an infinite source of negative mass. We further comment on the appropriate use of the Komar formula for the mass of the system to make it consistent with the fact that these solutions have vanishing Arnowitt-Desser-Misner mass.