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

TL;DR

This paper introduces an infinite family of exact, asymptotically flat solutions to Einstein's equations describing static, axially symmetric thin dust disks with positive energy density and finite mass, all expressed explicitly in oblate spheroidal coordinates.

Contribution

It provides a new set of explicit, regular solutions for static dust disks with an inner edge, expanding the known catalog of exact solutions in general relativity.

Findings

Solutions are explicitly computed and expressed in oblate spheroidal coordinates.

All solutions are asymptotically flat and regular everywhere.

Disks have positive, well-behaved energy densities and finite total mass.

Abstract

An infinite family of new exact solutions of the Einstein vacuum equations for static and axially symmetric spacetimes is presented. All the metric functions of the solutions are explicitly computed and the obtained expressions are simply written in terms of oblate spheroidal coordinates. Furthermore, the solutions are asymptotically flat and regular everywhere, as it is shown by computing all the curvature scalars. These solutions describe an infinite family of thin dust disks with a central inner edge, whose energy densities are everywhere positive and well behaved, in such a way that their energy-momentum tensor are in fully agreement with all the energy conditions. Now, although the disks are of infinite extension, all of them have finite mass. The superposition of the first member of this family with a Schwarzschild black hole was presented previously [G. A. Gonz\'alez and A. C. Guti\'errez-Pi\~neres, arXiv: 0811.3002v1 (2008)], whereas that in a subsequent paper a detailed analysis of the corresponding superposition for the full family will be presented.