Exact Static Axially Symmetric Thin Annular Dust Disks

TL;DR

This paper introduces a new family of exact solutions to Einstein's equations describing infinite, static, axially symmetric thin annular dust disks with positive energy density, finite mass, and physically consistent energy conditions.

Contribution

It provides explicit metric functions for a novel class of static axially symmetric dust disks using oblate spheroidal coordinates, expanding the set of known exact solutions in general relativity.

Findings

Disks have positive energy density everywhere

Disks possess finite total mass

Energy conditions are satisfied throughout the disks

Abstract

A new family of exact solutions of the Einstein field equations for static and axially simmetric 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. The solutions describe an infinite family of static axially symmetric thin annular dust disks. The disk are of infinite extension but with an inner annular edge. The energy densities of the disks are everywhere positive functions of the radial coordinate, equals to zero at the inner edge of the disk, having a maximun at a finite value of the radius and then vanishing at infinity. The disks have finite mass and their energy-momentum tensor agrees with all the energy conditions.