Counter-rotating dust rings around a static black hole

Abstract

Relativistic configurations describing counter-rotating dust discs with a central Schwarzschild black hole are analysed. It is shown that the corresponding solutions to the Einstein equations which belong to the Weyl class can be obtained from a boundary value problem for the Laplace equation where the boundary data are given at the two-dimensional dust ring. Using a result of Poole, one can prove existence and uniqueness of a solution to this problem. In adapted coordinates, the potential can be represented via eigenfunctions of the Lamé equation in the `flat ring' case which are not known explicitly. Following Kanwal, one can reformulate the boundary value problem in a system of Fredholm integral equations. As an example, a rigidly (counter-)rotating dust ring is considered. In the case of a dust ring stretching to infinity, it is possible to relate this problem via a Kelvin transformation to the case of a disc. An explicit solution is given for an isochrone dust ring.