Counterrotating Dust Disk Around a Schwarzschild Black Hole: New Fully Integrated Explicit Exact Solution

TL;DR

This paper presents the first fully integrated explicit exact solution of Einstein's equations for a superposition of a counterrotating dust disk with a Schwarzschild black hole, satisfying all energy conditions.

Contribution

It introduces a novel explicit solution describing a thin, finite-mass disk around a black hole, expanding the set of known solutions in general relativity.

Findings

Finite mass of the disk with all energy conditions satisfied

Total mass of the combined system is less than the disk alone

Solution is part of an infinite family of solutions

Abstract

The first fully integrated explicit exact solution of the Einstein field equations corresponding to the superposition of a counterrotating dust disk with a central black hole is presented. The obtained solution represents an infinite annular thin disk (a disk with an inner edge) around the Schwarszchild black hole. The mass of the disk is finite and the energy-momentum tensor agrees with all the energy conditions. Furthermore, the total mass of the disk when the black hole is present is less than the total mass of the disk alone. The solution can also be interpreted as describing a thin disk made of two counterrotanting dust fluids that are also in agreement with all the energy conditions. Additionally, as we will show shortly in a subsequent paper, the above solution is the first one of an infinite family of solutions.