Dirichlet Boundary Value Problems of the Ernst Equation

TL;DR

This paper presents a method to solve exterior Dirichlet boundary value problems for the Ernst equation using generalized Backlund solutions, confirming conjectures about gravitational fields of rotating dust disks.

Contribution

It introduces a generalized solution approach for the Ernst equation and proves its validity, advancing the understanding of gravitational fields in axisymmetric stationary spacetimes.

Findings

Validates the use of generalized Backlund solutions for the Ernst equation

Proves conjectures about gravitational fields of rotating disks of dust

Provides a new method for solving boundary value problems in general relativity

Abstract

We demonstrate how the solution to an exterior Dirichlet boundary value problem of the axisymmetric, stationary Einstein equations can be found in terms of generalized solutions of the Backlund type. The proof that this generalization procedure is valid is given, which also proves conjectures about earlier representations of the gravitational field corresponding to rotating disks of dust in terms of Backlund type solutions.