Boundary value problems for the stationary axisymmetric Einstein equations: a rotating disk

TL;DR

This paper applies a modern integrable PDE boundary value problem method to the Ernst equation in general relativity, specifically analyzing a rotating disk, and recovers known results through a linearizable boundary value problem approach.

Contribution

It introduces a Riemann-Hilbert problem formulation for the Ernst equation with a square root spectral parameter and demonstrates the linearizability of the rotating disk boundary condition.

Findings

Formulation of a Riemann-Hilbert problem on a two-sheeted Riemann surface.

Identification of the rotating disk boundary condition as linearizable.

Recovery of Neugebauer and Meinel's results using the new formalism.

Abstract

The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analyzing boundary value problems for integrable PDEs, and this method consists of two steps: (a) Construct an integral representation of the solution characterized via a matrix Riemann-Hilbert (RH) problem formulated in the complex $k$-plane, where $k$ denotes the spectral parameter of the associated Lax pair. This representation involves, in general, some unknown boundary values, thus the solution formula is {\it not} yet effective. (b) Characterize the unknown boundary values by analyzing a certain equation called the {\it global relation}. This analysis involves, in general, the solution of a nonlinear problem; however, for certain boundary value problems called linearizable, it is possible to determine the unknown boundary values using only linear operations. Here, we employ the above methodology for the investigation of certain boundary value problems for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurence of the spectral parameter in the form of a square root and this necessitates the introduction of a two-sheeted Riemann surface for the formulation of the relevant RH problem. As a concrete application of the general formalism, it is shown that the particular boundary value problem corresponding to the physically significant case of a rotating disk is a linearizable boundary value problem. In this way the remarkable results of Neugebauer and Meinel are recovered.