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

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 analysing boundary-value problems (BVPs) 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 not yet effective. (b) Characterize the unknown boundary values by analysing a certain equation called the global relation. This analysis involves, in general, the solution of a nonlinear problem; however, for certain BVPs 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 BVPs for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurrence 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 BVP corresponding to the physically significant case of a rotating disc is a linearizable BVP. In this way the remarkable results of Neugebauer and Meinel are recovered.