Analyticity Properties and Limiting Cases

Abstract

In Chap. 3 we have used the linear system for the Ernst equation to construct solutions via Riemann–Hilbert techniques. This was done on the Riemann surface of the spectral parameter, the physical coordinates were .xed in a way that they did not coincide with the singularities of the matrix of the linear system. In the present chapter we want to investigate the behavior of the found solutions in dependence of the physical coordinates, especially at the potential singularities which were so far excluded. This analysis allows usto identify a whole subclass of solutions which will be only singular at some contour which could be identi.ed withthe surface of some star or galaxy. Since solutions of astrophysical interest typically have an equatorial symmetry, a re.ection symmetry at the equatorial plane, we will identify theta-functional solutions withthis property. For the found subclass we investigate interesting limiting cases as the limit of large distancefromthe material source. This allows to identify asymptotically .at solutions which can describe isolated matter sources. We also study the static limit and the ‘solitonic‚ limit, in which the Riemann surface degenerates. In this vicinity the solutions can be given in terms of elementary functions, they belong either to the static Weyl class or the multi-black hole solutions.