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GR: Gravitation und Relativitätstheorie

GR 101: Eigenschaften von klassischen Lösungen

GR 101.1: Hauptvortrag

Montag, 20. März 2006, 14:00–14:40, K

Ernst Equation and Riemann Surfaces — •Christian Klein¹ and Olaf Richter² — ¹MPI für Mathematik in den Naturwissenschaften, Inselstr. 22-26, 04103 Leipzig — ²Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, 04009 Leipzig

In a general relativistic framework stars and galaxies in thermodynamical equilibrium lead to stationary axisymmetric spacetimes. Therefore it is of special physical interest that the Einstein equations in this case are equivalent to the completely integrable Ernst equation. The integrability of the Ernst equation implies the knowledge of many explicit exact solutions , the most prominent being the Kerr solution which describes the exterior of a rotating black hole. We discuss applications of the Ernst equation in various field of physics and mathematics as in the context of Yang-Mills-Higgs monopoles and Bianchi surfaces. Rich classes of solutions to integrable equations can be constructed with methods from the theory of Riemann surfaces which were originally introduced to generate periodic solutions to integrable wave equations such as the Korteweg-de Vries equation. The corresponding solutions to the Ernst equation, which contain the Kerr solution as a limit, are not periodic and are related to deformations of the underlying Riemann surface. We study these solutions to the Ernst equation in detail and discuss physical of this class.