Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces - art. no. 124018

We show that the class of hyperelliptic solutions to the Ernst equation (th e stationary axisymmetric Einstein equations in vacuum) previously discover ed by Korotkin and Meinel and Neugebauer can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical prope rties of these solutions that was begun in a previous Letter and supplies c omplete proofs. We identify a physically interesting subclass where the Ern st potential is everywhere regular except at a closed surface which might b e identified with the surface of a body of revolution. The corresponding sp acetimes an asymptotically flat and equatorially symmetric. This suggests t hat they could describe the exterior of an isolated body, for instance, a r elativistic star or a galaxy. Within this class, one has the freedom to spe cify a real function and a set of complex parameters which can possibly be used to solve certain boundary value problems for the Ernst equation. The s olutions can have ergoregions, a Minkowskian limit, and an ultrarelativisti c limit where the metric approaches the extreme Kerr solution. We give expl icit formulas for the potential on the axis and in the equatorial plane whe re the expressions simplify. Special attention is paid to the simplest nons tatic solutions (which are of genus 2) to which the rigidly rotating dust d isk belongs. [S0556-2821(98)03922-8].