Physically Realistic Solutions to the Ernst Equation on Hyperelliptic Riemann Surfaces

A class of solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) is discussed which is constructed via Riemann–Hilbert techniques on hyperelliptic Riemann surfaces. We identify a physically interesting subclass where the Ernst potential is everywhere regular except at a closed surface which might be identified with the surface of a body of revolution. The corresponding spacetimes are asymptotically flat and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance a relativistic star or a galaxy. Within this class, one has the freedom to specify 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 solutions can have ergoregions, a Minkowskian limit and an ultrarelativistic limit where the metric approaches the extreme Kerr solution. We give explicit formulae for the potential on the axis and in the equatorial plane where the expressions simplify because of the equatorial symmetry. Special attention is paid to the simplest non–static solutions (which are of genus two) to which the rigidly rotating dust disk belongs. PACS numbers: 04.20.Jb, 02.10.Rn, 02.30.Jr