AII remains a highly lethal condition. Mortality rates remain as high as they did decades ago due in part to advanced presentation and advanced age with multiple associated conditions and risk factors, all of which are independent predictors of adverse outcome.

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We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Neugebauer and Meinel can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical properties of these solutions that was begun in a Physical Review Letter, and supplies complete proofs. 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. Special attention is paid to the simplest non-static solutions (which are of genus two) to which the rigidly rotating dust disk belongs.

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We present a solution to the Ernst equation which represents an infinitesimally thin dust disk consisting of two streams of particles circulating with constant angular velocity in opposite directions. These streams have the same density distribution but their relative density may vary continuously. In the limit of only one component of dust, we get the solution for the rigidly rotating dust disk of Neugebauer and Meinel; in the limit of identical densities, the static disk of Morgan and Morgan is obtained. We discuss the Newtonian and the ultrarelativistic limit, the occurrence of ergospheres, and the regularity of the solution.PACS numbers: 04.20.JbIn astrophysics thin disks of collisionless matter, socalled dust, are discussed as models for certain galaxies (see, e.g., [1]) or for accretion disks. A fully relativistic treatment of these models is necessary if a black hole is present since black holes are genuinely relativistic objects. But the exact treatment of dust disks without a central object would provide deep insight both in the mathematical structure of the field equations and in the physics of rapidly rotating relativistic bodies, since dust disks can be viewed as a limiting case for extended matter sources. Corresponding exact solutions hold globallyin the vacuum and in the matter region-and can thus provide physically realistic test beds for numerical codes. Since Newtonian dust disks are known to be unstable and since there are hints by numerical work (see, e.g., [2]) that the same holds in the relativistic case, such solutions could be taken as exact initial data for numerical collapse calculations. Whereas the Newtonian theory of such disks is well established (see [1], and references therein), the same holds in the relativistic case only for static disks which can be interpreted as consisting of two counterrotating streams of matter with vanishing total angular momentum. The first disk of this type was considered by Morgan and Morgan [3]. Infinitely extended dust disks with finite mass were studied by Bičák, and Katz [4] in the static case and by Bičák and Ledvinka [5] in the stationary case. The first to construct the exact solution for a finite stationary dust disk were Neugebauer and Meinel [6] who gave the solution for the rigidly rotating dust disk which was first treated numerically by Bardeen and Wagoner [2]. They solved the corresponding boundary value problem for the Einstein equations with the help of a corotating coordinate system.In this Letter we present a class of new disk solutions to the Ernst equation where such a coordinate system cannot be used. The disks of finite radius r 0 consist of two counterrotating components of dust with respective density s 6 ͑r͒. The angular velocity of both streams of particles is of the same constant absolute value V but of different sign. The relative density g ͑s 1 2 s 2 ͒͑͞s 1 1 s 2 ͒ is a constant with respect to r and z which varies between one, the rigidly rotating dust disk [6], and zero, the static Morgan and Morgan disk [3]. Interesting...