This is the third in a series of papers on the construction of explicitsolutions to the stationary axisymmetric Einstein equations which can beinterpreted as counter-rotating disks of dust. We discuss the physicalproperties of a class of solutions to the Einstein equations for disks withconstant angular velocity and constant relative density which was constructedin the first part. The metric for these spacetimes is given in terms of thetafunctions on a Riemann surface of genus 2. It is parameterized by two physicalparameters, the central redshift and the relative density of the twocounter-rotating streams in the disk. We discuss the dependence of the metricon these parameters using a combination of analytical and numerical methods.Interesting limiting cases are the Maclaurin disk in the Newtonian limit, thestatic limit which gives a solution of the Morgan and Morgan class and thelimit of a disk without counter-rotation. We study the mass and the angularmomentum of the spacetime. At the disk we discuss the energy-momentum tensor,i.e. the angular velocities of the dust streams and the energy density of thedisk. The solutions have ergospheres in strongly relativistic situations. Theultrarelativistic limit of the solution in which the central redshift divergesis discussed in detail: In the case of two counter-rotating dust components inthe disk, the solutions describe a disk with diverging central density butfinite mass. In the case of a disk made up of one component, the exterior ofthe disks can be interpreted as the extreme Kerr solution.
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