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Abstract:
The general relativistic motion of a test particle near a rigidly rotating disk of dust is investigated. Circular orbits within the plane of the disk (centered on the rotation axis) are special cases of the geodesic motion. One finds that there is always a (stable or unstable) circular orbit for positive angular momentum and a given radius. However, for sufficiently relativistic disks there are regions within the plane of the disk in which a particle with negative angular momentum cannot follow a circular path. If the disk is still more strongly relativistic, then one finds circular orbits with negative energies of arbitrary magnitude. Within the theoretical construction of the Penrose effect, this property can be used to produce arbitrarily high amounts of energy. The study of Hamiltonian mechanics forms another topic of this article. It turns out that the stochastic behavior of the geodesics is related to the position of the region containing all the crossing points of the particle through the plane of the disk. If this region contains points lying inside the disk as well as points outside, the geodesic motion shows highly stochastic behavior. However, if the crossing region is completely inside or outside the disk, the motion proves to be nearly integrable. In these cases the corresponding Hamiltonian system is close to an integrable system of the so-called Liouville class.
