Timelike geodesic motions within the general relativistic gravitational field of the rigidly rotating disk of dust

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. O ne finds that there is always a (stable or unstable) circular orbit for pos itive angular momentum and a given radius. However, for sufficiently relati vistic disks there are regions within the plane of the disk in which a part icle with negative angular momentum cannot follow a circular path. If the d isk is still more strongly relativistic, then one finds circular orbits wit h negative energies of arbitrary magnitude. Within the theoretical construc tion of the Penrose effect, this property can be used to produce arbitraril y high amounts of energy. The study of Hamiltonian mechanics forms another topic of this article. It turns out that the stochastic behavior of the geo desics 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 conta ins points lying inside the disk as well as points outside, the geodesic mo tion shows highly stochastic behavior. However, if the crossing region is c ompletely inside or outside the disk, the motion proves to be nearly integr able. In these cases the corresponding Hamiltonian system is close to an in tegrable system of the so-called Liouville class. (C) 1998 American Institu te of Physics. [S0022-2488(98)03711- 6].