Rotating matter in general relativity -- stationary state I

Stationary rotating matter configurations in general relativity are considered. A formalism for general stationary space times is developed. Axisymmetric systems are discussed by the use of a nonholonomic and nonrigid frame in the three-space of the time-like Killing trajectories. Two symmetric and trace-free tensors are constructed. They characterize a class of matter states in which both the interior Schwarzschild and the Kerr solution are contained. Consistency relations for this class of perfect fluids are derived. Incompressible fluids characterized by these tensors are investigated, and one differentially rotating solution is found.