The dynamics of general relativistic isotropic stellar cluster models -- Do relativistic extensions of the Plummer model exist?

We show that the general relativistic theory of the dynamics of isotropic stellar clusters can be developed essentially along the same lines as the Newtonian theory. We prove that the distribution function can be derived from any isotropic momentum moment and that every higher-order moment of the distribution can be written as an integral over a zeroth-order moment. We propose a mathematically simple expression for the distribution function of a family of isotropic general relativistic cluster models and investigate their dynamical properties. In the Newtonian limit, these models obtain a distribution function of the form F(E) ~ (E-E_0)^alpha, with E binding energy and E_0 a constant that determines the model's outer radius. The slope alpha sets the steepness of the distribution function and the corresponding radial density and pressure profiles. We show that the field equations only yield solutions with finite mass for alpha<3.5. Moreover, in the limit alpha->3.5, only Newtonian models exist. In other words: within the context of this family of models, no general relativistic version of the Plummer model exists. The most strongly bound model within the family is characterized by alpha=2.75 and a central redshift z_c~0.55.