  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.
