Spherically Symmetric Equilibria for Self-Gravitating Kinetic or Fluid Models in the Nonrelativistic and Relativistic Case - A Simple Proof for Finite Extension

We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E0, and in the Euler case by the corresponding growth condition on the equation of state p = P (ρ) at ρ = 0. These purely local conditions are slight generalizations to known such conditions.