Relativistic self-gravitating gas in dynamical equilibrium

Static spherically symmetric solution of the Einstein's equations is found representing averaged properties of an infinite self-gravitating gas in the dynamical equilibrium. It depends upon three parameters: the core radius, the relativistic factor $0<z<1$, defining the density and the mass, and one structural parameter. In the relativistic limit, $z\to 1$, an open nucleus of focused gravitational fields develops with the size being a small fraction of the core radius whereas the outskirts of the self-gravitating gas remains non-relativistic. Inside the nucleus the self-gravitating gas is described by a universal perfect fluid with the relativistic one-dimensional equation of state. At $z=1$, the space and time develops a point like singularity at the center. A characteristic property of the nucleus is incompressibility. New particles added to the self-gravitating gas bounce back into the outskirts rather than fall into the center.