Ya. B. Zel'Dovich

Models of clusters of point masses with large central redshifts

ConclusionsIn the Newtonian case we have obtained an isotropic self-consistent distribution of gravitationally interacting point masses which satisfies the transport equation without collisions, and the gravitational equation for an arbitrary powerfunction density distribution ρ=βr−s, s<3.For ρ=βr−2 the analogous self-consistent solution was obtained for the anisotropic distribution function both in Newtonian and GTR cases.The GTR solutions with ρ=βr−2 have central redshifts which increase without limit in accordance with the law 1+z∼r−1/α as we approach the center. In the isotropic case, they appear to be stable when the mean velocities are much less than the velocity of light u<0.2c, α>21.The hydrodynamic GTR solution was found for a perfect gas at constant temperature (but variable T′=T(g00)1/2) which also has z→∞ for r→0.We should like to thank K. Thorne, L. Hazin, and M. Podurets for valuable discussions. K. Thorne was particularly helpful in supplying unpublished results on circular orbits obtained by American authors.