Self-similar space-times

An exact solution of Einstein's field equations is given representing an inhomogeneous sphere of matter in static isothermal equilibrium with a density profile of the form R^–2. A perturbation analysis shows that initial small density inhomogeneities grow self-similarly in a non-extreme configuration for which general relativistic effects are important. The perturbed configuration develops asymptotically like the expanding solution to Einstein's equations considered previously by Henriksen and Wesson, and the present solution is therefore identified as the initial stage of that evolving model. As a cosmology, the solutions in tandem provide an expanding cosmology that did not begin as a Big Bang, but evolved from an inhomogeneous primeval atom, tending at late epochs to a model with homogeneous density. The new static solution has applications also in other branches of astrophysics.