Solutions of Einstein’s equations relevant to the description of ‘‘bubbles’’ in the early universe

Abstract

A new class of spherically symmetric solutions of the Einstein field equations is presented. Their main features are that (i) the azimuthal metric coefficient depends on time only, and (ii) they possess similarity symmetry. The physical motivation for the study of such class of solutions is that according to recent investigations [R. N. Henriksen, A. G. Emslie, and P. S. Wesson, Phys. Rev. D 27, 1219 (1983); P. S. Wesson, Phys. Rev. D 34, 3925 (1986)], they can be relevant to ‘‘bubbles’’ of new phases, in phase transitions typical of inflationary universe models. The solutions have shear, are inhomogeneous, and may be interpreted as ‘‘mixtures’’ of perfect fluids. They have some adjustable parameters which can be used to assure the fulfillment of the energy conditions. There are two different types of solutions. One of them has similarity symmetry of the first kind and negative total pressure. These models can be used in a classical description of particle production phases in the early universe. The other type of solution has similarity of the second kind, i.e., it represents models with dimensional constraints. Explicit solutions representing mixtures of fluids with equations of state p=nρ and ρ=p of this type are given. They may be useful for cosmological models in closed universes. The dimensional constraints are found to be due to the ‘‘boundary conditions’’ in such universes. The specific characteristics of both types of solutions suggest that a transition from a particle-production phase to a radiation-dominated era can be described by means of bubbles of a ‘‘broken-symmetry’’ phase with positive pressure growing into a region of ‘‘unbroken-symmetry’’ phase with negative pressure.