Inflation and bubbles in general relativity

Following Israel’s study of singular hypersurfaces and thin shells in general relativity, the complete set of Einstein’s field equations in the presence of a bubble boundary SIGMA is reviewed for all spherically symmetric embedding four-geometries $M^{\pm }$. The mapping that identifies points between the boundaries ${\Sigma }^{+}$ and ${\Sigma }^{\mathrm{-}}$ is obtained explicitly when the regions $M^{+}$ and $M^{\mathrm{-}}$ are described by a de Sitter and a Minkowski metric, respectively. In addition, the evolution of a bubble with vanishing surface energy density is studied in a spatially flat Robertson-Walker space-time, for region $M^{\mathrm{-}}$ radiation dominated with a vanishing cosmological constant, and an energy equation in $M^{+}$ determined by the matching. It is found that this type of bubble leads to a ‘‘worm-hole’’ matching; that is, an infinite extent exterior of a sphere is joined across the wall to another infinite extent exterior of a sphere. Interior-interior matches are also possible. Under this model, solutions for a bubble following a Hubble law are analyzed. Numerical solutions for bubbles with constant tension are also obtained.