Developing the thin-wall method of Israel, we present a formalism to investigate bubble dynamics in generalized Einstein theories. We derive the equations of motion for a bubble, finding that the space-time inside a bubble is always inhomogeneous. Applying this formalism to extended inflation, we find the following two results: (1) Any true vacuum bubble expands, contrary to the results of Goldwirth-Zaglauer, who claim that bubbles created initially later collapse. We show that their initial conditions for collapsing bubbles are physically inconsistent. (2) Concerning the global space-time structure of the Universe in extended inflation, we show that worm-holes are produced as in old inflation, resulting in the multi-production of universes.
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