  We consider the effect of inhomogeneities on the rate of false vacuum decay.
Modelling the inhomogeneity by a black hole, we construct explicit Euclidean
instantons which describe the nucleation of a bubble of true vacuum centred on
the inhomogeneity. We find that inhomogeneity significantly enhances the
nucleation rate over that of the Coleman-de Luccia instanton -- the black hole
acts as a nucleation site for the bubble. The effect is larger than previously
believed due to the contributions to the action from conical singularities. For
a sufficiently low initial mass, the original black hole is replaced by flat
space during this process, as viewed by a single causal patch observer.
Increasing the initial mass, we find a critical value above which a black hole
remnant survives the process. This resulting black hole can have a higher mass
than the original black hole, but always has a lower entropy. We compare the
process to bubble-to-bubble transitions, where there is a semi-classical
Lorentzian description in the WKB approximation.
