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We study the negative modes of gravitational instantons representing vacuumdecay in asymptotically flat space-time. We consider two different vacuum decayscenarios: the Coleman-de Luccia $\mathrm{O}(4)$-symmetric bubble, and$\mathrm{O}(3) \times \mathbb{R}$ instantons with a static black hole. In spiteof the similarities between the models, we find qualitatively differentbehaviours. In the $\mathrm{O}(4)$-symmetric case, the number of negative modesis known to be either one or infinite, depending on the sign of the kineticterm in the quadratic action. In contrast, solving the mode equationnumerically for the static black hole instanton, we find only one negative modewith the kinetic term always positive outside the event horizon. The absence ofadditional negative modes supports the interpretation of these solutions asgiving the tunnelling rate for false vacuum decay seeded by microscopic blackholes.

Negative modes of Coleman-de Luccia and black hole bubbles