Recently, there are comparable revised interests in bubble nucleation seeded
by black holes. However, it is debated in the literature that whether one shall
interpret a static bounce solution in the Euclidean Schwarzschild spacetime
(with periodic Euclidean Schwarzschild time) as describing a false vacuum decay
at zero temperature or at finite temperature. In this paper, we show a
correspondence that the static bounce solution describes either a thermal
transition of vacuum in the static region outside of a Schwarzschild black hole
or a quantum transition in a maximally extended Kruskal-Szekeres spacetime,
corresponding to the viewpoint of the external static observers or the freely
falling observers, respectively. The Matsubara modes in the thermal
interpretation can be mapped to the circular harmonic modes from an O(2)
symmetry in the tunneling interpretation. The complementary tunneling
interpretation must be given in the Kruskal-Szekeres spacetime because of the
so-called thermofield dynamics. This correspondence is general for bubble
nucleation around horizons. We propose a new paradox related to black holes as
a consequence of this correspondence.Comment: 26 pages; v2: typos corrected; v3: references added, discussion on
AdS black holes added, to match the published version; v4(v5): Ref [37]
updated, footnote [10] added v6: two typos correcte