# Correspondence between Thermal and Quantum Vacuum Transitions around Horizons

## TL;DR

This paper establishes a correspondence between thermal and quantum vacuum transitions around horizons, clarifying interpretations of bubble nucleation in black hole spacetimes and revealing a new paradox related to black holes.

## Contribution

It demonstrates a general correspondence between thermal and quantum interpretations of vacuum transitions around horizons, linking static bounce solutions to different observer perspectives.

## Key findings

- Static bounce solutions describe either thermal or quantum transitions depending on the observer.
- Matsubara modes map to circular harmonic modes in tunneling interpretation.
- A new paradox related to black holes arises from this correspondence.

## Abstract

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.