Vacuum Decay around Black Holes formed from Collapse

TL;DR

The paper investigates vacuum decay in the presence of black holes formed by gravitational collapse, asking whether the process is quantum tunneling or thermally assisted. It develops a semiclassical thin-wall framework in a two-region geometry (interior Minkowski and exterior Schwarzschild) and shows that a single Schwarzschild-energy saddle dominates the decay, with interior collapse details washed out by horizon redshift. The resulting exponential suppression, $B_S$, depends nonmonotonically on the black-hole mass, approaching the flat-space value as $r_s\to0$ and attaining a finite minimum at finite mass, implying only weak catalysis by black holes. These results challenge the expectation of strong small-black-hole enhancement and have implications for primordial black-hole bounds on metastable vacua, including the Higgs potential, highlighting the need to reevaluate scenarios invoking thermal catalysis around collapsing black holes.

Abstract

We re-examine the problem of vacuum decay in the presence of spherically symmetric black holes. Within the semiclassical approximation, we study configurations describing a bubble of true vacuum propagating outside a black hole formed from gravitational collapse. We find that the saddle point is dominated by a single energy state and that the dependence on the initial conditions in the stellar interior vanishes exponentially fast at late stages of the collapse. Prescriptions are given for implementing the corresponding boundary conditions in the eternal black-hole geometry. We find that vacuum decay can only be weakly catalyzed by the black hole, as the suppression exponent attains a minimum at a finite black-hole mass. In the limit of vanishing black-hole mass, the suppression smoothly approaches the flat-space result.

Figures (4)

• Figure 1: Penrose diagram of a spherically symmetric spacetime formed from gravitational collapse. The interior region $I$ is matched to an exterior Schwarzschild geometry (S) across the stellar surface (solid line), which follows a timelike trajectory. The event horizon (EH) forms at late times.
• Figure 2: The quantity $\dot{r}^2$ as a function of $r/r_c$ for three different values of the energy $E$. For the critical energy $E = E_c$ the horizon is a turning point, for $E < E_{max}$ there are two turning points, and for $E = E_{max}$ the turning points degenerate into one.
• Figure 3: Exponential suppression around a Schwarzschild black hole compared to the one in flat space as a function of the black hole gravitational radius.
• Figure 4: Euclidean Schwarzschild geometry represented as a cigar. The radial direction runs along the axis, while the Euclidean time circle shrinks smoothly to zero size at the horizon. The black curve represents the Euclidean trajectory of the bubble, starting at the horizon (cross) and terminating at the outer turning point (red dot).