Quantum Critical Collapse Abhors a Naked Singularity

TL;DR

The paper tackles whether quantum effects alter the classical naked singularities predicted at the threshold of gravitational collapse. It develops a semiclassical framework based on the two-dimensional trace anomaly for the s-wave sector of the Einstein-scalar system, with a regular Boulware-like vacuum fixed by regularity in self-similar backgrounds. An exact one-loop backreaction calculation on CSS solutions (Garfinkle in $2+1$ and Roberts in $3+1$) reveals a universal quantum growing mode that competes with the classical unstable mode, shifting the critical point to $p^*_q<p^*$ and producing a finite mass gap $M_{\text{gap}}$, thereby enforcing horizon formation. Consequently, near-threshold dynamics become Type I-like, with exponential sensitivity in primordial black hole formation fractions and a shifted mass spectrum toward $M_{\text{gap}}$, offering concrete predictions and a path forward for semiclassical studies of gravitational collapse.

Abstract

Classical critical collapse yields naked singularities from smooth initial data, challenging cosmic censorship and shaping the spectrum of primordial black holes. We show that one-loop vacuum polarization near the threshold alters this outcome. In analytically tractable Einstein-scalar critical spacetimes, regularity uniquely selects a Boulware-like state whose stress tensor supplies a universal quantum growing mode. Its backreaction competes with the classical unstable mode, producing a shift of the critical point and a finite mass gap at the new threshold, thereby enforcing horizon formation even under arbitrary fine-tuning. In primordial collapse, the threshold shift enters exponentially into the formation fraction, while the gap truncates the low-mass tail -- effects that may reshape the predicted mass spectrum. These results provide the first consistent quantum treatment of critical collapse, offering definitive predictions for several long-standing problems.

Figures (2)

• Figure 1: Global structure of a CSS critical spacetime, in which the past light cone of the naked singularity naturally separates the spacetime into interior and exterior regions Gundlach_2003Martin-Garcia:2003xgmCicortas:2024hpkk. Our focus is on the interior region, where the singularity dynamically emerges from smooth initial data.
• Figure 2: (a) For the most physically relevant $n=4$ Garfinkle solution, we plot the logarithm of $M_{\text{EMOTS}}$ against the ratio defined in Eq. \ref{['eq:ratio']}. As $\mathcal{R}$ decreases, effectively moving toward $p \to p^\ast_q$, the mass function monotonically approaches a constant value, indicating a Type I behavior. (b) For the Roberts spacetime, note the Hawking mass in four dimensions is defined as $M \equiv \frac{\bar{r}}{2}[1-(\nabla \bar{r})^2]$. As $\mathcal{R}$ decreases, quantum corrections cause an abrupt, nonmonotonic change in the behavior of $M_{\text{EMOTS}}$.