Unveiling horizons in quantum critical collapse

TL;DR

This work provides a first-principles semiclassical analysis of scalar-field collapse in Einstein gravity by leveraging a robust anomaly-based one-loop framework and s-wave dimensional reduction to a 2D dilaton gravity. It identifies a unique, Boulware-like quantum state arising from regularity, and discovers a universal quantum growing mode that backreacts on the geometry, inducing horizon formation and a finite mass gap $M_{\text{gap}}$ that signals a quantum-modified Type I behavior near the classical Type II threshold. The results, demonstrated explicitly in solvable 2+1 Garfinkle and 3+1 Roberts spacetimes, show how quantum vacuum polarization can enforce cosmic censorship by shifting the critical threshold $p^*$ to $p^*_q$ and generating a mass gap independent of the fiducial length $\ell$, with the gap scaling as $M_{\text{gap}}\sim \hbar^{1/\omega_q}$ and the quantum Lyapunov exponent $\omega_q$ set by dimensionality. The study provides analytic control over quantum backreaction in strongly curved, time-dependent backgrounds and establishes a framework to extend semiclassical insights to more general critical-collapse scenarios, including potential connections to cosmology and holography. Its findings offer a concrete mechanism by which quantum effects shield naked singularities and contribute to the broader understanding of cosmic censorship in quantum gravity regimes.

Abstract

Critical gravitational collapse offers a unique window into regimes of arbitrarily high curvature, culminating in a naked singularity arising from smooth initial data -- thus providing a dynamical counterexample to weak cosmic censorship. Near the critical regime, quantum effects from the collapsing matter are expected to intervene before full quantum gravity resolves the singularity. Despite its fundamental significance, a self-consistent treatment has so far remained elusive. In this work, we perform a one-loop semiclassical analysis using the robust anomaly-based method in the canonical setup of Einstein gravity minimally coupled to a free, massless scalar field. Focusing on explicitly solvable critical solutions in both 2+1 and 3+1 dimensions, we analytically solve the semiclassical Einstein equations and provide definitive answers to several long-standing questions. We find that regularity uniquely selects a Boulware-like quantum state, encoding genuine vacuum polarization effects from the collapsing matter. Remarkably, the resulting quantum corrections manifest as a growing mode. Horizon-tracing analyses, incorporating both classical and quantum modes, reveal the emergence of a finite mass gap, signaling a phase transition from classical Type II to quantum-modified Type I behavior, thereby providing a quantum enforcement of the weak cosmic censorship. The most nontrivial aspect of our analysis involves dealing with non-conformal matter fields in explicitly time-dependent critical spacetimes. Along the way, we uncover intriguing and previously underexplored features of quantum field theory in curved spacetime.

Figures (39)

• Figure 1: In the left panel, we depict the generic phase space structure near the threshold of black hole formation. Each point represents an initial data set, with arrows indicating the corresponding solution curves. A one-parameter family of initial data, labeled by $p$, intersects the threshold and evolves toward the critical spacetime at $p=p^\ast$. The right panel offers a different perspective, where the direction perpendicular to the threshold represents the global scale. In this view, the scale-invariant Type II critical spacetime appears as a straight solid line. Only precisely fine-tuned initial data will asymptotically approach the critical solution with decreasing scale. Nearby data points, although initially drawn toward the critical solution, eventually deviate, leading either to black hole formation or to dispersion.
• Figure 2: Penrose diagrams of the exact critical spacetime and the supercritical regime featuring black hole formation are shown, with $\Sigma_0$ denoting the initial data surface. In the left panel, the critical solution exhibits a naked singularity, with a Cauchy horizon (CH) emanating along its future light cone. Beyond this CH, the spacetime admits no unique continuation. The geometry is self-similar in the interior region, obeying $g_{\mu \nu}(T,x^i) =e^{-2T}\tilde{g}_{\mu \nu}(x^i)$, with the self-similarity horizon (SSH) lying along the past light cone of the singularity. This region is matched to an asymptotically flat exterior across the junction surface. In the right panel, the supercritical regime depicts matter collapsing to form a black hole. The event horizon (EH) is a null surface, determined only when the infalling matter has stopped, with the final mass $m_f$ independent of $p$. In contrast, the apparent horizon (AH), upon its initial formation, exhibits the characteristic Choptuik scaling relation $m_i \propto (p - p^\ast)^\gamma$.
• Figure 3: Penrose diagrams of continuously self-similar (CSS, left panel) and discretely self-similar (DSS, right panel) critical spacetimes, each featuring a naked singularity as $T \to \infty$. In a self-similar spacetime, the geometry repeats itself under rescalings of spacetime coordinates. A CSS spacetime possesses a homothetic Killing vector, with the metric $g_{\mu \nu}(T,x^i)$ varying continuously in $T$. In contrast, a DSS spacetime exhibits periodicity in $T$, with a fixed period $\Delta$ that depends on the specific matter model and is typically determined numerically.
• Figure 4: A typical naked singularity problem consists of two regions separated by the self-similarity horizon: the interior fill-in region and the exterior-naked singularity region, as illustrated in the left panel. The right panel shows the global structure of the interior region, modeled by a CSS critical spacetime, which is the focus of the present work.
• Figure 5: The causal structure of Garfinkle spacetime for odd (left panel) and even $n$ (right panel). The thick black dot represents the kinematical point singularity at $T = \infty$. The primary distinction lies in the analytic continuation beyond the past light cone of the singularity, covering the region $x \in [-1,0)$. As can be seen from the curvature invariants \ref{['eq:GarfinkleRicci']}, for odd $n$ there is a spacelike curvature singularity located at $x = -1$. Throughout this paper, we focus exclusively on the interior region, which corresponds to $x \in [0,1]$.