Critical phenomena in gravitational collapse (Physics Reports)

TL;DR

This work surveys critical phenomena in gravitational collapse, showing that black-hole formation exhibits a threshold behavior marked by universal critical solutions (static or self-similar) and robust scaling laws. Through a dynamical-systems lens, it explains how near-threshold evolutions forget initial data details and approach a codimension-one attractor with a single unstable mode, yielding either a finite-mass type I transition or a mass-scaling type II transition $M\simeq C(p-p_*)^{\gamma}$, with DSS cases producing a wiggle in $\ln M$. The review also connects these GR phenomena to statistical mechanics via renormalization-group ideas, explores the GR-specific geometric structure of self-similarity, and discusses the genericity, higher-dimensional extensions, and astrophysical and semiclassical implications, including naked singularities and potential observational signatures. Overall, critical collapse provides deep insights into universality, scale-invariance, and the interplay between gravity and matter in extreme regimes. The results motivate ongoing work toward non-spherical, rotating, and charged collapses, as well as rigorous mathematical formulations of the underlying dynamical-system structure.

Abstract

In general relativity black holes can be formed from regular initial data that do not contain a black hole already. The space of regular initial data for general relativity therefore splits naturally into two halves: data that form a black hole in the evolution and data that do not. The spacetimes that are evolved from initial data near the black hole threshold have many properties that are mathematically analogous to a critical phase transition in statistical mechanics. Solutions near the black hole threshold go through an intermediate attractor, called the critical solution. The critical solution is either time-independent (static) or scale-independent (self-similar). In the latter case, the final black hole mass scales as $(p-p_*)^γ$ along any one-parameter family of data with a regular parameter $p$ such that $p=p_*$ is the black hole threshold in that family. The critical solution and the critical exponent $γ$ are universal near the black hole threshold for a given type of matter. We show how the essence of these phenomena can be understood using dynamical systems theory and dimensional analysis. We then review separately the analogy with critical phase transitions in statistical mechanics, and aspects specific to general relativity, such as spacetime singularities. We examine the evidence that critical phenomena in gravitational collapse are generic, and give an overview of their rich phenomenology.

Figures (6)

• Figure 1: The phase space picture for the black hole threshold in the presence of a critical point. The arrow lines are time evolutions, corresponding to spacetimes. The line without an arrow is not a time evolution, but a 1-parameter family of initial data that crosses the black hole threshold at $p=p_*$.
• Figure 2: The phase space picture in the presence of a limit cycle. The plane represents the critical surface. The circle (fat unbroken line) is the limit cycle representing the critical solution. Shown also are two trajectories in the critical surfaces and therefore attracted to the limit cycle, and two trajectories out of the critical surface and repelled by it.
• Figure 3: Choptuik's critical solution in coordinates adapted to DSS. The first-order matter variables $U$ and $V$ are defined in (\ref{['UVdef']}). $\mu=2m/r$ where $m$ is the Hawking mass defined in (\ref{['mdef']}) and $r$ the area radius defined in (\ref{['tr_metric']}). $\mu=1$ therefore signals an apparent horizon. $f=\alpha/a$ where $\alpha$ and $a$ are defined in (\ref{['tr_metric']}). The two axes are $0\le x\le 1$ and $0\le \tau\le \Delta\simeq 3.44$, where $x$ and $\tau$ are defined in (\ref{['x_tau']}). In particular $x=0$ is the center of spherical symmetry and $x=1$ is the past lightcone of the singularity. Note that the period of the matter variables is $\Delta$ with $U(x,\tau+\Delta/2)=-U(x,\tau)$, while that of the metric variables is $\Delta/2$. This figure is taken from critcont.
• Figure 4: Spacetime diagram of spherically symmetric critical solution (for example Choptuik's solution for the scalar field) with schematic indication of coordinate patches. Here the spacetime from the regular center $x=x_c$ to the past lightcone $x=x_p$ of the singularity has been covered with a coordinate patch where the surfaces of constant $\tau$ are spacelike (eg $\tau=-\ln(-t)$). The region between the past lightcone and the future lightcone (Cauchy horizon) $x=x_f$ has been covered with a patch where the $\tau$-surfaces are timelike (eg $\tau=-\ln r$). A part of the region to the future lightcone has been covered with a patch where the $\tau$-lines are ingoing null lines (eg $\tau=-\ln v$). The spoke-like lines are $x$-lines. This figure is taken from critcont.
• Figure 5: The global structure of spherically symmetric critical spacetimes up to the Cauchy horizon. Infinity has been conformally compactified. The curved lines are typical trajectories of the homothetic vector field (in a CSS critical solution) or lines that are invariant under the discrete isometry (in a DSS solution).