Critical phenomena in gravitational collapse

TL;DR

The review synthesizes how the threshold of black hole formation in general relativity exhibits universal, scale-invariant phenomena across a wide range of matter models and dimensions. By framing GR as a dynamical system with a codimension-one attractor, it derives mass-scaling laws $M \propto (p-p_*)^{\gamma}$ and distinguishes CSS and DSS regimes, including the associated periodic wiggles and phase-space structure. The canonical 3+1-dimensional scalar-field model (Choptuik) serves as a core example, with detailed analyses of the critical solution's global structure, naked-singularity implications, and extensions to charge, self-interactions, and nonspherical perturbations, as well as a broad survey of other spherical models (fluids, YM, sigma models, higher dimensions). Beyond spherical symmetry, the review discusses axisymmetric collapse, angular-momentum effects, and black-hole collisions, highlighting both supporting evidence for universality and important open questions, such as the robustness of critical behavior under non-spherical dynamics and the role of quantum effects. Together, these results illuminate the mechanism by which smooth initial data treads the brink to arbitrarily strong curvature, with implications for cosmic censorship, quantum gravity, and the dynamics of GR.

Abstract

As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term "critical phenomena". They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. Critical phenomena give a natural route from smooth initial data to arbitrarily large curvatures visible from infinity, and are therefore likely to be relevant for cosmic censorship, quantum gravity, astrophysics, and our general understanding of the dynamics of general relativity.

Figures (5)

• Figure 1: The phase space picture for the black hole threshold in the presence of a critical point. Every point correspond to an initial data set, that is, a 3-metric, extrinsic curvature, and matter fields. (In type II critical collapse these are only up to scale). The arrow lines are solution curves, corresponding to spacetimes, but the critical solution, which is stationary (type I) or self-similar (type II) is represented by a point. 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 $p=p_*$. The 2-dimensional plane represents an $(\infty-1)$-dimensional hypersurface, but the third dimension represents really only one dimension.
• Figure 2: A different phase space picture, specifically for type II critical collapse, and two 2-dimensional projections of the same picture. In contrast with Fig. \ref{['fig:dynsim']}, one dimension of the two representing the (infinitely many) decaying modes has been suppressed. The additional axis now represents a global scale which was suppressed in Fig. \ref{['fig:dynsim']}, so that the scale-invariant critical solution CS is now represented as a straight line (in red). Several members of a family of initial conditions (in blue) are attracted by the critical solution and then depart from it towards black hole formation (A or B) or dispersion (D). Perfectly fine-tuned initial data asymptote to the critical solution with decreasing scale. Initial conditions starting closer to perfect fine tuning produce smaller black holes, such that the parameter along the line of black hole endstates is $-\ln M_{BH}$. Two 2-dimensional projections of the same picture are also given. The horizontal projection of this figure is the same as the vertical projection of Fig. \ref{['fig:dynsim']}.
• Figure 3: The spacetime diagram of all generic DSS continuations of the scalar field critical solution, from critcont. The naked singularity is timelike, central, strong, and has negative mass. There is also a unique continuation where the singularity is replaced by a regular centre except at the spacetime point at the base of the CH, which is still a strong curvature singularity. No other spacetime diagram is possible if the continuation is DSS. The lines with arrows are lines of constant adapted coordinate $x$, with the arrow indicating the direction of $\partial/\partial\tau$ towards larger curvature.
• Figure 4: Conformal diagram of the critical solution matched to an asymptotically flat one. RC=regular centre, S=Singularity, SSH=self-similarity horizon. Curved lines are lines of constant coordinate $\tau$, while converging straight lines are lines of constant coordinate $x$. Let the initial data on the Cauchy surface CS be those for the exact critical solution out to the 2-sphere R, and let these data be smoothly extended to some data that are asymptotically flat, so that the future null infinity $\mathscr{I}^+$ exists. To the past of the matching surface MS the solution coincides with the critical solution. The spacetime cannot be uniquely continued beyond the Cauchy horizon CH. The redshift from point A to point B is finite by self-similarity, and the redshift from B to C is finite by asymptotic flatness.
• Figure 5: The final event horizon of a black hole is only known when the infall of matter has stopped. Radiation at 1 collapses to form a small black hole which settles down, but later more radiation at 2 falls in to give rise to a larger final mass. Fine-tuning of a parameter may result in $m_1\sim (p-p_*)^\gamma$, but the final mass $m_2$ would be approximately independent of $p$.