Type II Critical Collapse

Type II critical collapse is the scale-generating threshold regime: increasingly fine tuning makes the evolution reproduce a universal self-similar solution on ever smaller scales, and supercritical black holes have masses that approach zero in the classical model. This behavior requires a scale-free or asymptotically scale-free regime and a critical solution with exactly one growing physical perturbation.

Required background. The Einstein equation defines the constrained nonlinear flow, while differential equations supplies eigenmodes and asymptotic scaling.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon mass diagnostics; Black-Hole Perturbation Theory develops the unstable-mode spectrum; Formulations of General Relativity explains scale-adapted gauges; and Computational Relativity develops fine tuning and convergence.

Scale-free approach to the threshold

Let a regular one-parameter family $Z(p)$ cross the boundary between dispersion and black-hole formation at $p=p_*$. Take $p$ dimensionless, or normalize it by a fixed parameter scale. Assume a fixed matter model and symmetry class, transverse crossing, and a continuously or discretely self-similar critical solution $Z_*$. In logarithmic scale time $\tau=-\ln(L/L_0)$, smaller physical scales correspond to larger $\tau$.

Scale-invariant variables near the critical solution have the expansion

$$Z(\tau,x)=Z_*(\tau,x) +\sum_i C_i(p)e^{\lambda_i\tau}Z_i(\tau,x).$$

For continuous self-similarity, $Z_*$ and $Z_i$ can be independent of $\tau$ after gauge fixing. For discrete self-similarity they are periodic in $\tau$. Remove gauge directions and the neutral shift of phase along the DSS cycle. Exactly one independent real growing amplitude with $\lambda_0\gt0$ is assumed; all remaining physical modes decay. Transversality gives $C_0(p)\simeq\kappa(p-p_*)$.

The evolution follows $Z_*$ until the unstable term becomes order unity. Defining $\tau_*$ by $\lvert C_0\rvert e^{\lambda_0\tau_*}=\varepsilon$ yields the only dynamically generated length scale

$$L_*\propto e^{-\tau_*} \propto\lvert p-p_*\rvert^{1/\lambda_0}.$$

An observable produced by this core scales with the corresponding power of $L_*$ if no other scale re-enters. This proviso matters: the exterior initial data still have their original size and can retain mass that later accretes. The mass at formation and a much later accreted mass must not be fitted indiscriminately to the same one-scale law.

Black-hole mass and subcritical curvature

On the supercritical side in four dimensions, mass has dimensions of length divided by $G$, so

$$GM_{\rm BH}=C_{\mathcal F} (p-p_*)^\gamma, \qquad \gamma=\frac{1}{\lambda_0}.$$

Here $M_{\rm BH}$ is physical mass, so $GM_{\rm BH}$ is the generated length; equivalently one may absorb $G^{-1}$ into a dimensionful coefficient in a formula for $M_{\rm BH}$. The exponent belongs to the critical solution and matter model; the coefficient $C_{\mathcal F}$ depends on the initial-data family and parameterization. On the subcritical side, a curvature scalar of dimension $L^{-2}$ obeys a maximum-curvature scaling such as

$$\max\sqrt{\left|R_{abcd}R^{abcd}\right|} \propto\lvert p-p_*\rvert^{-2\gamma}$$

within the classical scaling regime, provided the chosen invariant is nonzero in the critical region. The absolute value avoids assuming that a Lorentzian curvature contraction is positive in every spacetime. This offers a horizon-independent check using both sides of the threshold; a Ricci scalar that vanishes identically in the chosen matter model would not work.

Choptuik discovered universality, echoing, and mass scaling for the spherically symmetric massless scalar field in Choptuik 1993, pp. 9–12. The dynamical-systems derivation and its assumptions are given in Gundlach and Martín-García 2007, §§2.1–2.3, pp. 4–8.

Universality and its boundary

Universality means that different regular one-parameter families crossing the same critical surface approach the same $Z_*$ and share $\gamma$ and, for discrete self-similarity, the echoing period. It does not mean that all matter models share one exponent. A perfect fluid’s equation-of-state parameter can change the critical solution and exponent, while dimensionful masses or potentials may become irrelevant only at sufficiently small scales.

Three checks are decisive. First, the fitted $p_*$ must stabilize with numerical resolution, because an error in threshold location bends a log–log scaling plot. Second, several initial-data families must share a slope while retaining different intercepts. Third, the same unstable eigenvalue must explain the departure time, mass law, and subcritical curvature law. Discrete echoing should appear in dimensionless local fields after scale and phase alignment.

The classical $M_{\rm BH}\to0$ limit is not a prediction about nature at arbitrarily short distances. Quantum gravity, a matter length scale, loss of continuum resolution, or another unstable mode can terminate the scaling window. A finite numerical dynamic range likewise cannot distinguish exact self-similarity from a long approximate regime.

Check your understanding

A type II solution has one unstable eigenvalue $\lambda_0=2.7$. Predict the mass exponent and the exponent for the maximum Kretschmann scalar $K=R_{abcd}R^{abcd}$ on the subcritical side.

Solution

The mass exponent is $\gamma=1/2.7\simeq0.3704$. Since $K$ has dimension $L^{-4}$, the maximum absolute value satisfies

$$|K|_{\max}\propto\lvert p-p_*\rvert^{-4\gamma} \simeq\lvert p-p_*\rvert^{-1.4815}.$$

Using $\sqrt{|K|}$ instead would give exponent $-2\gamma$.

References

• Choptuik, Matthew W. “Universality and Scaling in Gravitational Collapse of a Massless Scalar Field.” Physical Review Letters 70 (1993): 9–12. doi:10.1103/PhysRevLett.70.9.
• Gundlach, Carsten, and José M. Martín-García. “Critical Phenomena in Gravitational Collapse.” Living Reviews in Relativity 10 (2007): 5. doi:10.12942/lrr-2007-5.