Type I Critical Collapse

Type I critical collapse occurs when near-threshold data approach a finite-scale metastable solution before departing toward black-hole formation or a non-black-hole endpoint. Its characteristic scaling concerns the lifetime of that intermediate state, not a black-hole mass tending to zero. A finite mass gap, stationary or periodic critical solution, and one unstable mode are the linked signatures.

Required background. The Einstein equation defines the collapse flow on constrained data, while differential equations supplies linear modes and exponential growth.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon outcome tests; Black-Hole Perturbation Theory develops unstable spectra; Formulations of General Relativity explains gauge-fixed evolution on phase space; and Computational Relativity develops threshold searches.

A finite-scale threshold attractor

Consider a smooth one-parameter family of regular initial data $Z(p)$ that crosses the black-hole threshold transversely at $p=p_*$. Take $p$ dimensionless; for a dimensionful parameter use $(p-p_*)/p_0$ with a fixed parameter scale $p_0$ inside logarithms. Assume the matter model and symmetry class are fixed, and the threshold contains a stationary or time-periodic solution $Z_*$ with exactly one physical growing mode. After removing gauge directions and the neutral phase of a periodic solution, the remaining perturbations relevant to this attractor decay.

Near $Z_*$, write the evolution as

$$Z(t)=Z_*(t)+C_0(p)e^{\lambda_0t}Z_0(t) +\sum_{i\geq1}C_i(p)e^{\lambda_it}Z_i(t), \qquad \lambda_0>0,$$

where $t$ is a fixed physical clock, such as central proper time, and $\lambda_i$ has units of inverse time. For a stationary solution the profiles are time independent; for a periodic solution the $Z_i(t)$ are periodic Floquet profiles. Transverse crossing gives $C_0(p)=\kappa(p-p_*)+O((p-p_*)^2)$ with $\kappa\ne0$. Decaying modes erase most family-specific information while the solution approaches $Z_*$. The sign of the growing amplitude then selects the departure side.

Unlike a type II self-similar solution, $Z_*$ has a finite length and mass scale. A black hole formed after departure therefore need not become arbitrarily small as $p\to p_*^+$. The limiting mass can be a finite fraction of the critical configuration’s mass, and that fraction may depend on which member of a family of metastable solutions is selected.

Logarithmic lifetime law

Choose a small fixed departure amplitude $\varepsilon$. The near-critical phase ends at time $t_p$ when

$$\left\lvert C_0(p)\right\rvert e^{\lambda_0t_p} =\varepsilon.$$

Solving gives

$$t_p=-\frac{1}{\lambda_0}\ln\lvert p-p_*\rvert +\text{constant}+o(1).$$

For a stationary attractor this gives the leading lifetime law directly. For a periodic attractor it describes the growing-mode envelope; a pointwise departure criterion can add a bounded phase-dependent modulation rather than a correction that vanishes as $p\to p_*$. The slope is the inverse Lyapunov exponent of the single unstable mode. The additive constant depends on the initial-data family, parameter normalization, chosen departure criterion, and time coordinate. The slope is universal only after fixing the critical solution and time normalization.

Gundlach and Martín-García derive this law and distinguish the mass gap from type II scaling in Gundlach and Martín-García 2007, §2.4, p. 8. Einstein–Yang–Mills collapse provides a canonical example with an unstable static solution; the numerical evidence and coexistence with type II behavior are described in Choptuik, Chmaj, and Bizoń 1996, pp. 424–427.

Assumptions, checks, and crossover

The one-mode assumption is testable. Fit the lifetime slope for several initial-data families crossing the same threshold, and independently compute or perturbatively estimate $\lambda_0$. The profiles should approach the same $Z_*$ after aligning the time origin and any phase of a periodic solution. Constraint-violating and gauge modes must be excluded from the spectrum.

A dimensionful scale in the equations can support type I behavior, but does not force it everywhere in phase space. A massive scalar field, for example, can show type I behavior when a long-lived finite-scale configuration is selected and type II behavior when the evolution reaches scales at which the mass term is dynamically irrelevant. The boundary between those basins is an additional problem, not captured by the lifetime law.

The mass gap is also conditional. It means that along the specified near-threshold branch the black-hole mass approaches a nonzero scale. It does not provide a lower bound on black-hole masses obtainable from all initial data in the theory. Nor does a long-lived oscillating object establish criticality unless one unstable mode, transverse threshold crossing, and the logarithmic lifetime law are demonstrated.

Check your understanding

Two near-critical runs approach the same stationary critical solution and have $\lvert p_1-p_*\rvert=10^{-6}$ and $\lvert p_2-p_*\rvert=10^{-9}$. If $\lambda_0=0.5/T$, what lifetime difference is predicted?

Solution

The family-dependent constant cancels:

$$t_2-t_1 =-\frac{1}{\lambda_0}\ln\left(\frac{10^{-9}}{10^{-6}}\right) =\frac{3\ln10}{0.5/T} =6T\ln10\approx13.82T.$$

This comparison assumes both runs approach the same critical solution and use the same time normalization and departure criterion.

References

• Choptuik, Matthew W., Tadeusz Chmaj, and Piotr Bizoń. “Critical Behavior in Gravitational Collapse of a Yang–Mills Field.” Physical Review Letters 77 (1996): 424–427. doi:10.1103/PhysRevLett.77.424.
• 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.