Critical Collapse in Anti-de Sitter

Critical collapse in global AdS combines a local type-II threshold with a global sequence of boundary returns. In resolved numerical branches of four-dimensional spherical Einstein–massless-scalar evolution, a sufficiently small forming horizon approaches the asymptotically flat scalar exponent. This is numerical evidence within that model, not a theorem for arbitrary matter, dimension, or threshold type. Across thresholds, the number of reflections changes discontinuously, so a reliable computation must identify the branch before fitting a scaling law.

Helpful background. Black-Hole Geometry and Mechanics supplies marginal-surface diagnostics; Black-Hole Perturbation Theory supplies mode tests; Formulations of General Relativity supplies the constrained equations; Exact Solutions of Einstein’s Equations supplies Schwarzschild–AdS checks; Causality and Global Structure supplies the reflecting boundary; and Computational Relativity supplies convergence and benchmark design.

Prompt and delayed critical branches

Take four-dimensional Einstein gravity with $\Lambda=-3/L^2$, a real massless scalar, spherical symmetry, a regular center, and reflecting AdS boundary conditions. Let $\varepsilon$ be a dimensionless amplitude multiplying a fixed, normalized smooth profile. Large amplitudes may form an apparent horizon before the pulse reaches conformal infinity. Below the prompt threshold, the pulse reflects and refocuses; it can then collapse after one or more returns.

Write $\varepsilon_n$ for the threshold separating survival or local re-expansion past the $n$th focusing event from collapse at that event. On the supercritical side of one branch,

$$R_H=C_n(\varepsilon-\varepsilon_n)^\gamma F_n\!\left(\ln\frac{\varepsilon-\varepsilon_n}{\varepsilon_{\rm ref}}\right),$$

Here $\varepsilon_{\rm ref}\gt 0$ is an arbitrary dimensionless reference amplitude, $C_n$ has dimensions of length, and $F_n$ is dimensionless. Discrete self-similarity makes $F_n$ periodic in its logarithmic argument; its period depends on the echoing convention and the observable. In the resolved scaling window, spherical massless-scalar calculations find $\gamma\simeq0.37$. Bizoń and Rostworowski display the sequence of reflection branches and the local fits in Bizoń and Rostworowski 2011, Eq. (9), Fig. 1, p. 2.

The local reason for approximate universality is scale separation. For $R_H\ll L$, the dimensionless cosmological correction is $R_H^2/L^2$. The Schwarzschild–AdS horizon relation,

$$2GM_H=R_H\left(1+\frac{R_H^2}{L^2}\right),$$

then gives $M_H\simeq R_H/(2G)$, so radius and mass have the same leading critical exponent. Husain and collaborators varied negative $\Lambda$, pulse family, width, and location and found compatible exponent ranges; their equations, grid tests, and fit uncertainty are reported in Husain et al. 2003, Eqs. (19)–(24), Tables I–II, pp. 2–3, PDF.

This does not make the entire phase diagram independent of $\Lambda$. The AdS return time, branch locations, finite gaps between neighboring reflection branches, and stable islands are global. Martinon reviews this multibranch structure and its relation to type-II criticality in Martinon 2017, §3.1, pp. 13–16.

A constrained evolution to reproduce

Use dimensionless $t,x$ and rescale the scalar as $\varphi=\sqrt{4\pi G}\,\phi$. With

$$ds^2=\frac{L^2}{\cos^2x} \left(-Ae^{-2\delta}dt^2+A^{-1}dx^2+\sin^2x\,d\Omega^2\right),$$

define $\Phi=\partial_x\varphi$ and $\Pi=A^{-1}e^\delta\partial_t\varphi$. The continuum system is

$$\partial_t\Phi=\partial_x\!\left(Ae^{-\delta}\Pi\right), \qquad \partial_t\Pi=\frac{1}{\tan^2x} \partial_x\!\left(\tan^2x\,Ae^{-\delta}\Phi\right),$$

with radial equations

$$\partial_x A =\frac{1+2\sin^2x}{\sin x\cos x}(1-A) -\sin x\cos x\,A(\Phi^2+\Pi^2),$$ $$\partial_x\delta=-\sin x\cos x(\Phi^2+\Pi^2).$$

Regularity gives $A(t,0)=1$ and finite even $\varphi$ at the center. Fix the remaining time gauge by $\delta(t,0)=0$, so central proper time is $Lt$. Boundary proper time in the standard conformal representative instead satisfies $dT_\partial=L e^{-\delta(t,\pi/2)}dt$; collapse times from different gauges must be converted before comparison. The standard reflecting falloff has $\varphi=O((\pi/2-x)^3)$. One reproducible family is

$$\Phi(0,x)=0, \qquad \Pi(0,x)=\frac{2\varepsilon}{\pi} \exp\!\left[-\frac{4\tan^2x}{\pi^2\sigma^2}\right],$$

with $\sigma=1/16$ for the Bizoń–Rostworowski reference family and $\varepsilon$ varied. Record the first time $A$ falls below a declared horizon threshold, the corresponding areal radius $R_H=L\tan x_H$, the number of completed reflections, the conserved total mass, and a curvature diagnostic at the center. Treat $A=0$ as the continuum apparent-horizon condition; a finite numerical cutoff is only a stopping criterion.

The mass diagnostic follows directly from the geometry. Define the dimensionless mass aspect $\mathfrak m$ by

$$A=1-\frac{2\mathfrak m\cos^3x}{\sin x}, \qquad \partial_x\mathfrak m =\frac12 A(\Phi^2+\Pi^2)\tan^2x.$$

Its center value is zero, and the conserved physical total mass is $M_{\rm tot}=(L/G)\lim_{x\to\pi/2}\mathfrak m$. At $A=0$ this definition yields the horizon mass relation above. The independent momentum equation is

$$\partial_t A+2\sin x\cos x\,A^2e^{-\delta}\Phi\Pi=0.$$

Monitoring this equation tests more than re-evaluating radial constraints already solved by the algorithm Bizoń and Rostworowski 2011, Eqs. (4)–(9) and numerical method, pp. 1–2, PDF. The mass at the first apparent horizon is only the mass inside that sphere: later accretion can make the final black hole much heavier. Critical scaling must therefore name which mass it fits.

Validation and benchmark matrix

A credible rerun should make the following controls explicit.

Control	Minimum comparison	Acceptance evidence
Vacuum benchmark	$\Phi=\Pi=0$	$A=1$, $\delta=0$, zero mass, and no secular drift
Linear benchmark	One small normal mode	Frequency $\omega_jL=3+2j$ and bounded amplitude over many crossings
Prompt-collapse reference	A published Gaussian above $\varepsilon_0$	Same reflection count and compatible $R_H$ after translating normalization
Resolution study	At least $N$, $2N$, and $4N$ radial points	Constraint norm and mass drift decrease at the scheme’s designed order before horizon formation
Time tolerance	At least three tolerances, for example $10^{-8}$, $10^{-9}$, $10^{-10}$	Branch label and fitted $\gamma$ remain within the reported uncertainty
Independent implementation	Compare finite differences with a pseudospectral or double-null code	Threshold window, exponent, and reflection count agree after coordinate-invariant conversion

For each branch, bisect $\varepsilon_n$ separately and report the bracketing interval. Fit only points satisfying both $R_H/L\ll1$ and several-grid-point resolution of the smallest length scale. Vary the inner and outer fit endpoints; the resulting spread belongs in the exponent uncertainty. Monitor the unused momentum equation as an independent residual, not merely the radial equations solved at every time step.

The strongest reference-solution cross-check is formulation independence. The compactified Cauchy system above and the double-null dilaton reduction used by Husain and collaborators do not share the same coordinate grid or horizon diagnostic. Agreement of their small-scale exponent is therefore more informative than agreement between two resolutions of one code.

Failure modes and rerun boundary

Near threshold the pulse narrows exponentially. Under-resolution can either trigger a spurious horizon or add numerical dissipation that falsely prevents collapse. A fixed cutoff such as $A\lt 10^{-4}$ biases the smallest measurable $R_H$; boundary interpolation can change the energy at every reflection; and an incorrect branch label can turn the finite jump between branches into a false scaling violation. Long runtime also compounds tiny mass-flux errors.

No new numerical result is claimed on this page. The quoted exponent and branch behavior are literature benchmarks. A rerun supports a publishable number only if it supplies code and parameter versions, the complete initial profile and normalization, three-resolution and three-tolerance data, threshold brackets, constraint and mass histories, fit-window variation, and an independent implementation or reference solution. Without those items, the calculation is a pedagogical reproduction rather than new evidence.

Exercise: separate local and global scaling

Suppose a fit gives $R_H/L=10^{-3}$ near one threshold. Estimate the relative cosmological correction in the Schwarzschild–AdS mass relation and explain why the same estimate does not predict the location of the next reflection threshold.

Solution

The exact relation is $2GM_H=R_H(1+R_H^2/L^2)$. At $R_H/L=10^{-3}$, the fractional cosmological correction to the leading small-hole relation is $R_H^2/L^2=10^{-6}$. This supports using the asymptotically flat critical solution locally.

The next branch location depends on propagation to the timelike boundary, reflection, nonlinear phase evolution, and refocusing over distances of order $L$. Those are global effects and are not controlled by the small parameter evaluated at the already formed horizon. Local exponent universality and global threshold placement are therefore logically independent.

References

• Bizoń, Piotr, and Andrzej Rostworowski. “On Weakly Turbulent Instability of Anti-de Sitter Space.” Physical Review Letters 107 (2011): 031102. doi:10.1103/PhysRevLett.107.031102. Open PDF.
• Husain, Viqar, Gabor Kunstatter, Bruce Preston, and Michael Birukou. “Anti-de Sitter Gravitational Collapse.” Classical and Quantum Gravity 20 (2003): L23–L30. doi:10.1088/0264-9381/20/4/101. Open PDF.
• Martinon, Grégoire. “The Instability of Anti-de Sitter Space-Time.” arXiv:1708.05600v3 (2017), 58 pp. Abstract. Open PDF.