Critical Collapse Beyond Spherical Symmetry

Critical collapse beyond the canonical four-dimensional, neutral, asymptotically flat scalar model asks how codimension-one threshold attractors, self-similarity, and scaling respond to gravitational radiation, rotation, charge, dimension, angular modes, and a cosmological length. Several restricted systems support parts of that extension, but no single generic nonspherical universality class is known. For symmetry breaking the key issue is spectral: modes suppressed by symmetry may decay, add unstable directions, or lead to a different critical solution. Charge, dimension, and anti-de Sitter asymptotics instead test which couplings remain relevant at the shrinking departure scale.

Required background. The Einstein equation supplies the unrestricted nonlinear flow, while differential equations supplies angular modes and codimension.

Helpful background. Black-Hole Geometry and Mechanics supplies invariant outcome measures; Black-Hole Perturbation Theory develops nonspherical spectra; Formulations of General Relativity explains gauge and constraint modes; and Computational Relativity develops adaptive three-dimensional fine tuning.

Data and symmetries beyond the spherical model

A claim outside the baseline must name at least the matter and charge normalization, spacetime dimension, asymptotics and boundary conditions, allowed angular modes, imposed discrete or axial symmetries, and family of regular initial data. Axisymmetry permits $m=0$ deformation and, for rotating matter, axial angular momentum, but excludes every $m\ne0$ instability. Octant or reflection symmetry removes additional modes. Vacuum collapse adds genuine gravitational-wave degrees of freedom absent in spherical vacuum. A spherical charged or higher-dimensional computation extends the universality class along a different axis; it does not by itself answer the nonspherical stability question.

The threshold logic remains simple. A spherical critical solution continues to govern generic data only if its nonspherical physical modes decay. Each additional independent real growing amplitude increases the codimension by one; angular degeneracies must also be counted. Alternatively, an intrinsically nonspherical solution with one unstable mode may organize a different region of the threshold. Numerical evidence must separate those possibilities from gauge deformation and finite-resolution asymmetry.

Nonspherical collapse results through 2026-08-12

As of 2026-08-12, the evidence supports conditional persistence but leaves the unrestricted problem unresolved.

System and symmetry	Supporting evidence	Competing evidence or limitation	Current status
Massless real scalar field near the spherical DSS solution	Linear calculations find all nonspherical physical perturbations decaying; axisymmetric evolutions approach the spherical solution.	Numerical tuning resolves few echoes, and a linear result controls only the neighborhood of that solution.	Strong evidence that the spherical attractor remains codimension one for small general perturbations.
Axisymmetric vacuum gravitational waves	Three independent codes agree on Brill-wave threshold curvature; different families approach distinct threshold geometries.	Fine tuning is finite; approximate DSS occurs for some tested families and is absent or unresolved in others.	Strong numerical evidence against one universal vacuum attractor covering these families, rather than merely failed reproduction of an old result.
Rotating perfect fluids	Perturbation theory predicts angular-momentum scaling, and axisymmetric simulations support two-parameter mass/spin scaling in selected radiation-fluid families.	Axisymmetry excludes bar modes; equation of state and rotation profile remain formulation axes.	Conditional scaling is supported in the tested regime, not for generic rotating matter.
Spherical charged complex scalar field in four dimensions	Numerical evolutions recover neutral-like echoing and mass scaling while black-hole charge decays with a larger power than mass.	The calculation uses a particular Einstein–Maxwell–scalar normalization and spherical data; it does not admit electromagnetic or gravitational angular modes.	Charge is irrelevant at this studied critical solution, not in every charged-collapse universality class.
Spherical massless scalar field in $4\leq D\leq11$	Dimension-by-dimension simulations find DSS and universal length exponents within each tested dimension.	The exponent and echoing period vary with $D$; high-$D$ axis regularization is delicate, and the proposed extrema near $D\sim11$ were tentative.	Type II behavior persists in the tested dimensions, but there is no dimension-independent exponent.
Four-dimensional spherical scalar collapse with $\Lambda<0$	For two initial profiles and tested negative $\Lambda$, local horizon-radius scaling gave exponents compatible with the asymptotically flat value.	The runs cover a finite $\Lambda$ range and terminate at a horizon diagnostic; they do not classify repeated AdS reflections or generic boundary data.	Local near-threshold scale separation is supported; global AdS collapse is a distinct problem.
Fully three-dimensional data	Modern methods can evolve symmetry-breaking perturbations and extract multipoles.	The required fine tuning, adaptive dynamic range, constraint control, and mode separation remain exceptionally demanding.	No comprehensive generic phase diagram or established universal 3D critical solution.

For the scalar-field case, the nonspherical spectrum and its gauge treatment are given in Martín-García and Gundlach 1999, §VIII, pp. 15–18 and Table I, p. 19, PDF, while the numerical and perturbative evidence is summarized in Gundlach and Martín-García 2007, §§3.7 and 5, pp. 20–21 and 31–35. Abrahams and Evans’ axisymmetric vacuum result is reported in Abrahams and Evans 1993, pp. 2980–2983; its restricted families and short scaling range should remain visible when citing it.

More recent results sharpen two cells of the comparison without changing its cutoff. For the real scalar field, nonlinear axisymmetric evolutions recover damped deformations close to spherical symmetry, but sufficiently large imposed deformations grow and suggest two centers of collapse; the off-center region becomes under-resolved, so its later endpoint is not established Baumgarte 2018, §III.B and §IV, pp. 7–12, PDF.

For vacuum Brill waves, three independent codes agree on threshold curvature, while two of them allow closer tuning and comparison in null coordinates. The positive- and negative-amplitude families approach different geometries, with approximate echoing in the latter Baumgarte et al. 2023, Eqs. (3)–(6), Figs. 1–3, pp. 2–4, PDF. Independent agreement strengthens this evidence of family dependence; it does not turn finite numerical tuning into a theorem about all vacuum data.

Rotation and additional scaling variables

For a perfect fluid, rotation can be carried by an axial $\ell=1$ perturbation of a spherical critical solution. Let its dimensionless amplitude at entry be $q$ and its logarithmic growth exponent be $\lambda_J$. At departure it has amplitude $q e^{\lambda_J\tau_*}$. Since $GJ$ has dimension $L^2$, a first-order rotational contribution obeys

$$GJ_{\rm BH}\propto L_*^2q e^{\lambda_J\tau_*} \propto q\,|p-p_*|^{(2-\lambda_J)\gamma}.$$

Thus

$$\gamma_J=(2-\lambda_J)\gamma, \qquad \chi_{\rm BH}=\frac{J_{\rm BH}}{GM_{\rm BH}^2} \propto q\,|p-p_*|^{-\lambda_J\gamma}.$$

For $\lambda_J\lt0$, the spin tends to zero and $\gamma_J\gt2\gamma$, a stronger statement than merely $\gamma_J\gt\gamma$. For a radiation fluid the perturbative value $\lambda_J=-1/2$ gives $\gamma_J=(5/2)\gamma$ Baumgarte and Gundlach 2016, Eqs. (1)–(5), pp. 1–2, PDF. This first-order argument applies when an axial matter mode exists; angular momentum in perturbations of a real scalar spherical solution arises at higher order and needs its own calculation. A two-parameter family can also be organized by distance from threshold and rotation, yielding scaling functions beyond the leading small-$q$ law.

Baumgarte and Gundlach tested such mass and spin scaling for axisymmetric rotating radiation fluids in Baumgarte and Gundlach 2016, pp. 1–5. This supports the perturbative framework in that matter model. It does not demonstrate stability to $m\ne0$ modes or prove that a rapidly rotating branch remains governed by the same spherical solution.

An angular mode can also become weakly unstable. Then finite tuning may show a long apparently universal spherical phase before nonspherical departure. Detecting that scenario requires measuring mode amplitudes over multiple echoes rather than fitting mass alone.

Charge, dimension, and the AdS scale

In spherically symmetric collapse of a charged complex massless scalar field, Hod and Piran found that the near-critical core approached the neutral scalar behavior and that the black-hole charge obeyed a steeper scaling law than the mass. In their electromagnetic and geometrized-unit convention the fitted charge exponent was about $0.88$, larger than twice the mass exponent, so $\lvert Q_{\rm BH}\rvert/M_{\rm BH}\to0$ along the studied supercritical families. The equations, numerical checks, mass/echo evidence, and charge fit are separated in Hod and Piran 1997, §§II–IV, pp. 3–13, §§V.A–V.B, pp. 14–20, and §V.D, pp. 26–29. This is numerical evidence that charge is an irrelevant perturbation of that spherical critical solution, not a proof for arbitrary charged matter or nonspherical Maxwell modes.

Changing dimension alters even the dimensional conversion from a departure length to a mass. In $D$ spacetime dimensions, $G_D M$ has dimension $L^{D-3}$, so if $L_*\propto\lvert p-p_*\rvert^{\gamma_L}$ then the physical-mass exponent is

$$\gamma_M=(D-3)\gamma_L.$$

Sorkin and Oren evolved spherical massless-scalar collapse for $4\leq D\leq11$, extracted subcritical curvature scaling and echoing periods, and found dimension-dependent values rather than one universal number Sorkin and Oren 2005, §4, pp. 8–12 and Table 1, p. 12. Their discussion of a possible extremum near $D=11$ explicitly identifies high-dimensional numerical uncertainty and extrapolation Sorkin and Oren 2005, §5, pp. 12–18, PDF; it should not be promoted to an established critical dimension.

For anti-de Sitter asymptotics, distinguish a local type II threshold from the global sequence of reflections caused by the timelike boundary. Husain and collaborators evolved four-dimensional, spherically symmetric massless-scalar data with two profile families and several negative cosmological constants. Their fitted horizon-radius exponents were compatible with the $\Lambda=0$ value within quoted windows Husain et al. 2003, Eqs. (2)–(24), pp. L23–L28, Tables 1–2 and Figs. 1–2, pp. L27–L29. The natural explanation is scale separation: sufficiently close to threshold, $L_*\sqrt{\lvert\Lambda\rvert}\ll1$, so the cosmological term is subleading in the shrinking core. This finite numerical result does not prove exact $\Lambda$ independence or settle the multi-bounce turbulent and stable-island questions treated in Collapse and Dynamics in de Sitter and Anti-de Sitter Spacetimes.

Uncertainty and resolution criteria

The numerical uncertainty is unusually coupled to the physics. Cartesian-grid anisotropy seeds specific $m$ modes; gauge choices distort coordinate multipoles; mesh refinement must follow exponentially shrinking scales; and a small error in $p_*$ limits the number of echoes. Constraint-violating modes can grow under refinement interfaces. Comparisons need invariant curvature or matter multipoles on geometrically aligned surfaces and asymptotic wave modes, not coordinate-density shapes alone.

A decisive study would use several constraint-solved families, deliberately seed independent $(\ell,m)$ sectors, retain no unnecessary symmetry, and show at least three resolutions through the same number of echoes. It would extract convergent growth or decay exponents, mass and subcritical-curvature scaling, and the full dimensionless core profile. Independent formulations or codes should reproduce the critical solution and its physical spectrum.

We did not rerun the cited simulations for this page. The matrix describes published evidence, not locally reproduced thresholds.

The remaining questions are concrete: whether the scalar critical solution is nonlinearly stable to small unrestricted perturbations, which vacuum data approach each observed critical geometry, and whether a genuinely nonspherical one-mode attractor governs a generic threshold. The available results answer parts of these questions in restricted models; they do not establish the endpoint for arbitrary three-dimensional data.

Check your understanding

An axisymmetric critical solution has one growing axisymmetric mode and a very weak growing non-axisymmetric sector. What behavior can an axisymmetric one-parameter tuning show, and what changes when that sector is admitted?

Solution

Axisymmetry deletes the non-axisymmetric sector, so one tuning can reveal clean approach, echoing, and mass scaling within that restricted data class. Admitting the extra sector adds growing real amplitudes and requires additional tunings: for an $m=2$ sine/cosine pair, there are two extra amplitudes. With only one tuning, the evolution may resemble the axisymmetric threshold for several echoes and then depart. The full count depends on the sector and its degeneracies, not merely on how many eigenvalues are written down.

References

• Abrahams, Andrew M., and Charles R. Evans. “Critical Behavior and Scaling in Vacuum Axisymmetric Gravitational Collapse.” Physical Review Letters 70 (1993): 2980–2983. doi:10.1103/PhysRevLett.70.2980.
• Baumgarte, Thomas W. “Aspherical Deformations of the Choptuik Spacetime.” Physical Review D 98 (2018): 084012. doi:10.1103/PhysRevD.98.084012. Open PDF.
• Baumgarte, Thomas W., and Carsten Gundlach. “Critical Collapse of Rotating Radiation Fluids.” Physical Review Letters 116 (2016): 221103. doi:10.1103/PhysRevLett.116.221103.
• Baumgarte, Thomas W., Bernd Brügmann, Daniela Cors, Carsten Gundlach, David Hilditch, Anton Khirnov, Tomáš Ledvinka, Sarah Renkhoff, and Isabel Suárez Fernández. “Critical Phenomena in the Collapse of Gravitational Waves.” Physical Review Letters 131 (2023): 181401. doi:10.1103/PhysRevLett.131.181401. Open PDF.
• 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.
• Hod, Shahar, and Tsvi Piran. “Critical Behaviour and Universality in Gravitational Collapse of a Charged Scalar Field.” Physical Review D 55 (1997): 3485–3496. doi:10.1103/PhysRevD.55.3485.
• 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.
• Martín-García, José M., and Carsten Gundlach. “All Nonspherical Perturbations of the Choptuik Spacetime Decay.” Physical Review D 59 (1999): 064031. doi:10.1103/PhysRevD.59.064031.
• Sorkin, Evgeny, and Yonatan Oren. “On Choptuik’s Scaling in Higher Dimensions.” Physical Review D 71 (2005): 124005. doi:10.1103/PhysRevD.71.124005.