Rotating and Nonspherical Collapse

Rotation and nonspherical structure turn collapse from a competition between gravity and radial stress into a coupled problem of angular-momentum balance, mode growth, transport, and horizon geometry. This chapter develops the quantities that remain meaningful across that change, shows how symmetry-reduced calculations must be qualified, and ends with the open question of how spin and shape select an endpoint.

Helpful background. The Einstein equation supplies the nonlinear geometry–matter coupling, and Relativistic Matter and Compact Objects supplies rotating equilibria, equations of state, and transport mechanisms.

Enter this chapter

The working regime is four-dimensional, asymptotically flat general relativity with regular finite-energy initial data. Every quantitative result is conditional on its matter model, rotation law, perturbation family, boundary conditions, and imposed symmetries. Axisymmetry retains rotation and gravitational waves but excludes all $m\ne0$ modes; a three-dimensional evolution is required to test bars, spirals, and fragmentation.

The chapter follows a deliberate order:

1. Angular Momentum in Collapse distinguishes ADM, quasi-local, and matter-specific angular momentum and constructs a closed spin budget.
2. Axisymmetric Vacuum and Matter Collapse compares the two continuum systems and sets requirements for invariant computational evidence.
3. Nonspherical Modes and Collapse Thresholds relates mode eigenvalues to the codimension of a threshold.
4. Rotating Fluid Collapse develops a reproducible fluid–spacetime calculation with convergence and error controls.
5. Fragmentation and Bar-Mode Instabilities separates established model examples from disputed generalization.
6. Spin, Shape, and the Collapse Endpoint assembles the conditional endpoint map and its resolution criteria.

One budget, several notions of rotation

The initial ADM angular momentum is a charge of the asymptotic data. A horizon spin is quasi-local and exists only after a suitable closed surface is found. A fluid’s specific-angular-momentum distribution determines which parcels can enter the center without transport. Confusing those levels produces the common but incorrect inference that initial $J/(GM_{\mathrm{ADM}}^2)>1$ requires a superextremal horizon.

For an evolution that forms a black hole, the organizing relation is

$$J_{\mathrm{initial}}=J_H+J_{\mathrm{matter,out}}+J_{\mathrm{escaped}} +J_{\mathrm{grav,out}}+\delta J_{\mathrm{num}}.$$

Here $J_{\mathrm{escaped}}$ is the accumulated angular momentum carried through the chosen outer boundary by matter and radiation, counted once; $J_{\mathrm{grav,out}}$ allows for the gravitational field still outside the horizon. In a nonsymmetric strong-field region that last term depends on the adopted charge prescription: gravitational angular momentum has no unique local density. Exact axisymmetry makes the outgoing gravitational-wave flux of axial angular momentum zero, although waves can carry energy.

The analogous mass and baryon-number balances must close. The distribution $M_b(<j)$, the baryon mass below a chosen specific angular momentum $j$, adds information that the total $J$ omits: a low-$j$ core may form a horizon while high-$j$ matter makes a disk. Nonaxisymmetric pressure forces, magnetic stresses, and radiation can redistribute or remove angular momentum. Heating in an axisymmetric ideal-fluid shock changes pressure support, but does not by itself supply an axial torque.

Symmetry, thresholds, and mode competition

Near a self-similar threshold solution, let $\tau$ be dimensionless logarithmic scale time, increasing toward smaller spacetime scales. After gauge and constraint directions are removed, a physical linear perturbation has amplitude $C_i e^{\lambda_i\tau}$, where $C_i$ is fixed by the initial data and $\lambda_i$ is its eigenvalue. With this convention $\operatorname{Re}\lambda_i>0$ means growth. A solution with exactly one physical growing mode has a codimension-one stable manifold and can be reached by tuning one parameter. A growing nonspherical mode adds another unstable direction. Exact axisymmetry deletes that direction rather than proving it stable.

This distinction connects the analytic and computational halves of the chapter. Axisymmetric vacuum waves and rotating fluids are controlled test beds for strong focusing, gauge, constraints, and horizon finding. Abrahams and Evans found critical scaling and approximate echoing in a specified axisymmetric vacuum-wave family Abrahams and Evans 1993, pp. 2980–2983 and Figs. 1–3. More recent calculations using three independent codes agree on the curvature scaling of selected Brill-wave families and find approximate discrete self-similarity in some families, while the critical geometry depends on the family Baumgarte et al. 2023, Figs. 1–3 and discussion, pp. 2–4, PDF. These results do not establish a single universal vacuum critical solution. Three-dimensional perturbations then ask whether a bar or spiral grows before central collapse, shock heating, or angular-momentum transport changes the background.

The published evidence supports conditional possibilities: in Shibata’s axisymmetric $\Gamma=2$ families, larger angular momentum raised the prompt-collapse threshold, with important shock-heating and model qualifications Shibata 2000, § 4.1, pp. 336–338, and § 5, pp. 343–345. Selected quasi-toroidal models fragment; some fragments form horizons; and related fragments can rebound Zink et al. 2007, § V, pp. 12–26, and § VI, pp. 27–29. This evidence does not support a universal spin threshold, a universal bar threshold, or the implication that fragmentation always produces multiple black holes. Gundlach and Martín-García place nonspherical threshold modes in their phase-space setting in Gundlach and Martín-García 2007, §§ 3.7 and 5.1–5.3, pp. 20–21 and 31–34.

Standards for an endpoint claim

A numerical endpoint claim should declare the continuum formulation, initial-data solver and parameters, equation of state, atmosphere, symmetry, grid hierarchy, outer boundary, gauge, horizon finder, and extraction method. It should show at least three resolutions, constraint behavior, conservation and flux balances, invariant horizon outputs, and separate discretization from model uncertainty. A coordinate density maximum or lapse collapse is insufficient.

The frontier pages use an evidence cutoff of 2026-08-12. Their conclusions distinguish demonstrated examples from general claims: a broader theorem, a robust counterexample, or a converged nonlinear survey can change which endpoints are established under stated assumptions.

Review the chapter

A one-parameter axisymmetric sequence approaches a threshold solution. A separate three-dimensional perturbation calculation finds a weakly growing $m=2$ eigenmode. Explain why the axisymmetric threshold can be numerically sharp yet fail to be generic, and name three quantities needed to classify the later endpoint.

Solution

Axisymmetry removes the $m=2$ perturbation, so within that restricted phase space the solution can have one growing mode and a sharp codimension-one threshold. On an axisymmetric background, a growing complex $m=2$ amplitude represents two real directions, proportional to $\cos2\phi$ and $\sin2\phi$. Together with the original growing mode, these require three independent tunings in unrestricted data space, or two in a family whose imposed reflection symmetry fixes the bar orientation. One generic parameter does not suffice. Endpoint classification needs, for example, a located horizon with quasi-local mass and spin, the mass and angular momentum of exterior matter, and escaped energy/angular momentum, all in convergent closed budgets. Mode amplitudes and invariant horizon multipoles further distinguish bar growth, fragmentation, and relaxation.

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., 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.
• Shibata, Masaru. “Axisymmetric Simulations of Rotating Stellar Collapse in Full General Relativity: Criteria for Prompt Collapse to Black Holes.” Progress of Theoretical Physics 104 (2000): 325–358. doi:10.1143/PTP.104.325.
• Zink, Burkhard, Nikolaos Stergioulas, Ian Hawke, Christian D. Ott, Erik Schnetter, and Ewald Müller. “Non-Axisymmetric Instability and Fragmentation of General Relativistic Quasi-Toroidal Stars.” Physical Review D 76 (2007): 024019. doi:10.1103/PhysRevD.76.024019.