Spin, Shape, and the Collapse Endpoint

The endpoint of rotating, nonspherical collapse is selected by a competition among central compactness, the distribution and transport of angular momentum, pressure and shock heating, and unstable shape modes. A stationary Kerr black hole may be the late-time central remnant, but the route to it can include a disk, ejecta, rebound, fragmentation, or a long-lived star. No function of the initial total spin and one shape parameter predicts all of those outcomes.

Required background. The Einstein equation governs the nonlinear competition among geometry, matter, and radiation.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr and quasi-local endpoint tests; Black-Hole Perturbation Theory develops ringdown and multipoles; Formulations of General Relativity clarifies gauge-dependent shape; Relativistic Matter and Compact Objects supplies equations of state and rotating equilibria; and Computational Relativity develops three-dimensional endpoint calculations.

Mass, spin, and shape in the collapse endpoint

Assume regular, finite-energy, asymptotically flat initial data in four-dimensional general relativity. For fluid data, specify the equation of state, entropy, rotation law, magnetic or viscous stresses, and any imposed symmetry. “Spin” may mean $J/(GM_{\mathrm{ADM}}^2)$ for the complete slice, $J_H/(GM_H^2)$ for a horizon, or a local specific-angular-momentum profile. “Shape” may mean a coordinate axis ratio, an intrinsic surface multipole, or a matter quadrupole. These are not interchangeable.

A useful outcome map therefore has more than two coordinates:

$$\mathcal{O}=\mathcal{O} \bigl(M_b,M_{\mathrm{ADM}},J,M_b(<j),\mathrm{EOS}, \Omega(\varpi),S,\{A_m\}\bigr),$$

where $S$ denotes entropy information and $A_m$ the non-axisymmetric seed spectrum. This notation is a reminder of dependence, not a claim that finitely many scalars determine the Cauchy development.

Collapse outcomes through 2026-08-12

As of 2026-08-12, several conditional statements are well supported, while the general endpoint map remains unresolved.

Initial regime	Evidence for an outcome	Competing evidence or counterexample	Defensible scope
Sub-Kerr, axisymmetric rotating fluid in studied polytropic families	Prompt horizon formation can leave little high-specific-angular-momentum matter outside.	Stronger pressure depletion, shock heating, and different rotation laws shift the threshold and disk fraction.	Quantitative fractions belong to the declared family and symmetry.
Total $J/(GM_{\mathrm{ADM}}^2)>1$ in a material configuration	Collapse can redistribute angular momentum, leave an exterior reservoir, or avoid prompt horizon formation.	The initial global ratio is not a horizon spin and is not itself a censorship violation.	It constrains a budget, not a unique endpoint.
Strongly differential, quasi-toroidal rotation	One-armed or multi-armed modes can fragment the density and sometimes produce an off-center horizon.	Some fragments rebound, and axisymmetric collapse can outrun nonspherical growth.	Fragmentation is possible in demonstrated families, not generic.
A newly formed, distorted horizon	Area, spin, multipoles, and radiation may approach those of Kerr.	Exterior matter can continue accreting; finite-time ringdown agreement does not prove the entire exterior is stationary.	Kerr is tested through converging quasi-local and asymptotic observables.

Shibata’s parameter survey and qualifications are explicit in Shibata 2000, §5, pp. 19–22. Zink and collaborators’ coexistence of collapse and rebound in related fragmented models appears in Zink et al. 2007, §V.B and §V.G, pp. 13–25. Those results jointly refute both “rotation always prevents a horizon” and “a growing fragment always makes one.”

Uncertainty and failed shortcuts

The largest physical uncertainty is often transport. Hydrodynamic torques act on a dynamical time, magnetic winding and magnetorotational turbulence on field- and resolution-dependent times, and viscosity or neutrino cooling on still other scales. Omitting a channel is a regime assumption, not a small numerical error. The largest numerical uncertainties arise near shocks, stellar surfaces, refinement interfaces, and highly distorted horizons. Exact axisymmetry converts an unknown three-dimensional stability question into an excluded one.

Three shortcuts fail. First, $J/(GM^2)>1$ for the whole star does not imply a superextremal horizon. Second, a coordinate oblateness is not an invariant signal of fragmentation. Third, a lapse collapse is not proof of a black hole. The endpoint needs a closed mass–angular-momentum budget, a located marginal surface, invariant horizon geometry, radiation at multiple extraction radii, and convergence under symmetry-breaking perturbations.

What remains to be established

A substantially resolved map would span compactness, equation of state, rotation profile, magnetization, and controlled $m$-mode seeds with constraint-solved data. It would use at least three resolutions, independent atmosphere and boundary variations, invariant multi-horizon tracking, and evolution long enough to classify rebound, disk survival, or merger. Agreement by independent formulations or codes would separate physical boundaries from implementation-specific ones.

The present conclusion is deliberately conditional: spin and shape organize outcome families, but do not yet supply a universal phase diagram.

Check your understanding

Initial data have $J=1.1GM_{\mathrm{ADM}}^2$. A horizon later has $M_H=0.8M_{\mathrm{ADM}}$ and $J_H=0.6GM_{\mathrm{ADM}}^2$. Is the horizon superextremal, and where must the remaining angular momentum be sought?

Solution

Its dimensionless spin is $\chi_H=0.6/0.8^2=0.9375$, so it is subextremal. The remaining $0.5GM_{\mathrm{ADM}}^2$ must be accounted for by exterior matter, gravitational angular momentum still outside under the adopted charge prescription, and accumulated matter/radiation flux through the outer boundary, counted once, plus a numerical residual. If exact axisymmetry is imposed, the axial gravitational-wave flux vanishes. The initial total ratio never equaled the later horizon ratio.

References

• 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.