Fragmentation and Bar-Mode Instabilities

Rapidly and differentially rotating matter can amplify non-axisymmetric structure before a central horizon forms. A bar, one-armed spiral, or several density concentrations may then redistribute angular momentum or fragment the configuration. The unsettled question is not whether such modes exist, but which of them survives realistic initial data, microphysics, and fully resolved relativistic evolution long enough to determine the black-hole endpoint.

Required background. The Einstein equation supplies the nonlinear spacetime–matter evolution in which an instability must be tested.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon diagnostics; Black-Hole Perturbation Theory develops azimuthal mode decompositions; Formulations of General Relativity explains gauge and constraints; Relativistic Matter and Compact Objects develops rotating-star equilibria and instabilities; and Computational Relativity supplies nonlinear three-dimensional tests.

Competing instability mechanisms

On a chosen spatial slice, let $\rho_*=\rho_0W\sqrt{\gamma}$ be the conserved coordinate baryon density, where $W$ is the fluid Lorentz factor relative to the slice normal. A commonly monitored full-volume diagnostic is

$$C_m(t)=\int \rho_*(t,\varpi,\phi,z)e^{im\phi}\,d^3x, \qquad A_m=\frac{|C_m|}{C_0}.$$

Exponential growth of $A_2$ signals a bar-like mode; $A_1$ signals a one-armed displacement or spiral. The integral is coordinate dependent, so a physical case also needs pattern speeds, gravitational-wave modes, proper density geometry, constraints, and convergence. Classical uniformly rotating, nearly incompressible stars develop a dynamical bar only at large $\beta=T/\lvert W_{\rm grav}\rvert$, where $T$ is rotational kinetic energy and $W_{\rm grav}$ is gravitational binding energy, distinct from the Lorentz factor $W$. These energies also need a declared prescription in relativity. Strong differential rotation introduces corotation points, where a mode’s pattern speed equals the local angular velocity, and can support low-$\beta$ shear instabilities. In one Newtonian $N=3.33$ polytropic family, two independent hydrodynamics codes found a dominant dynamical $m=1$ instability at $\beta\geq0.14$, with higher harmonics and explicitly reported late-time numerical limitations Centrella et al. 2001, §§ 2–4, pp. L194–L196, and Figs. 1–4. A toroidal density maximum further favors off-center collapse.

Fragmentation is a nonlinear claim. Separate density maxima must persist under refinement and smaller seed perturbations; a horizon around one maximum must be located geometrically; and the remaining matter must be followed long enough to distinguish a temporary compression from a new black hole. A mode that grows fastest in a deliberately seeded run need not dominate generic truncation noise or astrophysical perturbations.

Bar-mode and fragmentation results through 2026-08-12

As of 2026-08-12, the evidence is conditional rather than universal.

Question	Supporting evidence	Counterevidence or limitation	Present conclusion
Can differential rotation trigger non-axisymmetric growth at low $\beta$?	Relativistic polytrope evolutions show growing $m=1$ and $m=2$ modes with corotation points.	Growth rates and unstable bands depend strongly on rotation law, toroidal structure, and seed.	Yes for demonstrated model families; no model-independent threshold is known.
Does fragmentation form an off-center black hole?	Selected quasi-toroidal evolutions developed a collapsing fragment, and some located an enclosing apparent horizon.	Other fragments rebounded; some runs could not locate a horizon or continue through the nonlinear stage.	Established as a possible channel, not as a generic endpoint.
Do several fragments make several black holes?	Seeded $m=2$ and $m=3$ runs produced multiple contracting concentrations.	Resolution, atmosphere treatment, and limited post-collapse evolution obstructed a closed multi-horizon budget.	Suggestive, but the broad astrophysical claim remains unresolved.

Zink and collaborators define their equilibrium families and perturbations in Zink et al. 2007, §§III–IV, pp. 4–11, report mode growth, corotation, and resolution tests in Zink et al. 2007, §V, pp. 12–26, PDF, and state the restrictions on black-hole conclusions in Zink et al. 2007, §VI, pp. 27–29, PDF. In particular, the same study contains both black-hole-forming and rebounding fragments. That internal counterexample rules out the shortcut “fragmentation implies a black hole.”

Sources of uncertainty

Dynamical growth, secular growth, and seed size

A dynamical instability grows without requiring viscosity or radiation reaction, typically on a time comparable to the rotation or fluid dynamical time. A secular instability needs a dissipative channel, such as viscosity or gravitational radiation reaction; its growth time depends on that channel. Thus the same visible bar shape does not identify the mechanism. The fluid’s rotation law and mode pattern speed must be compared with the proposed energy and angular-momentum transfer.

For a mode $\delta\rho\propto e^{st}\cos[m(\phi-\Omega_p t)]$, the above convention gives $C_m\propto e^{st+im\Omega_p t}$. Hence $s=d\ln A_m/dt$ measures growth when the normalization $C_0$ is constant, and $\Omega_p=m^{-1}d(\arg C_m)/dt$ measures the pattern speed while the mode remains linear. Loss of baryon mass from the integration domain must be separated from modal growth. A corotation point satisfies $\Omega(\varpi,z)=\Omega_p$; its existence is a diagnostic for shear-driven mechanisms, not by itself proof of instability.

If a mode starts at amplitude $A_{\rm seed}$ and becomes nonlinear at $A_{\rm nl}$, its growth time is

$$t_{\rm nl}-t_{\rm seed} =s^{-1}\ln\left(\frac{A_{\rm nl}}{A_{\rm seed}}\right).$$

This must be compared with the remaining time to central collapse. It also explains why a genuine instability can start later on a finer grid: reducing truncation noise reduces its seed. Resolution-dependent onset alone does not distinguish a physical unstable mode from a numerical one; a convergent growth rate and controlled seeding do.

Model and numerical limitations

The main formulation axis is the progenitor model: idealized $\Gamma=4/3$ polytropes and imposed differential-rotation laws do not span cooling, magnetic braking, viscosity, nuclear equations of state, or cosmological assembly. The numerical axis includes atmosphere torque, refinement seeding, outer boundaries, gauge, horizon finding, and the cost of resolving an off-center fragment. The interpretive axis is timescale competition: central axisymmetric collapse, $m=1$ growth, $m=2$ growth, shock heating, and angular-momentum transport can exchange order as parameters vary.

Evidence favoring bars in less toroidal configurations does not settle fragmentation. Conversely, a clean fragmented solution in an extreme quasi-toroidal family is a counterexample to impossibility but does not measure occurrence rates. Exact reflection or rotational symmetries can delete the mode that would otherwise win.

What would resolve the dispute

A strong resolution would use constraint-solved families spanning compactness, rotation law, equation of state, and perturbation spectra; at least three resolutions with decreasing seed and atmosphere floors; invariant horizon tracking through merger or rebound; and closed mass, angular-momentum, and waveform budgets. Independent codes should recover the same instability boundaries in dimensionless parameters. Astrophysical prevalence further requires formation calculations showing that progenitors actually enter those regions.

The unresolved step is to connect demonstrated fragmentation in selected models to converged multi-horizon outcomes and to plausible formation channels for their progenitors. The present evidence establishes possibility under specified conditions; it does not establish an astrophysical occurrence rate.

Check your understanding

A sequence shows $A_2$ growing at all resolutions, but the growth begins when the refinement boundary crosses the star and its onset time changes with grid spacing. Does this establish a bar instability?

Solution

Not by itself. The grid event may seed a genuine unstable mode or may create a numerical artifact. If the physical rate is $s\gt0$, a smaller seed delays nonlinear growth by $s^{-1}\ln(A_{\rm seed,old}/A_{\rm seed,new})$ without changing $s$. Repeat with controlled seeds, different refinement layouts, and higher resolution. A convergent growth rate and pattern speed, together with compatible geometric or asymptotic diagnostics, distinguish physical growth from a grid-dependent instability.

References

• Centrella, Joan M., Kimberly C. B. New, Lisa L. Lowe, and J. David Brown. “Dynamical Rotational Instability at Low $T/\lvert W\rvert$.” Astrophysical Journal Letters 550 (2001): L193–L196. doi:10.1086/319634.
• 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.