Rotating Fluid Collapse

Rotating-fluid collapse couples strong-field geometry to shocks, differential rotation, and angular-momentum transport. A trustworthy calculation must answer two separate questions: whether a marginally trapped surface forms, and how mass and angular momentum divide among the horizon, exterior matter, and radiation. Neither question is settled by a coordinate-density movie or by the initial ratio $J/(GM^2)$ alone.

Required background. The Einstein equation supplies the constrained spacetime evolution coupled to the fluid stress tensor.

Helpful background. Black-Hole Geometry and Mechanics develops quasi-local horizon mass and spin; Black-Hole Perturbation Theory supplies waveform and mode diagnostics; Formulations of General Relativity explains gauge and constraint propagation; Relativistic Matter and Compact Objects develops rotating equilibria and equations of state; and Computational Relativity supplies finite-volume evolution, mesh refinement, and error analysis.

The continuum and initial-data problem

Assume four-dimensional, asymptotically flat general relativity with a perfect fluid,

$$T^{ab}=(\rho_0 h)u^au^b+Pg^{ab}, \qquad \nabla_aT^{ab}=0, \qquad \nabla_a(\rho_0u^a)=0,$$

where $\rho_0$ is rest-mass density, $h=1+\epsilon+P/\rho_0$ is specific enthalpy, and the equation of state closes the system. The spacetime formulation, slicing, shift, constraint treatment, and outer boundary are part of the continuum specification—not interchangeable implementation details.

Initial data should be an equilibrium rotating star or a constraint-solved perturbation of one. Record the central density or enthalpy, gravitational and baryon masses, equation-of-state parameters, rotation law $\Omega(\varpi)$, polar-to-equatorial radius ratio, $T/|W|$, total $J$, and the perturbation that initiates collapse. Pressure depletion is a convenient numerical trigger, but its magnitude changes shock heating and therefore labels a family of physical initial data. Imposed axisymmetry excludes bar and spiral modes; reflection symmetry excludes odd-parity structure.

A calculation with closed budgets

Why the low-angular-momentum core matters

As a weak-field estimate, a pressureless parcel of specific angular momentum $j_{\rm N}=R^2\Omega$ orbiting an enclosed mass $M$ has radial acceleration $-GM/R^2+j_{\rm N}^2/R^3$. Its centrifugal scale is therefore

$$R_{\rm cent}\sim\frac{j_{\rm N}^2}{GM}.$$

This is a Newtonian diagnostic, not a relativistic horizon criterion. It nevertheless explains the essential mechanism: parcels with different $j$ meet rotational support at different radii. A low-$j$ core can continue inward while a high-$j$ envelope forms a torus even when the total $J$ is large. Near a forming black hole, replace this estimate by the actual relativistic evolution, horizon geometry, and pressure-sensitive fluid trajectories.

The relativistic baryon mass outside a found horizon is unambiguous once the slice and horizon are specified. Let $\Sigma_{\rm out}$ be the region of the spatial slice exterior to that horizon section:

$$M_{b,\mathrm{out}} =\int_{\Sigma_{\rm out}}\rho_0 W\sqrt{\gamma}\,d^3x, \qquad W=-u^an_a,$$

where $n^a$ is the slice normal and $\gamma$ the determinant of its spatial metric. This is not the gravitational mass or binding energy of the disk. Keeping that distinction prevents a baryon fraction from being inserted incorrectly into an ADM energy budget.

Evolving and accounting for the fluxes

Conservative finite-volume variables evolve baryon number and momentum across shocks. The grid must resolve the stellar surface, low-angular-momentum core, prospective horizon, and any exterior torus. State the atmosphere floor and reset rule, refinement boxes, finest proper resolution, outer-boundary radius, reconstruction and Riemann solver, time integrator, horizon finder, and gravitational-wave extraction method.

Invariant or asymptotically meaningful outputs include baryon mass, ADM mass and $J$, constraint norms, maximum curvature, horizon area, irreducible mass, quasi-local spin, radiated energy and angular momentum, and the exterior baryon mass just defined. A useful angular-momentum closure is

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

with every term evaluated using a declared prescription. Here $J_{\rm escaped}$ is the integrated matter and radiation flux already through the outer boundary, counted once; $J_{\rm grav,out}$ accounts for gravitational angular momentum still outside the horizon in the chosen balance prescription. It cannot generally be replaced by a local covariant density. In exact axisymmetry the net axial gravitational-wave terms vanish and the momentum constraint relates the remaining surface charges directly to matter angular momentum. The residual is a diagnostic, not a new physical reservoir. Shibata gives the conserved fluid variables and global diagnostics for an axisymmetric implementation in Shibata 2000, §2.1, Eqs. (2.15)–(2.18), pp. 9–10, then relates disk formation to the initial specific-angular-momentum spectrum in Shibata 2000, §5, pp. 19–22, PDF.

Benchmarks, convergence, and errors

Before collapse, evolve a stable rotating equilibrium for many dynamical times and demonstrate bounded constraints, convergent oscillation frequencies, and controlled losses of baryon mass and $J$. As a benchmark, recover a spherical collapse result as $\Omega\to0$ and compare the equilibrium oscillations with an independent implementation or perturbative reference solution. After horizon formation, compare the measured horizon circumference or multipoles with a Kerr black hole having the same quasi-local $M_H,J_H$; compare exterior matter’s $j$ with the Kerr innermost-stable-orbit scale only as a diagnostic, since pressure and self-gravity invalidate a test-particle classification.

Use at least three finest-grid spacings and repeat with a larger outer domain. Demonstrate convergence of horizon formation time in a geometrically defined time coordinate, final $M_H$ and $J_H$, baryon mass outside, waveform phase, and integrated budgets. Shocks and stellar surfaces generally reduce the convergence order. Quote discretization, finite-radius, atmosphere, and extraction errors separately. Varying the equation of state or rotation law measures model dependence, not numerical uncertainty.

Baiotti and collaborators document a three-dimensional rotating-star collapse test, including initial models and evolution setup in Baiotti et al. 2005, §§II–III, pp. 2–8 and horizon/ringdown diagnostics in Baiotti et al. 2005, §IV, pp. 8–15, PDF. Their results are benchmarks for comparable conventions, not universal endpoint fractions.

Failure modes and reproducibility boundary

Atmosphere resets can add mass or remove angular momentum; under-resolved primitive recovery can heat the core; gauge stretching can masquerade as delayed collapse; refinement boundaries can seed non-axisymmetric modes; a horizon finder can miss a distorted surface; and excision or puncture errors can contaminate the exterior. Exact axisymmetry can produce excellent convergence while excluding the fastest physical instability. Every endpoint statement must therefore name the symmetry and parameter range.

We did not rerun the cited simulations for this page. No locally generated waveform, horizon history, or threshold is claimed. A reproducible extension would publish constraint-solved initial data, code version, parameter file, equation-of-state table or formula, atmosphere prescription, refinement history, horizon and extraction settings, and machine-readable budgets at all resolutions.

Check your understanding

A run loses $0.4\%$ of baryon mass to atmosphere resets, while the reported disk contains $0.3\%$ of the initial baryon mass. Can the run establish that disk fraction? What check is required?

Solution

No. The known mass nonconservation is larger than the claimed signal and could create or erase it. The atmosphere floor and reset policy must be varied with resolution, the baryon flux through boundaries and the horizon must be integrated, and the exterior mass must converge to a value whose uncertainty is well below $0.3\%$.

References

• Baiotti, Luca, Ian Hawke, Pedro J. Montero, Frank Löffler, Luciano Rezzolla, Nikolaos Stergioulas, José A. Font, and Ed Seidel. “Three-Dimensional Relativistic Simulations of Rotating Neutron-Star Collapse to a Kerr Black Hole.” Physical Review D 71 (2005): 024035. doi:10.1103/PhysRevD.71.024035.
• 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.