Axisymmetric Vacuum and Matter Collapse

Axisymmetry removes dependence on the azimuthal coordinate while retaining gravitational radiation, rotation, and highly nonspherical geometry. It is therefore a powerful intermediate problem between spherical collapse and unrestricted three-dimensional evolution. The reduction is physical only for exactly axisymmetric data: it suppresses every $m\ne0$ instability and cannot establish the stability of an axisymmetric outcome to general perturbations.

Required background. The Einstein equation supplies the continuum system and its constraint equations.

Helpful background. Black-Hole Geometry and Mechanics supplies invariant horizon diagnostics; Black-Hole Perturbation Theory organizes nonspherical modes; Formulations of General Relativity develops hyperbolic evolution systems and gauge choices; Relativistic Matter and Compact Objects supplies fluid initial data and equations of state; and Computational Relativity develops discretization, extraction, and convergence tests.

Two continuum problems behind one symmetry

In vacuum the equations are $G_{ab}=0$. Axisymmetric gravitational-wave packets may focus through their own curvature, disperse, or form a black hole. A regular axis forbids conical defects, asymptotic flatness supplies the outer condition, and the initial three-metric and extrinsic curvature must satisfy the Hamiltonian and momentum constraints. The classic Brill-wave family provides one parameter controlling wave amplitude, but its coordinate profile is not an invariant measure of curvature.

For matter collapse one instead solves $G_{ab}=8\pi G T_{ab}$ together with conservation laws $\nabla_aT^{ab}=0$ and $\nabla_a(\rho_0u^a)=0$, plus an equation of state. Rotation introduces a conserved axial charge when the symmetry is exact. Vacuum and matter cases can share a spacetime formulation and horizon finder, but their characteristic scales, regularity failures, and physically relevant diagnostics differ. Abrahams and Evans found threshold behavior in a particular axisymmetric vacuum-wave family; their compact statement and numerical qualifications are in Abrahams and Evans 1993, pp. 2980–2983. That result is not a theorem about all axisymmetric vacuum data.

From initial data to invariant outputs

A concrete vacuum example explains why solving the constraints matters. On a time-symmetric slice, $K_{ij}=0$, choose cylindrical coordinates $(\varpi,z,\phi)$ and a Brill metric

$$dl^2=\psi^4\left[e^{2q}(d\varpi^2+dz^2)+\varpi^2d\phi^2\right].$$

The seed $q(\varpi,z)$ describes the distortion; it is not itself a freely propagating linear wave on the finished geometry. The vacuum Hamiltonian constraint $R^{(3)}=0$ determines $\psi$:

$$\left(\partial_\varpi^2+\frac1\varpi\partial_\varpi+\partial_z^2\right)\psi =-\frac{\psi}{4}\left(\partial_\varpi^2q+\partial_z^2q\right).$$

For example $q=A(\varpi/\sigma)^2e^{-(\varpi^2+z^2)/\sigma^2}$ has a dimensionless amplitude $A$, length scale $\sigma$, and $q=O(\varpi^2)$ on the axis. Impose $\partial_\varpi\psi=0$ there, regularity elsewhere, and $\psi\to1$ at infinity. Setting $A=0$ recovers $\psi=1$ and flat data. Nonzero $A$ requires the elliptic solve before evolution. The convention here uses $e^{2q}$; Baumgarte et al. 2023, Eqs. (3)–(5), p. 2, PDF uses $e^q$, so its seed is twice ours and its constraint coefficient is $1/8$ instead of $1/4$.

This time-symmetric, twist-free family has zero ADM angular momentum even though it is nonspherical and radiates gravitational energy. Rotation of matter and axisymmetry of gravitational waves are therefore separate concepts. More generally, exact axisymmetry forces the gravitational-wave flux of angular momentum about that axis to vanish; it does not suppress the energy flux.

A reproducible computation must declare the continuum formulation, gauge, axis treatment, boundary conditions, and symmetry implementation. For a vacuum wave, record the freely specified seed function, amplitude, length scale, constraint solver, domain size, refinement hierarchy, and whether axisymmetry is imposed in cylindrical coordinates or through a Cartesian reduction. For a fluid, additionally record the equilibrium model, rotation law, equation of state, pressure depletion or velocity perturbation, atmosphere floor, shock-capturing scheme, and primitive-variable recovery.

Coordinate density or lapse plots are useful debugging views but not final observables. The output set should include constraint norms; ADM mass and angular momentum; radiated energy and angular momentum extracted from curvature at several radii; existence and area of marginally trapped surfaces; quasi-local horizon mass and spin; and, for matter, baryon mass and the specific-angular-momentum distribution. The spherical or nonrotating limit supplies a benchmark. A second benchmark is an independently implemented stationary or oscillating star for the matter code, or a weak-wave convergence test for the vacuum code.

Shibata’s rotating-fluid study specifies its evolution variables and axisymmetric Cartesian implementation in Shibata 2000, §2.1, pp. 5–10, and reports resolution comparisons in Shibata 2000, §4.1, pp. 13–15, PDF. Its thresholds belong to the chosen $\Gamma=2$ and $\Gamma$-law model families; they are not equation-of-state-independent constants.

Convergence, uncertainty, and failure modes

At least three resolutions are needed to test a claimed convergence order. For an invariant scalar $Q$, compare $Q_h$, $Q_{h/r}$, and $Q_{h/r^2}$ and test whether

$$\frac{Q_h-Q_{h/r}}{Q_{h/r}-Q_{h/r^2}}\approx r^p$$

over an interval in which the same continuum feature is resolved. Near shocks, punctures, or refinement boundaries, the observed order may fall below the formal order. Quote a discretization uncertainty from the convergent regime, then list atmosphere, extraction-radius, boundary, and initial-data-family effects separately; those are not all reducible to one Richardson error bar.

High-risk failures include loss of regularity at the axis, gauge collapse mistaken for curvature collapse, reflected radiation from the outer boundary, horizon-finder failure near a highly distorted surface, under-resolved gradients, and apparent conservation caused by enforced symmetry. For fluids, artificial-atmosphere accretion and angular-momentum loss during primitive recovery can change the threshold. For vacuum waves, a curvature peak moving with gauge rather than with an invariant diagnostic can create a false impression of self-similarity.

We did not rerun the cited simulations for this page. Reproduction therefore means that the papers expose enough continuum equations, initial-data parameters, numerical methods, and diagnostics for a new implementation; numerical values quoted from them remain published results, not locally regenerated data.

Check your understanding

Three resolutions with refinement ratio $r=2$ give horizon masses $0.8120M$, $0.8060M$, and $0.8045M$. Estimate the observed order and state one reason not to use that estimate as the entire uncertainty budget.

Solution

The successive differences are $0.0060M$ and $0.0015M$, whose ratio is $4=2^2$; the observed order is therefore $p\simeq2$. This measures discretization behavior only if the three runs resolve the same solution. Finite outer boundaries, extraction radius, atmosphere effects, gauge choices, and model-family dependence require separate checks.

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.
• 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.