Nonspherical Modes and Collapse Thresholds

A collapse threshold is a hypersurface in the space of admissible initial data, while a nonspherical mode is a direction away from a chosen symmetric solution. Their relation is controlled by the spectrum of linear perturbations: a spherical threshold solution remains relevant to generic collapse only if every allowed nonspherical mode decays, apart from the single mode that points across the threshold.

Required background. The Einstein equation defines the nonlinear evolution and its linearization.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon-based outcome criteria; Black-Hole Perturbation Theory develops parity and harmonic decompositions; Formulations of General Relativity separates physical and constraint-violating modes; Relativistic Matter and Compact Objects supplies fluid mode spectra; and Computational Relativity explains threshold searches and mode extraction.

Perturbations as directions in phase space

For a spherical continuously self-similar background, let $x$ be a dimensionless radial coordinate and $\tau$ logarithmic scale time. A scalar component of a nearby solution has the schematic expansion

$$Z(\tau,x,\theta,\phi)=Z_*(x)+ \sum_{\ell m i} C_{\ell mi}\,e^{\lambda_{\ell i}\tau} Z_{\ell mi}(x)Y_{\ell m}(\theta,\phi)+\cdots .$$

Here $\tau$ increases toward smaller spacetime scales. Vector and tensor components require the corresponding spherical harmonics; a metric perturbation is not a collection of scalar harmonics alone. For a discretely self-similar background, both the background and eigenfunctions also depend periodically on $\tau$, and $\lambda_{\ell i}$ is a Floquet exponent. Gauge and constraint modes must be removed before interpreting it. A mode with $\operatorname{Re}\lambda\gt0$ grows; one with negative real part decays; a complex pair produces oscillatory growth or decay.

Spherical symmetry makes the eigenvalue independent of $m$ within a fixed $\ell$ and parity sector: rotations relate the $2\ell+1$ real angular components. On a merely axisymmetric background, $\ell$ can mix and eigenvalues may depend on $m$. Codimension counts independent real growing amplitudes, including these degeneracies, rather than merely the number of distinct eigenvalues.

Suppose precisely one independent real physical amplitude grows, with exponent $\lambda_0\gt0$. Initial data tuned by a parameter $p$ then remain near $Z_*$ until $C_0(p)e^{\lambda_0\tau}\sim1$. The set $C_0=0$ is locally codimension one and can separate dispersal from black-hole formation. If a nonspherical mode also has positive real part, reaching the same solution requires additional tunings; it is no longer the generic one-parameter threshold attractor.

Shape perturbations shift more than one threshold

For a two-parameter family $Z(p,\epsilon)$, with $\epsilon$ the amplitude of a nonspherical perturbation, the local threshold can be written $p=p_*(\epsilon)$. If reversing the perturbation is equivalent to a rotation or reflection, symmetry may force

$$p_*(\epsilon)=p_*(0)+a_2\epsilon^2+O(\epsilon^4),$$

whereas a perturbation without that equivalence can allow a linear term. This shift is family dependent. It should not be confused with the universal eigenvalues of a critical solution or with the nonlinear onset of a stellar bar mode.

The massless scalar critical solution provides a clean example: the calculated nonspherical modes decay, so the spherical solution is expected to retain codimension one near that solution in the larger phase space. The linear calculation and its numerical accuracy are presented in Martín-García and Gundlach 1999, §VIII, pp. 15–18, and Table I, p. 19, PDF. This is perturbative evidence around the critical solution, not a proof that arbitrarily aspherical data enter that neighborhood. Gundlach and Martín-García summarize the perturbative result and axisymmetric evolutions in Gundlach and Martín-García 2007, §3.7, pp. 20–21. Other matter models can possess additional unstable modes, and an imposed symmetry can hide them. The same review discusses the perturbative angular-momentum scaling problem and its limited domains in Gundlach and Martín-García 2007, §5.1, pp. 31–32, PDF.

Diagnostics that distinguish a mode from gauge

A threshold study should perturb the data with controlled $m$ and parity, then extract gauge-resistant quantities such as curvature multipoles on geometrically defined surfaces, matter-density multipoles using proper-volume weights, and asymptotic waveform modes. The fitted growth rate must be stable under resolution, gauge changes, extraction radius, and a change of initial-data family. Constraint residuals projected into the same harmonics expose spurious growing modes.

Two limits are indispensable. As the perturbation amplitude tends to zero, the measured threshold and spherical observables must approach their symmetric values. Under a rigid rotation of the initial perturbation, scalar outcomes must be unchanged while the $m$-mode phases transform predictably. Failure of either check invalidates a claim about physical shape dependence.

Linear analysis also has a boundary: it predicts departure directions only while every perturbation remains small. Once several modes reach order unity, shocks, horizon formation, mode coupling, and angular-momentum transport select the endpoint. A measured bar or fragment is therefore not explained by quoting a linear $\ell=2$ eigenvalue alone.

Check your understanding

An axisymmetric candidate threshold solution has one growing $m=0$ amplitude with $\lambda_0=2.5$, a decaying $m=0$ mode, and an unstable $m=2$ sector with rate $0.1$ for both its sine and cosine components. How many independent real amplitudes must be suppressed in unrestricted data? What changes in axisymmetric data or in a family fixing the $m=2$ phase?

Solution

There are three growing real amplitudes: one axisymmetric amplitude and the cosine and sine amplitudes of the $m=2$ deformation. Generic unrestricted data therefore require three tunings. Exact axisymmetry removes both non-axisymmetric amplitudes and leaves one tuning. A deliberately restricted family with a fixed $m=2$ phase retains only one real deformation amplitude and requires two tunings. For a spherical background the full $\ell=2$ degeneracy would need to be counted instead; assigning different eigenvalues to its $m=0$ and $m=2$ members would violate rotational symmetry.

References

• 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.
• Martín-García, José M., and Carsten Gundlach. “All Nonspherical Perturbations of the Choptuik Spacetime Decay.” Physical Review D 59 (1999): 064031. doi:10.1103/PhysRevD.59.064031.