Type II Critical Collapse of a Self-Gravitating Nonlinear $\sigma$-Model

TL;DR

This paper demonstrates the existence of type II critical collapse in SU(2) sigma-models coupled to gravity, revealing discretely self-similar behavior and universal scaling properties near black hole formation thresholds.

Contribution

It provides the first detailed numerical analysis of type II critical phenomena in SU(2) sigma-models, including the behavior of the critical solution and scaling exponents across a range of coupling constants.

Findings

Discretely self-similar critical solutions observed.

Critical exponent for mass scaling approximately 0.1185.

Critical behavior persists across a wide range of coupling constants.

Abstract

We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) $\sigma$-models coupled to gravity. Numerical investigations in spherical symmetry show discretely self-similar (DSS) behavior at the threshold of black hole formation for values of the dimensionless coupling constant $\ccbeta$ ranging from 0.2 to 100; at 0.18 we see small deviations from DSS. While the echoing period $\Delta$ of the critical solution rises sharply towards the lower limit of this range, the characteristic mass scaling has a critical exponent $\gamma$ which is almost independent of $\ccbeta$, asymptoting to $0.1185 \pm 0.0005$ at large $\ccbeta$. We also find critical scaling of the scalar curvature for near-critical initial data. Our numerical results are based on an outgoing-null-cone formulation of the Einstein-matter equations, specialized to spherical symmetry. Our numerically computed initial-data critical parameters $p^*$ show 2nd order convergence with the grid resolution, and after compensating for this variation in $p^*$, our individual evolutions are uniformly 2nd order convergent even very close to criticality.