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

Details Type II Critical Collapse of a Self-Gravitating Nonlinear $σ$-Model Type II Critical Collapse of a Self-Gravitating Nonlinear $σ$-Model

2000-02-18

arxiv.org/abs/gr-qc/0002067v3

Phys.Rev.D62:104007,2000

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

23 pages, includes 10 postscript figure files, uses REVTeX, epsf, psfrag, and AMS math fonts (amstex + amssymb); to appear in

Scientific paper

10.1103/PhysRevD.62.104007

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.

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