Animations of SU(2) EYM Collapse

The following sequences show the dynamics of spherically symmetric SU(2) Einstein-Yang-Mills collapse within the context of the "purely magnetic ansatz". All renderings show quantities (redundantly) color-mapped with the function being displayed.

Type II Critical Behaviour

In this case the initial data is of the form 
 W(r,0) = p * Gaussian(r)
Fine-tuning of p to the black-hole threshold generates the discretely self-similar "Type II" critical solution shown below, and there is no mass gap at threshold. The behaviour here is very similar to that seen in spherically-symmetric massless scalar collapse.
  1. Weak field evolution, W(r) vs r: (1.1MB MPEG)
  2. Near critical evolutions, (W - 1)/r vs log(r): (1.1MB MPEG)

Type I Critical Behaviour

In this case the initial data is of the form 
 W(r,0) = Kink(r) + p * dGaussian(r)/dr
where Kink(r) is a kink profile which interpolates between the two vacuum states W=+1, W=-1, and Gaussian(r) is a Gaussian. Fine-tuning of p to the black-hole threshold generates the static "Type I" critical solution (n=1 Bartnik-Mckinnon solution) shown below, and there is a minimum black hole mass at threshold (approximately equal to the mass of the static configuration). Note that "kinkiness" is "conserved" in the evolution in the sense that the solution which completely disperses has (asymptotically) W(r,t) = +1---i.e. the kink escapes to infinity, while the black hole solution has (again asymptotically, and in this case for r > r_h, the black hole radius), W(r,t) = -1---i.e. the kink falls into the black hole.
  1. Weak field evolution, W(r) vs r: (0.8MB MPEG)
  2. Interpolating evolutions (top, weak field; bottom, black hole formation) W(r) vs log(1+r): (1.6MB MPEG)
  3. Near-critical evolutions. Here the intermediate attractor is the static n=1 Bartnik-Mckinnon solution. W(r) vs log(1+r): (1.9MB MPEG). Note that the code used to generate these results uses Schwarzschild-like coordinates, in which black hole calculations necessarily crash ("blow up") on a dynamic time scale due to the development of a coordinate singularity. In particular, in the following sequence, the bottom evolution actually stops at about t=60.

Fine-Tuning to Form Hairy (Colored) Black Holes

In this case the initial data is of the form 
 W(r,0) = Kink(r; p_1, p_2)
where Kink(r) is a kink profile with two parameters, p_1 and p_2, governing the center and thickness of the kink, respectively. Solutions generated by varying p_1 and p_2 exhibit both types of transitions displayed above. In addition, in the super-critical regimes, fine tuning generates a one-parameter family of intermediate attractors (critical solutions, one-mode-unstable solutions), namely the static, hairy (or colored) black holes found by Bizon. (A convenient parametrization of the family is the area, or equivalently radius, r_h, of the horizon). These critical solutions separate "generalized Type I" black holes from "generalized Type II" black holes---the former end up with most of the mass of the YM "hair", while in the latter case, most of the "hair" is radiated away. ("Kinkiness" is again preserved.)
  1. Weak field evolution, W(r) vs r: (1.0MB MPEG)
  2. Near-critical evolutions. Here the intermediate attractor is a static colored black hole. W(r) vs log(1+r): (1.6MB MPEG) The code used for these computations uses maximal slicing and horizon excising techniques---this enables accurate, stable, long-time simulations of dynamic black-hole spacetimes.
For more details see gr-qc/9603051 (Type I and II behaviour) and gr-qc/9903081 (Tuning to hairy black holes).
Created/maintained by M.W. Choptuik: [mail] [URL]. 
Supported by NSERC, CIAR and NSF PHY9722068.
Copyright 1999 M.W Choptuik, E. Hirschmann and R.L. Marsa