Angular momentum at the black hole threshold

Assuming the existence of a spherically symmetric, self-similar solution of general relativity with matter that is locally an attractor of codimension one on the black hole threshold, we calculate the dependence of the black hole angular momentum on the initial data, near the black hole threshold and near spherical symmetry. As any smooth one-parameter family of initial data crosses the threshold, the black hole angular momentum vector rotates in space an infinite number of times while its length decreases as a power of distance to the black hole threshold. The new critical exponent and the “angular frequency” are calculated as μ ' 0.799 and ω ' 0.231 in the example of p = ρ/3 perfect fluid matter, where the assumption is known to hold.