### 5.1 Perturbative approach to angular momentum

We have already mentioned that when angular momentum is small, a critical exponent for can be
derived in perturbation theory. This has been done for the perfect fluid (see Section 4.2) in first-order
perturbation theory and for the massless scalar field (see Section 3.7), where second-order perturbation
theory in the scalar field is necessary to obtain an angular momentum perturbation in the stress-energy
tensor [87]. However, neither of the predicted angular momentum scaling laws has been verified in
numerical evolutions.
For a perfect fluid with EOS with 0 < k < 1/9, precisely one mode that carries angular
momentum is unstable, and this mode and the known spherical mode are the only two unstable modes of
the spherical critical solution. (Note by comparison that from dimensional analysis one would not
expect an uncharged critical solution to have a growing perturbation mode carrying charge.) The
presence of two growing modes of the critical solution is expected to give rise to interesting
phenomena [104]. Near the critical solution, the two growing modes compete. J and M of the final
black hole are expected to depend on the distance to the black hole threshold and the angular
momentum of the initial data through universal functions of one variable that are similar to
“universal scaling functions” in statistical mechanics (see also the end of Section 2.7). While
they have not yet been computed, these functions can in principle be determined from time
evolutions of a single 2-parameter family of initial data, and then determine J and M for all initial
data near the black hole threshold and with small angular momentum. They would extend the
simple power-law scalings of J and M into a region of initial data space with larger angular
momentum.