We carry out numerical simulations of the collapse of a complex rotating scalar field of the form $\mathrm{\Psi }(t,r,\theta )=e^{im\theta }\mathrm{\Phi }(t,r)$, giving rise to an axisymmetric metric, in $2+1$ spacetime dimensions with cosmological constant $\mathrm{\Lambda }<0$, for $m=0$, 1, 2, for four one-parameter families of initial data. We look for the familiar scaling of black hole mass and maximal Ricci curvature as a power of $|p-p_{*}|$, where $p$ is the amplitude of our initial data and $p_{*}$ some threshold. We find evidence of Ricci scaling for all families, and tentative evidence of mass scaling for most families, but the case $m>0$ is very different from the case $m=0$ we have considered before: the thresholds for mass scaling and Ricci scaling are significantly different (for the same family); scaling stops well above the scale set by $\mathrm{\Lambda }$, and the exponents depend strongly on the family. Hence, in contrast to the $m=0$ case, and to many other self-gravitating systems, there is only weak evidence for the collapse threshold being controlled by a self-similar critical solution and no evidence for it being universal.

• Received 16 February 2017

DOI:https://doi.org/10.1103/PhysRevD.95.084001