Understanding critical collapse of a scalar field

Abstract

I construct a spherically symmetric solution for a massless real scalar field minimally coupled to general relativity which is discretely self-similar (DSS) and regular. This solution coincides with the intermediate attractor found by Choptuik in critical gravitational collapse. The echoing period is $\ensuremath{\Delta}=3.4453\ifmmode\pm\else\textpm\fi{}0.0005$. The solution is continued to the future self-similarity horizon, which is also the future light cone of a naked singularity. The scalar field and metric are ${C}^{1}$ but not ${C}^{2}$ at this Cauchy horizon. The curvature is finite nevertheless, and the horizon carries regular null data. These are very nearly flat. The solution has exactly one growing perturbation mode, thus confirming the standard explanation for universality. The growth of this mode corresponds to a critical exponent of $\ensuremath{\gamma}=0.374\ifmmode\pm\else\textpm\fi{}0.001$, in agreement with the best experimental value. I predict that in critical collapse dominated by a DSS critical solution, the scaling of the black hole mass shows a periodic wiggle, which like $\ensuremath{\gamma}$ is universal. My results carry over to the free complex scalar field. Connections with previous investigations of self-similar scalar field solutions are discussed, as well as an interpretation of $\ensuremath{\Delta}$ and $\ensuremath{\gamma}$ as anomalous dimensions.