Critical phenomena in the Einstein-massless-Dirac system

We investigate the general relativistic collapse of spherically symmetric, massless spin-$\frac{1}{2}$ fields at the threshold of black hole formation. A spherically symmetric system is constructed from two spin-$\frac{1}{2}$ fields by forming a spin singlet with no net spin-angular momentum. We study the system numerically and find strong evidence for a type II critical solution at the threshold between dispersal and black hole formation, with an associated mass scaling exponent $\gamma \sim \mathrm{0.26.}$ Although the critical solution is characterized by a continuously self-similar (CSS) geometry, the matter fields exhibit discrete self-similarity with an echoing exponent $\Delta \sim \mathrm{1.34.}$ We then adopt a CSS ansatz and reduce the equations of motion to a set of ODEs. We find a solution of the ODEs that is analytic throughout the solution domain, and show that it corresponds to the critical solution found via dynamical evolutions.