Perturbations and critical behavior in the self-similar gravitational collapse of a massless scalar field

This paper studies the perturbations of the continuously self-similar critical solution of the gravitational collapse of a massless scalar field (Roberts solution). The perturbation equations are derived and solved exactly. The perturbation spectrum is found to be not discrete, but occupying a continuous region of the complex plane. The renormalization group calculation gives the value of the mass-scaling exponent $\beta =1.\mathrm{}$