Critical behaviour in gravitational collapse of radiation fluid --- A renormalization group (linear perturbation) analysis ---

A scenario is presented, based on renormalization group (linear perturbation) ideas, which can explain the self-similarity and scaling observed in a numerical study of gravitational collapse of radiation fluid. In particular, it is shown that the critical exponent $\beta$ and the largest Lyapunov exponent $\mathrm{R}\mathrm{e}\kappa$ of the perturbation is related by $\beta= ({\rm Re, } \kappa) ^{-1}$. We find the relevant perturbation mode numerically, and obtain a fairly accurate value of the critical exponent $\beta \simeq 0.3558019$, also in agreement with that obtained in numerical simulation.