Gravitational collapse of massless scalar field and radiation fluid

Several classes of conformally-flat and spherically symmetric exact solutions to the Einstein field equations coupled with either a massless scalar field or a radiation fluid are given, and their main properties are studied. It is found that some represent the formation of black holes due to the gravitational collapse of the matter fields. When the spacetimes have continuous self-similarity (CSS), the masses of black holes take a scaling form $M_{BH} \propto (P - P^{*})^{\gamma}$, where $\gamma = 0.5$ for massless scalar field and $\gamma = 1$ for radiation fluid. The reasons for the difference between the values of $\gamma$ obtained here and those obtained previously are discussed. When the spacetimes have neither CSS nor DSS (Discrete self-similarity), the masses of black holes always turn on with finite non-zero values.