We study the spherically symmetric gravitational collapse of massless scalar matter field in asymptotic flat spacetime in the Starobinsky $R^2$ gravity, one specific model in the $f(R)$ gravity. In the Einstein frame of $f(R)$ gravity, an additional scalar field arises due to the conformal transformation. We find that in addition to the usual competition between gravitational energy and kinetic energy in the process of gravitational collapse, the new scalar field brought by the conformal transformation adds one more competing force in the dynamical system. The dynamical competition can be controlled by tuning the amplitudes of the initial perturbations of the new scalar field and the matter field. To understand the physical reasons behind these phenomena, we analyze the gravitational potential behavior and calculate the Ricci scalar at center with the change of initial amplitudes of perturbations. We find rich physics on the formation of black holes through gravitational collapse in $f(R)$ gravity.

• Received 18 August 2016

DOI:https://doi.org/10.1103/PhysRevD.94.104013