Critical behavior near the singularity in a scalar field collapse

The critical behavior of a field near the singularity in the spherical symmetric scalar field collapse is studied. We consider the supercritical case in which a black hole singularity is formed by the strong imploding scalar held at some advanced time v. We find that the field evolution near the singularity can be divided into the following two stages. The spacetime structure near the onset of the first singularity is shown to be well approximated by a self-similar solution. In this self-similar stage the horizon mass linearly increases with v. After the self-similar stage ends, the logarithmic behavior becomes remarkable and the system evolves toward the Schwarzschild spacetime with the advanced time. According to this evolution, the strength of the curvature singularity decreases as I similar or equal to chi(-2(f+2)/(1+f)), where chi is the circumference radius and f runs from 0 to 1 with the advanced time v. In the final stage of gravitational collapse (v -->infinity), the scalar filed dies away as exp(-kv) inside the apparent horizon and the system smoothly approaches the static Schwarzschild spacetime. We find that the power-law behavior of the black hole mass is crucially related to the logarithmic behavior of the field. We also propose our main idea that the critical exponent beta of the mass power law is a decreasing function of v, which is due to the area law of the apparent horizon.