Spherical collapse in f(R) gravity

The spherical scalar collapse in f(R) gravity is studied numerically in Einstein frame. The dark-energy-oriented f(R) theory is a modification of general relativity at low-curvature scale. The scalar curvature decreases when the scalar field collapses to form a black hole. Correspondingly, gravity transits from general relativity to f(R) gravity, and the scalar degree of freedom f' is released from a coupled state to a light state. Compared to the gravitational force from the scalar sphere (which collapses to form a black hole in a later stage), the left barrier of the potential for f' is not steep enough. As a result, near the singularity of the black hole, f' will cross the minimum of the potential and then approach zero. As the singularity is approached, the equations of motion are dominated by time derivative terms; the contributions from other terms, including those from the scalar fields, are negligible. The spacetime and the scalar fields are described by Kasner solution. These results support the Belinskii, Khalatnikov, and Lifschitz (BKL) conjecture well. The final state of the collapse is discussed.