Interior Dynamics of Neutral and Charged Black Holes in f(R) Gravity

In this paper, we explore the interior dynamics of neutral and charged black holes in f (R) gravity. We transform f (R) gravity from the Jordan frame into the Einstein frame and simulate scalar collapses in flat, Schwarzschild, and Reissner-Nordstrom geometries. In simulating scalar collapses in Schwarzschild and Reissner-Nordstrom geometries, Kruskal and Kruskal-like coordinates are used, respectively, with the presence of f' and a physical scalar field being taken into account. The dynamics in the vicinities of the central singularity of a Schwarzschild black hole and of the inner horizon of a Reissner-Nordstrom black hole is examined. Approximate analytic solutions for different types of collapses are partially obtained. The scalar degree of freedom phi, transformed from f', plays a similar role as a physical scalar field in general relativity. Regarding the physical scalar field in f (R) case, when d phi/dt is negative (positive), the physical scalar field is suppressed (magnified) by phi, where t is the coordinate time. For dark energy f (R) gravity, inside black holes, gravity can easily push f' to 1. Consequently, the Ricci scalar R becomes singular, and the numerical simulation breaks down. This singularity problem can be avoided by adding an R-2 term to the original f (R) function, in which case an infinite Ricci scalar is pushed to regions where f' is also infinite. On the other hand, in collapse for this combined model, a black hole, including a central singularity, can be formed. Moreover, under certain initial conditions, f' and R can be pushed to infinity as the central singularity is approached. Therefore, the classical singularity problem, which is present in general relativity, remains in collapse for this combined model.