Abstract
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-Nordström geometries. In simulating scalar collapses in Schwarzschild and Reissner-Nordström 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-Nordström black hole is examined. Approximate analytic solutions for different types of collapses are partially obtained. The scalar degree of freedom Φ, 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 dΦ/dt is negative (positive), the physical scalar field is suppressed (magnified) by Φ, 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 R2 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.
Highlights
The internal structure of black holes and spacetime singularities are key topics in gravitation and cosmology [1,2,3,4,5], and are great platforms to explore the connection between classical and quantum physics
We transform the dynamical system in f (R) gravity from the Jordan frame into the Einstein frame
For convenience, we transform f (R) gravity from the current frame, which is usually called the Jordan frame, into the Einstein frame, in which the formalism can be formally treated as Einstein gravity coupled to a scalar field
Summary
The internal structure of black holes and spacetime singularities are key topics in gravitation and cosmology [1,2,3,4,5], and are great platforms to explore the connection between classical and quantum physics. Ever since the foundation of general relativity, people have been trying to go beyond it This endeavor arises from unifying gravitation and quantum mechanics, and addressing some cosmological problems, including the singularity problem in the early Universe and the dark energy problem in the late Universe. For a review of modified gravity theories, see Reference [7]. Gravitational collapses in some modified gravity theories have been studied numerically. Spherical collapse of a neutral scalar field in a given spherical, charged black hole in Brans-Dicke theory was investigated in Reference [17]. Spherical collapses of a charged scalar field in dilaton gravity and f (R) gravity were explored in References [18] and [19], respectively. Asymptotic analysis was implemented in the vicinity of the singularity of a formed black hole
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