Quantum f(R) gravity and AdS/CFT

Abstract

We propose to study the entanglement entropy in braneworld modified gravity. We show that the $d+1$-dimensional action $I_{d+1}$ is the dual of $d-1$% -dimensional entanglement entropy $S_{d-1}^{EE}$. Moreover, we remark that the generalization of action-entropy shows us a new form of $f(R)$ gravity which is in good agreement with the most choices of $f(R)$ gravity in the literature. We show also that there are two copies of the AdS spaces in holographic entanglement $f(R)$ gravity. We have proposed that the time is a holographic projection of the hidden $d$-brane on the visible $d$-brane. To determine the geometry of this holographic projection, we have used the $% AdS_{2}$ geometry of the black hole. Namely, the past of events is memorized over an $\mathbb{S}^{1}$. We show that the time projection is a stereographic projection from a time sphere $\mathbb{S}^{d-2}$ on the hidden $d$-brane to the visible $d$-brane. We suggest that there is a difference between the time in classical gravity and the time in quantum gravity.