Coarse Graining Holographic Black Holes in Higher Curvature Gravity

Abstract

We consider the holographic description of the dynamical black hole entropy in $f(R)$ higher curvature gravity proposed by Hollands-Wald-Zhang. On the bulk side, we show that the coarse-grained entropy (outer entropy) of a generalized marginally trapped surface corresponds precisely to the Wald entropy associated with this surface. To get this result, we first formulate the AdS/CFT correspondence in the Einstein frame and derive the correspondence between von Neumann entropy of the Einstein frame and the $f(R)$ frame. This facilitates the derivation of the correspondence between the two outer entropies in the two frames. Furthermore, we directly derive a focusing theorem associated with generalized expansion in $f(R)$ gravity. We then formulate how to construct the stationary null hypersurface for the generalized expansion and the junction condition to glue a hypersurface in $f(R)$ gravity. Combining these results, we derive the expression for the entropy in the $f(R)$ frame and identify its holographic dual.

Figures (4)

• Figure 1: (a) is the Schwinger-Keldysh construction for $\text{Tr}\rho(t)$. (b) is the Schwinger-Keldysh construction for the reduced density matrix $\rho_{A}(t)$.
• Figure 2: This diagram is about the path integral construction of reduced density matrix $\rho_{A}(t)$ in the bulk. The gravity correspondence of tracing the degree of freedom of $A^{c}$ is gluing the geometry across $\mathcal{R}_{A^{c}}$.
• Figure 3: This is an example of $f"(R)$. We use $R_{\alpha}$, $R_{\beta}$ denote the discrete zeros, $\alpha$ and $\beta$ are used to label different zeros. And use $I^{j}_{R}$ denote the interval zero, that is, if $R$ is belong to the interval $I^{j}_{R}$, then $f"(R)=0$. Here $j$ is used to label different interval zero.
• Figure 4: This is an example of $f"(R)$. We use $R_{\alpha}$, $R_{\beta}$ denote the discrete zeros, $\alpha$ and $\beta$ are used to label different zeros. And use $I^{j}_{R}$ denote the interval zero, that is, if $R$ is belong to the interval $I^{j}_{R}$, then $f"(R)=0$. Here $j$ is used to label different interval zero.