Coarse Graining Holographic Black Holes in Higher Curvature Gravity

We consider the holographic description of the dynamical black hole entropy in $f(R)$ higher curvature gravity which 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 the von Neumann entropy in 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 directly derive the expression for the outer entropy in the $f(R)$ frame and identify its holographic dual.