# Is the Ryu-Takayanagi (RT) formula calculating coarse-grained or fine-grained entropy?

I think it is computing the fine-grained entropy. However, I am confused by the case that when there is a black hole in the bulk. The Ryu-Takayanagi surface may include the horizon of the black hole in this case. However, the Bekenstein-Hawking entropy is a coarsed-grain entropy.
I have also heard that the RT formula is computing the coarse-grained entropy of the complementary region. What does this mean?

I cannot tell you how stoked I am to see a fellow AdS/CFT-er. For the sake of the answer and physicality, I will not work with Ryu-Takayanagi but instead the covariant description, given by the Hubeny-Rangamani-Takayanagi (HRT) prescription. That is, the entanglement entropy of a boundary subregion is computed by the area of a (minimal) extremal surface $$\mathcal{X}_{HRT}$$. To answer your question, HRT is a fine-grained quantity (now see the case for RT), and the essential way to see this is that this does not involve coarse-graining on thermal d.o.f. The Jaynes' approach to holographic entanglement entropy describes the maximization of some von Neumann entropy to find a coarse-graining procedure, so when you have to make sense of black hole entropy in a varying state, e.g. finding a holographic second law (using holographic screens), you would have to coarse-grain. This is referred to as the Engelhardt-Wall prescription, which is a very fascinating result. There are some other aspects to this, but I think this should be a good motivation for RT as well.
Edit: Let me also point out that RT + coarse-graining would not be useful. Primarily, since RT is the statement that the for a black hole, the RT surface $$\gamma _{A}$$ "wraps" around the black hole horizon, giving the entropy of the black hole. However, it does not have the bulk interpretation with different kinds of marginally trapped surfaces (called minimar surfaces; extremal surfaces like $$\mathcal{X}_{HRT}$$ are a class of minimar surfaces), meaning that one cannot give a nice generalized second law interpretation of the black hole. See Engelhardt and Wall and Engelhardt and Bousso's works on holographic screens inside AdS black holes. RT will not give such kind of a description (why?), and does not have any physical importance like HRT does.