Generalized Black Hole Entropy is von Neumann Entropy

As originally proposed by Bekenstein, the generalized black hole entropy is the sum of the area of the black hole (divided by Newton’s constant) and the von Neumann entropy of quantum fields in the exterior. However, in the semiclassical limit (i.e. Newton’s constant -> 0) the black hole entropy diverges. Furthermore, the entropy of quantum fields in the exterior of a black hole is infinite due to the strong UV entanglement across the horizon (associated with a so-called Type III algebra).

It was proposed by Susskind and Uglam that the sum of these two quantities is better defined than the individual terms. Remarkably, it was shown by Witten that the algebra of physical “gravitationally dressed” observables in the exterior of the black hole is well-defined (and given by a “Type II” algebra). Furthermore, the entropy of “semi-classical” states is equivalent to the generalized entropy. However, these arguments relied on the existence of an equilibrium thermal (KMS) state and thus do not apply to, for example, black holes formed from gravitational collapse, Kerr black holes, or black holes in asymptotically de Sitter space. 

We show that the algebra of observables in the “exterior” of any Killing horizon always contains a Type II factor "localized" on the horizon and, consequently, the entropy of semi-classical states is the generalized entropy. I will illustrate this with two examples of (1) a black hole in asymptotically flat spacetime and (2) black holes in asymptotically de Sitter. In all cases, the von Neumann entropy for semiclassical states is given by the generalized entropy.