More on area density of localization-entropy and problematization of black hole entropy

The holographic boundary projection method for the study of vacuumpolarization caused thermal aspects is extended to double cone localized algebras. As in the previous case of wedge algebras, this algebra in its restricted vacuum state is identical to its holographic projection onto its causal horizon and can be approximated from the inside in analogy to the thermodynamic limit approximation. The spectrum of the modular “Hamiltonian” of each approximand is discrete and converges against the desired continuous modular Hamiltonian. In the correspondence to the standard thermodynamic limit two of the length dimensions of the volume factor correspond to the transverse extension (the area factor l) of the holographic projection and the remaining third dimension passes to the inverse of the lightlike distance ε of the approximand from its limit; hence altogether the localization entropy behaves as l× 1 ε with the proportionality constant being determined by the holographic matter content. The inverse relationship used the conformal equivalence between short and long distances in an essential way. The result leads to a fresh look at the problematization of black hole