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Didn't Hawking propose that Black Holes were detectable because the boundary quantum uncertainty at a critical distance around a black hole would produce a net detectable outflow of photons? A black hole detector seems like it might be a keen working device to me. Or is the critical region around a black hole entropy-irrelevant?

It was once thought that black holes did raise problems for entropy and the second law. Does a black hole retain all the entropy of all the stuff that it eats and if so how does it show it? That kind of thing. Even Hawking thought that BHs violated the second law.

Kip Thorne*, after many convolutions, describes the bottom line as follows: A black hole's entropy is the logarithm of the number of ways that the hole could have been made. This entropy does not disappear but is retained in the surface area of the hole's event horizon as the evaporation-radiation particle "atmosphere."

The second law is obeyed. The dim glow of a black hole may be a free lunch to local systems, but only in the same way the sun is to local systems. The entropy of the universe goes up when the black hole eats something. It goes up more as the BH evaporates.

* Black Holes and Time Warps, W. W. Norton, 1994.

A black hole's entropy is the logarithm of the number of ways that the hole could have been made.

Hope you don't mind a totally amateurish question. Doesn't entropy have to be expressed in terms of dimensions of some sort, like energy/temperature, or temperature/energy? Can a logarithm have such dimensions?