Dirty black holes: Entropy as a surface term.

It is by now clear that the naive rule for the entropy of a black hole, (entropy) = 1/4 (area of event horizon), is violated in many interesting cases. Indeed, several authors have recently conjectured that in general the entropy of a dirty black hole might be given purely in terms of some surface integral over the event horizon of that black hole. A formal proof of this conjecture, using Lorentzian signature techniques, has recently been provided by Wald. This note performs two functions. Firstly, by extending a previous analysis due to the present author [Physical Review D48, ???? (15 July 1993)] it is possible to provide a rather different proof of this result — a proof based on Euclidean signature techniques. The proof applies both to arbitrary static [aspheric] black holes, and also to arbitrary stationary axisymmetric black holes. The total entropy is