  Considerable interest has recently been expressed in the entropy versus area
relationship for ``dirty'' black holes --- black holes in interaction with
various classical matter fields, distorted by higher derivative gravity, or
infested with various forms of quantum hair. In many cases it is found that the
entropy is simply related to the area of the event horizon: S = k
A_H/(4\ell_P^2). For example, the ``entropy = (1/4) area'' law *holds* for:
Schwarzschild, Reissner--Nordstrom, Kerr--Newman, and dilatonic black holes. On
the other hand, the ``entropy = (1/4) area'' law *fails* for: various types of
(Riemann)^n gravity, Lovelock gravity, and various versions of quantum hair.
The pattern underlying these results is less than clear. This paper
systematizes these results by deriving a general formula for the entropy: S =
{k A_H/(4\ell_P^2)}
  + {1/T_H} \int_\Sigma [rho - {L}_E ] K^\mu d\Sigma_\mu
  + \int_\Sigma s V^\mu d\Sigma_\mu. (K^\mu is the timelike Killing vector,
V^\mu the four velocity of a co--rotating observer.) If no hair is present the
validity of the ``entropy = (1/4) area'' law reduces to the question of whether
or not the Lorentzian energy density for the system under consideration is
formally equal to the Euclideanized Lagrangian. ****** To appear in Physical
Review D 15 July 1993 ****** [Stylistic changes, minor typos fixed, references
updated, discussion of the Born-Infeld system excised]
