Dirty black holes: Entropy versus area

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=${\mathit{kA}}_{\mathit{H}}$/(4${\mathit{l}}_{\mathit{P}}^2$). For example, the ‘‘entropy =(1/4) area’’ law holds for Schwarzschild, Reissner-Nordström, Kerr-Newman, and dilatonic black holes. On the other hand, the ‘‘entropy =(1/4) area’’ law fails for various types of (Riemann${)}^{\mathit{n}}$ gravity, Lovelock gravity, the Einstein-Hilbert-Born-Infield system, 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= ${\mathit{kA}}_{\mathit{H}}$ / 4${\mathit{l}}_{\mathit{P}}^2$ + 1 / ${\mathrm{T}}_{\mathit{H}}$ ${\mathcal{F}}_{\mathrm{\Sigma }}${ρL-${\mathcal{L}}_{\mathit{E}}$}${\mathit{K}}^{\mathrm{\mu }}$d${\mathrm{\Sigma }}_{\mathrm{\mu }}$+${\mathcal{F}}_{\mathrm{\Sigma }}$${\mathit{SV}}^{\mathrm{\mu }}$d${\mathrm{\Sigma }}_{\mathrm{\mu }}$. ($K sup mu— is the timelike Killing vector, ${\mathit{V}}^{\mathrm{\mu }}$ the four-velocity of a corotating 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.