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 paper performs two functions. First, by extending a previous analysis due to the present author [Phys. Rev. D 48, 583 (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 S=${\mathit{kA}}_{\mathit{H}}$/4${\mathit{l}}_{\mathit{P}}^2$+${\mathcal{F}}_{\mathit{H}}$S√2g ${\mathit{d}}^2$x. The integration runs over a spacelike cross section of the event horizon H. The surface entropy density scrS is related to the behavior of the matter Lagrangian under time dilations. Second, I shall consider the specific case of Einstein-Hilbert gravity coupled to an effective Lagrangian that is an arbitrary function of the Riemann tensor (though not of its derivatives). In this case a more explicit result is obtained: S=${\mathit{kA}}_{\mathit{H}}$/4${\mathit{l}}_{\mathit{P}}^2$+4π k/h/ ${\mathcal{F}}_{\mathit{H}}$ ∂L/∂${\mathit{R}}_{\mathrm{\mu }\nu \lambda \mathrm{\rho }}$ ${\mathit{g}}_{\mathrm{\mu }\lambda }^{\mathrm{\perp }}$${\mathit{g}}_{\nu \mathrm{\rho }}^{\mathrm{\perp }}$ √2g ${\mathit{d}}^2$x. The symbol ${\mathit{g}}_{\mathrm{\mu }\nu }^{\mathrm{\perp }}$ denotes the projection onto the two-dimensional subspace orthogonal to the event horizon. Although the derivation exhibited in this paper proceeds via Euclidean signature techniques the result can be checked against certain special cases previously obtained by other techniques, e.g., (Ricci${)}^{\mathit{n}}$ gravity, ${\mathit{R}}^{\mathit{n}}$ gravity, and Lovelock gravity.