1993/07/30 by Matt Visser · 1 citation
Physics and Astronomy · #hep-th #gr-qc
paper · pdf · doi:10.1103/physrevd.48.5697
published as Phys.Rev. D48 (1993) 5697-5705 · Plain LaTeX. 19 pages
arxiv created 1993/07/30 · arxiv updated 2016/09/06
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, a rather different proof of this result is presented --- a proof based on Euclidean signature techniques. The total entropy is S = 1/4 k AH / lP2 + ∫H S √(g) d2x. The integration runs over a spacelike cross-section of the event horizon H. The surface entropy density, S, is related to the behaviour of the matter Lagrangian under time dilations. Secondly, 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 = 1/4 k AH / lP2 + 4 pi k/hbar ∫H partial L / partial Rμνλρ g^⊥μλ g^⊥νρ √(g) d2x . The symbol g^⊥μν denotes the projection onto the two-dimensional subspace orthogonal to the event horizon.