Statistical Entropy of BTZ Black Hole in Higher Curvature Gravity

TL;DR

This paper demonstrates that the BTZ black hole in general diffeomorphism-invariant (higher curvature) gravity has a statistical entropy, computed via AdS$_3$/CFT and Cardy formula, that matches the geometrical Noether-charge entropy $S_{IW}$. By transforming to the Einstein frame and deriving the central charge $C=(l/2G)\, g^{\mu\nu} (\partial f/\partial R_{\mu\nu})$, the authors show the Virasoro eigenvalues scale as $\lambda=\Omega\tfrac{1}{2}(M+J)$ and $\tilde{\lambda}=\Omega\tfrac{1}{2}(M-J)$, leading to $S_C=\Omega\,\frac{\pi}{4G}[\sqrt{8Gl(M+J)}+\sqrt{8Gl(M-J)}]$, which coincides with the geometrical entropy computed from the Noether charge. An explicit example with $f=R+aR^2+bR_{\mu\nu}R^{\mu\nu}-2\Lambda$ shows the conformal factor $\Omega$ multiplies both entropies yet preserves their equality. The results support the universality of entropy in diffeomorphism-invariant theories and connect black hole thermodynamics to boundary CFT unitarity and cosmic censorship considerations, with potential extensions to higher dimensions.

Abstract

For the BTZ black hole in the Einstein gravity, a statistical entropy has been calculated to be equal to the Bekenstein-Hawking entropy. In this paper, the statistical entropy of the BTZ black hole in the higher curvature gravity is calculated and shown to be equal to the one derived by using the Noether charge method. This suggests that the equivalence of the geometrical and statistical entropies of the black hole is retained in the general diffeomorphism invariant theories of gravity. A relation between the cosmic censorship conjecture and the unitarity of the conformal field theory on the boundary of AdS is also discussed.