Euclidean and Noetherian entropies in AdS space

We examine the Euclidean action approach, as well as that of Wald, to the entropy of black holes in asymptotically AdS spaces. From the point of view of holography these two approaches are somewhat complementary in spirit and it is not obvious why they should give the same answer in the presence of arbitrary higher derivative gravity corrections. For the case of the ${\mathrm{AdS}}_5$ Schwarzschild black hole, we explicitly study the leading correction to the Bekenstein-Hawking entropy in the presence of a variety of higher derivative corrections studied in the literature, including the Type IIB $R^4$ term. We find a nontrivial agreement between the two approaches in every case. Finally, we give a general way of understanding the equivalence of these two approaches.

Figure 1

RG flow between two holographic descriptions.