On Euclidean and Noetherian Entropies in AdS Space
Suvankar Dutta, Rajesh Gopakumar
TL;DR
The paper investigates how two fundamental methods for computing black hole entropy—the Euclidean action approach and Wald's Noether-charge prescription—agree in asymptotically AdS spacetimes when higher-derivative corrections are present. It performs explicit leading alpha-prime corrections for the AdS5 Schwarzschild black hole, examining Gauss–Bonnet, R^2, and R^4 terms, and finds exact agreement between the two approaches in each case. A general argument is provided that connects the Euclidean background subtraction to Wald's Noether-charge framework via a Noetherian notion of mass in AdS, elucidating why the two methods must yield the same result in holographic settings. The results bolster the consistency of holographic entropy descriptions and hint at a deeper RG-flow interpretation linking the boundary (UV) and horizon (IR) pictures in AdS/CFT, with practical implications for computing quantum corrections to black hole entropy.
Abstract
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 $AdS_5$ 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 non-trivial agreement between the two approaches in every case. Finally, we give a general way of understanding the equivalence of these two approaches.