Hawking radiation from AdS black holes

We investigate Hawking radiation from black holes in $\mathrm{}(d+1\mathrm{})\mathrm{}$-dimensional anti–de Sitter space. We focus on s waves, make use of the geometrical optics approximation, and follow three approaches to analyze the radiation. First, we compute a Bogoliubov transformation between Kruskal and asymptotic coordinates and compare the different vacua. Second, following a method due to Kraus, Parikh, and Wilczek, we view Hawking radiation as a tunneling process across the horizon and compute the tunneling probability. This approach uses an anti–de Sitter version of a metric originally introduced by Painlevé for Schwarzschild black holes. From the tunneling probability one also finds a leading correction to the semiclassical emission rate arising from back reaction to the background geometry. Finally, we consider a spherically symmetric collapse geometry and the Bogoliubov transformation between the initial vacuum state and the vacuum of an asymptotic observer.