The Interior Volume of Kerr-AdS Black Holes

The interior volume of black holes as defined by Christodoulou and Rovelli exhibits many surprising features. For example, it increases with time, even under Hawking evaporation. For some black holes, the interior volume is not even a monotonic increasing function of its area, which means one cannot infer how large a black hole is by just looking from the outside. Such a notion of volume, however, turns out to be useful in the context of holography, as it seems to be dual to the complexity of the boundary field theory. In this study, we investigate the properties of the interior volume of 4-dimensional Kerr-AdS black holes, fixing either the mass parameter $M$ or the physical mass $E$, whilst varying the values of the cosmological constant. We found that the volume as a function of the radial coordinate features a "double lobe" while fixing $M$, whereas fixing $E$ yields behaviors that are qualitatively similar to the asymptotically flat case. We briefly comment on the holographic complexity of Kerr-AdS black holes.