A note on the mass of Kerr-AdS black holes in the off-shell generalized ADT formalism

In this note, the off-shell generalized Abbott-Deser-Tekin (ADT) formalism is applied to explore the mass of Kerr-anti-de Sitter (Kerr-AdS) black holes in various dimensions within asymptotically rotating frames. The cases in four and five dimensions are explicitly investigated. It is demonstrated that the asymptotically rotating effect may make the charge be non-integrable or unphysical when the asymptotic non-rotating timelike Killing vector associated with the charge is allowed to vary and the fluctuation of the metric is determined by the variation of all the mass and rotation parameters. To avoid such a dilemma, we can let the non-rotating timelike Killing vector be fixed or perform calculations in the asymptotically static frame. Our results further support that the ADT formalism is background-dependent.