The definition of mass in asymptotically de Sitter space-times

The definition of invariant quantities due to Wald and collaborators is used to calculate the mass of a black-hole in asymptotically de Sitter space-time. The construction relies on the existence of a time-like Killing vector on a sphere surrounding the mass but does not require going to an asymptotic region, in particular the mass can be calculated exactly on a sphere inside the cosmological horizon. The formalism requires varying the background metric solution by a perturbation that satisfies the linearized equations of motion but need not share the Killing symmetry of the solution and is therefore applicable when a gravitational wave within the cosmological horizon perturbs a background metric with a time-like Killing vector. (note)