BMS type symmetries at null-infinity and near horizon of non-extermal black holes

In this paper we consider a generally covariant theory of gravity, and extend the generalized off-shell ADT current such that it becomes conserved for field dependent (asymptotically) Killing vector field. Then we define the extended off-shell ADT current and the extended off-shell ADT charge. Consequently, we define the conserved charge perturbation by integrating from the extended off-shell ADT charge over a spacelike codimension two surface. Eventually, we use the presented formalism to find the conserved charge perturbation of an asymptotically flat spacetime. The conserved charge perturbation we obtain is exactly matched with the result of the paper \cite{6'}. These charges are as representations of the $BMS_4$ symmetry algebra. Also, we find that the near horizon conserved charges of a non-extremal black hole with extended symmetries are the Noether charges. For this case our result is also exactly matched with that of the paper \cite{15}.