Abstract
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 Ref. (Barnich and Troessaert, 12:105 2011). These charges are as representations of the $$\hbox {BMS}_4$$ 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 Ref. (Donnay et al., arXiv:1607.05703 [hep-th], 2016).
Highlights
Introduction3 we try to find the expression of the conserved charges associated to the near horizon symmetry of the non-extremal black hole solution of general relativity in four dimensions by the Noether method
We have considered a generally covariant theory of gravity, and found an off-shell conserved current Eq (5) by virtue of the Bianchi identities
The generalized off-shell ADT current Eq (11) is conserved for the asymptotically field-independent Killing vector fields and field-independent Killing vector fields admitted by spacetime everywhere
Summary
3 we try to find the expression of the conserved charges associated to the near horizon symmetry of the non-extremal black hole solution of general relativity in four dimensions by the Noether method. Our result for this case is exactly matched with that of Ref. The symplectic current (9) is conserved on-shell and it gives us conserved charges correspond to asymptotically field-independent Killing vectors. [54], the authors have generalized the off-shell ADT current as follows: JGμADT(g, δg; ξ ) = JAμDT(g, δg; ξ ) This current is conserved off-shell for the asymptotically fieldindependent Killing vector fields as well as field-independent. This formula is independent of δξ , we have shown that this formula is valid for the case in which ξ is field-dependent
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