Quasi-local conserved charges in the Einstein-Maxwell theory

TL;DR

This paper develops a method to define finite, conserved quasi-local charges in Einstein-Maxwell theory with negative cosmological constant by combining diffeomorphism and gauge transformations, addressing divergence issues in previous approaches.

Contribution

It introduces a field-dependent combined transformation approach to obtain finite conserved charges in Einstein-Maxwell theory, especially for asymptotically AdS$_{3}$ spacetimes.

Findings

Successfully defines finite conserved charges for asymptotically AdS$_{3}$ Einstein-Maxwell solutions.

Addresses divergence problems in previous quasi-local charge definitions.

Provides a new method applicable to field-dependent symmetry generators.

Abstract

In this paper we consider the Einstein-Maxwell theory and define a combined transformation composed of diffeomorphism and $U(1)$ gauge transformation. For generality, we assume that the generator $\chi$ of such transformation is field-dependent. We define the extended off-shell ADT current and then off-shell ADT charge such that they are conserved off-shell for the asymptotically field-dependent symmetry generator $\chi$. Then, we define the conserved charge corresponding to the asymptotically field-dependent symmetry generator $\chi$. We apply the presented method to find the conserved charges of the asymptotically AdS$_{3}$ spacetimes in the context of the Einstein-Maxwell theory in three dimensions. Although the usual proposal for the quasi local charges provides divergent global charges for the Einstein-Maxwell theory with negative cosmological constant in three dimensions, here we avoid this problem by introducing proposed combined transformation $\chi$.