Conserved charges of black holes in Weyl and Einstein–Gauss–Bonnet gravities

Abstract

An off-shell generalization of the Abbott-DeserTekin (ADT) conserved charge was recently proposed by Kim et al. They achieved this by introducing off-shell Noether currents and potentials. In this paper, we construct the crucial off-shell Noether current by the variation of the Bianchi identity for the expression of EOM, with the help of the property of Killing vector. Our Noether current, which contains an additional term that is just one half of the Lie derivative of a surface term with respect to the Killing vector, takes a different form in comparison with the one in their work. Then we employ the generalized formulation to calculate the quasi-local conserved charges for the most general charged spherically symmetric and the dyonic rotating black holes with AdS asymptotics in four-dimensional conformal Weyl gravity, as well as the charged spherically symmetric black holes in arbitrary dimensional Einstein-Gauss-Bonnet gravity coupled to Maxwell or nonlinear electrodynamics in AdS spacetime. Our results confirm those obtained through other methods in the literature.

Thermodynamics of static dyonic AdS black holes in the ω -deformed Kaluza–Klein gauged supergravity theory

“…From the off-shell ADT potential, one can easily construct the quasi-local charge. Note that the same result can be arrived at by varying Bianchi identity [28].…”

“…[26]. The quasi-local ADT formalism [20] is a novel way to calculate the conserved charge at finite spacetime domains and receives a lot of recent attention [27,28]. Speaking roughly, one can establish a one-to-one correspondence between the ADT potential and the off-shell linear Noether potential by considering an appropriate variation of metric.…”

AdS and Lifshitz black hole solutions in conformal gravity sourced with a scalar field

“…See also the Refs. [83,84] where the off-shell Noether current was constructed using the variation of the Bianchi identity. In those references the method was applied to compute the mass of some black hole solutions in conformal Weyl gravity, in Einstein Gauss-Bonnet gravity and in Horndeski theory.…”

Revisiting the ADT mass of the five-dimensional rotating black holes with squashed horizons

“…That is to say, the background metric is off-shell. However, strictly speaking, the original ADT current requires that the background metric is on-shell although the potential corresponding to such a current is in agreement with the off-shell generalised one in [37][38][39][40]. Thus, in order to fully obey the spirit of the usual ADT formalism [32][33][34][35][36], here we shall look for an appropriate on-shell reference metric to recalculate the ADT mass of these black holes.…”

“…In this section, to compare with the original ADT formulation, we shall make use of its off-shell generalisation 1 recently proposed in [37] and developed in [38][39][40] to compute the mass of the squashed rotating KK black holes. In contrast to the original ADT formalism, the off-shell generalised one possesses the dramatically different feature that all the fields are linearised about arbitrary background fields.…”