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

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## Abstract

An off-shell generalization of the Abbott–Deser–Tekin (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.

### Keywords

Black Hole Black Hole Solution Bianchi Identity Surface Term Static Black Hole## 1 Introduction

Modified gravity theories that involves higher curvature terms in the Lagrangian have been extensively investigated, generally motivated by the intriguing feature that these higher curvature terms render the gravity theories perturbatively renormalizable in the quantization process [1]. A very natural higher-order derivative modification of general relativity is the fourth-order theories of gravitation, which includes the well-known theories of Weyl gravity and Einstein–Gauss–Bonnet gravity. To the former, its Lagrangian contains the square of the Weyl tensor, so it is invariant under the local conformal transformation of the metric. The Lagrangian for Einstein–Gauss–Bonnet gravity includes up to the term with quadratic Riemann tensor, which can be thought of as the higher curvature correction to general relativity in the low energy limit of heterotic string theory. Due to the salient properties of the two gravity theories, together with the AdS/CFT correspondence, a lot of efforts have been made in seeking asymptotically AdS black hole solutions in Weyl gravity [2, 3] and Einstein–Gauss–Bonnet gravity [4, 5, 6, 7, 8], to provide various interesting backgrounds of spacetime. Generally speaking, after obtaining a black hole solution, an important task is to identify its conserved charges, such as the energy and the angular momentum.

Till now several approaches have been proposed to compute the conserved charges of asymptotically AdS solutions, such as the so-called counterterm subtraction approach [9, 10] generalized from the Brown–York method [11], the Ashtekar–Magnon–Das formalism [12, 13], the covariant phase space approach [14, 15], the method [16, 17, 18, 19] developed by Barnich et al. and the Abbott–Deser–Tekin (ADT) formalism [20, 21, 22, 23]. Particularly, the ADT formalism, which is defined by the Noether potential got through the linearized perturbation for the expression of EOM in a fixed background of AdS spacetime, has made some progress on computation scheme for conserved charges of asymptotically AdS black holes in fourth-order gravity theories. Since the background metric is a vacuum solution of the equation of motion (EOM), the Noether potential in ADT formalism is on-shell. Recently, in Ref. [24], Kim, Kulkarni and Yi proposed a quasi-local formulation of conserved charges by generalizing the on-shell Noether potential in the ADT formalism to off-shell level, as well as Refs. [16, 17, 18, 19] to incorporate a single parameter path in the space of solutions into their definition. These modifications make it more operable to evaluate the Noether potential in terms of the corresponding current. The generalized formalism for the quasi-local conserved charges provides a more efficient way to compute the ADT conserved charges for covariant theories of gravity, and it has been extended to the theory of gravity with a gravitational Chern–Simons term [25] and the gravity theory in the presence of matter fields [26]. In [27], it was utilized to obtain the mass of the three- and five-dimensional Lifshitz black holes. To compare with the original ADT formalism, it is meaningful to employ this generalized quasi-local formulation to study the conserved charges in higher-order derivative gravity theories.

In this paper, to provide a deep understanding on the generalized ADT formalism proposed in [24], we derive the off-shell Noether current that educes the Noether potential finally entering into the formulation of conserved charges from different perspective. Our derivation endows the off-shell Noether current with a natural connection with its corresponding potential. Then we extend this formalism to investigate the quasi-local conserved charges of charged (rotating) black holes with AdS asymptotics in the two typical fourth-order derivative gravity theories: conformal Weyl gravity and Einstein–Gauss–Bonnet gravity. The remainder of this paper goes as follows. In Sect. 2, we give a brief review of the method in [24]. However, unlike there, we derive the off-shell Noether current and its corresponding potential through the variation of the Bianchi identity for the expression of EOM. Our results are formally different from those in [24]. In Sect. 3, we first present the explicit expressions of the off-shell Noether potentials in Weyl gravity. Then these quantities are applied to compute the mass of the most general static black hole and both the mass and the angular momentum of the dyonic rotating black hole in four-dimensional Weyl gravity. In Sect. 4, we calculate the energy of the general charged spherically symmetric black hole in arbitrary dimensional Einstein–Gauss–Bonnet gravity, coupled to Maxwell or nonlinear electrodynamics in AdS spacetime. The general formalizations of the Noether potentials for Einstein–Gauss–Bonnet gravity are also given. The last section is for our conclusions.

## 2 The generalized ADT formalism

^{1}is a surface term. To preserve the diffeomorphism, \(\mathcal {E}_{\mu \nu }\) satisfies the Bianchi identity

^{2}In the following sections, we shall make use of Eq. (16) to calculate the mass and angular momenta of charged static and rotating black holes in Weyl and Einstein–Gauss–Bonnet gravities although Eq. (16) is defined in terms of the Lagrangian for pure gravity, without any matter fields. We can do this since the terms associated with the gauge fields fall off fast enough at asymptotic infinity to guarantee that the integration is finite.

## 3 Conserved charges of black holes in four-dimensional Weyl gravity

### 3.1 The conserved charge of the charged spherically symmetric black hole

### 3.2 Conserved charges of the dyonic rotating black holes

## 4 Conserved charges of black holes in Einstein–Gauss–Bonnet gravity

^{3}namely, they differ by a trivial current. In fact, it can be proved that the generalized ADT potential (14) matches that got from the covariant phase space approach [14, 15]. This means that the Noether potentials for Weyl gravity and Einstein–Gauss–Bonnet gravity can be rederived along the line of [44], which gives an explicit form of conserved charges for general higher derivative gravity theory. What is more, the generalized ADT potential (14) is also equivalent to the one by the method developed by Barnich et al. [16, 17, 18, 19] for the theories of general relativity and \((2n+1)\)-dimensional supergravity [26, 50].

From Eq. (44), one sees that the gauge field makes no contribution to the energy of the static black hole (42). This is attributed to that the term with the electric charge parameter \(q\) in function \(U(r)\) falls off much faster than the one with the mass parameter \(m\) when \(r\rightarrow \infty \) so that its contribution to the energy can be neglected. The same situation takes place for the charged spherically symmetric black hole in \(d\)-dimensional \((d>4)\) Einstein–Gauss–Bonnet gravity coupled to nonlinear electrodynamics in [45]. The metric of this black hole can be reexpressed as the same form as Eq. (42) except for the term associated with the electric field in \(U(r)\), whose contribution to \(U(r)\) is smaller than the one from the mass term. Therefore, the term including the mass parameter still plays a dominant role in determining the energy of the black hole. By utilizing the formulation of the conserved charge (16), combined with Eq. (43), we obtain the energy of the static black hole in the case for the nonlinear coupling of electrodynamics, the same as that in Eq. (44). It also matches the energy in [45].

At the end of this section, it is worth mentioning that the Gauss–Bonnet term \(L_{(\mathrm{GB})}\) in the Lagrangian (36) is a surface term in \(d=4\) dimensions, namely, it makes no contribution to the equation of motion. This implies that all the black hole solutions in four-dimensional Einstein gravity are also the ones in Einstein–Gauss–Bonnet gravity. We have applied the generalized ADT formalism (16) to compute the masses and angular momenta of the four-dimensional Kerr(–AdS) and Kerr–Newman black holes corrected by the Gauss–Bonnet term. However, our results show that this term makes no corrections to all the conserved charges, compared with their corresponding ones in Einstein gravity.

## 5 Conclusions and discussions

In this paper, we have extended the off-shell generalization of the conventional ADT formalism proposed in [24] to calculate the quasi-local conserved charges for the most general charged static 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 higher dimensional Einstein–Gauss–Bonnet gravity coupled to Maxwell or nonlinear electrodynamics in AdS spacetime. Our results confirm those through other methods in the literature. To do this, we first directly vary the Bianchi identity (3), together with the help of the property of a Killing vector, to get the off-shell Noether currents (8) and (12), which are formally different from the ones in [24]. But they are actually equal to each other since both the surface term \(\Theta ^{\mu }(g;\mathcal {L}_\xi g)\) and the Lie derivative of the surface term \(\Theta ^\mu (g;\delta g)\) with respect to the Killing vector \(\xi \) vanish for the generally diffeomorphism gravity theory. The Noether current (8) makes it natural to derive the corresponding off-shell Noether potential (15), without a further requirement of the property for the symplectic current. Next, we present the explicit expressions of the surface term and Noether potential for Weyl gravity as the ones in Eqs. (20) and (22). Utilizing these quantities, we obtain the mass (27) of the most general static black hole in four-dimensional Weyl gravity, as well as the mass and angular momentum in Eq. (35) for the dyonic rotating black hole, although a naive application of the original ADT method fails to give a finite result in the static case. Finally, as the case of Weyl gravity, we start by deriving the surface term and the off-shell Noether potential in Eq. (40) for the general Lagrangian (36) and then utilize them to gain the energy (44) of the general charged static asymptotically AdS black hole in higher dimensional Einstein–Gauss–Bonnet gravity coupled to Maxwell or nonlinear electrodynamics. The energy (44) is independent on the electric parameter due to a fast fall-off of the term related to electric field.

Weyl gravity and Einstein–Gauss–Bonnet gravity are two typical fourth-order derivative gravity theories. It has been proposed that the general fourth-order gravity admits a critical theory in [47, 48]. The application of the ADT formalism to the critical theory demonstrates that the mass and angular momenta of all asymptotically Kerr–AdS and Schwarzschild–AdS black holes vanish at the critical point [40, 48]. We expect to learn whether the generalized formulation of the ADT charge supports this or not.

Although the formulation (16) for the quasi-local conserved charge is defined by only taking into account of the contribution from the pure gravity part, our analysis on charged black holes implies that it may be applicable to the black holes with matter fields, if the terms including matter fields in the given metric fall off fast enough at asymptotic infinity to ensure that the formulation (16) is convergent. Otherwise the effect of matter fields must be considered [26]. For instance, the formulation (16) fails to give a finite mass when it is utilized to the charged rotating Gödel-type black hole [49] in five-dimensional minimal supergravity since the \((t,r)\) component of the Noether potential (14) is divergent at infinity if the contribution from gauge field is omitted. To get finite conserved charges for the Gödel-type black hole, the effect of the gauge field has to be incorporated into the definition like in [50]. What is more, even if the conserved charge through the expression (16) is well defined in the presence of matter fields, it is possible for one to omit a finite value^{4} from the actions of the matter fields. In order to overcome these, the contribution from the matter fields has to be taken into account in future work.

## Footnotes

## Notes

### Acknowledgments

JJP would like to thank Professors Rong-Gen Cai, Yu Tian, Shuang-Qing Wu, and Xiao-Ning Wu for valuable discussions. He is also grateful to the Institute of Theoretical Physics, Chinese Academy of Sciences, for hospitality during the visit when this work was done. This work was supported by the Natural Science Foundation of China under Grant Nos. 11275157 and 11247225.

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