Till now, exact rotating black hole solutions with a cosmological constant have been found in various dimensions within the context of Einstein gravity. In 1968, Carter first found a generalization of the four-dimensional (4D) rotating Kerr black hole with a cosmological constant.^[1] Since this black hole has asymptotically de Sitter (dS) or anti-de Sitter (AdS) boundary conditions, it is usually called a Kerr–dS or Kerr–AdS black hole in the literature. Many years later, Hawking, Hunter and Taylor–Robinson found the five-dimensional (5D) generalization of the 4D Kerr–(A)dS black hole, as well as the solutions with just one nonzero angular momentum parameter in all dimensions.^[2] In fact, the 5D Kerr–(A)dS black hole can also be regarded as a generalization of the 5D Ricci-flat rotating Myers–Perry black hole^[3] including a cosmological constant. Subsequently, Gibbons, Lü Page, and Pope further constructed the general Kerr–(A)dS black holes with arbitrary angular momenta in all higher dimensions,^[4,5] which exactly satisfy the vacuum Einstein field equation with a cosmological constant. For the Kerr–AdS black holes, inspired by string theory, especially by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, as well as black hole thermodynamics, much work has been done on their diverse aspects.^[6–27]

Recently, in Ref. [28], relieving the restriction that the background spacetime satisfies the field equations, Kim, Kulkarni, and Yi proposed a quasi-local formalism of conserved charges within the framework of generic covariant pure gravity theories by constructing an off-shell ADT current to generalize the conventional on-shell Noether potential in the usual ADT formulation^[29–32] to the off-shell level, as well as following the Barnich–Brandt–Compere (BBC) method^{[22,23,33,34]} to incorporate a single parameter path in the space of solutions into their definition. Since the current and potential in the modified approach is off-shell, one may refer to it as the off-shell generalized ADT formalism. In contrast with the usual one, the off-shell generalization makes it more operable to derive the Noether potential from the corresponding current and the procedure of computation becomes more convenient to manipulate. Owing to these, it provides another fruitful way to evaluate the ADT charges for various theories of gravity. For several developments and applications of the off-shell generalized ADT formalism see the works.^[35–43]

As usual, after obtaining the Kerr–AdS black hole solutions, it is of great necessity to identify their mass and angular momenta. Because of the asymptotically AdS structure, the usual Arnowitt–Deser–Misner (ADM) formalism, as well as the Komar integral, fails to produce their mass. So seeking other feasible approaches is desirable. Fortunately, some works^[12–27] have succeeded in yielding the conserved charges of the Kerr–AdS black holes through a series of methods, such as the Ashtekar–Magnon–Das (AMD) formalism,^[26,27] the (off-shell) ADT formulation, the BBC method, and so on. For example, in Ref. [21], the usual ADT formulation has been applied to obtain the mass of Kerr–AdS black holes in various dimensions. To realize this, the perturbation of the metric is set as the divergence between the metric and a fixed reference background, which is the spacetime got through letting the parameter associated with the mass in the original metric be zero. In Refs. [22] and [23], the same background was also adopted to calculate the conserved charges of the Kerr–AdS black holes via the BBC method. There the fluctuation of the metric only depends on the parameter related to the mass rather than all the solution parameters. Besides, the potential used to define the conserved charges coincides with that in the usual ADT formulation. Due to these, the BBC method adopted in Refs. [22] and [23] is essentially in accordance with the usual ADT one.

Notably, the usual ADT formulation and the BBC approach are background-dependent. Actually, in Refs. [21], [22], and [23] the reference spacetimes adopted to calculate the mass of Kerr–AdS black holes in all dimensions are non-rotating at infinity. Otherwise, both the two methods may fail to yield physical results if the timelike Killing vector associated with the mass is chosen as the usual one . For the off-shell generalized ADT formalism, we also wonder what will happen if it is used to deal with the mass of the Kerr–AdS black holes in an asymptotic rotating frame. On the other hand, in the light of the fact that the mass together with the angular momenta enters into the first law of thermodynamics for black holes as thermodynamical variables, the rotation parameters should be the members to determine the fluctuation of the metric for the Kerr-AdS black holes. However, in Refs. [22] and [23], only the mass parameter was regarded as a variable to fluctuate the metric, while the rotation parameters were fixed. In view of the above-mentioned issues, unlike in Refs. [21], [22], and [23], we shall utilize the off-shell generalized ADT formalism to explore the mass of the Kerr–AdS black holes in a more general manner. Namely, we do this under the conditions that the background spacetime is asymptotically rotating and the fluctuation of the metric is determined by the variation of all the mass and rotation parameters. The results demonstrate that the way that the Kerr–AdS black holes behave at infinity, rotating or not, plays a key role in determining whether the off-shell generalized ADT formalism can produce their physically meaningful mass.

The remainder of this work is arranged as follows. Sections 2 and 3 are devoted to investigating the off-shell generalized ADT mass of the Kerr–AdS black holes in four and five dimensions respectively. In Section 4, the mass of the Kerr–AdS black holes in arbitrary dimensions is calculated by generalizing the results in the 4D and 5D cases. The last section is our conclusions.

As is well known, the theory of Einstein gravity in D dimensions is described by the Einstein–Hilbert Lagrangian

We now go on to compute the mass of the 4D Kerr-AdS black hole via the off-shell generalized ADT method proposed in Ref. [28]. With the choice of a Killing vector , the definition of the ADT conserved charge related to the Einstein–Hilbert Lagrangian (1) is read off as

For the 4D Kerr–AdS black hole described by the metric (3), the fluctuation of the spacetime is determined by the infinitesimal change of both the parameters rather than the single mass parameter m like in Refs. [22] and [23], that is

A straightforward computation then shows that

Since the 4D Kerr black hole is rotating at infinity, the Killing vector corresponding to its mass should be chosen as the asymptotic non-rotating timelike Killing vector , which is just the linear combination of the Killing vector and . With the help of Eq. (A1) in Appendix A, we have

The 5D Kerr-AdS black hole, which behaves like an asymptotic space and possesses two independent rotations, is an exact solution of the field equation (2) in five dimensions. This black hole was first constructed in Ref. [2], and it can be seen as a special case of the general solutions in arbitrary higher dimensions, found in Refs. [4] and [5]. For the sake of convenience on our analysis, the metric for the 5D Kerr–AdS black hole takes the form

In Eq. (14), both the coordinates ϕ and ψ range from , while x takes the value [0, 1]. Significantly, when and , which depend on the four parameters , disappear and the coordinate x is substituted by a new one θ through the relation , the metric (14) returns to the usual form of the 5D Kerr–AdS black hole in the literature, whose angular velocities along the ϕ and ψ directions are and respectively at infinity. Unlike this, in the more general cases , the angular velocities of the metric (14) at infinity become and .

In order to calculate the off-shell generalized ADT mass of the 5D Kerr–AdS black hole (14), the fluctuation of the metric is determined by the infinitesimal variation of the parameters as

In the above equation, the quantity has the property , which implies that the contribution to the mass from the first term in Eq. (17) can be neglected. Another avenue to cancel the contribution from the r² term is to let the perturbation of the metric be independent of the rotation parameters a and b, like in Refs. [22] and [23]. This holds for the cancelation of the contribution from the r term in Eq. (8) as well. Further making use of the definition (5) for the off-shell ADT conserved charges, one obtains

in which, . Note that equation (22) is similar to Eq. (13) for the 4D Kerr–AdS black hole. As before, in order to make the quantity coincide with the physically meaningful mass , a rather simple way is to set and , that is, the angular velocities of the 5D Kerr–AdS black hole vanish at infinity.

A remark is in order here. In the above discussions, we have demonstrated that it is a better choice to evaluate the mass of the 4D and 5D Kerr–AdS black holes in an asymptotically non-rotating frame. However, equation (A3) in Appendix A allows the possibility to compute their mass in a rotating frame at infinity when the asymptotic angular velocity is independent of the mass parameter. For instance, to gain the mass for the 4D Kerr–AdS black hole (3) with , one can set that the variation of the metric only depends on that of the single parameter m and the Killing vector corresponding to the mass is chosen as the asymptotic non-rotating timelike Killing vector . The former makes the charge be integrable. Its integral yields , while . Thus the linear combination of and gives rise to . On the other hand, in the case where the parameters within the metric form (14) for the 5D Kerr–AdS black hole, if the fluctuation of the metric is still determined by the parameter m rather than the ones , and the timelike Killing vector is set as , one obtains . Besides, in the more general cases, if the Killing vectors are fixed, the terms proportional to the angular momenta in Eqs. (13) and (22) vanish, yielding the physical mass of the 4D and 5D Kerr–AdS black holes in the general asymptotically rotating frames.

In the present section, we deal with the off-shell generalized ADT mass for the general Kerr–AdS black holes in D dimensions by generalizing the analysis for the 4D and 5D ones. The general Kerr–AdS black holes in dimensions, where ϵ = 1 when D is even and ϵ = 0 when D is odd, were constructed in Refs. [4] and [5]. They possess N independent rotations in N orthogonal 2-planes, characterized by the parameters a_iʼs and the azimuthal angles ϕ_iʼs. The metric for the D-dimensional Kerr–AdS black hole is read off as

We turn our attention to evaluating the off-shell generalized ADT mass of the Kerr–AdS black holes in D dimensions. As is shown in both the 4D and 5D cases, we choose the asymptotic non-rotating timelike Killing vector associated with the mass as the one , where the timelike Killing vector and the spacelike Killing vectors , while the perturbations of the metric rely on the infinitesimal changes of all the parameters through

Equation (26) can be regarded as the generalization of Eqs. (13) and (22) in D dimensions. By adopting Eq. (B2) in Appendix B to calculate the component of the off-shell ADT potential, this equation has been checked to hold in cases. Furthermore, to make , one has to cancel the contribution arising from the variation of the asymptotic velocities ʼs. To realize this, a rather simple method is to perform the coordinate transformations to keep the Kerr–AdS black holes static at infinity. This is to say, as long as the symmetry related to the mass is generated by the usual timelike Killing vector and the fluctuations of the metric depend on the variation of all the mass and rotation parameters, only under the condition that these black holes are asymptotically non-rotating, can the off-shell generalized ADT formalism yield their physical mass.

For the Kerr–AdS black holes in the asymptotic rotating frame with the angular velocities ʼs irrelevant to the mass parameter m, when the perturbations of the metric are only dependent on the variation of this parameter, together with the relevant Killing vector given by the asymptotic non-rotating timelike one , one is able to obtain the mass according to Eq. (26).

Besides, if the Killing vector in Eq. (6) is supposed to be fixed, that is , giving rise to that the conventional ADT potential rather than enters into the definition of the conserved charge , one observes that the terms with angular momenta in Eq. (26) disappear, yielding . Its integral gives the mass of the D-dimensional Kerr–AdS black holes with general asymptotic angular velocities, which is consistent with the AMD mass.^[19,26–28]

In this paper, we have made use of the off-shell generalized ADT formalism to compute the mass of the asymptotically-rotating Kerr–AdS black holes in all dimensions. Without the requirement to fix the Kiling vector, for 4D and 5D Kerr–AdS black holes in a general asymptotic rotating frame, when the timelike Killing vectors associated with the mass are chosen as the asymptotic non-rotating timelike one and the perturbations of the metric are determined by the variation of all the mass and angular momentum parameters, the conserved charges take the forms in Eqs. (13) and (22) respectively. A similar equation (26) holds for the Kerr–AdS black holes in diverse dimensions. All the equations show that the charges are non-integrable or unphysical because of the appearance of the general asymptotic angular velocities. Thus, in order to obtain charges coinciding with the physically meaningful mass in the literature, the spacetime has to be non-rotating at infinity. In this sense, all the results support that the (off-shell) ADT formalism depends on the reference background.

By comparison, in an asymptotic rotating frame with angular velocities that are only dependent of the rotation parameters, one can obtain the physical mass of the Kerr–AdS black holes, however, it is required that the fluctuations of the metric merely rely on the mass parameter and the related Killing vector is set as the asymptotic non-rotating timelike one. Besides, in the more general asymptotic rotating frame, one can still get the physical mass if the asymptotic non-rotating timelike Killing vector is assumed to be fixed. In the light of the above, we suggest that one had better perform calculations in an asymptotic non-rotating frame when the off-shell generalized ADT formalism is applied to compute the mass of asymptotically AdS black holes. This also holds for the usual ADT formulation and the BBC method since they are essentially equivalent to the off-shell generalized ADT formalism for the Einstein gravity theory.

It should be emphasized that the cosmological constant is fixed in our calculations so that it is not involved in the perturbation of the metric. Otherwise, the charges of the Kerr–AdS black holes are non-integrable. To overcome this, the off-shell generalized ADT formalism has to be modified by reevaluating the contribution from the cosmological constant.^[38] Accordingly, the mass of the Kerr–AdS black holes should be reconsidered with the varying cosmological constant.