Total Conserved Charges of Kerr-Newman Spacetimes in Gravity Theory Using a Poincaré Gauge Version of the Teleparallel Equivalent of General Relativity

Abstract

The total conserved charges of several tetrad spacetimes, generating the Kerr-Newman (KN) metric, are calculated using the approach of invariant conserved currents generated by an arbitrary vector field that reproduces a diffeomorphism on the spacetime. The accompanying charges of some tetrads give the known value of energy and angular momentum, while those of other tetrads give, in addition to the unknown format charges, a divergent entity. Therefore, regularized expressions are considered also to get the commonly known form of conserved charges of KN.

1. Introduction

To find new black hole solutions in the Einstein gravitational theory, one needs to couple the Einstein Hilbert action to matter fields that have associated conserved charges. The charges must be those of point particles, or one will obtain solutions describing extended objects instead of black holes. Thus, one needs to consider vector fields, and the simplest one is an Abelian vector field 𝐴_𝜇. Axially symmetric solutions of the Einstein field equations are used to describe a wide range of black holes appearing in the universe. The most promising of these solutions is the Kerr spacetime that describes an axially symmetric, rotating black hole [1]. Generalizations of the Kerr black holes are the charged Kerr-Newman, Kerr-de Sitter spacetime that incorporates a nonvanishing cosmological constant, Kerr-Taub-NUT spacetime that includes the NUT charge, or some combination of these. All of these spacetimes could be seen as special cases of the family of electrovac spacetimes of Petrov type D provided by the Plebański-Demiański class of solutions [2]. They are distinguished by seven parameters: the mass, the rotation around the symmetry axis, the electric and magnetic charges, the NUT charge, the cosmological constant, and the acceleration of the gravitating source.

The definition of conserved charges in general relativity (GR) is an important and a delicate issue to some extent, which concerns the definition of the energy-momentum tensor of the gravitational field. Perturbatively, GR gives the unique energy-momentum tensor [3, 4]. Many researchers (starting from Einstein himself) have tried in vain to find a fully general-covariant energy-momentum tensor for the gravitational field that is reducible to the weak-field limit. Therefore, one has to restrict with energy-momentum pseudotensors for the gravitational field, which are covariant only under a limited set of coordinate transformations. This, in fact, would be one of the properties of the gravitational field linked to the principle of equivalence of gravitation and inertia that says that all the physical effects from the gravitational field can be locally eradicated by selecting a locally inertial coordinate system [5].

The main result of the lack of a totally general-covariant energy-momentum tensor for the gravitational field is the nonlocalizability of the gravitational energy; just the total energy of a spacetime is well defined (and conserved), since the integral of the energy-momentum pseudotensor over a finite volume will be dependent on the choice of coordinates. Some researcher find this is unacceptable, and, therefore, the search for the general-covariant tensor goes on.

It is well established in the laws of physics that symmetry principles and conservation laws play key roles in understanding physical phenomena [6–8]. Various physical problems can be solved remarkably easier, and one might gain deeper insight of their nature when symmetry is taken into consideration. Some fields of study in modern physics have developed relatively quickly through the understanding of the principles of symmetry, for instance, the gauge symmetries in quantum field theory and elementary particle physics [9–11].

The existence of symmetries in a physical system means that there exists one or more transformations that leave the physical system invariant. Since physical systems can be fully described by their Lagrangians/Hamiltonians, then it is natural to expect the symmetry transformations to be canonical ones. The number of symmetry transformations that can be generated from a system depends on the number of conserved quantities associated with that system [6, 7, 12].

A number of methods are currently used in the calculations of energy and conserved charges that provide reliable results. Gibbons et al. [5] compute the energy of Kerr-AdS black holes firstly indirectly, through integrating the first law, then using the conformal definition of Ashtekar et al. [13, 14], and they indicate that both of them agree. In addition to the Regge-Teitelboim method [15], many others are present: the approach of Abbott and Deser [16–18], the spinor definition [19, 20] based on the electric Weyl tensor, covariant phase space methods [21–24], cohomological techniques [25, 26], the KBL approach [27–29], Noether methods [30–36], the counter term subtraction method [37, 38] (more references on this in [23], for example), improved surface integrals [39, 40], and regularization of the Euclidean action [41, 42]. Various examples proposing a universal approach for carefully revising geometric field theories in the sense of the language of Cartan connections are given [43].

The aim of this work is to apply the definition of conserved currents and charges for models with quasi-invariant Lagrangians (Lagrangians that change by a total derivative under local Lorentz transformations) to KN spacetimes, which are considered as exact solutions to the Einstein field equations. In Section 2, the language of exterior forms is used to give the scheme approach to the teleparallel approach. In Section 2, an axially symmetric spacetime, KN, in TEGR is given. The total conserved currents of this spacetime are calculated, and an unacceptable form of angular momentum is obtained as a result. In Section 3, regularized conserved charges are provided and are used to recalculate the conserved charges of the first KN spacetime. Also in Section 3, a second KN spacetime is provided, and calculation of the total conserved charges is given. Unfortunately, the conserved quantity is divergent in azimuthal direction [44–47]. Even when the regularized conserved charges are used, to recalculate the conserved charges of the second coframe, curious result is obtained in azimuthal direction. In Section 4, conserved charges using the Poincaré gauge (PG) version of TEGR are employed. The PG version is used to recalculate the conserved charges of the second KN spacetime, and a consistent result is derived. In Section 4, a third KN spacetime is given to calculate its total conserved charges. The conserved quantity of this coframe, when it is calculated using the total conserved charges, is divergent in the azimuthal direction [48, 49]. Using the PG version, the total charges of the third KN spacetime are recalculated, and a satisfactory result is obtained. Also in Section 4, fourth and fifth KN spacetimes are provided, and the PG version is used to calculate their total conserved charges. The final section is devoted to the main results.

1.1. Notation

The Latin indices 𝑖,𝑗,…, are used for local holonomic spacetime coordinates, and the Greek indices 𝛼,𝛽,…, label (co)frame components. Particular frame components are denoted by hats, 0, 1, and so forth. As usual, the exterior product is denoted by ∧, while the interior product of a vector 𝜉 and a 𝑝-form Ψ is denoted by 𝜉⌋Ψ. The vector basis dual to the frame 1-forms 𝜗^𝛼 is denoted by 𝑒_𝛼, and they satisfy 𝑒_𝛼⌋𝜗^𝛽=𝛿_𝛼^𝛽. Using local coordinates 𝑥^𝑖, we have 𝜗^𝛼=ℎ^𝛼_𝑖𝑑𝑥^𝑖 and 𝑒_𝛼=ℎ^𝑖_𝛼𝜕_𝑖 where ℎ^𝛼_𝑖 and ℎ^𝑖_𝛼 are the covariant and contravariant components of the tetrad field. We define the volume 4-form by 𝜂=^def.𝜗0∧𝜗1∧𝜗2∧𝜗3. Furthermore, with the help of the interior product, we define 𝜂=_𝛼^def.𝑒_𝛼1⌋𝜂=𝜖3!^{𝛼𝛽𝛾𝛿}𝜗^𝛽∧𝜗^𝛾∧𝜗^𝛿, (1) where 𝜖_{𝛼𝛽𝛾𝛿} is completely antisymmetric with 𝜖₀₁₂₃=1. One has 𝜂=_𝛼𝛽^def.𝑒_𝛽⌋𝜂_𝛼=1𝜖2!^{𝛼𝛽𝛾𝛿}𝜗^𝛾∧𝜗^𝛿,𝜂=_𝛼𝛽𝛾^def.𝑒_𝛾⌋𝜂_𝛼𝛽=1𝜖1!^{𝛼𝛽𝛾𝛿}𝜗^𝛿, (2) which are bases for 3-, 2-, and 1-forms, respectively. Finally, 𝜂=_{𝛼𝛽𝜇𝜈}^def.𝑒_𝜈⌋𝜂_𝛼𝛽𝜇=𝑒_𝜈⌋𝑒_𝜇⌋𝑒_𝛽⌋𝑒_𝛼⌋𝜂, (3) is the Levi-Civita tensor density. The 𝜂-forms satisfy the useful identities as follows: 𝜗^𝛽∧𝜂=_𝛼^def.𝛿^𝛽_𝛼𝜂,𝜗^𝛽∧𝜂=_𝜇𝜈^def.𝛿^𝛽_𝜈𝜂_𝜇−𝛿^𝛽_𝜇𝜂_𝜈,𝜗^𝛽∧𝜂=_𝛼𝜇𝜈^def.𝛿^𝛽_𝛼𝜂_𝜇𝜈+𝛿^𝛽_𝜇𝜂_𝜈𝛼+𝛿^𝛽_𝜈𝜂_𝛼𝜇,𝜗^𝛽∧𝜂=_{𝛼𝛾𝜇𝜈}^def.𝛿^𝛽_𝜈𝜂_𝛼𝛾𝜇−𝛿^𝛽_𝜇𝜂_𝛼𝛾𝜈+𝛿^𝛽_𝛾𝜂_𝛼𝜇𝜈−𝛿^𝛽_𝛼𝜂_𝛾𝜇𝜈. (4) The line element 𝑑𝑠=^2def.𝑔_𝛼𝛽𝜗^𝛼⊗𝜗^𝛽 is defined by the spacetime metric 𝑔_𝛼𝛽.

2. Brief Review of Teleparallel Gravity and Conserved Currents

Teleparallel geometry can be viewed as a gauge theory of translation [50–62]. The coframe 𝜗^𝛼 plays the role of the gauge translational potential of the gravitational field. GR can be reformulated as the teleparallel theory. Geometrically, teleparallel gravity can be considered as a special case of the metric-affine gravity in which 𝜗^𝛼 and the local Lorentz connection are subject to the distant parallelism constraint 𝑅_𝛼^𝛽=0 [63–71]. In this geometry, the torsion 2-form 𝑇^𝛼=𝐷𝜗^𝛼=𝑑𝜗^𝛼+Γ_𝛽^𝛼∧𝜗^𝛽=12𝑇^𝛼_𝜇𝜈𝜗^𝜇∧𝜗^𝜈=12𝑇^𝛼_𝑖𝑗𝑑𝑥^𝑖∧𝑑𝑥^𝑗 (5) arises as the gravitational gauge field strength, Γ_𝛼^𝛽 is the Weitzenböck 1-form connection, 𝑑 is the exterior derivative, and 𝐷 is the exterior covariant derivative. The torsion 𝑇^𝛼 can be decomposed into three irreducible pieces [72]: the tensor part, the trace, and the axial trace given, respectively, by 𝑇=^(1)𝛼def.𝑇^𝛼−⁽²⁾𝑇^𝛼−⁽³⁾𝑇^𝛼,with₍₂₎𝑇=^𝛼def.13𝜗^𝛼∧𝑇,where𝑇=𝑒_𝛽⌋𝑇^𝛽,𝑒_𝛼⌋𝑇=𝑇^𝜇_𝜇𝛼,vectorsoftraceoftorsion₍₃₎𝑇=^𝛼def.13𝑒^𝛼⌋𝑃,with𝑃=𝜗^𝛽∧𝑇_𝛽,𝑒_𝛼⌋𝑃=𝑇^𝜇𝜈𝜆𝜂_{𝜇𝜈𝜆𝛼},axialoftraceoftorsion. (6) The Lagrangian of the TEGR has the form 1𝑉=−𝑇2𝜅^𝛼∧^∗(1)𝑇_𝛼−2⁽²⁾𝑇_𝛼−12⁽³⁾𝑇_𝛼. (7)𝜅=8𝜋𝐺/𝑐³, 𝐺 is the Newtonian constant, 𝑐 is the speed of light, and ∗ denotes the Hodge duality in the metric 𝑔_𝛼𝛽 which is assumed to be flat Minkowski metric 𝑔_𝛼𝛽=𝑜_𝛼𝛽=diag(+1,−1,−1,−1) that is used to raise and lower local frame (Greek) indices. (The effect of adding the non-Riemannian parity odd pseudoscalar curvature to the Hilbert-Einstein-Cartan scalar curvature was studied by many authors (cf. [73–77] and references therein).)

The variation of the total action with respect to the coframe gives the field equations in the form 𝐷𝐻_𝛼−𝐸_𝛼=Σ_𝛼,whereΣ=_𝛼^def.𝛿𝐿_mattter𝛿𝜗^𝛼 (8) is the symmetric energy-momentum tensor 3-form of matter which is considered as the source. In accordance with the general Lagrange-Noether scheme [59, 78], one derives from (7) the translational momentum 2-form and the canonical energy-momentum 3-form as follows: 𝐻=_𝛼^def.−𝜕𝑉𝜕𝑇^𝛼=1𝜅∗⁽¹⁾𝑇_𝛼−2⁽²⁾𝑇_𝛼−12⁽³⁾𝑇_𝛼,𝐸=_𝛼^def.𝜕𝑉𝜕𝜗^𝛼=𝑒_𝛼⌋𝑉+𝑒_𝛼⌋𝑇^𝛽∧𝐻_𝛽. (9)(10) Due to geometric identities [72], the Lagrangian (7) can be recast as 1𝑉=−2𝑇^𝛼∧𝐻_𝛼. (11) The presence of the connection field Γ^𝛼_𝛽 plays an important regularizing role due to the following.

First. The theory becomes explicitly covariant under the local Lorentz transformations of the coframe; that is, the Lagrangian (7) is invariant under the change of variables 𝜗^′𝛼=Λ^𝛼_𝛽𝜗^𝛽,Γ^′_𝛼^𝛽=Λ^𝜇_𝛼Γ_𝜇^𝜈Λ^−1𝛽_𝜈−Λ^−1𝛽_𝛾𝑑(Λ)^𝛾_𝛼. (12) Due to the noncovariant transformation law of Γ_𝛼^𝛽, see (12), if a connection vanishes in a given frame, it will not vanish in any other frame related to the first by a local Lorentz transformation. When Γ_𝛼^𝛽=0 for all frames, which is the tetrad gravity, the Lagrangian (7) is invariant under the local Lorentz group.

Second. Γ_𝛼^𝛽 plays an important role in the teleparallel framework. This role represents the inertial effects which arise from the choice of the reference system [79]. The contributions of this inertial in many cases lead to unphysical results for the total energy of the system. Therefore, the role of the teleparallel connection is to separate the inertial contribution from the truly gravitational one. Since the teleparallel curvature is zero, the connection is a pure gauge; that is, Γ=_𝛼^𝛽def.Λ^𝛽_𝛾⁻¹𝑑Λ^𝛾_𝛼. (13) The Weitzenböck connection always has the form (13). The translational momentum has the form [72, 80] 𝐻_𝛼=1Γ2𝜅^𝛽𝛾∧𝜂_𝛼𝛽𝛾,𝐸_𝛼=12𝑒_𝛼⌋𝑑𝜗^𝛽∧𝐻_𝛽−𝑑𝜗^𝛽∧𝑒_𝛼⌋𝐻_𝛽,Γ=_𝛼^𝛽def.Γ_𝛼^𝛽−𝐾_𝛼^𝛽, (14) where Γ_𝛼^𝛽 is the purely Riemannian connection and 𝐾^𝜇𝜈 is the contorsion 1-form which is related to the torsion through the relation 𝑇=^𝛼def.𝐾^𝛼_𝛽∧𝜗^𝛽. (15)

The teleparallel model (7) belongs to the class of quasi-invariant theories. In fact, one can verify that under a change of the coframe 𝜗^′𝛽=(Λ(𝑥)^𝛼_𝛽)𝜗^′𝛽, the Lagrangian changes by a total derivative as follows: 𝑉𝜗^′=𝑉(𝜗)−1𝑑Λ2𝜅^−1𝛼𝛽𝑑Λ^𝛽_𝛾∧𝜂^𝛾_𝛼. (16) In addition to (16), it is easy to verify that (14) changes like 𝐻^′_𝛼𝜗^′=Λ^−1𝛽_𝛼𝐻_𝛽(𝜗)−1𝑑Λ2𝜅^−1𝛽𝛼Λ^−1𝜈_𝛾𝑑Λ^𝛾_𝜇∧𝜂_𝛽^𝜇_𝜈,𝐸^′_𝛼𝜗^′=Λ^−1𝛽_𝛼𝐸_𝛽(𝜗)+𝑑Λ^−1𝛽_𝛼∧𝐻_𝛽(𝜗)−1𝑑Λ2𝜅^−1𝛽𝛼Λ^−1𝜈_𝛾𝑑Λ^𝛾_𝜇∧𝜂_𝛽^𝜇_𝜈. (17)

The total conserved charge in the tetrad gravity theory is given by [48, 49] 𝒬(𝜉,𝜗)=_𝜕𝑆𝜉^𝛼𝐻_𝛼=12𝜅_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾, (18) where 𝜉^𝛼=𝜉⌋𝜗^𝛼. Under a local Lorentz transformation, (18) transforms as 𝒬^′()=𝒬()−1𝜉,𝜗𝜉,𝜗2𝜅_𝜕𝑆𝜉^𝛼Λ^−1𝜈_𝛾𝑑Λ^𝛾_𝜇∧𝜂_𝛼^𝜇_𝜈=12𝜅_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾−Γ^𝛽𝛾∧𝜂_𝛼𝜇𝜈, (19) where Γ^𝜈_𝜇=(Λ⁻¹)^𝜈_𝛾𝑑Λ^𝛾_𝜇. The vector 𝜉=𝜁^𝑖𝜕_𝑖 does not depend on the choice of the frame, while its components, that is, 𝜉^𝛼, transform as a vector. Now let use apply (18) to different coframes.

3. Total Conserved Charges of the Kerr-Newman Spacetimes

3.1. First Coframe

The covariant form of the first charged rotating tetrad, that is, KN, field having axial symmetry in spherical coordinates (𝑡,𝑟,𝜃,𝜙) can be written asℎ^𝛼_𝑖𝐼=𝒜ℬ𝒞−0002𝑀𝑟−𝑞²𝑎sin𝜃cos𝜃√−𝒜𝒞𝒜ℬsin𝜃cos𝜙−sin𝜃sin𝜙√𝒜−𝒞𝒜sin𝜃cos𝜃2𝑀𝑟−𝑞²𝑎sin²𝜃√−𝒜𝒞𝒜ℬcos𝜃cos𝜙−cos𝜃sin𝜙√𝒜𝒞𝒜sin²𝜃0𝒜ℬsin𝜙−cos𝜙√𝒜0. (20)

𝒜, ℬ, and 𝒞 are three functions of 𝑟 and 𝜃 having the form 𝒜=𝑟²+𝑎²cos²𝜃,ℬ=𝑟²−2𝑀𝑟+𝑎²+𝑞²,𝒞=𝑟²+𝑎²𝑟²+𝑎²cos²𝜃+2𝑀𝑟−𝑞²𝑎²sin²𝜃, (21) where 𝑀, 𝑎, and 𝑞 are mass, rotation and charge parameters, respectively [44–47]. An asymptotically flat spacetime is considered in this study, and a boundary condition, that for 𝑟→∞ the tetrad (20) approaches the tetrad of Minkowski spacetime in Cartesian coordinates, is imposed. The metric tensor 𝑔=_𝑖𝑗^def.𝑜_𝜇𝜈ℎ^𝜇_𝑖ℎ^𝜈_𝑗 associated with the tetrad field (20) has the form 𝑑𝑠²=1−2𝑀𝑟−𝑞²𝒜𝑑𝑡²−𝒜ℬ𝑑𝑟²−𝒜𝑑𝜃²−𝑟²+𝑎²+2𝑀𝑟−𝑞²𝑎²sin²𝜃𝒜sin²𝜃𝑑𝜙²−22𝑀𝑟−𝑞²𝑎sin²𝜃𝒜𝑑𝑡𝑑𝜙, (22) which is the KN metric written in the Boyer-Lindquist coordinates [44–47].

In spherical local coordinates, the first rotating frame is taken to be described by the coframe ^K𝜗^N_𝐼^𝛼=Λ^𝛼_𝛾𝜗_𝐼^𝛾,where𝜗_𝐼^𝛾=ℎ^𝛾_𝑖𝐼𝑑𝑥^𝑖. (23)

The local Lorentz transformation, Λ^𝛼_𝛾, is given by the matrix [44–47] Λ^𝛼_𝛾=10000sin𝜃cos𝜙cos𝜃cos𝜙−sin𝜙0sin𝜃sin𝜙cos𝜃sin𝜙cos𝜙0cos𝜃−sin𝜃0. (24) Using (23) and (24), one gets ^K𝜗^N_𝐼⁰=−𝒜ℬ𝒞𝑑𝑡,𝜗^KN_𝐼¹=𝒜ℬ𝑑𝑟,𝜗^KN_𝐼²=√𝒜𝑑𝜃,_KN𝜗_𝐼³=sin𝜃𝒞𝑑𝜙+2𝑀𝑟−𝑞²𝑑𝑡√.𝒜𝒞 (25)

The nonvanishing components of the Riemannian connection, of the first coframe, have the form Γ10=1𝒜²√𝒞𝑎sin²𝜃𝑀3𝑟⁴+𝑟²𝑎²+𝑎²cos²𝜃𝑟²−𝑎²−𝑟𝑞²𝒜+𝑟²+𝑎²−𝑟𝑑𝜙₂+𝑎²𝑀𝑟²−𝑎²cos²𝜃−𝑟𝑞²,Γ12𝑎𝑑𝑡=−²sin𝜃cos𝜃𝑑𝑟+𝑟ℬ𝑑𝜃𝒜√ℬ,Γ13=sin𝜃√ℬ√𝒞𝒜²𝑟𝒜²−𝑎²sin²𝜃𝑀𝑟²−𝑎²cos²𝜃−𝑞²𝑟+𝑟𝑑𝜙𝑀𝑎²−𝑎²cos²𝜃−𝑞²𝑟,Γ20𝑐𝑑𝑡=−2𝑀𝑟−𝑞²𝑎²√ℬsin𝜃cos𝜃√𝒞𝒜²𝑎sin²,Γ23=𝜃𝑑𝜙+𝑑𝑡cos𝜃√𝒞𝒜²𝒜²𝑟²+𝑎²+2𝑀𝑟−𝑞²𝑎²sin²𝜃×2𝑟²+𝑎²1+cos²𝜃−𝑑𝜙2𝑀𝑟−𝑞²𝑎𝑟²+𝑎²,Γ30=𝑑𝑡𝑎sin𝜃2√ℬ𝒞𝒜2𝑀3𝑟⁴+𝑟²𝑎²+𝑎²cos²𝜃𝑟²−𝑎²−𝑞²4𝑟³+2𝑟𝑎²1+cos²𝜃−𝑑𝑟2𝑀𝑟−𝑞²𝑎²sin𝜃cos𝜃ℬ𝑑𝜃. (26) Using (25), the nonvanishing components of the vector 𝜉^𝛼 have the form 𝜉⁰=𝒜ℬ𝒞𝜁⁰,𝜉¹=𝒜ℬ𝜁¹,𝜉²=√𝒜𝜁²,𝜉³=√𝒞𝜁³+2𝑀𝑟−𝑞²𝑎𝜁⁰sin𝜃√.𝒜𝒞 (27) If we take tetrad (20), as well as the trivial Weitzenböck connection, Γ^𝛼_𝛽=0, we finally get the nonvanishing components of the translation momentumthat is, 𝐻𝛼, 𝐻0=−sin𝜃𝑑𝜃∧𝑑𝜙4𝑀𝑟+𝑎²sin²𝜃−2𝑞²𝐻34𝜋𝑟+⋯,=−𝑎sin²𝜃𝑑𝜃∧𝑑𝜙2𝑟²+6𝑀𝑟−𝑎²cos²𝜃−4𝑞²4𝜋𝑟²+⋯, (28) where ⋯ means terms which are multiplied by 𝑑𝜃∧𝑑𝑟, 𝑑𝜃∧𝑑𝑡, 𝑑𝑟∧𝑑𝜙, and so forth. By using the identities of [36], that is, (A18), one can directly prove that, for solutions with vanishing torsion, the Killing equation ℒ_𝜁𝜗^𝛼=0 implies also that ℒ_𝜁Γ_𝛼^𝛽=0. Therefore, for the Kerr-Newman solution we have ℒ_𝜁Γ_𝛼^𝛽=0 for 𝜁=𝜉⁰𝜕₀+𝜉³𝜕_𝜙, with constants 𝜉⁰ and 𝜉³. For a vector field 𝜉=𝜉^𝛼𝑒_𝛼=𝜁^𝑖𝜕_𝑖 with constant holonomic components, 𝜁^𝑖, in the coordinate system used in (20), the direct evaluation of the integral (18) yields the following infinite total conserved charge: 𝒬=_𝜕𝑆𝜉^𝛼𝐻_𝛼=𝑞𝑀−²𝜁2𝑟⁰−𝑟3+𝑀𝑎𝜁³. (29)

For the coframe ^K𝜗^N_𝐼^𝛼=(Λ^𝛼_𝛾)𝜗_𝐼^𝛾 determined by (25), the conserved charges corresponding to the diffeomorphism generated by the shift along the time coordinate has a usual value 𝑀−𝑞²/2𝑟 of the total energy of the configuration. Whereas, for the vector field along the azimuthal coordinate we get (−𝑟𝑎/3−𝑀𝑎), which is a nonfamiliar term [80]. This is because tetrad (20) is nonholonomic. Another reason is that the Riemannian connection Γ(𝜗^′) does not vanish either at spatial infinity or when the physical quantities, 𝑀=0, 𝑎=0, and 𝑞=0, are vanishing. The nonvanishing of the Riemannian connection may be responsible for the divergence of the integral of the conserved charges.

4. Total Charge of KN Spacetime Using Regularized Conserved Invariant Charge

One possible solution of the above unfamiliar result is the use of regularized form of total charges 𝒬()=1𝜉,𝜗2𝜅_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾−Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾, (30) where Γ^𝛽𝛾 has the form Γ_𝛼^𝛽=Λ^−1𝛽_𝛾𝑑Λ^𝛾_𝛼. (31)

The most appropriate local Lorentz transformation which serves for this purpose has the form (24). The nonvanishing components of Γ^𝛽𝛾 have the form Γ12132=−𝑑𝜃,Γ=−sin𝜃𝑑𝜙,Γ3=−cos𝜃𝑑𝜙. (32) Using (26), (27), and (32) in (30) and the fact that the vector field 𝜉 is with constant holonomic components, we get the total conserved quantities in the form 𝒬(𝜉,𝜗)=_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾−Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾=𝑞𝑀−²𝜁2𝑟⁰−𝑀𝑎𝜁³. (33) It is observed that the last term in (30) acts as a regularizing term in removing divergence.

4.1. Second Coframe

The covariant form of the second KN tetrad field can be written asℎ^𝛼_𝑖𝐼𝐼=^{−𝑎2sin2𝜃−𝒜1ℬ𝒜00−𝑎𝒜1sin2𝜃ℬ−𝒜1𝒜−𝑎ℬ−𝒜1sin𝜃sin𝜙𝒜𝑟sin𝜃cos𝜙√ℬ𝒜1𝑎cos𝜃cos𝜙−sin𝜃sin𝜙2sin2𝜃√ℬ−𝒜13/2𝒜}_{𝑎ℬ−𝒜1sin𝜃cos𝜙𝒜𝑟sin𝜃sin𝜙√ℬ𝒜1𝑎cos𝜃sin𝜙−sin𝜃cos𝜙2sin2𝜃√ℬ−𝒜13/2𝒜0𝒜1ℬcos𝜃−𝑟sin𝜃0}, (34)

where 𝒜₁ has the form 𝒜₁=√𝑟²+𝑎². (35) Tetrad (34) has the same associated metric of tetrad (20). The second rotating frame is described by the coframe components ^K𝜗^N𝛼_𝐼𝐼=Λ^𝛼_𝛾𝜗^𝛾_𝐼𝐼,where𝜗^𝛾_𝐼𝐼=ℎ^𝛾_𝑖𝐼𝐼𝑑𝑥^𝑖. (36) Using (24) and (34), one can get ^K𝜗^N0_𝐼𝐼=−𝑎𝒜₁sin²𝜃ℬ−𝒜₁𝑑𝜙+𝑎²sin²𝜃−𝒜₁ℬ𝑑𝑡𝒜_KN,𝜗¹_𝐼𝐼=−𝑟sin²𝜃+𝒜₁cos²𝜃𝑑𝑟+sin𝜃cos𝜃√ℬ𝒜₁−𝑟𝑑𝜃√ℬ_KN,𝜗²_𝐼𝐼=−sin𝜃cos𝜃𝑟−𝒜₁𝑑𝑟+ℬ𝑟sin²𝜃+cos²𝜃𝒜₁𝑑𝜃√ℬ_KN,𝜗³_𝐼𝐼𝒜=−sin𝜃^13/2−𝑎²sin²𝜃√ℬ𝑑𝜙+𝑎√ℬ−𝒜₁𝑑𝑡𝒜. (37) The two frames (25) and (37) are related through the local Lorentz transformation; that is, ^K𝜗^N₁^𝛼=Λ𝜗₁^𝛼_𝛾^KN₂^𝛾, (38) where (Λ₁^𝛼_𝛾) has the formΛ₁^𝛼_𝛾=𝒜−100−²𝒜₃−𝑟⁵+2𝑟⁴𝑀−𝑟³𝑞²−𝑟𝑎⁴cos²𝜃𝑎sin𝜃𝑟²𝑟⁴−2𝑟³𝑀+𝑟²𝑞²+𝑎⁴cos²𝜃𝒜0−10000−10₂−𝑟𝑎sin𝜃𝑟𝒜₂00−1, (39)

where 𝒜₂=𝑟²−2𝑀+𝑞²,𝒜₃=𝑟⁴+𝑎⁴cos²𝜃. (40)

Coframe (37) has the following asymptotic nonvanishing components of Riemannian connection up to 𝑂(1/𝑟⁴): Γ10≅−4𝑀𝑟−3𝑞²𝑎sin²𝜃𝑑𝜙−2𝑀𝑟−𝑞²𝑑𝑡2𝑟³1+𝑂𝑟⁴,Γ12≅−2𝑟²−2𝑀𝑟+𝑞²𝑑𝜃2𝑟³+1𝑟⁴,Γ13𝑟≅−sin𝜃2𝑟²−2𝑀𝑟+𝑞²𝑑𝜙+2𝑀𝑎𝑐𝑑𝑡2𝑟³+1𝑟⁴,Γ20≅2𝑀𝑟−𝑞²𝑎sin𝜃cos𝜃𝑑𝜙2𝑟³+1𝑟⁴,Γ23≅−cos𝜃2𝑟³𝑑𝜙−4𝑀𝑎𝑑𝑡2𝑟³+1𝑟⁴. (41) (Terms like 𝑀𝑎², 𝑀², 𝑀𝑞², …, are neglected in these calculations.) The asymptotic nonvanishing components of the vector 𝜉^𝛼 have the form 𝜉⁰≅2𝑟²−2𝑀𝑟+𝑞²𝜁⁰+2𝑀𝑎𝑟sin²𝜃𝜁³2𝑟²+1𝑟³,𝜉¹≅−2𝑟²+𝑀𝑟−𝑎²sin²𝜃−𝑞²𝜁¹+𝑎²sin𝜃cos𝜃𝜁²2𝑟²𝜁¹+1𝑟³,𝜉²≅−2𝑟²+𝑎²cos²𝜃𝜁2𝑟²+1𝑟³,𝜉³=sin𝜃2𝑀𝑎𝜁⁰−2𝑟³+𝑟𝑎²𝜁³2𝑟²+1𝑟³. (42)

Using coframe (37) as well as the trivial Weitzenböck connection, one can get asymptotically the nonvanishing components of the translation momentum, that is, 𝐻𝛼, 𝐻0≅sin𝜃16𝜋𝑟4𝑀𝑟−4𝑟²−𝑎²−2𝑞²1𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟³,𝐻3≅𝑎sin²𝜃𝑀𝑟−𝑞²8𝜋𝑟²1𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟³. (43) The direct evaluation of the integral (18), using (42) and (43) and assuming the fact that the vector field 𝜉 is with constant holonomic components, yields the following infinite total conserved charge: 𝒬=_𝜕𝑆𝜉^𝛼𝐻_𝛼𝑞=−𝑟−2𝑀−²𝑟+𝑎²𝜁4𝑟⁰−𝑎𝑀𝜁³. (44) For the tetrad ^K𝜗^N𝛼_𝐼𝐼=(Λ^𝛼_𝛾)𝜗^𝛾_𝐼𝐼 determined by (37), the conserved charge corresponding to the diffeomorphism generated by the shift along the time coordinate has a divergent value −(𝑟−2𝑀−𝑞²/𝑟+𝑎²/4𝑟) of the total energy of the configuration whereas, for the vector field along the azimuthal coordinate, we get −𝑎𝑀, which is a well-known value [80]. It is worth to mention that when one uses (19) to calculate the total conserved quantities related to coframe (37), one also does not get the standard value; that is, 𝑞𝒬=𝑀−²+𝑎2𝑟²𝜁6𝑟⁰−2𝑀+𝑟3𝑎𝜁³. (45) The only gain obtained from the use of (19), when it is used to calculate the total charge, is in fact that it removes the divergence from the time coordinate and gives the correct value of energy along the time coordinate; however, the value of the angular momentum along the azimuthal coordinate becomes unfamiliar and divergent. Therefore, we are going to use other conserved charges to overcome the previous problem.

5. Total Conserved Charge Using the PG Version of TEGR and Discussion on the Choice of the Frame

Obukhov et al. [48, 49] show that teleparallel gravity can be naturally defined as a particular case of PG gravity [63], and from the Lagrangian of this theory, that is, the Lagrangian of PG gravity, they derived the invariant conserved charges for the teleparallel model to have the form 𝒬₁()=1𝜉,𝜗2𝜅_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾−Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾−Ξ^𝛼𝛽𝜂_𝛼𝛽, (46) where Ξ^𝛼𝛽 is defined as Ξ=^𝛼_𝛽^def.𝜉⌋Γ^𝛼_𝛽+Θ^𝛼_𝛽,Γ^𝛼_𝛽=Λ_𝐼^−1𝛼_𝛾𝑑Λ^𝛾_𝐼𝛽,Θ^𝛼𝛽=𝑒^[𝛼⌋ℓ_𝜉𝜗^𝛽], (47) where Λ^𝛾_𝐼𝛽 is an appropriate choice of local Lorentz transformation. (We will denote the antisymmetric part by the square bracket [], 𝐴_[𝜇𝜈]=(1/2)(𝐴_𝜇𝜈−𝐴_𝜈𝜇).) (The Lie derivative is given on exterior forms by ℓ_𝜉=𝑑𝜉⌋+𝜉⌋𝑑.) Obukhov et al. [48, 49] show that the last term in (46) acts as a regularized term for the calculation of the total conserved charge.

Equation (46) is employed to calculate the total conserved charges (Table 1) associated with frame (38). The appropriate matrix of local Lorentz transformation, (Λ_𝐼^𝛼_𝛾), has the form _𝐼Λ^𝛼_𝛾def.=100𝑎sin𝜃𝑟−𝑎sin𝜃sin𝜙𝑟sin𝜃cos𝜙cos𝜃cos𝜙−sin𝜙𝑎sin𝜃cos𝜙𝑟.sin𝜃sin𝜙cos𝜃sin𝜙cos𝜙0cos𝜃−sin𝜃0 (48) Using (48), we get the nonvanishing components of Γ^𝛼_𝛽Γ01=−𝑎sin²𝜃𝑑𝜙𝑟0,Γ2=−𝑎sin𝜃cos𝜃𝑑𝜙𝑟,Γ03=−−𝑎(𝑟cos𝜃𝑑𝜃−sin𝜃𝑑𝑟)𝑟²,Γ12132=−𝑑𝜃,Γ=−sin𝜃𝑑𝜙,Γ3=−cos𝜃𝑑𝜙, (49) and the nonvanishing components of Θ^𝛼𝛽 have the form Θ⁰¹=−2𝑎sin²𝜃𝜁³𝑟,Θ⁰²=−𝑎sin2𝜃𝜁³𝑟,Θ⁰³=2𝑎𝑟cos𝜃𝜁²−sin𝜃𝜁¹𝑟²,Θ¹²=−2𝜁²,Θ¹³=−2sin𝜃𝜁³,Θ²³=−2cos𝜃𝜁³. (50) Finally, the nonvanishing components of Ξ^𝛼𝛽 have the form Ξ⁰¹=−𝑎𝜁³sin²𝜃𝑟,Ξ⁰²=−𝑎𝜁³sin𝜃cos𝜃𝑟,Ξ¹³=−𝜁³sin𝜃,Ξ²³=−𝜁³cos𝜃. (51) Using (41), (42), (49), and (51) in (46), we finally get the total conserved charge of coframe (37) in the form 𝒬₁()≅1𝜉,𝜗2𝜅_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾−Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾+Ξ^𝛼𝛽𝜂_𝛼𝛽=𝑞𝑀−²𝜁2𝑟⁰−𝑀𝑎𝜁³. (52) Equation (52) shows that the total energy and the total angular momentum of coframe (37) have their commonly known form.

Table 1 

Comparison between the results of the total conserved charges and the other definitions.

5.1. Third Coframe

The covariant form of the third KN tetrad can be written asℎ^𝛼_{𝑖𝐼𝐼𝐼}=ℬ𝒜00−𝑎sin²𝜃ℬ𝒜𝑎sin𝜃sin𝜙√𝒜sin𝜃cos𝜙𝒜ℬcos𝜃cos𝜙√𝒜sin𝜃sin𝜙𝑟²+𝑎²√𝒜−𝑎sin𝜃cos𝜙√𝒜sin𝜃sin𝜙𝒜ℬcos𝜃sin𝜙√𝒜−sin𝜃cos𝜙𝑟²+𝑎²√𝒜0cos𝜃𝒜ℬ−sin𝜃√𝒜0. (53)

Tetrad field (53) has the same associated metric of tetrad field (20). Using the spherical local coordinates (𝑡,𝑟,𝜃,𝜙), the third rotating frame is described by the coframe components ^K𝜗^N𝛼_𝐼𝐼𝐼=Λ^𝛼_𝛾𝜗^𝛾_𝐼𝐼𝐼,where𝜗^𝛼_𝐼𝐼𝐼=ℎ^𝛼_{𝑖𝐼𝐼𝐼}𝑑𝑥^𝑖. (54) Using (24) and (53) we get 𝜗_𝐼𝐼𝐼0=−ℬ𝒜𝑐𝑑𝑡+𝑎sin²𝜃𝑑𝜙,𝜗_𝐼𝐼𝐼1=𝒜ℬ𝜗𝑑𝑟,_𝐼𝐼𝐼2=√𝒜𝑑𝜃,𝜗_𝐼𝐼𝐼3=sin𝜃𝑟²+𝑎²𝑑𝜙+𝑎𝑑𝑡√𝒜. (55)

The two frames (25) and (55) are related through local Lorentz transformation; that is, ^K𝜗^N_𝐼^𝛼=Λ𝜗₂^𝛼_𝛾^KN𝛾_𝐼𝐼𝐼, (56) where (Λ₂^𝛼_𝛾) has the form Λ₂^𝛼_𝛾=𝑟²+𝑎²√𝒞𝑎√00ℬsin𝜃√𝒞𝑎√01000010ℬsin𝜃√𝒞𝑟00²+𝑎²√𝒞. (57) The nonvanishing components of the corresponding Riemannian connection of the coframe (55) read Γ10=−𝑎sin²𝜃𝑟²(𝑟+𝑀)+𝑎²cos²𝜃(𝑟−𝑀)−𝑟𝑞²−𝑀𝑟𝑑𝜙₂−𝑎²cos²𝜃−𝑟𝑞²𝑑𝑡×𝒜²⁻¹,Γ12𝑎=−²sin𝜃cos𝜃𝑑𝑟+𝑟ℬ𝑑𝜃𝒜√ℬ,Γ13√=−𝑟sin𝜃ℬ𝑑𝜙𝒜,Γ02=𝑎sin𝜃cos𝜃√ℬ𝑑𝜙𝒜,Γ23=−cos𝜃𝒞𝑑𝜙−2𝑀𝑟−𝑞²𝑎𝑐𝑑𝑡𝒜²,Γ30=𝑟𝑎sin𝜃𝑑𝑟−cos𝜃ℬ𝑑𝜙𝒜√ℬ. (58)

The nonvanishing components of the vector 𝜉^𝛼 have the form 𝜉⁰=−ℬ𝒜𝜁⁰+𝑎sin²𝜃𝜁³,𝜉¹=𝒜ℬ𝜁¹,𝜉²=𝒜𝜁²,𝜉³=sin𝜃√𝒜𝑎𝜁⁰+𝑟²+𝑎²𝜁³. (59) If we take coframe (55), as well as the trivial Weitzenböck connection, one can get asymptotically the nonvanishing components of the translation momentum 𝐻0≅−sin𝜃2𝑀𝑟−𝑞²𝑑𝜃∧𝑑𝜙14𝑟𝜋+⋯+𝑂𝑟²,𝐻31=𝑂𝑟². (60) The direct evaluation of integral (18) yields the following anomalous conserved charge: 𝒬=_𝜕𝑆𝜉^𝛼𝐻_𝛼=𝑞𝑀−²𝜁2𝑟⁰−4𝑎𝑀𝜁³3. (61)

It is worth mentioning that when one uses (19) to calculate the conserved quantities related to coframe (55), one can get (−(4𝑀−𝑟)𝑎𝜁³)/3 in the azimuthal direction, and the conserved quantity along the time coordinate does not change. Therefore, we are going to use (46) to calculate the total conserved quantities related to coframe (55). The appropriate matrix of local Lorentz transformation is given by (48). Using the data calculated for the second coframe, that is, using (50), (51), (58), and (59) in (46), we get the conserved charge related to coframe (55) as given by (52).

5.2. Fourth Rotating Frame

The covariant form of the fourth rotating tetrad field having axial symmetry in spherical coordinates can be written asℎ^𝛼_𝑖4=𝐴₁𝐴₂𝐴₃𝐴₄𝐵₁sin𝜃cos𝜙𝐵₂sin𝜃cos𝜙𝐵₃cos𝜃cos𝜙𝐵₄𝐶sin𝜙sin𝜃₁sin𝜃sin𝜙𝐶₂sin𝜃sin𝜙𝐶₃cos𝜃sin𝜙𝐶₄𝐷cos𝜙sin𝜃₁cos𝜃𝐷₂cos𝜃𝐷₃sin𝜃𝐷₄cos𝜃, (62)

where 𝐴_𝑖, 𝐵_𝑖, 𝐶_𝑖, and 𝐷_𝑖 read 𝐴₁=1−𝑀𝑟−𝑞²/2𝒜,𝐴₂=𝑀𝑟−𝑞²/2ℬ,𝐴₃=0,𝐴₄=𝑀𝑟−𝑞²/2𝑎sin²𝜃𝒜,𝐵₁=𝑀𝑟−𝑞²/2𝒜,𝐵₂=1𝑞ℬcos𝜙𝑟𝛼−𝑀𝑟−²2cos𝜙,𝐵₃=𝛼,𝐵cos𝜙₄=𝛽−𝑀𝑟−𝑞²/2𝑎sin²𝜃cos𝜙/𝒜,𝐶sin𝜙₁=𝑀𝑟−𝑞²/2𝒜,𝐶₂=1𝑞Υsin𝜙𝑟𝛽−𝑀𝑟−²2,𝐶sin𝜙₃=𝛽sin𝜙,𝐶₄=−𝛼+𝑀𝑟−𝑞²/2𝑎sin²𝜃sin𝜙/𝒜,𝐷cos𝜙₁=𝑀𝑟−𝑞²/2𝒜,𝐷₂=1+𝑀𝑟−𝑞²/2ℬ,𝐷₃𝐷=−𝑟,₄=−𝑀𝑟−𝑞²/2𝑎sin²𝜃𝒜, (63) where 𝛼 and 𝛽 are defined by 𝛼=𝑟cos𝜙+𝑎sin𝜙,𝛽=𝑟sin𝜙−𝑎cos𝜙. (64)

Using the spherical local coordinates (𝑡,𝑟,𝜃,𝜙) the fourth rotating frame is described by the coframe components ^K𝜗^N𝛼_𝐼𝑉=Λ^𝛼_𝛾𝜗^𝛾_𝐼𝑉,where𝜗^𝛾_𝐼𝑉=ℎ^𝛾_𝑖𝐼𝑉𝑑𝑥^𝑖, (65) Using (24) and (62), we get 𝜗_𝐼𝑉0=𝐴₁𝑐𝑑𝑡+𝐴₂𝑑𝑟+𝐴₃𝑑𝜃+𝐴₄𝜗𝑑𝜙,_𝐼𝑉1=𝐵₁sin𝜃cos𝜙𝑐𝑑𝑡+𝐵₂sin𝜃cos𝜙𝑑𝑟+𝐵₃cos𝜃cos𝜙𝑑𝜃+𝐵₄𝜗sin𝜃sin𝜙𝑑𝜙,_𝐼𝑉2=𝐶₁sin𝜃sin𝜙𝑐𝑑𝑡+𝐶₂sin𝜃sin𝜙𝑑𝑟+𝐶₃cos𝜃sin𝜙𝑑𝜃+𝐶₄𝜗sin𝜃cos𝜙𝑑𝜙,_𝐼𝑉3=𝐷₁cos𝜃𝑐𝑑𝑡+𝐷₂cos𝜃𝑑𝑟+𝐷₃sin𝜃𝑑𝜃+𝐷₄cos𝜃𝑑𝜙. (66) The local Lorentz transformation Λ^𝛼_𝛾 is given by the matrix (24). The two frames (25) and (66) are related through local Lorentz transformation; that is, ^K𝜗^N_𝐼^𝛼=Λ𝜗₃^𝛼_𝛾^KN𝛼_𝐼𝑉, (67) where (Λ₃^𝛼_𝛾) has the formΛ₃^𝛼_𝛾=ℱ₁√𝒜ℬ2ℱ₂√𝒞−𝑀₁ℱ₃√𝒜ℬsin𝜃cos𝜃ℱ₂√𝒞−𝑀₁ℱ₄√𝒜ℬℱ₂√𝒞𝑀₁ℱ₅√𝒜ℬsin𝜃ℱ₂√𝒞−𝑀₁√−𝒜ℬsin𝜃cos𝜃ℱ₆2√ℱ𝒜ℬ2√𝒜⁷ℬsin𝜃ℱ₈2√0ℱ𝒜ℬ₉√𝒜−sin𝜃cos𝜃ℱ₁₀√𝒜cos𝜃ℱ₁₁√𝒜−𝑀₁𝑎sin𝜃√−𝒜𝒞cos𝜃ℱ₁₂2√−𝒜𝒞sin𝜃ℱ₁₃2√ℱ𝒜𝒞2√𝒜¹⁴𝒞, (68)

with 𝑀₁=𝑞𝑀𝑟−²2,ℱ₁=2𝑟⁴−2𝑀𝑟³+2𝑟²𝑎²cos²𝜃+2𝑟²𝑎²−4𝑀𝑎²𝑟cos²𝜃+2𝑀𝑟𝑎²+2𝑎⁴cos²𝜃+𝑞²𝑟²+𝑎²,ℱcos2𝜃₂=𝑟⁴−2𝑀𝑟³+𝑟²𝑎²cos²𝜃+𝑟²𝑎²−2𝑀𝑎²𝑟cos²𝜃+𝑎⁴cos²𝜃+𝑞²ℱ𝒜,₃=𝑟²sin2𝜙−1−𝑎²ℱsin2𝜙+1−2𝑟𝑎cos2𝜙,₄=𝑟²+𝑎²sin2𝜙sin²𝜃+cos²𝜃+2𝑎𝑟sin²ℱ𝜃cos2𝜙,₅=−𝑟²+𝑎²ℱcos2𝜙+2𝑎𝑟sin2𝜙,₆=2𝑟(𝑟−𝑀)1−sin2𝜙+2𝑎²+2𝑎𝑟cos2𝜙+𝑞²,ℱ1+sin2𝜙₇=𝑀2𝑟−2𝑟²−𝑞²sin2𝜙sin²𝜃+cos²𝜃−2𝑎𝑟sin²𝜃cos2𝜙−2𝑎²cos²ℱ𝜃,₈=2𝑟²−2𝑀𝑟+𝑞²ℱcos2𝜙+2𝑟𝑎cos2𝜙,₉=𝑟sin²𝜃+sin2𝜙cos²𝜃−𝑎cos²ℱ𝜃cos2𝜙,₁₀ℱ=𝑟1−sin2𝜙+𝑎cos2𝜙,₁₁ℱ=𝑟cos2𝜙+𝑎sin2𝜙,₁₂=2𝑟³cos2𝜙+𝑎sin2𝜙2𝑟²+2𝑀𝑟−𝑞²sin²𝜃+2𝑎²cos²𝜃−𝑎2𝑀𝑟−𝑞²sin²𝜃−2𝑎𝑟cos²,ℱ𝜃cos2𝜙₁₃=2𝑟³cos2𝜙+𝑎sin2𝜙2𝑟²+2𝑀𝑟−𝑞²sin²𝜃+𝑎²cos²𝜃+𝑎2𝑀𝑟−𝑞²cos²𝜃+𝑎cos²,ℱ𝜃cos2𝜙₁₄=2𝑟³sin2𝜙−𝑎cos2𝜙2𝑟²+2𝑀𝑟−𝑞²sin²𝜃+𝑎²cos²𝜃+2𝑎²𝑟sin2𝜙cos²𝜃. (69) The nonvanishing components of the Riemannian connection of the coframe (66) up to order 𝑂(1/𝑟³) read Γ10≅𝑞𝑀−²×)2𝑟sin2𝜃1−sin2𝜙(𝑑𝑟+𝑐𝑑𝑡+𝑟cos2𝜙𝑑𝜙+𝑟sin²𝜃+sin2𝜙cos²𝜃𝑑𝜃×2𝑟²⁻¹1+𝑂𝑟³,Γ12≅𝑞2𝑟−2𝑀−²𝑞2𝑟sin2𝜙𝑑𝜃+𝑀−²×()2𝑟×sin2𝜃cos2𝜙𝑑𝜙2𝑟⁻¹1+𝑂𝑟³,Γ131≅−2𝑟²×𝑞sin𝜃𝑎sin2𝜃𝑑𝑟+2𝑀−²2𝑟𝑟sin²𝜃cos2𝜙𝑑𝜃+sin2𝜃3𝑟²𝑞−𝑟𝑀−²2𝑟sin²𝜃11−sin2𝜙𝑑𝜙+𝑂𝑟³,Γ20𝑞≅−𝑀−²×2𝑟cos²𝜃+sin²𝜃sin2𝜙(𝑑𝑟+𝑐𝑑𝑡)+𝑟sin2𝜃2×1−sin2𝜙𝑑𝜃+𝑟sin²𝜃cos2𝜙𝑑𝜙×𝑟²⁻¹1+𝑂𝑟³,Γ231≅−2𝑟²𝑞2𝑎sin𝜃𝑑𝑟−2𝑀−²2𝑟𝑟cos𝜃cos2𝜙𝑑𝜃+2sin𝜃3𝑟²−𝑞𝑀−²𝑟2𝑟sin²𝜃+sin2𝜙cos²𝜃1𝑑𝜙+𝑂𝑟³,Γ30𝑞≅−𝑀−²()×𝑟2𝑟sin𝜃cos2𝜙𝑑𝑟+𝑐𝑑𝑡−𝑟cos𝜃cos2𝜙𝑑𝜃+sin𝜃sin2𝜙𝑑𝜙₂₋₁1+𝑂𝑟³. (70)

The nonvanishing components of the vector 𝜉^𝛼 up to order 𝑂(1/𝑟) have the form 𝜉⁰≅12()𝜁2𝑟𝑀−𝑟⁰+2𝑀𝜁¹−2𝑎𝑀sin²𝜃𝜁³,𝜉¹≅−1𝑟𝑀𝜁⁰+(𝑀+𝑟)𝜁¹+𝑎sin²𝜃(𝑟−𝑀)𝜁³,𝜉²≅−𝜁²𝑟−𝑎sin𝜃cos𝜃𝜁³,𝜉³≅1𝑟𝑎sin𝜃𝜁¹+𝑎𝑟cos𝜃𝜁²−𝑟²sin𝜃𝜁³. (71) Using coframe (66) as well as the trivial Weitzenböck connection, the nonvanishing components of the translation momentum read 𝐻0≅−𝑀−𝑞²/2𝑟sin𝜃14𝜋𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟²,𝐻1≅𝑀−𝑞²/2𝑟sin𝜃14𝜋𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟²,𝐻2≅𝑎²sin²𝜃cos𝜃14𝜋𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟²,𝐻3=−𝑎sin²𝜃14𝜋𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟². (72) The direct evaluation of the integral (18) for vector fields 𝜉 with constant components 𝜁 yields the following infinite total conserved charge: 𝒬=_𝜕𝑆𝜉^𝛼𝐻_𝛼=𝑞𝑀−²𝜁2𝑟⁰−(2𝑀−𝑟)𝑎𝜁³3. (73) When we apply (19) to calculate the total conserved quantity of the coframe (66), we get 𝑞𝒬=𝑀−²𝜁2𝑟⁰−2𝑎𝑀𝜁³3, (74) which is also not the standard result. Using the same arguments discussed for the second and third coframes, one can get the correct value of the total conserved charges when (46) is employed.

5.3. Fifth Coframe

The covariant form of the fifth KN tetrad field having axial symmetry in spherical coordinates may be written asℎ^𝛼_𝑖𝑉=−ℱ𝒜00−2𝑀𝑟−𝑞²𝑎sin²𝜃√0𝒜ℱ𝒜ℬsin𝜃cos𝜙sin𝜃sin𝜙√𝒜𝒜ℬℱ0sin𝜃cos𝜃𝒜ℬcos𝜃cos𝜙cos𝜃sin𝜙√𝒜−𝒜ℬℱsin²𝜃0−𝒜ℬsin𝜙cos𝜙√,𝒜0whereℱ=𝑟²−2𝑀𝑟+𝑎²cos²𝜃+𝑞². (75)

Tetrad (75) has the same associated metric of tetrad (20). Using the spherical local coordinates (𝑡,𝑟,𝜃,𝜙), the fifth rotating frame is described by the following coframe components: ^K𝜗^N_𝑉^𝛼=Λ^𝛼_𝛾𝜗_𝑉^𝛾,where𝜗_𝑉^𝛾=ℎ^𝛾_𝑖𝑉𝑑𝑥^𝑖. (76) Using (24) and (76), the coframe of the fifth KN read 𝜗_𝑉0=−2𝑀𝑟−𝑞²𝑎sin²𝜃𝑑𝜙+ℱ𝑑𝑡√,𝜗𝒜ℱ_𝑉1=𝒜ℬ𝑑𝑟,𝜗_𝑉2=√𝜗𝒜𝑑𝜃,_𝑉3=sin𝜃𝒜ℬℱ𝑑𝜙. (77) The two frames (25) and (77) are related through the local Lorentz transformation; that is, ^K𝜗^N_𝐼^𝛼=Λ𝜗₄^𝛼_𝛾^KN_𝑉^𝛾, (78) where (Λ₄^𝛼_𝛾) has the form Λ₄^𝛼_𝛾=−𝒜√ℬ√𝒞ℱ00−2𝑀𝑟−𝑞²𝑎sin𝜃√𝒞ℱ010000102𝑀𝑟−𝑞²𝑎sin𝜃√𝒜√𝒞ℱ00ℬ√.𝒞ℱ (79) Coframe (77) has the following nonvanishing components of Riemannian connection: Γ10=√ℬ2𝒜²√ℱ𝑎sin²𝜃2𝑀𝑟²−2𝑀𝑎²cos²𝜃+2𝑟𝑞²−𝑟𝑑𝜙2𝑀²−𝑎²cos²𝜃−2𝑟𝑞²,Γ12𝑎𝑑𝑡=−²sin𝜃cos𝜃𝑑𝑟+𝑟ℬ𝑑𝜃𝒜√ℬ,Γ13=−sin𝜃√ℱ𝒜²×𝑟𝒜²−𝑀𝑟²2𝑟²+𝑎²1+cos²𝜃−𝑎⁴sin²𝜃cos²𝜃+𝑞²𝑟𝑟²+𝑎²𝑑𝜙−𝑎𝑀𝑟²−𝑎²cos²𝜃−𝑟𝑞²,Γ20=𝑑𝑡2𝑀𝑟−𝑞²𝑎sin𝜃cos𝜃√ℱ𝒜²𝑟²+𝑎²,Γ23√𝑑𝜙+𝑎𝑑𝑡=−cos𝜃ℬ√ℱ𝒜²×𝒜²+2𝑀𝑟−𝑞²𝑎²sin²𝜃𝑑𝜙+𝑎2𝑀𝑟−𝑞²,Γ30=𝑎𝑐𝑑𝑡sin𝜃𝑀𝑟²−𝑀𝑎²cos²𝜃−𝑟𝑞²−𝑑𝑟2𝑀𝑟−𝑞²×√cos𝜃ℬ𝑑𝜃ℬ𝒜𝒜−2𝑀𝑟−𝑞²⁻¹. (80) The nonvanishing components of the vector 𝜉^𝛼 have the form 𝜉⁰=2𝑀𝑟−𝑞²𝑎𝜁³sin²𝜃+𝑟²−2𝑀𝑟−𝑎²cos²𝜃−𝑞²𝜁⁰√,𝜉𝒜ℬ₁=𝒜ℬ𝜁¹,𝜉²=√𝒜𝜁²,𝜉³=𝒜𝑟²−2𝑀𝑟+𝑎²+𝑞²√ℬ𝜁³. (81)

If we take coframe (77), as well as the trivial Weitzenböck connection, we finally get the translation momentum, that is, 𝐻𝛼, 𝐻0=sin𝜃8𝜋𝑟4𝑀𝑟−𝑎²1−3cos²𝜃−2𝑞²1𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟²,𝐻3=𝑎sin²𝜃𝑟²−𝑀𝑟+𝑞²4𝜋𝑟²1𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟³. (82) The direct evaluation of the integral (18) for vector fields 𝜉 with constant components 𝜁 yields the following infinite total conserved charges: 𝒬=_𝜕𝑆𝜉^𝛼𝐻_𝛼=𝑞𝑀−²𝜁2𝑟⁰−𝑀−𝑟3𝑎𝜁³. (83) When we use (19) to calculate the total conserved charge related to coframe (77), one also did not get the standard value; that is, 𝑞𝒬=𝑀−²𝜁2𝑟⁰−𝑎𝑀𝜁³3. (84) Using the same arguments discussed for the pervious coframes, one can get the correct value of the total conserved charge when (46) is hired.

6. Discussion and Conclusion

Within the framework of MAG, the total Lagrangian is formed from geometrical part plus matter part; that is, ℒ_total=𝑉(𝑔_𝛼𝛽,𝜗^𝛼,𝑄_𝛼𝛽,𝑇^𝛼,𝑅_𝛼^𝛽)+𝐿_matter(𝑔_𝛼𝛽,𝜗^𝛼,Ψ,𝐷Ψ) with 𝑄_𝛼𝛽 is the nonmetricity and Ψ is the matter fields. Baekler and Hehl [81] have obtained a stationary axially symmetric exact solution of the vacuum field equation for a specific gravitational Lagrangian. Their black hole solution embodies a Kerr-de Sitter metric and the post-Riemannian structures of torsion and nonmetricity and is characterized by mass, angular momentum, and shear charge which measure the violation of the Lorentz invariance. When the cosmological constant vanishes, that is, 𝜆=0, the solution given by (61) derived in [81] will be identical with the third coframe defined by 𝜗^𝑅₃^𝛼=(Λ₁^𝛼_𝛾)(ℎ^𝛾_𝑖)₃𝑑𝑥^𝑖.

Careful comparison between purely tetrad formulation and PG theory reveals an interesting observation: both approaches are not identical. Namely, in the formulation of pure tetrad, one is working with the coframe 𝜗^𝛼 as the only field variable. It is determined completely (up to local Lorentz transformation) from the equations of motion. However, the resulting conserved current and charge are not invariant under local Lorentz transformations. Moreover, in general, the total conserved quantities are divergent. Therefore, one must select a tetrad field that meets certain conditions in order to obtain physically acceptable charges. The situation in the PG theory is to some extent complementary. Namely, the total conserved current and charge are explicitly invariant under both diffeomorphisms and local Lorentz transformations. However, for their computation, one needs, beside the tetrad, to know the connection and the Lagrange multiplier that meet their requirements. Each of these variables is not determined through the field equations in a unique local Lorentz covariant way. The lack of uniqueness in the Lagrange multiplier can be avoided through the imposition of generalized symmetry conditions on the field configurations. But in the selection of the flat connection, there still remains a freedom compatible to the freedom in the choice of the coframe in the tetrad formulation.

The main results of this study are as follows:(i)we use five coframes; their associated metrics grant the same spacetime, that is, KN spacetime;(ii)one of these coframes, first coframe, has its total conserved charge divergent when the conserved charge in the tetrad gravity theory is used ([48, 49] Equation (4⋅11)). This is because of the fact that this coframe is nonholonomic and its Weitzenböck connection is not vanishing neither when 𝑟→∞ nor when the physical quantities are vanishing; (iii)when a regularized conserved charge is used, within purely tetrad formulation, the divergent quantity related to the first coframe, emerged in the time coordinate direction, is removed, and consistent results of total energy and angular momentum are achieved;(iv)the total conserved charges of the second coframe lack the standard form. This is because of the arguments mentioned above regarding the first coframe. Even when the regularized expression is used, the conserved charge is not the familiar one. Therefore, another definition for the total conserved charge ([48, 49] Equation (4⋅25)), “PG version of conserved charge,” is used and the desired outcome is achieved;(v)the total charges of the third, fourth, and fifth coframes are calculated by using the conserved and regularized conserved charge, and it turns out that they suffer from the same disadvantage of the second coframe. When the Poincaré conserved charge is used, satisfactory outcomes are obtained;(vi)we may at this point suggest that in the context of tetrad theory of gravitation, there are many definitions of the total conserved of energy and angular momentum. The most suitable one relies on the employment of the coframe;(vii)Nester et al. [82–84] considered the quasilocal center of mass in tetrad gravity. They used the covariant Hamiltonian formalism, which is given by the quasilocal quantities Hamiltonian term limits, along with the symplectic covariant expressions of the limits of the Hamiltonian. They show that the Hamiltonian boundary term of the tetrad theory plays a significant role in angular momentum and plays a key role in obtaining the correct center-of-mass moment;(viii)Blagojević [85] has assumed that spacetime behaves asymptotically as Minkowski space M4 and has obtained the corresponding global Poincaré generators. The surface terms related to the improved form of these generators appeared to represent the relevant conserved charges energy, momentum, and angular momentum. Mielke and Wallner [86] discussed the Lagrangian form of the energy-momentum of some exact solutions in the teleparallel limit of the PG theory;(ix)the results achieved in the present study can be used to justify some proposals employed in the literature for the energy-momentum in TEGR [82–84, 87–89] and to extend the concepts of energy-momentum and angular momentum from TEGR to the general teleparallel theory.

Acknowledgment

This work is partially supported by the Egyptian Ministry of Scientific Research under project ID 24-2-12.