Kerr-Taub-NUT General Frame, Energy, and Momentum in Teleparallel Equivalent of General Relativity

Abstract

A new exact solution describing a general stationary and axisymmetric object of the gravitational field in the framework of teleparallel equivalent of general relativity (TEGR) is derived. The solution is characterized by three parameters “the gravitational mass 𝑀, the rotation 𝑎, and the NUT 𝐿.” The vierbein field is axially symmetric, and the associated metric gives the Kerr-Taub-NUT spacetime. Calculation of the total energy using two different methods, the gravitational energy momentum and the Riemannian connection 1-form Γ_𝛼^𝛽, is carried out. It is shown that the two methods give the same results of energy and momentum. The value of energy is shown to depend on the mass 𝑀 and the NUT parameter 𝐿. If 𝐿 is vanishing, then the total energy reduced to the energy of Kerr black hole.

1. Introduction

One of the generalizations of Kerr solution in general relativity (GR) is done by introducing an extra parameter, the so-called gravitomagnetic monopole moment or magnetic mass. The resulting solution describes the spacetime of a localized stationary and axisymmetric object known as the Kerr-Taub-NUT (Newman-Unti-Tamburino) solution of the vacuum Einstein field equations [1]. This solution belongs to a general class of metrics that admits separability of variables in the Hamilton-Jacobi equations and contains three physical parameters: the gravitational mass 𝑀 (gravitoelectric charge), the magnetic mass 𝐿 (NUT charge), and the rotation parameter 𝑎. The presence of the NUT charge in the spacetime destroys its asymptotic structure making it asymptotically nonflat.

Attempts at identifying an energy momentum density for gravity has led to various energy momentum complexes which are pseudotensors [2]. Pseudotensors are not covariant objects that is, they essentially depend on the reference frame and thus cannot provide a true physical local gravitational energy momentum density. Hence, the pseudotensor approach has been largely abandoned (cf., [3, 4]).

Teleparallel theories are interesting for several reasons: first of all, GR can be viewed as a particular theory of teleparallelism, and, thus, teleparallelism could be considered, at the very least, as a different point of view that can lead to the same results [5]. Second, in this framework, one can define an energy momentum complex for the gravitational field that is a true tensor under general coordinate transformations, but not under local Lorentz transformation. This is the reason why teleparallelism was reconsidered by Møller when he was studying the problem of defining an energy momentum tensor for the gravitational field [6, 7]. The idea was taken over by Pellegrini and Plebański that constructed the general Lagrangian for these theories [8]. The third reason why these theories are interesting is that they can be seen as gauge theories of the translation group (not the full Poincaré group), and, thus, they give an alternative interpretation of GR [9–16].

A well-posed and mathematically consistence expression for the gravitational energy has been developed [17]. It arises in the realm of the Hamiltonian formulation of the TEGR [17] and meets several crucial requirements for any acceptable definition of gravitational energy. The gravitational energy momentum 𝑃^𝑎 obtained in the framework of the TEGR has been investigated in the context of several distinct configurations of the gravitational filed. For asymptotically flat spacetimes, 𝑃⁰ yields the ADM energy [18].

The objective of the present work is to derive a new exact general axially symmetric solution in TEGR for the gravitational field and then to calculate the total energy using two different methods, the gravitational energy momentum that is coordinate independent and the other is the Riemannian connection 1-form Γ_𝛼^𝛽. In Section 2, the derivation of the field equations of the gravitational field is briefly reviewed. A summary of the derivation of energy and angular momentum using the Hamiltonian formulation is presented in Section 2. In Section 3, the general tetrad field with sixteen unknown functions of 𝑟 and 𝜙 is applied to the field equations. Solving the resulting partial differential equations, a new exact analytic solution is derived, that is, new solution within the context of TEGR. In Section 4, calculation of the energy and spatial momentum using the definitions provided in Section 2 is achieved. In Section 5, the language of forms is employed and a brief review of the field equations and the superpotential used in the calculations of energy and spatial momentum are presented. Using the Riemannian connection 1-form, calculation of energy and momentum is provided. The last section is devoted to discussion.

2. The TEGR for Gravitation

In a spacetime with absolute parallelism, the parallel vector field ℎ_𝑎^𝜇 identifies the nonsymmetric affine connectionΓ=^𝜆_𝜇𝜈^def.ℎ_𝑎^𝜆ℎ^𝑎_𝜇,𝜈, (2.1) where ℎ_{𝑎𝜇,𝜈}=𝜕_𝜈ℎ_𝑎𝜇 (spacetime indices 𝜇,𝜈,…, and SO (3,1) indices 𝑎,𝑏,… run from 0 to 3. Time and space indices are indicated to 𝜇=0,𝑖, and 𝑎=(0),(𝑖)). The curvature tensor defined by Γ^𝜆_𝜇𝜈 is identically vanishing, however. The metric tensor 𝑔_𝜇𝜈 is given by𝑔_𝜇𝜈=𝑂_𝑎𝑏ℎ^𝑎_𝜇ℎ^𝑏_𝜈, (2.2) with the Minkowski metric 𝑂_𝑎𝑏=diag(+1,−1,−1,−1) (Latin indices are raising and lowering with the aid of 𝑂_𝑎𝑏 and 𝑂^𝑎𝑏).

Lagrangian density of the gravitational field in the TEGR, in the presence of matter fields, is given by [17] (throughout this paper, we use the relativistic units, 𝑐=𝐺=1 and 𝜅=8𝜋): ℒ_𝐺=√−𝑔𝐿_𝐺√=−−𝑔𝑇16𝜋^𝑎𝑏𝑐𝑇_a𝑏𝑐4+𝑇^𝑎𝑏𝑐𝑇_𝑏𝑎𝑐2−𝑇^𝑎𝑇_𝑎−𝐿_𝑚√=−−𝑔Σ16𝜋^𝑎𝑏𝑐𝑇_𝑎𝑏𝑐−𝐿_𝑚, (2.3) where 𝑔=det(𝑔_𝜇𝜈). The tensor Σ^𝑎𝑏𝑐 is defined byΣ=^{𝑎𝑏𝑐def.}14𝑇^𝑎𝑏𝑐+𝑇^𝑏𝑎𝑐−𝑇^𝑐𝑎𝑏+12𝑂^𝑎𝑐𝑇^𝑏−𝑂^𝑎𝑏𝑇^𝑐. (2.4)𝑇^𝑎𝑏𝑐 and 𝑇^𝑎 are the torsion tensor and the basic vector field defined by𝑇=^𝑎_𝜇𝜈^def.ℎ^𝑎_𝜆𝑇^𝜆_𝜇𝜈=𝜕_𝜇ℎ^𝑎_𝜈−𝜕_𝜈ℎ^𝑎_𝜇,𝑇=^𝜇def.𝑇^𝜈_𝜈^𝜇,𝑇=^𝑎def.ℎ^𝑎_𝜇𝑇^𝜇=𝑇^𝑏_𝑏^𝑎. (2.5)(2.6) The quadratic combination Σ^𝑎𝑏𝑐𝑇_𝑎𝑏𝑐 is proportional to the scalar curvature 𝑅, except for a total divergence term [19]. 𝐿_𝑚 represents the Lagrangian density for matter fields.

The gravitational field equations for the system described by 𝐿_𝐺 are the following:ℎ_𝑎𝜆ℎ_𝑏𝜇𝜕_𝜈√−𝑔Σ^𝑏𝜆𝜈−√−𝑔Σ_𝑎^𝑏𝜈𝑇_𝑏𝜈𝜇−14ℎ_𝑎𝜇𝑇_𝑏𝑐𝑑Σ^𝑏𝑐𝑑=12𝜅√−𝑔𝑇_𝑎𝜇, (2.7) where𝛿𝐿_𝑚𝛿ℎ^𝑎𝜇≡√−𝑔𝑇_𝑎𝜇. (2.8) It is possible to prove by explicit calculations that the left-hand side of the symmetric part of the field equations (2.7) is exactly given by [17]√−𝑔2𝑅_𝑎𝜇−12ℎ_𝑎𝜇𝑅. (2.9) The axial-vector part of the torsion tensor 𝐴_𝜇 is defined by𝐴=_𝜇^def.16𝜖_{𝜇𝜈𝜌𝜎}𝑇^𝜈𝜌𝜎=13𝜖_{𝜇𝜈𝜌𝜎}𝛾^𝜈𝜌𝜎,where𝜖=_{𝜇𝜈𝜌𝜎}^def.√−𝑔𝛿_{𝜇𝜈𝜌𝜎}, (2.10) with 𝛾_𝜈𝜌𝜎=𝑂^𝑎𝑏ℎ_𝑎𝜈ℎ_{𝑏𝜌;𝜎} being the contorsion tensor and 𝛿_{𝜇𝜈𝜌𝜎} is completely antisymmetric and normalized as 𝛿₀₁₂₃=−1.

The definition of the gravitational energy momentum 𝑃^𝑎 four-vector has the form [17]𝑃^𝑎=−_𝑉𝑑³𝑥𝜕_𝑖Π^𝑎𝑖, (2.11) where 𝑉 is an arbitrary volume of the three-dimensional space. In the configuration space, we haveΠ^𝑎𝑖2=−𝜅√−𝑔Σ^𝑎0𝑖,with𝜕_𝜈√−𝑔Σ^𝑎𝜆𝜈=𝜅2√−𝑔ℎ^𝑎_𝜇𝑡^𝜆𝜇+𝑇^𝜆𝜇,where𝑡^𝜆𝜇=12𝜅4Σ^𝑏𝑐𝜆𝑇^𝜇_𝑏𝑐−𝑔^𝜆𝜇Σ^𝑏𝑐𝑑𝑇_𝑏𝑐𝑑. (2.12)

Maluf and Ulhoa [20] defined 𝐿^𝑎𝑏=2_𝑉𝑑³𝑥𝑀^[𝑎𝑏],where𝑀^𝑎𝑏=√−𝑔Σ4𝜋^𝑏0𝑎−Σ^𝑎0𝑏, (2.13) as the 4-angular momentum of the gravitational field for an arbitrary volume 𝑉 of the three-dimensional space.

3. Exact Analytic Solution

Let us begin with the general tetrad field which can be written in the polar coordinates asℎ^𝑖_𝜇=𝐴₁𝑟,𝜙𝐴₂𝑟,𝜙𝐴₃𝑟,𝜙𝐴₄𝐵𝑟,𝜙₁𝑟,𝜙sin𝜃cos𝜙𝐵₂𝑟,𝜙sin𝜃cos𝜙𝐵₃𝑟,𝜙cos𝜃cos𝜙𝐵₄𝐶𝑟,𝜙sin𝜙sin𝜃₁𝑟,𝜙sin𝜃sin𝜙𝐶₂𝑟,𝜙sin𝜃sin𝜙𝐶₃𝑟,𝜙cos𝜃sin𝜙𝐶₄𝐷𝑟,𝜙cos𝜙sin𝜃₁𝑟,𝜙cos𝜃𝐷₂𝑟,𝜙cos𝜃𝐷₃𝑟,𝜙sin𝜃𝐷₄𝑟,𝜙cos𝜃, (3.1) where 𝐴_𝑖(𝑟,𝜙),𝐵_𝑖(𝑟,𝜙),𝐶_𝑖(𝑟,𝜙),and𝐷_𝑖(𝑟,𝜙),𝑖=1,…,4, are unknown functions of 𝑟 and 𝜙. Applying (3.1) to the field equations (2.7), we get a set of nonlinear partial differential equations. (These calculations have been checked using Maple software 13.) Due to the lengthy of writing the output partial differential equations, we will write only the solution that satisfies such differential equations. Here, we list the boundary conditions used to derive the solution. The parallel vector field should have the expression ℎ^𝑘_𝜇=𝛿^𝑘_𝜇+𝑀𝑙^𝑘𝑙_𝜇−𝐿²2𝑚^𝑘𝑚_𝜇, (3.2) where 𝑀 and 𝐿 are free parameters, 𝑙_𝜇 and 𝑚_𝜇 are quantities satisfying the conditions𝑂^𝜇𝜈𝑙_𝜇𝑙_𝜈=0,𝑂^𝜇𝜈𝑚_𝜇𝑚_𝜈=0,𝑂^𝜇𝜈𝑙_𝜇𝑚_𝜈=0, (3.3)𝑙^𝑘 and 𝑚^𝑘 are defined by𝑙^𝑘=𝛿^𝑘_𝜇𝑂^𝜇𝜈𝑙_𝜈,𝑚^𝑘=𝛿^𝑘_𝜇𝑂^𝜇𝜈𝑚_𝜈. (3.4) The solution satisfying the above condition is found to have the following form.

The Exact Solution
If the arbitrary functions take the following values: 𝐴₁=1−𝑀𝑟+𝐿𝐿₁Ω,𝐴₂=𝑀𝑟+𝐿𝐿₁Υ,𝐴₃𝐴=0,₄=𝑀𝑟𝜒+𝐿𝑎𝐿₁1+cos²𝜃+2𝑟²cos𝜃Ω,𝐵₁=𝑀𝑟+𝐿𝐿₁Ω,𝐵₂=1𝑟Υcos𝜙¹𝛼−𝑀𝑟+𝐿𝐿₁cos𝜙,𝐵₃=𝛼cos𝜙,𝐵₄=𝛽−𝑀𝑟+𝐿𝐿₁𝜒cos𝜙/Ω,𝐶sin𝜙₁=𝑀𝑟+𝐿𝐿₁Ω,𝐶₂=1𝑟Υsin𝜙¹𝛽−𝑀𝑟+𝐿𝐿₁sin𝜙,𝐶₃=𝛽,𝐶sin𝜙₄=−𝛼+𝑀𝑟+𝐿𝐿₁𝜒sin𝜙/Ωcos𝜙,𝐷₁=𝑀𝑟+𝐿𝐿₁Ω,𝐷₂=1+𝑀𝑟+𝐿𝐿₁Υ,𝐷₃=−𝑟₁,𝐷₄=−𝑀𝑟+𝐿𝐿₁𝜒Ω, (3.5) where Ω, Υ, 𝐿₁, 𝛼, 𝛽, 𝑟₁, and 𝜒 are defined by Ω=^def.𝑟²+𝐿₁²,Υ=^def.𝑟²+𝑎²−2𝑀𝑟−𝐿²,𝐿₁^def.𝛼=𝐿+𝑎cos𝜃,_def.=𝑟₁cos𝜙+𝑎sin𝜙,𝛽=^def.𝑟₁sin𝜙−𝑎cos𝜙,𝑟=₁^def.𝑟²(),𝜒+𝐿𝐿+2𝑎cos𝜃_def.=𝑎sin²𝜃−2𝐿cos𝜃. (3.6)𝑀, 𝑎, and 𝐿 are the gravitational mass, the angular momentum of the rotating source, and the NUT parameter [1, 21].
The parallel vector field (3.1), using solution (3.5), is axially symmetric in the sense that it is form invariant under the transformations 𝜙⟶𝜙+𝛿𝜙,ℎ_𝜇⁽⁰⁾⟶ℎ_𝜇⁽⁰⁾,ℎ_𝜇⁽¹⁾⟶ℎ_𝜇⁽¹⁾cos𝛿𝜙−ℎ_𝜇⁽²⁾sin𝛿𝜙,ℎ_𝜇⁽²⁾⟶ℎ_𝜇⁽²⁾sin𝛿𝜙+ℎ_𝜇⁽²⁾cos𝛿𝜙,ℎ_𝜇⁽³⁾⟶ℎ_𝜇⁽³⁾. (3.7)
Solution (3.5) satisfies the field equations (2.7), and the associated metric has the following form: 𝑑𝑠²=1ΩΥ−𝑎²sin²𝜃𝑑𝑡²−ΩΥ𝑑𝑟²−Ω𝑑𝜃²−1ΩΩ+𝑎𝜒²sin²𝜃−Υ𝜒²𝑑𝜙²−2Ω𝜒Υ−𝑎Ω+𝑎𝜒sin²𝜃𝑑𝑡𝑑𝜙, (3.8) which is the Kerr-Taub-NUT black hole in the Boyer-Lindquist coordinate [22, 23].
The earlier obtained solutions Schwarzschild spacetime could be generated as special solutions of the tetrad (3.1) using (3.5) by putting 𝐿=0, 𝑎=0, and kerr spacetime can be obtained when 𝐿=0 [21, 23].

4. Energy Associated with the Axially Symmetric Solution

Now, we are going to calculate the energy content of the tetrad field (3.1) using (3.5). The nonvanishing components of the tensor Σ^𝑎𝑏𝑐 needed to the calculation of energy have the form Σ¹⁰¹=𝑎2𝑟⁵𝑎𝐿𝐿₁+𝑎𝑀𝑟sin²𝜃−4𝑎𝐿²cos²,Σ𝜃−2𝑀𝑟𝐿cos𝜃₂₀₁=𝑎2𝑟⁵sin𝜃𝐿𝑟sin²𝜃+2𝐿𝑀cos²𝜃−𝑀𝑎cos𝜃sin²𝜃,Σ³⁰¹=𝐿2𝑟⁵sin²𝜃𝐿²cos𝜃−𝑎𝐿₁sin²𝜃,Σ⁴⁰¹1=−4𝑟⁵𝑎²5𝐿²+2𝑀𝑟sin²𝜃+4𝑎²𝑟²−9𝐿²cos²𝜃+4𝐿𝑟{𝑟−𝑀}𝑎cos𝜃+𝑟𝐿₁−4𝑟⁴. (4.1) Using (3.6) in (2.11), we finally obtain (we subtract Σ^𝜇0𝑘_{𝑀=0,𝑎=0,𝐿=0} in (4.2) to remove divergences from the calculations of energy) 𝑃⁽⁰⁾=𝐸=−_𝑆→∞𝑑𝑆_𝑘Π^(0)𝑘1=−4𝜋_𝑆→∞𝑑𝑆_𝑘√−𝑔ℎ_𝜇⁽⁰⁾Σ^𝜇0𝑘−Σ^𝜇0𝑘_{𝑀=0,𝑎=0,𝐿=0}𝐿≅𝑀+²𝑟−𝐿²𝑀𝑟²−𝐿²5𝑎²+𝐿²𝑟³1+𝑂𝑟⁴. (4.2) The total energy, when 𝑟→∞, gives the ADM (Arnowitt-Deser-Misner) mass.

The nonvanishing components needed to calculate the spatial momentum have the form (terms like 𝑀²,𝐿³,𝐿³𝑀,𝑀²𝑎,…, etc. are neglected in this calculations) Σ⁽¹⁾⁰¹≅sin²𝜃cos𝜙4𝜋𝑟𝑀𝑟+𝐿²1+𝑂𝑟²,Σ⁽²⁾⁰¹≅−sin²𝜃sin𝜙4𝜋𝑟𝑀𝑟+𝐿²1+𝑂𝑟²,Σ⁽³⁾⁰¹≅−sin²𝜃cos𝜙4𝜋𝑟𝑀𝑟+𝐿²1+𝑂𝑟². (4.3) Using (4.3) in (2.11), we finally get the spatial momentum in the form𝑃₁=−_𝑆→∞𝑑𝑆_𝑘Π^(1)𝑘1=−4𝜋_𝑆→∞𝑑𝑆_𝑘𝑒Σ^(1)0𝑘=0,bysamemethod𝑃₂=0,𝑃₃≅1𝑟. (4.4)

The nonvanishing components of the angular-momentum are given by 𝑀⁽⁰⁾⁽¹⁾≅1𝑟,𝜃,𝜙8𝜋𝑟6𝐿𝑎cos𝜃+2𝑟𝑀+𝑎²+3𝐿²cos²𝜃cos𝜙+𝑟𝑎cos²+𝜃+1sin𝜙2𝑟²−2𝐿²−2𝑀𝑟−𝑎²1−4𝐿𝑎cos𝜃cos𝜙+𝑂𝑟²,𝑀⁽⁰⁾⁽²⁾≅1𝑟,𝜃,𝜙8𝜋𝑟6𝐿𝑎cos𝜃+2𝑟𝑀+𝑎²+3𝐿²cos²𝜃sin𝜙−𝑟𝑎cos²+𝜃+1cos𝜙2𝑟²−2𝐿²−2𝑀𝑟−𝑎²1−4𝐿𝑎cos𝜃sin𝜙+𝑂𝑟²,𝑀⁽⁰⁾⁽³⁾𝑟,𝜃,𝜙≅sin𝜃6𝐿𝑎cos²𝜃+2𝑟𝑀+3𝐿²cos𝜃−2𝐿ℎ18𝜋𝑟+𝑂𝑟²,𝑀⁽¹⁾⁽²⁾𝑟,𝜃,𝜙≅sin𝜃3𝑀𝑎cos²𝜃+2[𝑀−𝑟]𝐿cos𝜃−𝑀𝑎18𝜋𝑟+𝑂𝑟²,𝑀⁽¹⁾⁽³⁾𝑟,𝜃,𝜙≅sin²𝜃[3𝑀𝑎cos𝜃+2𝐿𝑀−2𝐿𝑟]sin𝜙+𝐿𝑎cos𝜙18𝜋𝑟+𝑂𝑟²,𝑀⁽²⁾⁽³⁾𝑟,𝜃,𝜙≅sin²𝜃[3𝑀𝑎cos𝜃+2𝐿𝑀−2𝐿𝑟]cos𝜙−𝐿𝑎sin𝜙18𝜋𝑟+𝑂𝑟². (4.5) Using (4.5) in (2.13), we get 𝐿⁽⁰⁾⁽¹⁾=^𝜋₀₀^2𝜋∞₀𝑑𝜃𝑑𝜙𝑑𝑟𝑀⁽⁰⁾⁽¹⁾=0, (4.6) which is a consistent result. By the same method, we finally obtain𝐿⁽⁰⁾⁽²⁾=𝐿⁽⁰⁾⁽³⁾=𝐿⁽¹⁾⁽²⁾=𝐿⁽¹⁾⁽³⁾=𝐿⁽²⁾⁽³⁾=0. (4.7)

We show by explicit calculation that the energy momentum tensor which is a coordinate independent does not give a consistent result of the angular momentum when applied to the tetrad field given by (3.1)! The same problem appeared in ([24], equation (44)) when calculation of angular momentum of the BTZ black hole has been implemented. We may claim that the unusual result of the angular momentum is related to the use of expression (2.13). May be possible solution to this problem is to use other definition or to calculate the total conserved quantities of solution (3.5). This will be our forthcoming research work.

5. Brief Review of Teleparallel Gravity Using the Language of Forms

5.1. Notation

We use the Latin indices 𝑖,𝑗,…, 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 define𝜂=_𝛼^def.𝑒_𝛼1𝜂=𝜖3!^{𝛼𝛽𝛾𝛿}𝜗^𝛽∧𝜗^𝛾∧𝜗^𝛿, (5.1) where 𝜖_{𝛼𝛽𝛾𝛿} is completely antisymmetric tensor with 𝜖₀₁₂₃=1𝜂=_𝛼𝛽^def.𝑒_𝛽𝜂_𝛼=1𝜖2!^{𝛼𝛽𝛾𝛿}𝜗^𝛾∧𝜗^𝛿,𝜂=_𝛼𝛽𝛾^def.𝑒_𝛾𝜂_𝛼𝛽=1𝜖1!^{𝛼𝛽𝛾𝛿}𝜗^𝛿, (5.2) which are bases for 3-, 2-, and 1-forms, respectively. Finally,𝜂=_{𝛼𝛽𝜇𝜈}^def.𝑒_𝜈𝜂_𝛼𝛽𝜇=𝑒_𝜈𝑒_𝜇𝑒_𝛽𝑒_𝛼𝜂 (5.3) is the Levi-Civita tensor density. The 𝜂-forms satisfy the useful identities: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. (5.4) The line element 𝑑𝑠=^2def.𝑔_𝛼𝛽𝜗^𝛼⨂𝜗^𝛽 is defined by the spacetime metric 𝑔_𝛼𝛽.

Teleparallel geometry can be viewed as a gauge theory of translation [10–16, 25–28]. In this geometry, 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 the coframe 1-form 𝜗^𝛼 and the local Lorentz connection are subject to the distant parallelism constraint 𝑅_𝛼^𝛽=0 [29–38]. In this geometry, the torsion 2-form𝑇^𝛼=𝐷𝜗^𝛼=𝑑𝜗^𝛼+Γ_𝛽^𝛼∧𝜗^𝛽=12𝑇^𝛼_𝜇𝜈𝜗^𝜇∧𝜗^𝜈=12𝑇^𝛼_𝑖𝑗𝑑𝑥^𝑖∧𝑑𝑥^𝑗 (5.5) arises as the gravitational gauge field strength, Γ_𝛼^𝛽being the Weitzenböck connection 1-form, 𝑑 is the total derivative, and 𝐷 is the covariant derivative. The torsion 𝑇^𝛼 can be decomposed into three irreducible pieces: the tensor part, the trace, and the axial trace, given, respectively, by [9, 11–16], for example,𝑇=^(1)𝛼def.𝑇^𝛼−⁽²⁾𝑇^𝛼−⁽³⁾𝑇^𝛼,with₍₂₎𝑇=^𝛼def.13𝜗^𝛼∧𝑇,where𝑇=𝑒_𝛽𝑇^𝛽,𝑒_𝛼𝑇=𝑇^𝜇_𝜇𝛼,vectorsoftraceoftorsion,₍₃₎𝑇=^𝛼def.13𝑒^𝛼𝑃,with𝑃=𝜗^𝛽∧𝑇_𝛽,𝑒_𝛼𝑃=𝑇^𝜇𝜈𝜆𝜂_{𝜇𝜈𝜆𝛼},axialoftraceoftorsion. (5.6) The Lagrangian of the teleparallel equivalent using the language of forms has the form1𝑉=−𝑇2𝜅^𝛼∧^∗(1)𝑇_𝛼−2⁽²⁾𝑇_𝛼−12⁽³⁾𝑇_𝛼 (5.7)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 variation of the total action with respect to the coframe gives the field equations in the from [9]𝐷𝐻_𝛼−𝐸_𝛼=Σ_𝛼,whereΣ=_𝛼^def.𝛿𝐿_mattter𝛿𝜗^𝛼 (5.8) is the canonical energy momentum current 3-form of matter which is considered as the source. In accordance with the general Lagrange-Noether scheme [10, 25], one derives from (5.7) the translational momentum 2-form and the canonical energy momentum 3-form: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_𝛼𝑉+𝑒_𝛼𝑇^𝛽∧𝐻_𝛽. (5.9) Due to geometric identities [39], the Lagrangian (5.7) can be recast as1𝑉=−2𝑇^𝛼∧𝐻_𝛼. (5.10) The presence of the connection field Γ^𝛼_𝛽 plays an important regularizing role due to the following [40].

First
The theory becomes explicitly covariant under the local Lorentz transformations of the coframe, that is, the Lagrangian (5.7) is invariant under the change of variables𝜗^′𝛼=Λ^𝛼_𝛽𝜗^𝛽,Γ^′_𝛼^𝛽=(Λ)^𝜇_𝛼Γ_𝜇^𝜈Λ^−1𝛽_𝜈−Λ^−1𝛽_𝛾𝑑(Λ)^𝛾_𝛼. (5.11) Due to the noncovariant transformation law of Γ_𝛼^𝛽 as shown by (5.11), 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.

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 [9]. 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 pure gauge, that isΓ_𝛼^𝛽=Λ^−1𝛽_𝛾𝑑(Λ)^𝛾_𝛼. (5.12) The Weitzenböck connection always has the form (5.12). The translational momentum has the form [9]𝐻_𝛼=1Γ2𝜅^𝛽𝛾∧𝜂_𝛼𝛽𝛾,Γ=_𝛼^𝛽def.Γ_𝛼^𝛽−𝐾_𝛼^𝛽, (5.13) with 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. (5.14)

Using the spherical local coordinates (𝑡,𝑟,𝜃,𝜙), the Kerr-Taub-NUT, using (3.1), frame is described by the coframe components:𝜗^𝛿=(ℎ)^𝛿_𝑖𝑑𝑥^𝑖,𝜗0=𝐴₁𝑐𝑑𝑡+𝐴₂𝑑𝑟+𝐴₃𝑑𝜃+𝐴₄𝜗1𝑑𝜙,=𝐵₁sin𝜃cos𝜙𝑐𝑑𝑡+𝐵₂sin𝜃cos𝜙𝑑𝑟+𝐵₃cos𝜃cos𝜙𝑑𝜃+𝐵₄𝜗2sin𝜃sin𝜙𝑑𝜙,=𝐶₁sin𝜃sin𝜙𝑐𝑑𝑡+𝐶₂sin𝜃sin𝜙𝑑𝑟+𝐶₃cos𝜃sin𝜙𝑑𝜃+𝐶₄𝜗3sin𝜃cos𝜙𝑑𝜙,=𝐷₁cos𝜃𝑐𝑑𝑡+𝐷₂cos𝜃𝑑𝑟+𝐷₃sin𝜃𝑑𝜃+𝐷₄cos𝜃𝑑𝜙. (5.15)

If we take coframe (5.15), as well as the Riemannian connection Γ_𝛼^𝛽 and substitute into (5.13), we finally get (⋯ means terms which are multiply by 𝑑𝜃∧𝑑𝑟, 𝑑𝜃∧𝑑𝑡, 𝑑𝑟∧𝑑𝜙, etc.). 𝐻0≅sin𝜃4𝑟𝜋𝑀𝑟+𝐿𝐿₁1𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟². (5.16) Using (5.16) to compute the total energy at a fixed time in the 3-space with a spatial boundary 2-dimensional surface 𝜕𝑆={𝑟=𝑅,𝜃,𝜙}, we obtain𝐸=_𝜕𝑆𝐻0=𝐿𝑀+²𝑅1+𝑂𝑅², (5.17) which is identical with (4.2).

Turn our attention to the spatial momentum, we get the following nonvanishing components 𝐻𝛼, 𝛼=1,2,3,𝐻1=cos𝜃𝑀𝑟+𝐿𝐿₁14𝑟𝜋𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟²,𝐻2=2𝐿𝑎sin²𝜃18𝑟𝜋𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟²,𝐻3=2𝐿cos𝜃(𝑟−𝑀)+𝑀𝑎sin²𝜃18𝑟𝜋𝑑𝜃∧𝑑𝜙+⋯+𝑂𝑟². (5.18) Using (5.18) in (5.13), we finally get the spatial momentum in the form𝑃₁=_𝜕𝑆𝐻11=𝑂𝑅,bysamemethod𝑃₂=𝑃₃1=𝑂𝑅. (5.19)

6. Main Results and Discussion

In this paper, the equations of the gravitational field in the TEGR are considered, applying the most general tetrad (3.1) with sixteen unknown function of 𝑟 and 𝜙 to the field equations (2.7). Exact analytic new solution is obtained satisfying (3.2). This solution is a general one from which other solutions can be generated (Schwarzschild and Kerr spacetimes) by appropriate choice of the arbitrary functions of the tetrad equation (3.1) using (3.5), that is, putting 𝐿=0, 𝑎=0, and kerr spacetime can be obtained when 𝐿=0 [21, 23]. The tetrad field is axially symmetric as shown in (3.7).

By using the definition of the torsion tensor provided by (2.5) and appling it to the tetrad field (3.1), using (3.5), one can show that the torsion of flat spacetime is vanishing identically. Therefore, the gravitational energy momentum given by (2.11) is employed. Use of (2.11) is not limited to Cartesian coordinate. Accordingly, (2.11) is applied to the tetrad field (3.1), using (3.5), and the energy content (4.2) is derived. Equation (4.2) indicates clearly that the energy content is shared by the gravitational mass 𝑀 and the NUT parameter 𝐿 of the Kerr-Taub-NUT black hole. This result is agreed with what has been obtained before [22, 23]. Setting the NUT parameter 𝐿 equal zero, expression (4.2) gives the energy of the Kerr black hole [23]. With the assistance of gravitational energy momentum, calculation of the spatial momentum has been accomplished and results are consistent [18]. Calculations of the angular momentum components are achieved using definition (2.13). The calculations using this definition are not agreed with the previous one, since all the components are vanishing [41]!

To use another definition to calculate energy and angular momentum of tetrad (3.1), the language of forms is employed and the Riemannian connection, that is, Γ_𝛼^𝛽, is presented. The formulas of energies as shown by (4.2) and (5.17) indicate that the two methods, the energy momentum tensor and the translational momentum 2-form 𝐻_𝛼, gave the same value of energy and spatial momentum. Consistence of the results of two methods is due to the following.(i)Within the context of tetrad theories of gravity, asymptotically flat spacetimes may be featured by the asymptotic boundary condition 𝑒_𝑎𝜇≅𝜂_𝑎𝜇+12ℎ_𝑎𝜇1𝑟 (6.1) and by the condition 𝜕_𝜇𝑒^𝑎_𝜇=𝑂(1/𝑟²) in the asymptotic limit 𝑟→∞. It is obvious that the tetrad field (3.1) using (3.5) satisfied this condition.(ii)Components of torsion related to solution (3.5) are vanishing when the physical quantities are set equal to zero, that is, 𝑀, 𝑎, 𝐿. This is a prerequisite to get eligible results [17].(iii)When the coframe 𝜗^𝛼, taken as the only field variable in the formulation of purely tetrad and in order to obtain physical results, one should select a tetrad field that meets certain conditions.(a)The asymptotic behavior of Riemannian connection Γ_𝛽^𝛼 must behave like 𝑂(1/𝑟) [9].(b)When the physical quantities, that is, 𝑀, 𝑎, 𝐿, are set equal to zero then Riemannian connection is vanishing, that is, (Γ_𝛽^𝛼)_{𝑀,𝑎,𝐿=0}→0.(iv)Definition (2.13), related to the calculation of angular momentum, does not seem working properly for the Kerr-NUT spacetime. This problem needs an extensive study, and this will be our future work.

Acknowledgment

The author acknowledges the Egyptian Relativity Group (http://www.erg.eg.net/), in which he is a member.