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

Authors: Gamal G. L. Nashed
Journal: Advances in High Energy Physics
 (2012)

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## 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 massM, 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 Γα̃β, 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.


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## Body

## 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 Pa 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 derivea 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 Γα̃β. 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 fieldhaμ identifies the nonsymmetric affine connection(2.1)Γλμν=def.haλhaμ,ν,
where haμ,ν=∂νhaμ  (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 Γλμν  is identically vanishing, however. The metric tensor gμν is given by(2.2)gμν=Oabhaμhbν,
with the Minkowski metric Oab=diag(+1,-1,-1,-1)   (Latin indices are raising and lowering with the aid of Oab and Oab).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): (2.3)LG=-gLG=--g16π(TabcTabc4+TabcTbac2-TaTa)-Lm=--g16πΣabcTabc-Lm,
where g=det(gμν). The tensor Σabc is defined by(2.4)Σabc=def.14(Tabc+Tbac-Tcab)+12(OacTb-OabTc).Tabc and Ta are the torsion tensor and the basic vector field defined by(2.5)Taμν=def.haλTλμν=∂μhaν-∂νhaμ,(2.6)Tμ=def.Tννμ,Ta=def.haμTμ=Tbba.
The quadratic combination ΣabcTabc is proportional to the scalar curvature R, except for a total divergence term [19]. Lm represents the Lagrangian density for matter fields.The gravitational field equations for the system described byLG are the following:(2.7)haλhbμ∂ν(-gΣbλν)--g(ΣbνaTbνμ-14haμTbcdΣbcd)=12κ-gTaμ,
where(2.8)δLmδhaμ≡-gTaμ.
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](2.9)-g2[Raμ-12haμR].
The axial-vector part of the torsion tensor Aμ is defined by(2.10)Aμ=def.16ϵμνρσTνρσ=13ϵμνρσγνρσ,whereϵμνρσ=def.-gδμνρσ,
with γνρσ=Oabhaνhbρ;σ being the contorsion tensor and δμνρσ is completely antisymmetric and normalized as δ0123=-1.The definition of the gravitational energy momentumPa four-vector has the form [17](2.11)Pa=-∫Vd3x∂iΠai,
where V is an arbitrary volume of the three-dimensional space. In the configuration space, we have(2.12)Πai=-2κ-gΣa0i,with∂ν(-gΣaλν)=κ2-ghaμ(tλμ+Tλμ),wheretλμ=12κ(4ΣbcλTbcμ-gλμΣbcdTbcd).Maluf and Ulhoa  [20] defined (2.13)Lab=2∫Vd3xM[ab],whereMab=-g4π(Σb0a-Σa0b),
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 as(3.1)(hiμ)=(A1(r,ϕ)A2(r,ϕ)A3(r,ϕ)A4(r,ϕ)B1(r,ϕ)sin⁡θcos⁡ϕB2(r,ϕ)sin⁡θcos⁡ϕB3(r,ϕ)cos⁡θcos⁡ϕB4(r,ϕ)sin⁡ϕsin⁡θC1(r,ϕ)sin⁡θsin⁡ϕC2(r,ϕ)sin⁡θsin⁡ϕC3(r,ϕ)cos⁡θsin⁡ϕC4(r,ϕ)cos⁡ϕsin⁡θD1(r,ϕ)cos⁡θD2(r,ϕ)cos⁡θD3(r,ϕ)sin⁡θD4(r,ϕ)cos⁡θ),
where Ai(r,ϕ),Bi(r,ϕ),Ci(r,ϕ),andDi(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 (3.2)hkμ=δμk+Mlklμ-L22mkmμ,
where M and L are free parameters, lμ and mμ are quantities satisfying the conditions(3.3)Oμνlμlν=0,Oμνmμmν=0,Oμνlμmν=0,lk and mk are defined by(3.4)lk=δkμOμνlν,mk=δkμOμνmν.
The solution satisfying the above condition is found to have the following form.The Exact Solution
If the arbitrary functions take the following values:(3.5)A1=1-(Mr+LL1)Ω,A2=(Mr+LL1)Υ,A3=0,A4=Mrχ+L(aL1[1+cos⁡2θ]+2r2cos⁡θ)Ω,B1=(Mr+LL1)Ω,B2=1Υcos⁡ϕ(r1α-(Mr+LL1)cos⁡ϕ),B3=αcos⁡ϕ,B4=β-((Mr+LL1)χcos⁡ϕ/Ω)sin⁡ϕ,C1=(Mr+LL1)Ω,C2=1Υsin⁡ϕ(r1β-(Mr+LL1)sin⁡ϕ),C3=βsin⁡ϕ,C4=-α+(Mr+LL1)χsin⁡ϕ/Ωcos⁡ϕ,D1=(Mr+LL1)Ω,D2=1+(Mr+LL1)Υ,D3=-r1,D4=-(Mr+LL1)χΩ,
where Ω, Υ, L1, α, β, r1, and χ are defined by
(3.6)Ω=def.r2+L12,Υ=def.r2+a2-2Mr-L2,L1=def.L+acos⁡θ,α=def.r1cos⁡ϕ+asin⁡ϕ,β=def.r1sin⁡ϕ-acos⁡ϕ,r1=def.r2+L(L+2acos⁡θ),χ=def.asin⁡2θ-2Lcos⁡θ.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
(3.7)ϕ¯⟶ϕ+δϕ,h(0)¯μ⟶h(0)μ,h(1)¯μ⟶h(1)μcos⁡δϕ-h(2)μsin⁡δϕ,h(2)¯μ⟶h(2)μsin⁡δϕ+h(2)μcos⁡δϕ,h(3)¯μ⟶h(3)μ.
Solution (3.5) satisfies the field equations (2.7), and the associated metric has the following form:
(3.8)ds2=1Ω{Υ-a2sin⁡2θ}dt2-ΩΥdr2-Ωdθ2-1Ω{(Ω+aχ)2sin⁡2θ-Υχ2}dϕ2-2Ω{χΥ-a(Ω+aχ)sin⁡2θ}dtdϕ,
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 Σabc needed to the calculation of energy have the form
(4.1)Σ101=a2r5({aLL1+aMr}sin⁡2θ-4aL2cos⁡2θ-2MrLcos⁡θ),Σ201=a2r5sin⁡⁡θ(Lrsin⁡2θ+2LMcos⁡2θ-Macos⁡θsin⁡2θ),Σ301=L2r5sin⁡2θ(L2cos⁡θ-aL1sin⁡2θ),Σ401=-14r5(a2{5L2+2Mr}sin⁡2θ+4a2(r2-9L2)cos⁡2θ+4Lr[{r-M}acos⁡θ+rL1]-4r4).
Using (3.6) in (2.11), we finally obtain  (we subtract ΣM=0,a=0,L=0μ0k in (4.2) to remove divergences from the calculations of energy) (4.2)P(0)=E=-∮S→∞dSkΠ(0)k=-14π∮S→∞dSk-gh(0)μ(Σμ0k-Σμ0kM=0,a=0,L=0)≅M+L2r-L2Mr2-L2(5a2+L2)r3+O(1r4).
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 likeM2,L3,L3M,M2a,…, etc. are neglected in this calculations)  (4.3)Σ(1)01≅sin⁡2θcos⁡ϕ4πr(Mr+L2)+O(1r2),Σ(2)01≅-sin⁡2θsin⁡ϕ4πr(Mr+L2)+O(1r2),Σ(3)01≅-sin⁡2θcos⁡ϕ4πr(Mr+L2)+O(1r2).
Using (4.3) in (2.11), we finally get the spatial momentum in the form(4.4)P1=-∮S→∞dSkΠ(1)k=-14π∮S→∞dSkeΣ(1)0k=0,by same methodP2=0,P3≅(1r).The nonvanishing components of the angular-momentum are given by(4.5)M(0)(1)(r,θ,ϕ)≅18πr({6Lacos⁡θ+2rM+a2+3L2}cos⁡2θcos⁡ϕ+ra{cos⁡2θ+1}sin⁡ϕ+{2r2-2L2-2Mr-a2-4Lacos⁡θ}cos⁡ϕ)+O(1r2),M(0)(2)(r,θ,ϕ)≅18πr({6Lacos⁡θ+2rM+a2+3L2}cos⁡2θsin⁡ϕ-ra{cos⁡2θ+1}cos⁡ϕ+{2r2-2L2-2Mr-a2-4Lacos⁡θ}sin⁡ϕ)+O(1r2),M(0)(3)(r,θ,ϕ)≅sin⁡⁡θ(6Lacos⁡2θ+[2rM+3L2]cos⁡θ-2Lh)8πr+O(1r2),M(1)(2)(r,θ,ϕ)≅sin⁡⁡θ(3Macos⁡2θ+2[M-r]Lcos⁡θ-Ma)8πr+O(1r2),M(1)(3)(r,θ,ϕ)≅sin⁡2θ([3Macos⁡θ+2LM-2Lr]sin⁡ϕ+Lacos⁡ϕ)8πr+O(1r2),M(2)(3)(r,θ,ϕ)≅sin⁡2θ([3Macos⁡θ+2LM-2Lr]cos⁡ϕ-Lasin⁡ϕ)8πr+O(1r2).
Using (4.5) in (2.13), we get (4.6)L(0)(1)=∫0π∫02π∫0∞dθdϕdr[M(0)(1)]=0,
which is a consistent result. By the same method, we finally obtain(4.7)L(0)(2)=L(0)(3)=L(1)(2)=L(1)(3)=L(2)(3)=0.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 indicesi,j,…, for local holonomic spacetime coordinates and the Greek indices α,β,…, label (co)frame components. Particular frame components are denoted by hats, 0̂,1̂, 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 ϑα is denoted by eα, and they satisfy eα⌋ϑβ=δαβ. Using local coordinates xi, we have ϑα=hiαdxi and eα=hαi∂i where hiα and hαi are the covariant and contravariant components of the tetrad field. We define the volume 4-form by η=def.ϑ0̂∧ϑ1̂∧ϑ2̂∧ϑ3̂. Furthermore, with the help of the interior product, we define(5.1)ηα=def.eα⌋η=13!ϵαβγδϑβ∧ϑγ∧ϑδ,
where ϵαβγδ is completely antisymmetric tensor with ϵ0123=1(5.2)ηαβ=def.eβ⌋ηα=12!ϵαβγδϑγ∧ϑδ,ηαβγ=def.eγ⌋ηαβ=11!ϵαβγδϑδ,
which are bases for 3-, 2-, and 1-forms, respectively. Finally,(5.3)ηαβμν=def.eν⌋ηαβμ=eν⌋eμ⌋eβ⌋eα⌋η
is the Levi-Civita tensor density. The η-forms satisfy the useful identities:(5.4)ϑβ∧ηα=def.δαβη,ϑβ∧ημν=def.δνβημ-δμβην,ϑβ∧ηαμν=def.δαβημν+δμβηνα+δνβηαμ,ϑβ∧ηαγμν=def.δνβηαγμ-δμβηαγν+δγβηαμν-δαβηγμν.
The line element ds2=def.gαβϑα⨂ϑβ is defined by the spacetime metric gαβ.Teleparallel geometry can be viewed as a gauge theory of translation [10–16, 25–28]. In this geometry, the coframe ϑα 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 ϑα and the local Lorentz connection are subject to the distant parallelism constraint Rαβ=0 [29–38]. In this geometry, the torsion 2-form(5.5)Tα=Dϑα=dϑα+Γβα∧ϑβ=12Tμναϑμ∧ϑν=12Tijαdxi∧dxj
arises as the gravitational gauge field strength, Γαβbeing the Weitzenböck connection 1-form, d is the total derivative, and D is the covariant derivative. The torsion Tα can be decomposed into three irreducible pieces: the tensor part, the trace, and the axial trace, given, respectively, by [9, 11–16], for example,(5.6)Tα(1)=def.Tα-(2)Tα-(3)Tα,withTα(2)=def.13ϑα∧T,whereT=(eβ⌋Tβ),eα⌋T=Tμαμ,vectorsoftraceoftorsion,Tα(3)=def.13eα⌋P,withP=(ϑβ∧Tβ),eα⌋P=Tμνλημνλα,axialoftraceoftorsion.
The Lagrangian of the teleparallel equivalent using the language of forms has the form(5.7)V=-12κTα∧*(Tα(1)-2Tα(2)-12Tα(3))κ=8πG/c3, G is the Newtonian constant, c is the speed of light, and * denotes the Hodge duality in the metric gαβ which is assumed to be flat Minkowski metric gαβ=Oαβ=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](5.8)DHα-Eα=Σα,whereΣα=def.δLmattterδϑα
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:(5.9)Hα=def.-∂V∂Tα=1κ*(Tα(1)-2Tα(2)-12Tα(3)),Eα=def.∂V∂ϑα=eα⌋V+(eα⌋Tβ)∧Hβ.
Due to geometric identities [39], the Lagrangian (5.7) can be recast as(5.10)V=-12Tα∧Hα.
The presence of the connection field Γαβ 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 variables(5.11)ϑ′α=Λαβϑβ,Γ′αβ=(Λ)μαΓμν(Λ-1)βν-(Λ-1)βγd(Λ)γα.
Due to the noncovariant transformation law of Γαβ 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
Γαβ  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 is(5.12)Γαβ=(Λ-1)βγd(Λ)γα.
The Weitzenböck connection always has the form (5.12). The translational momentum has the form [9](5.13)H̃α=12κΓ̃βγ∧ηαβγ,Γαβ=def.Γα̃β-Kαβ,
with Γα̃β is the purely Riemannian connection and Kμν is the contorsion 1-form which is related to the torsion through the relation
(5.14)Tα=def.Kαβ∧ϑβ.Using the spherical local coordinates(t,r,θ,ϕ), the Kerr-Taub-NUT, using (3.1), frame is described by the coframe components:(5.15)ϑδ=(h)δidxi,ϑ0̂=A1cdt+A2dr+A3dθ+A4dϕ,ϑ1̂=B1sin⁡⁡θcos⁡ϕcdt+B2sin⁡⁡θcos⁡ϕdr+B3cos⁡⁡θcos⁡ϕdθ+B4sin⁡⁡θsin⁡ϕdϕ,ϑ2̂=C1sin⁡⁡θsin⁡ϕcdt+C2sin⁡⁡θsin⁡ϕdr+C3cos⁡⁡θsin⁡ϕdθ+C4sin⁡⁡θcos⁡ϕdϕ,ϑ3̂=D1cos⁡⁡θcdt+D2cos⁡⁡θdr+D3sin⁡⁡θdθ+D4cos⁡⁡θdϕ.If we take coframe (5.15), as well as the Riemannian connection  Γα̃β and substitute into (5.13), we finally get (⋯ means terms which are multiply by dθ∧dr, dθ∧dt, dr∧dϕ, etc.). (5.16)H̃0̂≅sin⁡θ4rπ[(Mr+LL1)](dθ∧dϕ)+⋯+O(1r2).
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 obtain(5.17)Ẽ=∫∂SH̃0̂={M+L2R}+O(1R2),
which is identical with (4.2).Turn our attention to the spatial momentum, we get the following nonvanishing componentsH̃α̂, α̂=1,2,3,(5.18)H̃1̂=cos⁡θ(Mr+LL1)4rπ(dθ∧dϕ)+⋯+O(1r2),H̃2̂=2Lasin⁡2θ8rπ(dθ∧dϕ)+⋯+O(1r2),H̃3̂=2Lcos⁡θ(r-M)+Masin⁡2θ8rπ(dθ∧dϕ)+⋯+O(1r2).
Using (5.18) in (5.13), we finally get the spatial momentum in the form(5.19)P1=∫∂SH̃1̂=O(1R),bysamemethodP2=P3=O(1R).


## 5.1. Notation

We use the Latin indicesi,j,…, for local holonomic spacetime coordinates and the Greek indices α,β,…, label (co)frame components. Particular frame components are denoted by hats, 0̂,1̂, 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 ϑα is denoted by eα, and they satisfy eα⌋ϑβ=δαβ. Using local coordinates xi, we have ϑα=hiαdxi and eα=hαi∂i where hiα and hαi are the covariant and contravariant components of the tetrad field. We define the volume 4-form by η=def.ϑ0̂∧ϑ1̂∧ϑ2̂∧ϑ3̂. Furthermore, with the help of the interior product, we define(5.1)ηα=def.eα⌋η=13!ϵαβγδϑβ∧ϑγ∧ϑδ,
where ϵαβγδ is completely antisymmetric tensor with ϵ0123=1(5.2)ηαβ=def.eβ⌋ηα=12!ϵαβγδϑγ∧ϑδ,ηαβγ=def.eγ⌋ηαβ=11!ϵαβγδϑδ,
which are bases for 3-, 2-, and 1-forms, respectively. Finally,(5.3)ηαβμν=def.eν⌋ηαβμ=eν⌋eμ⌋eβ⌋eα⌋η
is the Levi-Civita tensor density. The η-forms satisfy the useful identities:(5.4)ϑβ∧ηα=def.δαβη,ϑβ∧ημν=def.δνβημ-δμβην,ϑβ∧ηαμν=def.δαβημν+δμβηνα+δνβηαμ,ϑβ∧ηαγμν=def.δνβηαγμ-δμβηαγν+δγβηαμν-δαβηγμν.
The line element ds2=def.gαβϑα⨂ϑβ is defined by the spacetime metric gαβ.Teleparallel geometry can be viewed as a gauge theory of translation [10–16, 25–28]. In this geometry, the coframe ϑα 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 ϑα and the local Lorentz connection are subject to the distant parallelism constraint Rαβ=0 [29–38]. In this geometry, the torsion 2-form(5.5)Tα=Dϑα=dϑα+Γβα∧ϑβ=12Tμναϑμ∧ϑν=12Tijαdxi∧dxj
arises as the gravitational gauge field strength, Γαβbeing the Weitzenböck connection 1-form, d is the total derivative, and D is the covariant derivative. The torsion Tα can be decomposed into three irreducible pieces: the tensor part, the trace, and the axial trace, given, respectively, by [9, 11–16], for example,(5.6)Tα(1)=def.Tα-(2)Tα-(3)Tα,withTα(2)=def.13ϑα∧T,whereT=(eβ⌋Tβ),eα⌋T=Tμαμ,vectorsoftraceoftorsion,Tα(3)=def.13eα⌋P,withP=(ϑβ∧Tβ),eα⌋P=Tμνλημνλα,axialoftraceoftorsion.
The Lagrangian of the teleparallel equivalent using the language of forms has the form(5.7)V=-12κTα∧*(Tα(1)-2Tα(2)-12Tα(3))κ=8πG/c3, G is the Newtonian constant, c is the speed of light, and * denotes the Hodge duality in the metric gαβ which is assumed to be flat Minkowski metric gαβ=Oαβ=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](5.8)DHα-Eα=Σα,whereΣα=def.δLmattterδϑα
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:(5.9)Hα=def.-∂V∂Tα=1κ*(Tα(1)-2Tα(2)-12Tα(3)),Eα=def.∂V∂ϑα=eα⌋V+(eα⌋Tβ)∧Hβ.
Due to geometric identities [39], the Lagrangian (5.7) can be recast as(5.10)V=-12Tα∧Hα.
The presence of the connection field Γαβ 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 variables(5.11)ϑ′α=Λαβϑβ,Γ′αβ=(Λ)μαΓμν(Λ-1)βν-(Λ-1)βγd(Λ)γα.
Due to the noncovariant transformation law of Γαβ 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
Γαβ  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 is(5.12)Γαβ=(Λ-1)βγd(Λ)γα.
The Weitzenböck connection always has the form (5.12). The translational momentum has the form [9](5.13)H̃α=12κΓ̃βγ∧ηαβγ,Γαβ=def.Γα̃β-Kαβ,
with Γα̃β is the purely Riemannian connection and Kμν is the contorsion 1-form which is related to the torsion through the relation
(5.14)Tα=def.Kαβ∧ϑβ.Using the spherical local coordinates(t,r,θ,ϕ), the Kerr-Taub-NUT, using (3.1), frame is described by the coframe components:(5.15)ϑδ=(h)δidxi,ϑ0̂=A1cdt+A2dr+A3dθ+A4dϕ,ϑ1̂=B1sin⁡⁡θcos⁡ϕcdt+B2sin⁡⁡θcos⁡ϕdr+B3cos⁡⁡θcos⁡ϕdθ+B4sin⁡⁡θsin⁡ϕdϕ,ϑ2̂=C1sin⁡⁡θsin⁡ϕcdt+C2sin⁡⁡θsin⁡ϕdr+C3cos⁡⁡θsin⁡ϕdθ+C4sin⁡⁡θcos⁡ϕdϕ,ϑ3̂=D1cos⁡⁡θcdt+D2cos⁡⁡θdr+D3sin⁡⁡θdθ+D4cos⁡⁡θdϕ.If we take coframe (5.15), as well as the Riemannian connection  Γα̃β and substitute into (5.13), we finally get (⋯ means terms which are multiply by dθ∧dr, dθ∧dt, dr∧dϕ, etc.). (5.16)H̃0̂≅sin⁡θ4rπ[(Mr+LL1)](dθ∧dϕ)+⋯+O(1r2).
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 obtain(5.17)Ẽ=∫∂SH̃0̂={M+L2R}+O(1R2),
which is identical with (4.2).Turn our attention to the spatial momentum, we get the following nonvanishing componentsH̃α̂, α̂=1,2,3,(5.18)H̃1̂=cos⁡θ(Mr+LL1)4rπ(dθ∧dϕ)+⋯+O(1r2),H̃2̂=2Lasin⁡2θ8rπ(dθ∧dϕ)+⋯+O(1r2),H̃3̂=2Lcos⁡θ(r-M)+Masin⁡2θ8rπ(dθ∧dϕ)+⋯+O(1r2).
Using (5.18) in (5.13), we finally get the spatial momentum in the form(5.19)P1=∫∂SH̃1̂=O(1R),bysamemethodP2=P3=O(1R).


## 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,  Γα̃β,  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 Hα, 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(6.1)eaμ≅ηaμ+12haμ(1r)
and by the condition ∂μeaμ=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 coframeϑα, 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 connectionΓβ̃α 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, (Γβ̃α)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.



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*Source: 475460-2012-05-31.xml*