Total Conserved Charges of Kerr-Newman Spacetimes in Gravity Theory Using a Poincaré Gauge Version of the Teleparallel Equivalent of General Relativity
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.
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 . 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 . 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 . 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 .
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.  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 , 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 , 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 .
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.
The Latin indices , are used for local holonomic spacetime coordinates, and the Greek indices ,,, label (co)frame components. Particular frame components are denoted by hats, , , and so forth. As usual, the exterior product is denoted by , while the interior product of a vector and a -form is denoted by . The vector basis dual to the frame 1-forms is denoted by , and they satisfy . Using local coordinates , we have and where and are the covariant and contravariant components of the tetrad field. We define the volume 4-form by . Furthermore, with the help of the interior product, we define where is completely antisymmetric with . One has which are bases for 3-, 2-, and 1-forms, respectively. Finally, is the Levi-Civita tensor density. The -forms satisfy the useful identities as follows: The line element is defined by the spacetime metric .
2. Brief Review of Teleparallel Gravity and Conserved Currents
Teleparallel geometry can be viewed as a gauge theory of translation [50–62]. The coframe 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 and the local Lorentz connection are subject to the distant parallelism constraint [63–71]. In this geometry, the torsion 2-form arises as the gravitational gauge field strength, is the Weitzenböck 1-form connection, is the exterior derivative, and is the exterior covariant derivative. The torsion can be decomposed into three irreducible pieces : the tensor part, the trace, and the axial trace given, respectively, by The Lagrangian of the TEGR has the form , is the Newtonian constant, is the speed of light, and denotes the Hodge duality in the metric which is assumed to be flat Minkowski metric 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 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: Due to geometric identities , the Lagrangian (7) can be recast as The presence of the connection field 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 Due to the noncovariant transformation law of , 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 for all frames, which is the tetrad gravity, the Lagrangian (7) is invariant under the local Lorentz group.
Second. plays an important role in the teleparallel framework. This role represents the inertial effects which arise from the choice of the reference system . 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, The Weitzenböck connection always has the form (13). The translational momentum has the form [72, 80] where is the purely Riemannian connection and is the contorsion 1-form which is related to the torsion through the relation
The teleparallel model (7) belongs to the class of quasi-invariant theories. In fact, one can verify that under a change of the coframe , the Lagrangian changes by a total derivative as follows: In addition to (16), it is easy to verify that (14) changes like
The total conserved charge in the tetrad gravity theory is given by [48, 49] where . Under a local Lorentz transformation, (18) transforms as where . The vector does not depend on the choice of the frame, while its components, that is, , 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 can be written as
, , and are three functions of and having the form where , , and are mass, rotation and charge parameters, respectively [44–47]. An asymptotically flat spacetime is considered in this study, and a boundary condition, that for the tetrad (20) approaches the tetrad of Minkowski spacetime in Cartesian coordinates, is imposed. The metric tensor associated with the tetrad field (20) has the form 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
The nonvanishing components of the Riemannian connection, of the first coframe, have the form Using (25), the nonvanishing components of the vector have the form If we take tetrad (20), as well as the trivial Weitzenböck connection, , we finally get the nonvanishing components of the translation momentumthat is, , where means terms which are multiplied by , , , and so forth. By using the identities of , that is, (A18), one can directly prove that, for solutions with vanishing torsion, the Killing equation implies also that . Therefore, for the Kerr-Newman solution we have for , with constants and . For a vector field with constant holonomic components, , in the coordinate system used in (20), the direct evaluation of the integral (18) yields the following infinite total conserved charge:
For the coframe determined by (25), the conserved charges corresponding to the diffeomorphism generated by the shift along the time coordinate has a usual value of the total energy of the configuration. Whereas, for the vector field along the azimuthal coordinate we get , which is a nonfamiliar term . This is because tetrad (20) is nonholonomic. Another reason is that the Riemannian connection does not vanish either at spatial infinity or when the physical quantities, , , and , 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 where has the form
The most appropriate local Lorentz transformation which serves for this purpose has the form (24). The nonvanishing components of have the form Using (26), (27), and (32) in (30) and the fact that the vector field is with constant holonomic components, we get the total conserved quantities in the form 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 as
where has the form Tetrad (34) has the same associated metric of tetrad (20). The second rotating frame is described by the coframe components Using (24) and (34), one can get The two frames (25) and (37) are related through the local Lorentz transformation; that is, where has the form
Coframe (37) has the following asymptotic nonvanishing components of Riemannian connection up to : (Terms like , , , , are neglected in these calculations.) The asymptotic nonvanishing components of the vector have the form
Using coframe (37) as well as the trivial Weitzenböck connection, one can get asymptotically the nonvanishing components of the translation momentum, that is, , The direct evaluation of the integral (18), using (42) and (43) and assuming the fact that the vector field is with constant holonomic components, yields the following infinite total conserved charge: For the tetrad determined by (37), the conserved charge corresponding to the diffeomorphism generated by the shift along the time coordinate has a divergent value of the total energy of the configuration whereas, for the vector field along the azimuthal coordinate, we get , which is a well-known value . 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, 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 , 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 where is defined as where is an appropriate choice of local Lorentz transformation. (We will denote the antisymmetric part by the square bracket , .) (The Lie derivative is given on exterior forms by .) 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, , has the form Using (48), we get the nonvanishing components of and the nonvanishing components of have the form Finally, the nonvanishing components of have the form Using (41), (42), (49), and (51) in (46), we finally get the total conserved charge of coframe (37) in the form Equation (52) shows that the total energy and the total angular momentum of coframe (37) have their commonly known form.
5.1. Third Coframe
The covariant form of the third KN tetrad can be written as
Tetrad field (53) has the same associated metric of tetrad field (20). Using the spherical local coordinates , the third rotating frame is described by the coframe components Using (24) and (53) we get
The two frames (25) and (55) are related through local Lorentz transformation; that is, where has the form The nonvanishing components of the corresponding Riemannian connection of the coframe (55) read
The nonvanishing components of the vector have the form If we take coframe (55), as well as the trivial Weitzenböck connection, one can get asymptotically the nonvanishing components of the translation momentum The direct evaluation of integral (18) yields the following anomalous conserved charge:
It is worth mentioning that when one uses (19) to calculate the conserved quantities related to coframe (55), one can get 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 as
where , , , and read where and are defined by
Using the spherical local coordinates the fourth rotating frame is described by the coframe components Using (24) and (62), we get The local Lorentz transformation is given by the matrix (24). The two frames (25) and (66) are related through local Lorentz transformation; that is, where has the form
with The nonvanishing components of the Riemannian connection of the coframe (66) up to order read
The nonvanishing components of the vector up to order have the form Using coframe (66) as well as the trivial Weitzenböck connection, the nonvanishing components of the translation momentum read The direct evaluation of the integral (18) for vector fields with constant components yields the following infinite total conserved charge: When we apply (19) to calculate the total conserved quantity of the coframe (66), we get 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 as
Tetrad (75) has the same associated metric of tetrad (20). Using the spherical local coordinates , the fifth rotating frame is described by the following coframe components: Using (24) and (76), the coframe of the fifth KN read The two frames (25) and (77) are related through the local Lorentz transformation; that is, where has the form Coframe (77) has the following nonvanishing components of Riemannian connection: The nonvanishing components of the vector have the form
If we take coframe (77), as well as the trivial Weitzenböck connection, we finally get the translation momentum, that is, ,