Physical Quantities of Reissner-Nordström Spacetime with Arbitrary Function and Regularized Procedure

Abstract

We use the covariant teleparallel approach to compute the total energy of a spherically symmetric frame with an arbitrary function, that is, ℑ(𝑟). We show how the total energy is always effected by the inertia. When use is made of the pure gauge connection, teleparallel gravity always yields the physically relevant result. We also calculate the total conserved charge and show how inertia spoils the physics in the time coordinate direction. Therefore, a regularized expression is employed to get a plausible value of energy. Finally, we use the Euclidean continuation method, in the context of TEGR, to calculate the energy, Hawking temperature, entropy, and first law of thermodynamics.

1. Introduction

Teleparallel theories are interesting for several reasons: first of all, general relativity (GR) can be considered as a particular theory of teleparallelism, and thus it could be viewed from a different viewpoint that can lead to the same results [1]. Second, one can define an energy-momentum tensor for the gravitational field that is a true tensor under all 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 [2, 3], but he did not succeed in defining a true tensor that is invariant under general coordinate transformations. The idea was taken over by Pellegrini and Plebański that constructed the general Lagrangian for these theories [4]. 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 [5–16].

Geometrically, teleparallel models are described by the Weitzenböck spacetime that is characterized by vanishing curvature and nonzero torsion. The tetrad (or a frame) is the basic field variable which can be treated as a gauge potential corresponding to the group of local translations. Then, torsion is interpreted as the corresponding gauge field strength. As a result, a gravitational teleparallel Lagrangian is straightforwardly constructed from the quadratic torsion invariants.

Mathematically, there are infinitely many tetrads since a reference frame of an observer can obviously be constructed in an infinitely many ways. In particular, from a given tetrad field ℎ_𝑖^𝜇, we can obtain a continuous family of tetrads by performing the local Lorentz transformation ℎ^′_𝑖^𝛼=Λ^𝛼_𝛽ℎ_𝑖^𝛽, where the elements of the Lorentz matrix Λ^𝛼_𝛽(𝑥) are arbitrary functions of the spacetime coordinates [17].

An important difference between Einstein theory and teleparallel theories is that it is possible to distinguish gravitational field from inertia [18]. With the purpose of getting a deeper insight into the covariant teleparallel formalism, Lucas et al. [17] reanalyze the computation of the total energy of two models. Obukhov et al. [19] computed the energy and momentum transported by exact plane gravitational-wave solutions of Einstein equations using the teleparallel equivalent formulation of Einstein’s theory.

The existence of symmetries in a physical system means that there exist one or more transformations that leave the physical system invariant. Since physical systems can be fully described by their Lagrangians/Hamiltonians, it is natural to expect the symmetry transformations to be canonical ones. The number of symmetry transformations that can be generated from the system depends on how many conserved quantities of the associated system are considered. It has been known that conserved quantities of a physical system are linked to the generator of the associated symmetry transformations [20–22].

A number of methods for calculating energy and conserved charges that provide reliable result, are used in the present time. Using the Lagrange-Noether approach, one can derive the conserved currents that arise from the invariance of the classical action under transformations of fields. However, in Riemannian geometry, one cannot find symmetries that can be used to generate the conserved energy-momentum currents. Only one can speak about the energy of asymptotically flat spacetime. Earlier analysis of this problem can be found in details in [23, 24]. In the framework of the TEGR, it is possible to make definite statements about the energy and momentum of the gravitational field. This fact constitutes the major motivation for considering this theory. In the 3+1 formulation of the TEGR [25], and by imposing Schwinger’s time gauge condition [26] for the tetrad field by fixing ℎ⁰_(𝑎)=0 which implies ℎ_𝑎⁽⁰⁾=0, it is found that the Hamiltonian and vector constraints contain each one a divergence in the form of scalar and vector densities, respectively, that can be identified with the energy and momentum densities of the gravitational field [27, 28]. Gibbons et al. [29] compute the energy of Kerr-AdS black holes first indirectly through integrating the first law and then using the conformal definition of Ashtekar et al. [30, 31], and they indicate that both of them agree. In addition to the Regge-Teitelboim method [32], some other various definitions are the following: the approach of Abbott and Deser [33–35], the spinor definition [36, 37] based on the electric Weyl tensor, covariant phase space methods [38–41], cohomological techniques [42, 43], the KBL approach [44–46], Noether methods [47–52], and the “counter term subtraction method” [53, 54] (and more references on this in [40], for example) improved surface integrals [55] and regularization of the Euclidean action [56, 57].

Recently, the definition of conserved currents and charges for models with “quasi-invariant” Lagrangians is applied for Kerr-NUT spacetimes [58]. Within the framework of MAG, a stationary axially symmetric exact solution of the vacuum field equations is obtained for a specific gravitational Lagrangian ([59] and the references therein). In the teleparallel gravity, Nester et al. [60] have considered the quasilocal center of mass (COM). They have used the covariant Hamiltonian formalism, in which quasilocal quantities are given by the Hamiltonian boundary term, along with the covariant asymptotic Hamiltonian boundary expressions. Consideration of the COM not only gives the most restrictive asymptotic conditions on the variables but also gives strong constraints on the acceptable expressions [61]. A relation between spinor Lagrangian and teleparallel theory is established [61]. In metric-affine generalization of teleparallelism, it has been shown that there is an inconsistency in the coupling of spinors. Many discussions have been given for this in [62]. It's the aim of the present study to calculate energy, conserved charges, and thermodynamical quantities of a spherically symmetric frame with an arbitrary function. This calculation is done using two different procedures. In the first one, only Riemannian connection is considered, Γ_𝛼^𝛽, while in the second procedure, Weitzenböck connection Γ_𝛼^𝛽 is taken into account besides the Riemannian connection.

In Section 2, we use the language of exterior forms to give an outline of the teleparallel approach. In Section 3, we calculate the total energy of the spherically symmetric model and show that it is affected by inertia. Therefore, we use the covariant formalism and show that the Weitzenböck connection acts as a regularizing tool and always provides the physically meaningful result. In Section 4, we calculate the total conserved charge and show how inertia affects the final result. In Section 5, we use the regularized expression of conserved charge to recalculate the total charge, and plausible result is obtained. In Section 6, we calculate the energy using the Euclidean continuation method and also calculate Hawking temperature, entropy, and first law of thermodynamics. The final section is devoted for the main results and discussion.

Notation. The Latin indices 𝑖,𝑗,… for local holonomic spacetime coordinates and the Greek indices 𝛼,𝛽,… label (co)frame components are used. 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=𝜗0∧𝜗1∧𝜗2∧𝜗3. Furthermore, with the help of the interior product, we define 𝜂_𝛼=𝑒_𝛼1⌋𝜂=𝜖3!^{𝛼𝛽𝛾𝛿}𝜗^𝛽∧𝜗^𝛾∧𝜗^𝛿, (1) where 𝜖_{𝛼𝛽𝛾𝛿} is completely antisymmetric with 𝜖₀₁₂₃=1.

Consider 𝜂_𝛼𝛽=𝑒_𝛽⌋𝜂_𝛼=1𝜖2!^{𝛼𝛽𝛾𝛿}𝜗^𝛾∧𝜗^𝛿,𝜂_𝛼𝛽𝛾=𝑒_𝛾⌋𝜂_𝛼𝛽=1𝜖1!^{𝛼𝛽𝛾𝛿}𝜗^𝛿, (2) which are bases for 3-, 2-, and 1-forms, respectively. Finally, 𝜂_{𝛼𝛽𝜇𝜈}=𝑒_𝜈⌋𝜂_𝛼𝛽𝜇=𝑒_𝜈⌋𝑒_𝜇⌋𝑒_𝛽⌋𝑒_𝛼⌋𝜂=𝜖_{𝛼𝛽𝛾𝛿} (3) is the Levi-Civita tensor density. The 𝜂-forms satisfy the useful identities: 𝜗^𝛽∧𝜂_𝛼=𝛿^𝛽_𝛼𝜂,𝜗^𝛽∧𝜂_𝜇𝜈=𝛿^𝛽_𝜈𝜂_𝜇−𝛿^𝛽_𝜇𝜂_𝜈,𝜗^𝛽∧𝜂_𝛼𝜇𝜈=𝛿^𝛽_𝛼𝜂_𝜇𝜈+𝛿^𝛽_𝜇𝜂_𝜈𝛼+𝛿^𝛽_𝜈𝜂_𝛼𝜇,𝜗^𝛽∧𝜂_{𝛼𝛾𝜇𝜈}=𝛿^𝛽_𝜈𝜂_𝛼𝛾𝜇−𝛿^𝛽_𝜇𝜂_𝛼𝛾𝜈+𝛿^𝛽_𝛾𝜂_𝛼𝜇𝜈−𝛿^𝛽_𝛼𝜂_𝛾𝜇𝜈. (4)

The line element 𝑑𝑠²=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 [63–68]. 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 [69–79]. 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 being 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 [17]: the tensor part, the trace, and the axial trace given, respectively, by ⁽¹⁾𝑇^𝛼=𝑇^𝛼−⁽²⁾𝑇^𝛼−⁽³⁾𝑇^𝛼,with₍₂₎𝑇^𝛼=13𝜗^𝛼∧𝑇,where𝑇=𝑒_𝛽⌋𝑇^𝛽,𝑒_𝛼⌋𝑇=𝑇^𝜇_𝜇𝛼,polartraceoftorsion₍₃₎𝑇^𝛼=13𝑒^𝛼⌋𝑃,with𝑃=𝜗^𝛽∧𝑇_𝛽,𝑒_𝛼⌋𝑃=𝑇^𝜇𝜈𝜆𝜂_{𝜇𝜈𝜆𝛼},axialtraceoftorsion. (6) The Lagrangian of the teleparallel equivalent of general relativity (TEGR) has the form (The effect of adding the non-Riemannian parity odd pseudoscalar curvature to the Hilbert-Einstein-Cartan scalar curvature was studied by many authors (cf. [80–84] and the references therein.)) 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 variation of the total action with respect to the coframe which gives the field equations in the form 𝐷𝐻_𝛼−𝐸_𝛼=Σ_𝛼,whereΣ_𝛼=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 [55, 69], one derives from (7) the translational momentum 2-form and the canonical energy-momentum 3-form (In the absence of Weitzenböck connection, the contortion 𝐾^𝛼_𝛽 will be identical with Riemannian connection.) 𝐻_𝛼=−𝜕𝑉𝜕𝑇^𝛼=1𝜅∗⁽¹⁾𝑇_𝛼−2⁽²⁾𝑇_𝛼−12⁽³⁾𝑇_𝛼=1Γ2𝜅^𝛽𝛾∧𝜂_𝛼𝛽𝛾,𝐸_𝛼=𝜕𝑉𝜕𝜗^𝛼=𝑒_𝛼⌋𝑉+𝑒_𝛼⌋𝑇^𝛽∧𝐻_𝛽=12𝑒_𝛼⌋𝑑𝜗^𝛽∧𝐻_𝛽−𝑑𝜗^𝛽∧𝑒_𝛼⌋𝐻_𝛽, (9)(10) where 𝐾^𝛽𝛾 is the contortion 1-form which is related to the torsion through the relation 𝑇^𝛼=𝐾^𝛼_𝛽∧𝜗^𝛽. (11) Due to geometric identities [17], the Lagrangian (7) can be recast as 1𝑉=−2𝑇^𝛼∧𝐻_𝛼. (12) 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^−1𝜇_𝛼Γ_𝜇^𝜈Λ^𝛽_𝜈+Λ^𝛽_𝛾𝑑Λ^−1𝛾_𝛼. (13) Due to the noncovariant transformation law of Γ_𝛼^𝛽, see (13), 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 only quasi-invariant; that is, it changes by a total divergence.

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 [17]. This means that when one changes the reference system, that is, uses different tetrad which reproduced the same metric, the effect of the inertia will change. 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. (14) The Weitzenböck connection always has the form (14). The relation between Weitzenböck and Riemannian connections has the form [17, 19] Γ_𝛼^𝛽=Γ_𝛼^𝛽−𝐾_𝛼^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^′𝛼=(Λ(𝑥)^𝛼_𝛽)𝜗^𝛽, the Lagrangian changes by a total derivative: 𝑉𝜗^′=𝑉(𝜗)−1𝑑Λ2𝜅^−1𝛼𝛽𝑑Λ^𝛽_𝛾∧𝜂^𝛾_𝛼. (16)

In addition to (16), it is easy to verify that (9) and (10) change like 𝐻^′_𝛼𝜗^′=Λ^−1𝛽_𝛼𝐻_𝛽(𝜗)−1Λ2𝜅^−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 [61] 𝒬(𝜉,𝜗)=_𝜕𝑆𝜉^𝛼𝐻_𝛼=12𝜅_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾, (18) where 𝜉^𝛼=𝜉⌋𝜗^𝛼. Equation (18) is obviously reference frame dependent like Freud’s superpotential for the Einstein pseudotensor [85]. Under a local Lorentz transformation, (18) transforms as 𝒬𝜉,𝜗^′=𝒬()−1𝜉,𝜗2𝜅_𝜕𝑆𝜉^𝛼Λ^−1𝜈_𝛾𝑑Λ^𝛾_𝜇∧𝜂_𝛼^𝜇_𝜈=12𝜅_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾−Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾, (19) where Γ_𝜇^𝜈 is given by (14). 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. Equation (19) which depends on a tensorial object (the difference between 2 connections) is more satisfactory. In this study, we call it regularization but in the quasilocal literature it is described as a choice of [86].

3. Reissner-Nordström Spacetime with Arbitrary Function

In the spherical local coordinates (𝑡,𝑟,𝜃,𝜙), a spherically symmetric spacetime with arbitrary function is described by [87]ℎ^𝛼_𝑖=𝛽𝛼₁𝛽₁𝛼₁sin𝜃cos𝜙𝛽₁𝑟sin𝜃sin𝜙𝛽₁𝑟𝛼₁sin𝜃cos𝜃−𝛽₁𝛼₁𝑓𝜃,𝜙−𝛼₁sin𝜃cos𝜙1+𝛽₂−𝑟sin𝜃sin𝜙𝛽₂+cos𝜃cos𝜙−𝑟sin𝜃cos𝜃𝛽₂+sin𝜙−𝛽₁𝛼₁𝑓₁𝜃,𝜙−𝛼₁sin𝜃sin𝜙+cos𝜙𝛽₃−𝑟sin𝜙sin𝜃𝛽₃+cos𝜃−𝑟sin𝜃cos𝜃𝛽₃𝛽+cos𝜙₁𝛼₁𝑓₂𝜃,𝜙−𝛼₁cos𝜃1+tan𝜃cos𝜙𝛽₄−𝑟sin𝜃sin𝜙𝛽₄+1𝑟sin𝜃cos𝜃𝛽₄, (20)

where 𝛼₁,𝛽,𝛽₁,𝛽₂,𝛽₃,𝛽₄,𝑓(𝑟,𝜃),𝑓₁(𝑟,𝜃) and 𝑓₂(𝑟,𝜃) have the form 𝛼₁=1−2𝑀𝑟+𝑞²𝑟^2−(1/2),𝛽=𝑟ℑ^′−1²1−2𝑐₁/𝑟+𝑐₂²/𝑟^21/2,𝛽₁=𝛽²−1,𝛽₂=𝑓,𝛽𝛽−1𝜃,𝜙₃=𝛽−1𝑓₁𝜃,𝜙,𝛽₄=𝛽−1𝑓₂,𝑓=𝜃,𝜙𝜃,𝜙sin²𝜃cos²,𝑓𝜙+cos𝜃sin𝜙sin𝜃cos𝜙−1₁=,𝑓𝜃,𝜙cos𝜃cos𝜙+sin𝜃sin𝜙sin𝜃cos𝜙+cos𝜃sin𝜙₂𝜃,𝜙=sin𝜃sin𝜃sin𝜙−cos𝜙cos𝜃, (21) where 𝑐₁ and 𝑐₂ are two constants and 𝑀 and 𝑞 are the mass and the electric charge parameters, respectively. The metric associated with tetrad (20) has the Reissner Nordström spacetime [88, 89]. When 𝑐₂=0 and 𝑞=0, one returns to the Schwarzschild spacetime [90]. Tetrad (20) has been studied extensively in [88, 89] when 𝑞=𝑐₁=𝑐₂=0, but with different notation. In those studies, it is shown that the value of energy did not coincide with the ADM value [90, 91]. Some explanation related to the asymptotic behavior of ℑ has been given to explain how to make energy coincides with ADM [92]. Also, in [88, 89], it is shown that the value of energy in general depends on ℑ. In the present study, it is our goal to recalculate the value of energy related to tetrad (20) using the method developed in [17]. It is of interest to note that any physical tetrad must behave as ℎ_𝛼𝑖=𝑂_𝛼𝑖+𝑒_𝛼𝑖1𝑟, (22) with 𝑂_𝛼𝑖 as Minkowski’s metric tensor and 𝑒_𝛼𝑖(1/𝑟) as the first term in the asymptotic expansion of 𝑔_𝜇𝜈. Asymptotically flat spacetimes are defined by the above unnumerated equation together with 𝜕_𝛽ℎ_𝛼𝑖=𝑂(1/𝑟²).

3.1. Calculation of Energy

The necessary component of the superpotential 2-form needed to the energy calculation takes the form (We will not write the nonvanishing components of Riemannian connection and Weitzenböck connection due to their huge form.): 𝐻0=𝑟sin𝜃8𝜋1−sin²𝜃cos²𝜙𝛽−cos𝜃sin𝜙1+sin𝜃cos𝜙𝛽−1+21−𝛼₁𝑑𝜃∧𝑑𝜙. (23) Computing 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 𝐸=_𝜕𝑆𝐻0=𝑅332+𝛽1−𝛼₁. (24) From (24), when 𝛽=1, we obtain the value of energy up to (1/𝑅²) in the form: 1𝐸=𝑅1−1−(2𝑀/𝑟)+𝑞²/𝑟²𝑞≈𝑀−²+2𝑅3𝑀²−2𝑅3𝑀𝑞²−5𝑀³2𝑅²1+𝑂𝑅³+⋯. (25)

Generally, (24) shows that the standard energy value for Reissner Nordström spacetime depends on the arbitrary function ℑ(𝑟) and on the two constants 𝑐₁and𝑐₂. By giving a specific value of ℑ(𝑟), we get a different value of the energy.

3.2. Regularization of Energy

To overcome the contribution of inertia to physical quantities that appeared in the preceding subsection, we are going to use the regularization framework that is based on the covariance property, that is, taking into account the Weitzenböck connection Γ_𝛼^𝛽 provided by (14). The local Lorentz transformation introduced in [17] depends on the spatial coordinate because the energy is divergent, that is, depends on the radial coordinate 𝑟; however, in this study, we will introduce another local Lorentz transformation because the form of energy is not in the known form. If 𝛽=1, that is, ℑ(𝑟)=constant and 𝑐₁=𝑐₂=𝑞=0, then tetrad (20) will coincide with tetrad (4.10) studied in [17] which gives the correct value of energy as shown after (24) and also in [17]. This means that the main reason for the contribution of inertia comes from the arbitrary function ℑ(𝑟) and also from the two constants 𝑐₁ and 𝑐₂. Therefore, if we succeed in rewriting tetrad (20) as a composition of diagonal tetrad, local Lorentz transformation, which depends on the spatial coordinate, and another local Lorentz transformation, which depends on the arbitrary function, then we can use the resulting local Lorentz transformations as a regularizing tool that can remove inertia appearing in the previous subsection.

Tetrad (20) can be put into the following form: ℎ^𝛼_𝑖=Λ₁^𝛼_𝛾Λ₂^𝛾_𝛿ℎ^𝛿_𝑖𝑅, (26) where (Λ₁^𝛼_𝛾) is a local Lorentz transformation given by Λ₁^𝛼_𝛾=10000sin𝜃cos𝜙cos𝜃cos𝜙−sin𝜙0sin𝜃sin𝜙cos𝜃sin𝜙cos𝜙0cos𝜃−sin𝜃0 (27) and (Λ₂^𝛾_𝛿) is another local Lorentz transformation which has the formΛ₂^𝛾_𝛿=𝛽𝛽₁sin𝜃cos𝜙𝛽₁sin𝜃sin𝜙𝛽₁cos𝜃−𝛽₁sin𝜃cos𝜙1+1−𝛽sin²𝜃cos²𝜙1−𝛽sin²𝜃sin𝜙cos𝜙1−𝛽sin𝜃cos𝜃cos𝜙−𝛽₁sin𝜃sin𝜙1−𝛽sin²𝜃sin𝜙cos𝜙1+1−𝛽sin²𝜃sin²𝜙1−𝛽sin𝜃cos𝜃sin𝜙−𝛽₁cos𝜃1−𝛽sin𝜃cos𝜃cos𝜙1−𝛽sin𝜃cos𝜃sin𝜙1+1−𝛽cos²𝜃. (28)

Equation (28) in general is a boost transformation which depends on 𝛽, and (Λ₁^𝛼_𝛾) is a special case of Euler’s angle, and finally (ℎ^𝛾_𝑖)_𝑅 is the square root of the Reissner-Nordström metric. The trivial case to undo the effect of 𝛽 is to put it equal 1. The coframe associated with tetrad (20) has the form 𝜗^𝛼=ℎ^𝛼_𝑖𝑑𝑥^𝑖=Λ₁^𝛼_𝛾Λ₂^𝛾_𝛿𝜗^𝛿_𝑅, (29) where 𝜗^𝛿_𝑅=(ℎ^𝛿_𝑖𝑅)𝑑𝑥^𝑖 has the form 𝜗0_𝑅=1𝛼₁𝑐𝑑𝑡,𝜗_𝑅1=𝛼₁𝜗2𝑑𝑟,_𝑅=𝑟𝑑𝜃,𝜗3_𝑅=𝑟sin𝜃𝑑𝜙. (30)

The most appropriate form of local Lorentz transformation which is used in the calculation of the Weitzenböck connection, given by (14), is given by (Λ)^𝛼_𝛽=Λ₁^𝛼_𝛾Λ₂^−1𝛾_𝛿Λ₁^−1𝛿_𝛽,whereΛ₁^𝛼_𝛾,Λ₂^𝛼_𝛾 (31) are defined in (27) and (28). This transformation is the only suitable one that can serve to remove the inertia and leave only the physical quantities as will be shown below. It is interesting to note that the use of the above equation is to return to the fact that Lagrangian (7) is not invariant under local Lorentz transformation as shown by (16). Due to this noninvariance, there exists an extra term which appears as shown by (17). To remove such term, the most suitable connection which help for such purpose is constructed out from the above equation. Calculating the necessary components of the superpotential 2-form, we finally get 𝐻0=𝑟𝛽sin𝜃1−1/𝛼₁.4𝜋𝑑𝜃∧𝑑𝜙 (32) Using (32) 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 finally obtain the asymptotic value of the energy up to (1/𝑅²) in the form 1𝐸=𝑅𝛽1−1−(2𝑀/𝑅)+𝑞²/𝑅²𝑞≈𝑀−²+2𝑅3𝑀²+𝑐2𝑅¹𝑀𝑅1+𝑂𝑅²+⋯, (33) which is the value of energy of Reissner Nordström spacetime up to order 𝑂(1/𝑅) [88, 89, 93, 94]. Equation (33) shows in a clear way that whatever the value of 𝛽, there will be no contribution to 𝑂(1/𝑅⁰). The contribution begins from 𝑂(1/𝑅) as is clear from (33) when ℑ∼const. or ℑ∼1/𝑅, and so forth.

4. Total Conserved Charge of Reissner-Nordström Spacetime with Arbitrary Function

We are going to calculate the necessary quantities needed to compute the conserved charge. The nonvanishing components of Riemannian connection of tetrad (20), up to 𝑂(1/𝑟³) when 𝛽=1, have the form Γ01×≈−𝑀sin𝜃cos𝜙1+2sin𝜙cos𝜃1−sin𝜃cos𝜙+2cos²𝜙sin²𝜃×𝑟²⁻¹1𝑑𝑡+𝑂𝑟³,Γ02≈𝑀sin𝜃2sin𝜃cos𝜃cos𝜙sin²𝜙−sin𝜙1−2cos²𝜙sin²𝜃+2cos𝜃cos²𝜙×𝑟²⁻¹1𝑑𝑡+𝑂𝑟³,Γ03≈−𝑀cos𝜃1−2cos²𝜙sin²𝜃+2sin𝜃sin𝜙cos𝜙sin²𝜃×𝑟²⁻¹1𝑑𝑡+𝑂𝑟³. (34) The components of the vector 𝜉^𝛼 have the form 𝜉⁰𝜁=−⁰𝑟²−2𝑀𝑟+𝑞²𝑟²−2𝑐₁𝑟+𝑐₂²,𝜉¹=−𝜁₀cos𝜃sin𝜙1−sin𝜃cos𝜙−cos²𝜙sin²𝜃×𝑟²−2𝑀𝑟+𝑞²2𝑐₁𝑟−𝑐₂²×𝑟𝑟²−2𝑀𝑟+𝑞²𝑟²−2𝑐₁𝑟+𝑐₂²⁻¹,𝜉²=−𝜁₀cos𝜃cos𝜙+sin𝜃sin²𝜙+sin𝜙cos𝜙sin²𝜃×𝑟²−2𝑀𝑟+𝑞²2𝑐₁𝑟−𝑐₂²×𝑟𝑟²−2𝑀𝑟+𝑞²𝑟²−2𝑐₁𝑟+𝑐₂²⁻¹,𝜉³=𝜁₀×𝑟sin𝜃cos𝜃cos𝜙−sin𝜃sin𝜙₂−2𝑀𝑟+𝑞²2𝑐₁𝑟−𝑐₂²×𝑟𝑟²−2𝑀𝑟+𝑞²𝑟²−2𝑐₁𝑟+𝑐₂²⁻¹. (35) Finally, the nonvanishing components of the asymptotic form of the superpotential 2-form 𝐻𝛼 up to 𝑂(1/𝑟) have the form 𝐻0=sin𝜃𝑟2𝑟𝑐₁+3𝑐₁²−𝑐₂²×cos²𝜙sin²𝜃+sin𝜙cos𝜃1+sin𝜃cos𝜙+2𝑞²−4𝑀𝑐₁−𝑟−2𝑀²−𝑐₂²+𝑐₁3𝑐₁𝐻1=+2𝑟𝑑𝜃∧𝑑𝜙+⋯,sin𝜃√𝑐₁𝑟12𝑐₁𝑐₁+𝑟−8𝑀𝑐₁−3𝑐₂²×cos𝜙sin𝜃cos𝜙sin𝜃+cos𝜃sin𝜙+sin𝜙cos𝜃8𝑀𝑐₁+𝑐₂²−4𝑐₁𝑐₁+𝑟−4𝑐₁𝑐₁+𝑟+𝑐₂²𝐻2=𝑑𝜃∧𝑑𝜙+⋯,sin𝜃√𝑐₁𝑟8𝑀𝑐₁−12𝑐₁𝑐₁+𝑟−3𝑐₂²cos𝜙sin𝜙sin²𝜃+sin𝜃cos𝜃sin²𝜙8𝑀𝑐+3𝑐₂²−12𝑐₁𝑐₁+𝑟×cos𝜙cos𝜃8𝑀𝑐₁−4𝑐₁𝑐₁+𝑟+𝑐₂²×𝐻3=𝑑𝜃∧𝑑𝜙+⋯,sin𝜃√𝑐₁𝑟sin𝜙8𝑀𝑐₁−4𝑐₁𝑐₁+𝑟+𝑐₂²−8𝑀𝑐₁−12𝑐₁𝑐₁+𝑟+3𝑐₂²×cos𝜃cos𝜃sin𝜙+sin𝜃cos𝜙𝑑𝜃∧𝑑𝜙+⋯. (36) Using (35) as well as (36) in (18), we finally get the total conserved charges, up to 𝑂(1/𝑟), in the following form: 𝒬=_𝜕𝑆𝜉^𝛼𝐻_𝛼≈𝑀+2𝑐₁3−3𝑞²+2𝑐₂²𝜁6𝑟⁰1+𝑂𝑟. (37) For the tetrad determined by (20), the conserved charge corresponding to the diffeomorphism generated by the shift along the time coordinate has a nonusual value of the total energy of the configuration.

5. Regularization of Conserved Charge

The regularized form of the total conserved charge which can solve the above problem has the form [71] 𝒬()=1𝜉,𝜗2𝜅_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾−Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾, (38) where Γ^𝛽𝛾 has the form (13).

Again the most appropriate local Lorentz transformation which is used in the calculation of Weitzenböck connection has the form (Λ)^𝛼_𝛽=Λ₁^𝛼_𝛾Λ₂^−1𝛾_𝛿Λ₁^−1𝛿_𝛽. (39) Using the above data, that is, Riemannian connection and Weitzenböck connection in (38), we get up to 𝑂(1/𝑟)𝒬(𝜉,𝜗)=_𝜕𝑆𝜉^𝛼Γ^𝛽𝛾−Γ^𝛽𝛾∧𝜂_𝛼𝛽𝛾=𝑞𝑀−²𝜁2𝑟⁰1+𝑂𝑟². (40) Therefore, the last term in (38) improves the unusual coefficients in (37). It is of interest to note that the calculation of regularized total conserved charges is more accurate than the regularization of energy; this is due to the fact that the latter remove completely the contribution of inertia.

6. Thermal Properties of Reissner-Nordström Spacetime with Arbitrary Function

In GR, thermodynamical quantities are calculated using the Euclidean continuation of metric [95]. However, in TEGR, these quantities are calculated through the divergence term, which appears in the Lagrangian [90]. This term has no effect on the field equation.

Gibbons and Hawking [95] discussed the thermal properties of the Schwarzschild solution, for which the line element takes the positive-definite standard form 𝑑𝑠²=+1−2𝑀𝑟𝑑𝜏²+1−2𝑀𝑟⁻¹𝑑𝑟²+𝑟²𝑑Ω², (41) after Euclidean continuation of the time variable, 𝑡=−𝑖𝜏. By using the transformation 𝑥=4𝑀(1−2𝑀/𝑟)^1/2, the line element squared becomes 𝑑𝑠²𝑥=+4𝑀²𝑑𝜏²+𝑟²4𝑀²²𝑑𝑥²+𝑟²𝑑Ω², (42) which shows that 𝜏 can be regarded as an angular variable with period 8𝜋𝑀. Now, the Euclidean section of the Schwarzschild solution is the region defined by 8𝜋𝑀≥𝜏≥0 and 𝑥>0, where the metric is positive definite, asymptotically flat, and nonsingular. They calculated the Euclidean action, 𝐼̂, of GR from the surface term as 𝐼̂=4𝜋𝑀²=𝜒²,16𝜋 (43) where 𝜒=8𝜋𝑀=𝑇⁻¹ with 𝑇 being interpreted as the absolute temperature of the Schwarzschild black hole.

For a canonical ensemble, the energy is given by [95–97] ∑𝐸=^𝑛𝐸_𝑛𝑒^{−𝜒𝐸𝑛}∑_𝑛𝑒^{−𝜒𝐸𝑛}𝜕=−𝜕𝜒log𝑍, (44) where 𝐸_𝑛 is the energy in the 𝑛th state and 𝑍 is the partition function, which is in the tree approximation related to the Euclidean action of the classical solution by 𝐼̂=−log𝑍. (45) Use of (43) and (45) in (44) gives 𝜒𝐸=8𝜋=𝑀. (46)

They also calculated the entropy of the Schwarzschild black hole to obtain 𝑆=−_𝑛𝑃_𝑛log𝑃_𝑛=𝜒𝐸+log𝑍=4𝜋𝑀²=14𝐴, (47) where 𝑃_𝑛=𝑍⁻¹𝑒^{−𝜒𝐸𝑛} and 𝐴 is the area of the event horizon of the Schwarzschild black hole.

Now, let us apply Hawking and Gibbons procedure to tetrad (20). The Euclidean action is given by [90] 𝐼̂=−1√2𝜅𝑔𝑅−2⁽²⁾𝑇^𝜇_;𝜇𝑑⁴𝑥=1𝜅√𝑔⁽²⁾𝑇^𝜇_;𝜇𝑑⁴𝑥, (48) where 𝑅 is the Riemann-Christoffel scalar curvature, which is vanishing for the Reissner Nordström spacetime, and ⁽²⁾𝑇^𝜇 is the basic vector field of the torsion given by (6).

Tetrad (20) after using Euclidean continuation, 𝑡=−√−1𝜏, takes the following form:ℎ^𝛼_𝑖𝐸=−𝛽₅𝛼₁−𝛽₁𝛼₁𝛽sin𝜃cos𝜙−¹𝛼₁𝛽sin𝜃sin𝜙−¹𝛼₁𝛽sin𝜃cos𝜃₁𝑓𝜃,𝜙𝛼₁𝛼₁sin𝜃cos𝜙1+𝛽₆𝑟sin𝜃sin𝜙𝛽₆+cos𝜃cos𝜙𝑟sin𝜃cos𝜃𝛽₆𝛽+sin𝜙₁𝑓₁𝜃,𝜙𝛼₁𝛼₁sin𝜃sin𝜙+cos𝜙𝛽₇𝑟sin𝜙sin𝜃𝛽₇cos𝜃𝑟sin𝜃cos𝜃𝛽₇−𝛽+cos𝜙₁𝛼₁𝑓₂𝜃,𝜙−𝛼₁cos𝜃1+tan𝜃cos𝜙𝛽₈−𝑟sin𝜃sin𝜙𝛽₈+1−𝑟sin𝜃cos𝜃𝛽₈, (49)

where 𝐸 refers to the Euclidean continuation and 𝛽₅=2−𝛽^21/2,𝛽₆=1−𝛽₅𝑓,𝛽𝜃,𝜙₇=1−𝛽₅𝑓₁𝜃,𝜙,𝛽₈=1−𝛽₅𝑓₂𝜃,𝜙. (50) The Euclidean line element squared of tetrad (49) takes the form 𝑑𝑠²=1𝛼₁𝑑𝜏²+𝛼₁𝑑𝑟²+𝑟²𝑑𝜃²+sin²𝜃𝑑𝜙², (51) which is the Euclidian Reissner Nordström spacetime [88, 89]. Using (6), we get the following: ℎ=𝑟²sin𝜃,₍₂₎𝑇¹=𝛽+11+sin𝜃sin𝜙cos𝜃cos𝜙+sin𝜃sin𝜙+cos𝜃sin𝜙+𝑟𝛼₂−2𝛼₁×𝑟𝛼₁⁻¹,𝛼₂=𝛼^′₁, (52) where ℎ=√𝑔 is the determinant of tetrad (49). There are other nonvanishing components of ⁽²⁾𝑇^𝜇, that is, ⁽²⁾𝑇⁰,⁽²⁾𝑇², and ⁽²⁾𝑇³, but these components are not necessary in our computations. The volume integral (48), (𝜃=0⋯𝜋, 𝜙=0⋯2𝜋, 𝜏=0⋯𝜒), for tetrad (49), is calculated to give 𝐼̂=1𝜅√𝑔⁽²⁾𝑇^𝜇_;𝜇𝑑⁴𝑥≈2𝜋𝑟²1+2𝛽−6/𝛼₁−3𝑟𝛼₂3𝛼₁1+𝑞²/𝑟², (53) where 𝜒=4𝜋𝑟/(1−(𝑞²/𝑟²)). The largest positive root of 𝑔^𝑟𝑟=0, that is, (𝑟²−2𝑀𝑟+𝑞²)=0⇒𝑟_±=𝑀±𝑀²−𝑞². Now, one can write mass and charge in terms of horizons, that is, 𝑀=(𝑟₊+𝑟₋)/2 and 𝑞=√𝑟₋𝑟₊. Equation (53) shows in a clear way that the Euclidean action depends on the arbitrary function ℑ which makes the energy also depends on this arbitrary function. Therefore, we need to regularize (48). This regularization lies in the redefinition of the nonsymmetric connection. This redefinition takes the form Γ=^𝑖_𝑗𝑘^def.ℎ_𝛼^𝑖ℎ^𝛼_𝑗,𝑘+ℎ_𝛼^𝑖Λ₂^𝛽𝛼Λ_{2𝛽𝛾,𝑘}ℎ^𝛾_𝑗,𝑇=^𝜇_𝜈𝜆^def.Γ^𝜇_𝜈𝜆−Γ^𝜇_𝜆𝜈,⁽²⁾𝑇=_𝜈^def.𝑇^𝜇_𝜈𝜇. (54) Using (54) in (48), we get 𝐼̂=1𝜅√𝑔⁽²⁾𝑇^𝜇_;𝜇𝑑⁴𝑥≈𝜋𝑟₊²𝑞1−²𝑟₊². (55) From (55), (44), and (45), we get the energy up to 𝑂(1/𝑟₊) in the form 𝑞𝐸≈𝑀−²2𝑟₊1+𝑂𝑟₊², (56) which is consistent with (33) and (40).

Using (55) and (56) in (47), we get the entropy of tetrad (49) in the form 𝑆≈𝜋𝑟₊². (57)

The Hawking temperature of tetrad (49) has the form [98] 𝑞𝑇=1−²/𝑟₊²4𝜋𝑟₊. (58) From (56), (57), and (58), we get 𝑑𝑀=𝑇𝑑𝑆. (59) Equation (59) shows that the first law of thermodynamic is satisfied.

7. Discussion and Conclusion

Within the framework of MAG, the total Lagrangian is formed from a geometrical part plus a 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 [59] 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 Lorentz invariance. When the cosmological constant vanishes, that is, 𝜆=0, the solution given by (57) derived in [59] will be identical with the third coframe defined by 𝜗^𝑅₃^𝛼=(Λ₁^𝛼_𝛾)(ℎ^𝛾_𝑖)₃𝑑𝑥^𝑖.

To test the regularizing property of teleparallelism, we have used the superpotential 2-form to calculate energy of a spherically symmetric tetrad with arbitrary function. We have done such calculations when the Weitzenböck connection is taken into account and also when it is not taken into account. It has been shown that the energy depends on the arbitrary function in the trivial case and when it takes specific values, which make the local Lorentz transformations identical with the Kronecker delta; the total energy becomes physically acceptable [91, 94]. As Weitzenböck connection is taken into account and recalculation of energy is carried out, we have shown that it acts as a regularizing tool which separates inertia from the physical quantities.

In the context of TEGR, the total conserved charge of the spherically symmetric tetrad with arbitrary function is calculated. It has been shown how inertia spoils the total conserved charge along the time coordinate. Therefore, regularized expression to recalculate the total conserved charge is used, and a plausible result is obtained for energy. Finally, the energy of the tetrad field with arbitrary function is calculated using the Euclidean continuation method in the context of TEGR. It has been shown that energy depends on the arbitrary function. Therefore, a redefinition of the Riemannian connection is provided to recalculate energy. A satisfactory result which is consistent with the previous two methods is achieved. The thermodynamical quantities associated with the tetrad with arbitrary function are calculated, and satisfactory results are obtained [99]. The main conclusion of the present work is that Weitzenböck connection which acts as a regularized term works correctly for the spherically symmetric case. At present, it is not clear if this conclusion is correct for axially symmetric spacetimes. This needs a further study.

Blagojević [100] 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 [101] discussed a Lagrangian form of the energy-momentum of some exact solutions in the teleparallel limit of PG theory.

We make a comparison between the possible values which reproduce the previous spacetimes of Schwarzschild and Reissner Nordström as shown in Table 1. Also, we give a comparison between the results of the total energy for taking into account Weitzenböck connection and for not taking into account Weitzenböck connection and the pervious results of the total energy for the same spacetime with arbitrary function as shown in Table 2.

Table 1 

Comparison between the results of the arbitrary function ℑ(𝑟), other constants, and the associated energy.

Table 2 

Comparison between the results of energy using different translational momentum.

Acknowledgment

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