Total Energy of Charged Black Holes in Einstein-Maxwell-Dilaton-Axion Theory

Abstract

We focus on the energy content (including matter and fields) of the Møller energy-momentum complex in the framework of Einstein-Maxwell-Dilaton-Axion (EMDA) theory using teleparallel gravity. We perform the required calculations for some specific charged black hole models, and we find that total energy distributions associated with asymptotically flat black holes are proportional to the gravitational mass. On the other hand, we see that the energy of the asymptotically nonflat black holes diverge in a limiting case.

1. Introduction

There are many interesting theories aiming to investigate gravitational effects: general relativity and teleparallel gravity. In these theories, calculating the energy-momentum distribution is an old and interesting problem. One can construct a teleparallel equivalent of the general relativity by assuming that curvature and torsion give the equivalent descriptions of the gravitational interactions.

In order to avoid the singularities in the general relativity and to give a general definition of energy momentum, Møller obtained a new expression by using the teleparallel gravity [1, 2]. Pellegrini and Plebanski found the Lagrangian formulation of the teleparallel theory, and this formalism was developed further by Møller [3, 4]. The gauge theory studied in detail by Hayashi [5] for the translational group is formulated by Hayashi and Nakano [6]. After that Hayashi [7] underlined the connection between this theory and the teleparallel theory. Later, Hayashi and Shirafuji [8] tried to unify these two theories. Hehl et al. [9] discussed a generalization of Einstein’s gravitational theory with spin and torsion. In the near past Mikhail et al. [10] used the method of the superpotential in the case of the spherical symmetry for the Møller’s energy-momentum expression in the teleparallel theory. By using the Møller’s super potential method Shirafuji et al. [11, 12] found that energy is equivalent to the gravitational mass. Maluf [13] showed that the energy is equivalent to the gravitational mass by constructing a connection between the gravitational energy related closely to the Sparling two-forms and the teleparallel equivalence of the general relativity. de Andrade et al. [14] obtained a teleparallel equivalence of the Kaluza-Klein theory in five dimensions. Respectively, Pereira et al. [15] and Zhang [16] found the tetrad and torsion fields for the axial symmetric Kerr spacetime and alternative Kerr spacetime by using the teleparallel theory. A new interpretation for torsion in connection with the gravitational interaction is given by Arcos and Pereira [17] as a review. It is pointed out that the teleparallel gravity successfully was applied to describe the rotation spin effect as another application of spin and axial torsion interaction [18]. About the spacetime torsion one can examine the work in [19]. Nashed [20] calculated total energy for some special situations of the most general spherical symmetric and nonsingular black hole solutions in the teleparallel gravity by using the Møller’s super potential technique.

The study carried out by Vargas [21] for the Friedman-Robertson-Walker (FRW) universe possessing a cartesian geometry has been a starting point for the other studies done after that. Later, the energy densities for the Einstein, Bergman-Thomson, and Landau-Lifshitz energy momentum expressions written in cartesian coordinates are obtained in the teleparallel theory. Therefore the teleparallel equivalence of general relativistic case is obtained by Nashed [22] for some solutions of the models of the universe having charged and spherical symmetric solutions. Furthermore, the energy-momentum and angular momentum are calculated for two different fields by using energy-momentum tensor [23]. In the framework of Riemann’s geometry it is found that the energy density in Kerr-Nut space-time model is proportional to the gravitational mass by using Møller’s energy-momentum complex [24]. During the last five years it is stressed that the energy-momentum densities calculated both in general relativity theory and teleparallel theory are equivalent. In this point of view it is concluded that the teleparallel theory is an alternative one for the general relativity.

This paper is organized as follows. In the next section we introduce Møller energy-momentum complex in the teleparallel theory. In the Section 4, the total energy is found for the EMDA theory. Last section is devoted to discussion. Throughout this paper, Latin indices (𝑖,𝑗,𝑘,…) denote the vector numbers and Greek indices (𝜇, 𝜈, 𝜎,…) represent the vector components. All indices run from 0 to 3, and we use units in which 𝐺=1 and 𝑐=1.

2. The Møller Energy-Momentum Complex

The metric tensor can be written in the tetrad form: 𝑔^𝜇𝜈=𝜂^𝑖𝑗ℎ_𝑖^𝜇ℎ_𝑗^𝜈, (2.1) where 𝜂^𝑖𝑗 is the Minkowski metric defined by Diag{−1,+1,+1,+1}. The torsion tensor in Møller’s theory is 𝑇^𝜇_𝜈𝜆=Γ^𝜇_𝜆𝜈−Γ^𝜇_𝜈𝜆, (2.2) and here Γ^𝜇_𝜆𝜈 is Weitzenböck connection [25] given by Γ^𝜇_𝜈𝜆=ℎ_𝑎^𝜇𝜕_𝜆ℎ^𝑎_𝜈. (2.3)

A general expression for an energy-momentum complex was found by Møller by using the method of infinitesimal transformations in the superpotential ℑ_𝛽^𝜇𝜈 form [10]: 𝑀_𝛽^𝜇≡√−𝑔𝑇_𝛽^𝜇+𝑡_𝛽^𝜇=ℑ_𝛽;𝜈^𝜇𝜈. (2.4) Here, 𝑇_𝛽^𝜇, 𝑡_𝛽^𝜇 energy-momentum tensor and energy-momentum pseudotensor arise from matter and gravitational field, respectively. ℑ_𝛽^𝜇𝜈 is given by ℑ_𝛽^𝜇𝜈=√−𝑔𝑃2𝜅^{𝜒𝜌𝜎𝜏𝜇𝜈}Φ^𝜌𝑔^𝜎𝜒𝑔_𝛽𝜏−𝜆𝑔_𝜏𝛽𝜉^𝜒𝜌𝜎−(1−2𝜆)𝑔_𝜏𝛽𝜉^𝜎𝜌𝜒. (2.5) Here 𝜆 is a free dimensionless parameter. Φ^𝜌 is defined by Φ_𝜌=𝜉^𝜎_𝜌𝜎, (2.6) and 𝜉_𝛼𝛽𝜇=ℎ_𝑖𝛼𝑒^𝑖_𝛽;𝜇 is the con-torsion tensor. 𝑃_𝜒𝜌𝜎^𝜏𝜇𝜈 is the tensor of the form 𝑃^𝜏𝜇𝜈_𝜒𝜌𝜎=𝛿_𝜒^𝜏𝛿_𝜌^𝜇𝛿_𝜎^𝜈−𝛿_𝜎^𝜇𝛿_𝜌^𝜈+𝛿_𝜌^𝜏𝛿_𝜎^𝜇𝛿_𝜒^𝜈−𝛿_𝜒^𝜇𝛿_𝜎^𝜈−𝛿_𝜎^𝜏𝛿_𝜒^𝜇𝛿_𝜌^𝜈−𝛿_𝜌^𝜇𝛿_𝜒^𝜈. (2.7)

The total energy is given by the surface integral below 𝐸=lim_{𝑟→∞𝑟=const.}ℑ₀^0𝛼𝜂_𝛼𝑑𝑆, (2.8) where 𝑑𝑆 is the surface element and 𝜂_𝛼 is the unit 3-vector normal to the surface.

3. Energy Contents of Charged Black Holes in EMDA Theory

In Einstein’s frame, Einstein-Hilbert-Maxwell action coupled with a string in four dimensions is given [26]: 𝑆=𝑑⁴𝑥√−𝑔12𝜅𝑅−2𝜕_𝜇𝜑𝜕^𝜇1𝜑−2𝜔𝜑𝜕_𝜎𝜁𝜕^𝜎𝜁−𝛼𝜑,𝜁𝐹_𝛾𝜂𝐹^𝛾𝜂−𝛽𝜑,𝜁𝐹_𝛾𝜂𝐹^𝛾𝜂, (3.1) where 𝜅 is the four-dimensional gravitational coupling constant, 𝑅 is the curvature scalar, and 𝐹_𝜇𝜈 is the field strength of the Maxwell field. Here 𝜑, 𝜁 are scalar and pseudoscalar fields, respectively. Additionally, 𝛼 and 𝛽 functions describe how the Maxwell field is coupled with 𝜑 and 𝜁. Also, the Maxwell field strength is defined as 𝐹^𝛾𝜂=(1/2)𝜖^{𝛾𝜂𝜖𝜏}𝐹_𝜖𝜏 (𝜖 is the fourth Levi-Civita tensor).

Taking 𝜔(𝜑)=𝑒^2𝑎𝜑, 𝛼(𝜑)=𝑒^−𝑎𝜑, and 𝛽(𝜁)=𝑏𝜁 for the four dimensional Einstein-Maxwell theory the generalized action coupled with massless scalar dilaton 𝜑 and pseudoscalar axion 𝜁 is given as 𝑆=𝑑⁴𝑥√1−𝑔12𝜅𝑅−2𝜕_𝜇𝜑𝜕^𝜇1𝜑−2𝑒^2𝑎𝜑𝜕_𝜎𝜁𝜕^𝜎𝜁−𝑒^−𝑎𝜑𝐹_𝛾𝜂𝐹^𝛾𝜂−𝑏𝜁𝐹_𝛾𝜂𝐹^𝛾𝜂, (3.2) where 𝑎 and 𝑏 are two constant-free parameters.

The general metric for this theory is written as 𝑑𝑠²=−𝐺²(𝑟)𝑑𝑡²+1𝐺²(𝑟)𝑑𝑟²+𝐹²(𝑟)𝑑𝜃²+sin²𝜃𝑑𝜙. (3.3) For this line element, the metric tensor and its inverse are obtained as 𝑔_𝜇𝜈=diag−𝐺²(𝑟),1𝐺²(𝑟),𝐹²(𝑟),𝐹²(𝑟)sin²𝜃,𝑔^𝜇𝜈−1=diag𝐺²(𝑟),𝐺²(𝑟),1𝐹²(𝑟),1𝐹²(𝑟)sin²𝜃. (3.4) Having spherical symmetry for the general form of the tetrad and using the coordinate transformation, the tetrad components can be written in a matrix form as ℎ_𝑎^𝜇=1𝐺(𝑟)(𝑟)0000𝐺sin𝜃cos𝜙cos𝜃cos𝜙𝐹(𝑟)−sin𝜙𝐹(𝑟)(𝑟)sin𝜃0𝐺sin𝜃sin𝜙cos𝜃sin𝜙𝐹(𝑟)cos𝜙𝐹(𝑟)(𝑟)sin𝜃0𝐺cos𝜃−sin𝜃𝐹(𝑟)0. (3.5)

3.1. Asymptotically Flat Black Holes

For the asymptotically flat black holes, the line element (3.3) becomes [26] 𝑑𝑠²=−𝑟−𝑟₊𝑟−𝑟₋𝑟−𝑟₀^2−2𝑛𝑟+𝑟₀^2𝑛𝑑𝑡²+𝑟−𝑟₀^2−2𝑛𝑟+𝑟₀^2𝑛𝑟−𝑟₊𝑟−𝑟₋𝑑𝑟²+𝑟+𝑟₊^2𝑛𝑟−𝑟₀^2𝑛𝑑Ω², (3.6) where 𝑟_±, 𝑟₀ are constants, 0≤𝑛≤1, and 𝑑Ω²=𝑑𝜃²+sin²𝜃𝑑𝜙.

Case 𝑎=𝑏=1
Taking 𝑎=𝑏=1 in the action (3.2) the metric becomes [26] 𝑑𝑠²=−1−2𝑚𝑟𝑑𝑡²+1−2𝑚𝑟⁻¹𝑑𝑟²+𝑟𝑟−2𝑟₀𝑑Ω². (3.7) Taking 𝐺²(𝑟)=(1−(2𝑚/𝑟)) and 𝐹²(𝑟)=𝑟(𝑟−2𝑟₀) in (3.5) and then using it in (2.5) one can obtain the Freud superpotential as follows: ℑ₀⁰¹=2sin𝜃𝜅2𝑚+𝑟₀−2𝑟₀𝑟−𝑟+(𝑟−2𝑚)^1/2𝑟−2𝑟₀^1/2. (3.8) Using this result in the energy integral (2.8) the energy density is found as 𝐸(𝑟)𝑒=2𝑚+^−𝜑0𝑄²4𝑚1−2𝑚𝑟−𝑟+(𝑟−2𝑚)^1/2𝑒𝑟−^−𝜑0𝑄²2𝑚^1/2, (3.9) where 𝜑₀ is a scalar field when 𝑟→∞. Finally the total energy when 𝑟→∞ for the asymptotically flat black holes (𝑎=𝑏=1) is 𝐸=𝑚. (3.10) Here 𝑟₀=𝑄²𝑒^−𝜑0/(2𝑚), 𝑄_𝑒 is electrical charge, 𝑄_𝑚 is magnetic charge, and the 𝑄 is total charge (𝑄²=𝑄²_𝑒+𝑄²_𝑚).

Case 𝑎=1, 𝑏≪1
In this case the line element is [26] 𝑑𝑠²=−𝑟−𝑟₊𝑟−𝑟₋𝑟²−𝑟²₀𝑑𝑡²+𝑟²−𝑟²₀𝑟−𝑟₊𝑟−𝑟₋𝑑𝑟²+𝑟²−𝑟²₀𝑑Ω². (3.11) If one can use 𝐺²(𝑟)=(𝑟−𝑟₊)(𝑟−𝑟₋)/(𝑟²−𝑟²₀) and 𝐹²(𝑟)=(𝑟²−𝑟²₀) in (3.5), is (2.5) obtained as follows: ℑ₀⁰¹=2sin𝜃𝜅𝑟−𝑟₊^(1/2)𝑟−𝑟₋^1/2−𝑟𝑟−𝑟₊𝑟−𝑟₋𝑟²−𝑟²₀. (3.12) The energy density obtained by (2.8) is 𝐸(𝑟)=𝑟²−2𝑚𝑟+𝑄²_𝑒+𝑄²_𝑚𝑒^−𝜙0−𝑟²₀^1/2𝑄−𝑟1−2𝑚𝑟−²_𝑒+𝑄²_𝑚𝑒^−𝜙0𝑟²−𝑟²₀, (3.13) where 𝑟₀=(𝑄²_𝑒−𝑄²_𝑚)𝑒^−𝜙0/2𝑚 and 𝑟_±=𝑚±[𝑚²+𝑟²₀−(𝑄²_𝑒+𝑄²_𝑚)𝑒^−𝜙0]^1/2. One can easily see that, for 𝑄_𝑚=0, 𝑄_𝑒=𝑄 and under the coordinate transformation 𝑟±𝑟₀→𝑟, the case transforms in to a Garfinkle-Horowitz-Strominger (GHS) dilaton black hole [27]. Therefore the total energy for GHS dilaton black hole is [28] 𝑄𝐸=𝑚−²𝑒^−𝜑0𝑟. (3.14) The total energy obtained by taking 𝑄_𝑒=𝑄_𝑚=0 (Schwarzschild solution) for electrically or magnetically charged black holes is Schwarzschild solution 𝐸=𝑚. (3.15)

Case 𝑎=𝑏≫1
In this case the line element is written by [26] 𝑑𝑠²≈−1−2𝑚𝑟−2𝑟₀𝑑𝑡²+1−2𝑚𝑟−2𝑟₀⁻¹𝑑𝑟²+𝑟−2𝑟₀²𝑑Ω². (3.16) Using 𝐺²(𝑟)=(1−(2𝑚/(𝑟−2𝑟₀))) and 𝐹(𝑟)=(𝑟−2𝑟₀) in the (3.5), (2.5) is obtained as ℑ₀⁰¹=2sin𝜃𝜅2𝑚+𝑟−2𝑟₀1−2𝑚𝑟−2𝑟₀^1/2−1. (3.17) The energy density is obtained as 𝐸(𝑟)𝑎=2𝑚+𝑟−²𝑒^−𝑎𝜙𝑄²2𝑚₀1−4𝑚𝑚₀2𝑟𝑚₀−𝑎²𝑒^−𝑎𝜙𝑄^21/2−1, (3.18) and the total energy for 𝑎=𝑏≫1 is found as 𝐸=𝑚, (3.19) where 𝑟₀≈𝑎²𝑒^−𝑎𝜙𝑄²/4𝑚₀ and 𝑚₀≈𝑚+𝑟₀.

3.2. Asymptotically Nonflat Black Holes

When we consider asymptotically non-flat black holes, the line element (3.3) becomes [26] 𝑑𝑠²𝑟=−2𝑟₀^2𝑛1−2𝑚(1−𝑛)𝑟𝑑𝑡²+2𝑟₀𝑟^2𝑛1−2𝑚(1−𝑛)𝑟⁻¹𝑑𝑟²+𝑟²2𝑟₀𝑟^2𝑛𝑑Ω², (3.20) where 𝑟₊=2𝑚/(1+𝑛), 𝑟₋=0, and 𝑛=1/(1+𝑎²).

Case 𝑎=𝑏=1
Now, the metric has the form [26] 𝑑𝑠²=−𝑟−4𝑚2𝑟₀𝑑𝑡²+𝑟−4𝑚2𝑟₀⁻¹𝑑𝑟²+2𝑟𝑟₀𝑑Ω². (3.21) Considering 𝐺²(𝑟)=((𝑟−4𝑚)/2𝑟₀) and 𝐹²(𝑟)=2𝑟𝑟₀ in (3.5) and then using it in (2.5) one can obtain the necessary component of the Freud superpotentials as follows: ℑ₀⁰¹=2sin𝜃𝜅𝑟2𝑚−2+𝑟^1/2(𝑟−4𝑚)^1/2. (3.22) Substituting this result into the (2.8) the energy distribution is found as 𝐸(𝑟)𝑟=2𝑚−2+𝑟^1/2(𝑟−4𝑚)^1/2. (3.23) When we take the limit 𝑟→∞, we see that the energy distribution diverges.

Case 𝑎=𝑏≪1
At this point the metric becomes [26] 𝑑𝑠²𝑟=−2𝑟₀²1−2𝑚𝑎²𝑟𝑑𝑡²+2𝑟₀𝑟²1−2𝑚𝑎²𝑟⁻¹𝑑𝑟²+4𝑟²₀𝑑Ω². (3.24) If we use 𝐺²(𝑟)=(𝑟/2𝑟₀)²[1−(2𝑚/𝑎²𝑟)] and 𝐹(𝑟)=2𝑟₀ in (3.5) and (2.5), the calculated component of the Freud superpotentials is ℑ₀⁰¹=2𝑟sin𝜃𝜅1−2𝑚𝑎²𝑟^1/2. (3.25) From (2.8) the energy is found 𝐸(𝑟)=𝑟1−2𝑚𝑎²r^1/2. (3.26) At large distances, the total energy diverges.

Case 𝑎=𝑏≫1
Here the metric becomes [26] 𝑑𝑠²𝑟=−2𝑟₀^2/𝑎21−2𝑎²𝑚𝑎²−1𝑟𝑑𝑡²+2𝑟₀𝑟^2/𝑎21−2𝑎²𝑚𝑎²−1𝑟⁻¹𝑑𝑟²+𝑟²2𝑟₀𝑟^2/𝑎2𝑑Ω². (3.27) If we have 𝐺(𝑟)=(𝑟/2𝑟₀)^1/𝑎2[1−(2𝑎²𝑚/(𝑎²−1)𝑟)]^1/2 and 𝐹(𝑟)=𝑟(2𝑟₀/𝑟)^1/𝑎2 in (3.5), then (2.5) is obtained as follows: ℑ₀⁰¹=2sin𝜃𝜅𝑟^{(𝑎2+1)/𝑎2}1−2𝑎^𝑚𝑎²−1𝑟^1/21−1−2𝑎²𝑚𝑎²−1𝑟^1/2𝑟^1/𝑎2. (3.28) Using Freud superpotential in the (2.8) the obtained energy distribution is 𝐸(𝑟)=𝑟^{(𝑎2+1)/𝑎2}1−2𝑎^𝑚𝑎²−1𝑟^1/21−1−2𝑎²𝑚𝑎²−1𝑟^1/2𝑟^1/𝑎2. (3.29) When 𝑎→∞ and 𝑟→∞, the total energy becomes 𝐸=𝑚. (3.30)

Case |𝑎|≠|𝑏|
For this final case the metric becomes [26] 𝑑𝑠²=−𝑟−𝑟₊𝑟−𝑟₋2𝑟𝑟₀𝑑𝑡²+2𝑟𝑟₀𝑟−𝑟₊𝑟−𝑟₋𝑑𝑟²+2𝑟𝑟₀𝑑Ω². (3.31) If we consider 𝐺²(𝑟)=(𝑟−𝑟₊)(𝑟−𝑟₋)/2𝑟𝑟₀ and 𝐹²(𝑟)=2𝑟𝑟₀ the component of the Freud superpotentials is calculated as ℑ₀⁰¹=sin𝜃𝜅𝑟₊+𝑟₋+2𝑟−𝑟₊^1/2𝑟−𝑟₋²−𝑟₊𝑟₋𝑟−𝑟. (3.32) Now the energy distribution is 𝐸(𝑟)𝑞=2𝑚−^2𝑒𝑞²_𝑚2𝑟𝑟²₀−𝑟2+𝑟²𝑞−4𝑚𝑟+^2𝑒𝑞²_𝑚𝑟²₀^1/2. (3.33) It can be easily checked that, at large distances, the total energy diverges. Here, 𝑞_𝑒 and 𝑞_𝑚 are electromagnetic charges which are related to components of the electromagnetic field strength given by 𝐹_𝑡𝑟=(1/2𝑞_𝑒)𝑑𝑡∧𝑑𝑟 and 𝐹_𝜃𝜑=𝑞_𝑚sin𝜃𝑑𝜃∧𝑑𝜑.

4. Summary and Discussion

According to Lessner [29] perspective, the Møller energy-momentum complex can be evaluated in any coordinate system. Hence, this framework is the most powerful one in calculating the energy and momentum distributions associated with spacetime.

In the present work, in order to compute the energy distribution associated with some specific black holes in the Einstein-Maxwell-Dilaton Axion theory, we focus on the Møller energy-momentum distribution in the teleparallel gravity. For the asymptotically flat charged black holes, for all the cases (𝑎=𝑏=1, 𝑎=𝑏≫1, and 𝑎=1, 𝑏≪1), it is found that the energy distribution depends on the mass 𝑚 and the charge 𝑄. The corresponding teleparallel Møller total energy is obtained proportional to gravitational mass (Schwarzschild mass) when lim_𝑟→∞𝐸(𝑟)=𝑚. (4.1) It is given the plot for the Möller energy versus radius, in the case of the asymptotically flat black holes in Figure 1. According to graph it is seen that the energy of the case 𝑎=𝑏=1 approaches to the gravitational mass faster than the others.

Figure 1 

The plot for the Möller energy versus radius, in the case of the asymptotically flat black holes.

Considering asymptotically non-flat charged black holes, for both the cases 𝑎=𝑏=1 and 𝑎=𝑏≪1 the energies diverge. Only for the case 𝑎=𝑏≫1 the total energy is proportional to 𝑚. The energy distribution for the case |𝑎|≠|𝑏| diverges as well. For small values of 𝑎 and 𝑏 it is seen that the energy distributions tend toward infinity. If the values of 𝑎 and 𝑏 are large enough the corresponding teleparallel energy will be equal to the gravitational mass.

These results agree well with the previous results [30–35] obtained by using the general relativity version of the Møller energy-momentum complex. The energy is confined to the region of nonvanishing energy-momentum tensor of matter and all nongravitational fields [36]. The results are quite important in the theory of teleparallel gravity, since this theory provides more satisfactory solution of the energy-momentum problem than general relativity [10].

Acknowledgment

This work is supported by the project DÜBAP-07-02-017.