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

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.


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 Møller Energy-Momentum Complex
The metric tensor can be written in the tetrad form: where η ij is the Minkowski metric defined by Diag{−1, 1, 1, 1}.The torsion tensor in Møller's theory is and here Γ μ λν is Weitzenb öck connection 25 given by

2.3
A general expression for an energy-momentum complex was found by Møller by using the method of infinitesimal transformations in the superpotential β μν form 10 : Here, T β μ , t β μ energy-momentum tensor and energy-momentum pseudotensor arise from matter and gravitational field, respectively.β μν is given by Here λ is a free dimensionless parameter.Φ ρ is defined by and ξ αβμ h iα e i β;μ is the con-torsion tensor.P χρσ τμν is the tensor of the form The total energy is given by the surface integral below where dS is the surface element and η α is the unit 3-vector normal to the surface.

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 : 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 γη τ F τ 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 where a and b are two constant-free parameters.
The general metric for this theory is written as For this line element, the metric tensor and its inverse are obtained as F 2 r sin 2 θ .

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

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Case a b 1 Taking a b 1 in the action 3.2 the metric becomes 26 Taking G 2 r 1 − 2m/r and F 2 r r r − 2r 0 in 3.5 and then using it in 2.5 one can obtain the Freud superpotential as follows: Using this result in the energy integral 2.8 the energy density is found as where ϕ 0 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 r 0 Q 2 e −ϕ 0 / 2m , Q e is electrical charge, Q m is magnetic charge, and the Case a 1, b 1 In this case the line element is 26

3.11
If one can use G 2 r r − r r − r − / r 2 − r 2 0 and F 2 r r 2 − r 2 0 in 3.5 , is 2.5 obtained as follows:

3.12
The energy density obtained by 2.8 is One can easily see that, for Q m 0, Q e Q and under the coordinate transformation r ± r 0 → 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

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

3.16
Using G 2 r 1 − 2m/ r − 2r 0 and F r r − 2r 0 in the 3.5 , 2.5 is obtained as 3.17 The energy density is obtained as and the total energy for a b 1 is found as where r 0 ≈ a 2 e −aφ Q 2 /4m 0 and m 0 ≈ m r 0 .

Asymptotically Nonflat Black Holes
When we consider asymptotically non-flat black holes, the line element 3.3 becomes 26

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Case a b 1 Now, the metric has the form 26

3.21
Considering G 2 r r − 4m /2r 0 and F 2 r 2rr 0 in 3.5 and then using it in 2.5 one can obtain the necessary component of the Freud superpotentials as follows:

3.22
Substituting this result into the 2.8 the energy distribution is found as

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 dr 2 4r 2 0 dΩ 2 .

3.24
If we use G 2 r r/2r 0 2 1 − 2m/a 2 r and F r 2r 0 in 3.5 and 2.5 , the calculated component of the Freud superpotentials is

3.25
From 2.8 the energy is found

3.26
At large distances, the total energy diverges.

Case a b 1
Here the metric becomes 26

3.28
Using Freud superpotential in the 2.8 the obtained energy distribution is When a → ∞ and r → ∞, the total energy becomes E m.

Case |a| / |b|
For this final case the metric becomes 26

3.31
If we consider G 2 r r − r r − r − /2rr 0 and F 2 r 2rr 0 the component of the Freud superpotentials is calculated as Now the energy distribution is

3.33
Advances in High Energy Physics 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 m sin θdθ ∧ dϕ.

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 M öller 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.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 .

a 1 Radius Energy 5 a = 1 , 1 a = 1 , 1 Figure 1 :
Figure 1: The plot for the M öller energy versus radius, in the case of the asymptotically flat black holes.