Gravitational entropy of stringy charged black holes in teleparallel gravity

In order to resolve several theoretical and practical flaws in general relativity GR, a variety of modified theories of gravity have been proposed. One exciting strategy is to modify gravity’s geometrical nature. The teleparallel theory of gravity accomplishes this. In this paper, we study the gravitational energy GE and gravitational pressure of stringy charged black holes SCBH namely within the basic framework of the teleparallel equivalent of general relativity GR. We determine GE bounded by the event horizon of the black hole BH and the radial pressure RP over it. Furthermore, we examine the gravitational entropy of SCBH which is affected by BH’s mass m and charge Q.


Introduction
In actuality, the charged BHs in general relativity GR are represented by the Reissner-Nordstrom solution.In string theory, the stringy charged black holes SCBHs solution for charged BHs works well when the curvature is lower than the Planck scale.Although the two solutions are equivalent, there are many significant differences between them.For example, the inner horizon is not recorded in the Reissner-Nordstrom scale when there is a dilation field.But, the transition among stark singularities of BHs occurs when combined with a non-Reissner-Nordstrom statewidening charge's existence.What's significant is that the upper limit of Q/m contrary the horizon's SCBH [1][2][3][4].
The teleparallel equivalent of general relativity GR (see references [5,6]) is well known to have extraordinary advantages, such as the ability to investigate conserved currents [7] and non-steroidal solutions [8].Additionally, it has been confirmed that TEGR represents an appropriate framework for investigating gravitational waves [9,10] and non-singular BHs [11] and treating dark matter problems [12,13].
Vargas [14] derives the total energy of the universe in TG, which includes the energy of both matter and gravitational fields.It is demonstrated that, for a closed universe, the three dimensionless coupling constants of TG, the pseudo-tensor employed, and the total energy all vanish independently.Abedi and Salti [15] are investigating localized energy and multiple field modified gravity in TG.
One of the most important features of BH physics is that they behave like thermodynamic TD systems which have both entropy and temperature.The first achievement in the field of BH TDs was established by Bekenstein [16].He was the author who actually introduced the second law of TD which emphasizes that a BH must have an entropy proportional to the area of its horizon.Bardeen, Carter and Hawking [17] explained the four laws of BH mechanics, following the same approach as ordinary TDs.
The energy momentum EM densities of SCBH were first described by Vibhadra and Parikh [18] in Einstien's prescription followed by Xulu [19], adopting Tolman's EM complex.Their energy distribution result was the same.The EM density of this space-time has been determined by Gad [20] using Mo ¨ller's formula.
In the context of the new GR, Nashed and Mourad [21] have investigated EM for a fixed light beam.Gad and Mourad [22] successfully determined the energy distribution of the Kantowski-Sachs solution in TEGR.Mourad [23,24] discussed gravitational energy momentum GEM, GEM flux, angular momentum AM, and Dirac spin in a cylindrically and spherically axially symmetric solutions.In [25,26] spherically symmetric de Sitter DS and anti-de Sitter ADS of BHs with Smeared Matter Distribution were studied.In recent papers [27][28][29][30], TDs of BHs for different classes of space-time have been studied.
The next sections of the present paper are organized in the following manner: in Sect.2, GEM and the gravitational pressure of SCBH are investigated.In Sect.3, the gravitational entropy of SCBH solution is studied.Finally, in Sect.4, our conclusions are presented.

The GEM and Gravitational Pressure of SCBH Solution
Although the theory of teleparallel gravity is completely identical to general relativity, it is presented in terms of Weitzenbock geometry rather than Riemann geometry.The tetrad field h a l may be used to determine the torsion tensor T a lm in the form, [31] Equations of motion for the h l a that characterizes gravity as a result of the curvature of space-time in terms of mass and energy, which are called field equations are given by.1 where k ¼ 1 16p and K refers to the cosmological constant.Over an arbitrary volume V, the GEM can be written as follows [32] where P ai ¼ À4khR a0i refers to the momentum canonically conjugated to h ai and Ào i P ai gives the EM density.And The quantity / ðiÞj is given by The flux of momentum a ¼ 1; 2; 3 ð Þ , signifies a force.À/ ðiÞj depicts the pressure acting in the iÀ direction on an area element with jÀ directed orientation, since dS j represents area element, [33].It is important to remember that the equations of motion are the source for all the definitions in this section (2).
The selection of tetrad should fulfill the relations We consider the metric of SCBH which is given by [34] where where / 0 is the asymptotic value of the dilaton field, Q and m are the charge and mass parameters respectively.
To closely approach the metric with tetrad fields, we shall select a set of quadrilateral fields, which are the fundamental field variables of TEGR, and these fields are naturally interpreted as frames of reference composed with observers in space-time [35].The tetrad field corresponding to the metric ( 7) is given by The following are the torsion tensor's non-vanishing components: The nonzero components of R abc tensor are given by Now finding total energy by substituting the above component into (4) for a ¼ ð0Þ in the form To better understand the relationship between energy and the radius of BHs in the context of a change in the values of both mass and charge of BHs, equation ( 12) was taken into consideration.From Figs. 1 and 2, we notice that the energy decreases rapidly, then quickly rises again, and then tends to almost stationary with an increase in r.We also note that as any of the charge Q or mass m of the BH increases with the stability of the other, the value of energy similarly increases.Changes in dilaton field values have a less effect on energy distribution, as seen in Fig. 3. Now, / ðiÞj components can be used to construct the radial pressure RP when j ¼ 1.Using equation ( 6) we have, Fig. 2 Energy distribution of ( 12) for various mass m values, when Q ¼ 1 and / 0 ¼ 0:1 Fig. 1 Energy distribution of ( 12) for various charge Q values, when m ¼ 1 and / 0 ¼ 0:1 Substituting the above components in the following relation Then, The RP is given by From the previous equation and by setting a ¼ 0, it is clear that this result agrees with each of the Ref. [27] and by applying the Brown-York quasi local approach [35] Figures 4 and 5 illustrate that the RP value grows rapidly at first, then at a slower rate until it approaches a virtually constant state with an increase in r.When the BHs charge rises while its mass remains constant, the RP rises as well.When the charge remains constant while the mass grows, the opposite occurs (Figs. 6 and 7).

The Gravitational Entropy of SCBH Solution
The entropy of a BH is the amount of entropy that must be assigned to a BH in order to fulfill the laws of thermodynamics, according to observers outside the BH.BH entropy is a concept that has a geometric root but has many physical consequences.The first law of TD is the conservation of energy.It simply reads where r 0 is the radius of a spherical hyper-surface threedimension of integration.
Substituting equations ( 12) and (19) in equation ( 21) and integrating, then the gravitational entropy is given by In Fig. 7, the relationship between entropy and distance from the center of the BH is shown for various of parameter a at constant value of the temperature T, the entropy increases as r increases for all a values.When a decreases, the entropy increases and the relation between a and r is close to being linear.Figures 8, 9 and 10 illustrate the effect of differing values of the mass, the charge of BH and the dilaton field / 0 , on the relationship between entropy and the r-coordinate.When all other factors are held constant, entropy increases when the mass of BH or the dilaton field increases, and decreases when BH's charge increases.

Conclusions
The first result that we obtained is the energy of SCBH represented by equation (12).This result reduces to the Schwarzschild BH energy when a ¼ 0. Figures 1, 2 and 3 show that energy decreases with increasing of the charge of BH and increases with increasing of the mass of BH and the dilaton field.Another interesting result in this study is the Radial pressure RP of SCBH as a function of the parameters alpha and BH radius r, which is represented by equation (19).Regardless of the values of charge, mass, or the dilaton field, the value of the RP declines little by little as the radius of the BH rises.But for small values of BH radius r, it becomes clear to what extent the RP is affected by changing the values of the other parameters as shown in Figs. 4, 5 and 6.
Finally, the entropy for SCBH in teleparallel gravity TG is calculated.We generated a formula for the entropy using notations as energy and pressure that can be described in TEGR.From the result obtained in equation ( 22), we observed that entropy is affected by various of parameters, including the BHs mass and charge, as well as the radius of the hyper-surface of integration, r 0 .The entropy of BH decreases as its mass increases, and vice versa if the BH's charge increases, as shown in Figs. 8 and 9. Figure 10 illustrates that the dilaton field has a smaller effect on entropy.As shown in Fig. 7, the lower the a parameter, the closer the relationship between entropy and radius goes to   (22) for various values of the dilaton field / 0 , when Q ¼ 1; m ¼ 1 be a linear relationship.Our results in this paper agree with the results in [27,28].
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).
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