Higher Dimensional Charged Black Hole Solutions in Gravitational Theories
We present, without any assumption, a class of electric and magnetic flat horizon -dimension solutions for a specific class of , all of which behave asymptotically as Anti-de-Sitter spacetime. The most interesting property of these solutions is that the higher dimensions black holes, , always have constant electric and magnetic charges in contrast to what is known in the literature. For , we show that the magnetic field participates in the metric on equal foot as the electric field participates. Another interesting result is the fact that the Cauchy horizon is not identical with the event horizon. We use Komar formula to calculate the conserved quantities. We study the singularities and calculate the Hawking temperature and entropy and show that the first law of thermodynamics is always satisfied.
The most effective gravitational theory in the last century is the theory of general relativity (GR). This theory is a fully accepted one that depicts the macroscopic geometrical properties of spacetime. Using an isotropic and homogeneous symmetry, the field equations of GR give Friedmann equations which depict the evolution of the universe with radiation and then matter dominated epochs. However, recent observations indicate that our universe goes through a phase of accelerated expansion [1–3]. This fact cannot be explained in the frame of GR using ordinary matter as a source. Another issue that GR cannot explain is the cosmological era which is known as inflation . This phase of the universe is believed to have occurred before the radiation era which could relax some issues of cosmology like horizon, flatness singularities, and so on . Moreover, using baryonic matter, GR is not able to discuss the observed density limited by the fitting of the standard cold dark matter (CDM) of the Wilkinson Microwave Anisotropy Probe results for 7 years of observations data (WMAP7) , the recent measurements of Baryon Acoustic Oscillations (BAO) , and the Hubble constant . Therefore, GR needs to impose an extra component known as the dark matter (DM) which constitutes about 23% of the energy content of the universe . In spite of the fact that there are many possible roots of such component [9–18], DM is assumed in a form of thermal relics which naturally freeze-out with the right abundance in the extensions of the standard model of particles [19–31]. Coming experiments enable us to distinguish between large number of candidates and model by direct and indirect detection prepared for their search [32–34], or even at large hadron collider (LHC) where they could be produced [35–42].
Another puzzling issue is the one of the accelerated expansion of our universe. Many explanations have been setup to demonstrate such phenomena. Among these explanations is the one which assumes the validity of GR and suggests the presence of extra fluid called dark energy (DE). The equation of state of DE takes the form (where , , and are pressure, density, and a dimensionless parameter, resp.) with to create an accelerated cosmic expansion era [43–45]. There is another model which could explain the DE which includes the cosmological constant in the field equations of GR and assumes the equation of state to have the form . However, such model suffers from the discrepancy which comes from the fact that if we postulate the cosmological constant to represent the quantum vacuum energy then its value has higher orders of magnitude than those of observations . It has been shown that, in the Palatini formalism of , and are dimension parameters, the term cannot lead to an early time inflation .
To overcome the problem of acceleration a modification of GR has been considered by modifying Einstein-Hilbert action [48–62]. Some examples of these modifications are as follows:(i)Brans-Dicke theory whose interaction is considered by GR tensor and scalar field (ii)String theory which include Gauss-Bonnet term (iii)Lovelock theories which are of second derivatives at most .(iv)The gravitational theory whose field equations are of second order [66–80](v)The one known as theories which we focused the present study on 
It is shown that gravitational theories are able to describe the whole story of cosmology starting from inflation to the present accelerated expansion epoch . Many applications of gravitational theories have been carried out [83, 84]. Also local tests on have been achieved to constrain theories [85–88]. To study modified theories of gravity one requires to assure or reject their validity by deriving solutions that could investigate the evolution of the universe [89, 90] and the occurrence of GR astrophysical prediction. There are many BH derived in the frame of by assuming a constant scalar curvature, [91–93]. It is the aim of the present study to abandon this condition in a class of and try to derive -dimension solutions for a flat horizon metric spacetime.
The arrangements of this study are as follows: In Section 2, gradients elements of Maxwell- gravitational theory are presented. In Section 3, a metric spacetime with one unknown and two unknown functions is presented and applied to the charged field equation of . Exact classes of charged black holes are derived in Section 3. In Section 4, the relevant physics of these classes is discussed by calculating the singularities. In Section 5, we calculate the conserved charges related to each class by using Komar method. In Section 6, we calculate the thermodynamical quantities like Hawking temperature and entropy, and so on. Also in this section, we have shown that the first law of thermodynamics is always satisfied for all the solutions derived in this study. The main results are discussed in the final section.
2. Fundamentals of Maxwell- Gravitational Theories
The Lagrangian of theory has the form with representing the gravitational Lagrangian given by with being the cosmological constant and being the -dimensional gravitational constant defined as , where is the Newton’s constant in -dimensions (the units for the -dimensional gravitational constant are , where is the gravitational constant in 4-spacetime dimensions and is a unit of length). Here refers to the volume of -dimensional unit sphere that is defined aswhere is the gamma function of the argument (for , one gets ), is the Ricci scalar of the spacetime, is the determinant of the metric, and is the analytic function of the considered theory. In this study refers to the action of Maxwell field that is defined as with being the 1-form electromagnetic potential .
By carrying out variations of (2) with respect to the metric tensor and the vector potential one can obtain the following field equations of gravitational theory [95, 96]: with being the Ricci tensor defined bywhere is the Christoffel symbols second kind and the square brackets mean The D’Alembert operator is defined as , where is the covariant derivatives of the vector and . In this study is defined as which is the energy momentum tensor of the electromagnetic field.
The trace of (5) yields Now we are going to discuss some basic property of the above theories.
2.1. -Gravitational Theories and Their Viable Conditions
The most important conditions and restrictions  that are usually put on gravitational theories to give consistency on both of gravitational and cosmological evolutions are as follows.
(a) The first condition is given by which represents the stability condition for curvature . Condition (11) represents the existence of a matter dominated era in cosmological evolution. The relevant physics of (11) is that if the constant, has a defined value, then is fixed by the sign of .
(c) The condition ensures the recovery of GR behavior at early times.
3. Analytic Solution in
In this section, we are going to apply the field equations of Maxwell- with cosmological constant to two different metric spacetimes having flat horizon:
3.1. Flat Horizon Metric with One Variable
Let us assume the first metric spacetime possessing one unknown function has the form where is an unknown function of the radial coordinate, . Using (13) we get the Ricci scalar in the formwhere and . The nonvanishing components of the Maxwell- field equations, (5) and (6), when take the form (the detailed calculations of the Ricci curvature tensor are given in Appendix B)where and with and being two unknown functions related to the electric and magnetic charges of the system and defined from the gauge potential asThe solution of the differential equations (15) has the formwhere and , are constants. Solution (17) is decomposed into two parts one for and the other for . The reason for this decomposition is the electromagnetic field. In the four dimensions there is a charged solution; however, for there is no charged solution. In fact this is in conflict with the spherically symmetric case . It is important to mention here that solution (17) when will reduce to the well known AdS/dS solution in the case of . However, in the case of this solution does not allow the parameter to be vanishing; therefore, it has no analogue in GR. In the noncharged case and when solution (17) coincides with what is know in GR. However, when and as long as and have no relation between them solution (17) has no analogue in GR but when it will coincide with what is known in GR; that is, it behaves as Ads/dS.
The metric of solutions (17) has the form
3.2. Metric with Two Variables
The metric of a flat horizon with two unknown functions has the form where and are two unknown functions of the radial coordinate, . Using (19) we get the Ricci scalar in the form Using (19) we get the nonvanishing components of the field equations, (5) and (6), when in the form The above system of differential equations (21) has the following solution:where . We must mention here that solution (22) has the same property of solution (17); that is, when it will reduce to the well known AdS/dS solution in the case of . However, in the case of this solution does not allow the parameter to be vanishing; therefore, this solution has no analogue in GR. In the noncharged case and when and solution (22) coincides with what is known in GR. However, when and as long as and have no relation between them solution (22) has no analogue in GR but when it will coincide with GR solution that behaves as Ads/dS. The metric spacetimes of solutions (22) have the formwhere , are constants.
4. Physical Properties of the Analytic Solutions
4.1. Metric with One Variable
From (18) we can deduce the following properties.
(i) In case of 4 dimensions we get from which it is clear that the metric behaves asymptotically as dS/AdS. Equation (24) shows that the effect of the higher dimension curvature is related to the electric field as well as the magnetic field and also (ii) In case of more than 4 dimensions we getEquation (26) shows that the dimensional parameter must not be equal to zero; otherwise, we will have a singular metric. Also the spacetime of metric (26) behaves as dS/AdS and when the cosmological constant takes the form then, (26) reduces to which is asymptotically dS/AdS and cannot reduce to GR.
4.2. Metric with Two Variables
From (23), we can show the following.
(iii) In case of 4 dimensions we getFirst set of equations (29) shows that and when the constant then the metric behaves as dS/AdS. For the second set, it is allowed to put and when the constant then the second equation of (29) behaves as dS/AdS.
(iv) In case of more than 4 dimensions we get from (23) the two setsEquation (30) shows that the dimensional parameter must not be equal to zero; otherwise, we will have a singular metric. The asymptotes of (30) behave as dS/AdS. It is important to stress that metric (30) cannot reduce to that of GR and hence we can say that solution (22) is a new solution. Using condition given by (27) in (30) we get We can conclude from the above discussion of the metrics given by (30) or (31) that there is no charged solution for the form of . Also spacetime metrics of (30) or (31) instruct us that the dimension parameter must not be equal zero.
Now, let us explain the singularities and the horizons of solutions (17) and (22). For this reason, we have to find at which values of do the functions and turn out to be zero or infinity, due to the fact that singularities may be coordinate ones which are physical singularities. Usually, to study singularities one calculates all the invariants constructed from Riemann tensor and its contractions. The curvature invariants that arise from solution (17), in case of 4 dimensions, have the formAnd in case of more than 4 dimensions we getEquation (32), for the 4 dimensions case, shows the following.
(a) ; otherwise, we will have a singularity for both invariants and .
(b) Also (32) tells us that there is a singularity at which can not be removed for the invariants and .
When and by using (17), we can show the following.
(c) We have a true singularity at .
Repeating the same calculations for solution (22) we get for We can apply the same discussion applied for solution (17). In case of we getwhere , , and are lengthy functions of . Using (27) in (37) we getwhich indicates that the parameter must not be equal to zero.
The horizons of solutions (17) and (22) are the zeros of the metric ; therefore, all the above singularities are far from these horizons. The study of perturbations of solutions (17) and (22) is an important issue to study their stability and then discuss the formulation of weak cosmic censorship. This problem will be discussed elsewhere.
5. Total Conserved Charge
In this section, we are going to study the conserved quantities of the solutions derived in Section 3. For this purpose we are going to make a brief review of the geometry used in this calculations. The Lagrangian of Einstein-Cartan theory is defined by  (the fundamental entities of this theory Appendix A) where is the one-form coframe and is the two-form curvature tensor. Carrying out variation of (39) with respect to the coframes and lead to [100, 101] which are the canonical energy-momentum and rotational gauge field momentum, respectively. The translational momentum and the spin 2-forms are defined as The conserved quantity of the gravitational field has the form  with being the Hodge duality, being a vector field , and being parameters , . For the solutions having spinless matter or vacuum ones, the torsion is vanishing, , and therefore the total charge of (42) takes the form This invariant conserved quantity was given before in [102–106].
Using (44) in (42) we get The total derivative of (45) givesCalculating the inverse of (46) and using it in (46) and (43) after applying the Hodge-dual to the output of , we get conservation of the charge in the form where is the volume of the unit ()-sphere; we have substitute and have used the regularization method, by putting the physical quantities vanishing, and subtract the resulting expression from the original one to remove any divergence term  (physical quantities mean the constants , , , and for solution (17); constants , , , and of first solution of (22); and constants 13, 15, 18, and of the second solution of (22)).
Applying the same above techniques we get for the first solution of (22) in case of where and for the second one we get where .
In case we get where
6. Thermodynamics of Black Holes
In this section, we are going to study the thermodynamical quantities of solutions (17) and (22). The temperature of Hawking of any solution can be derived by requiring the singularity at the horizon to be vanishing in the Euclidean continuation of the black hole solutions. One can obtain the temperature of the outer event horizon at , for solution (17) in case of in the form and in case of we get
For the first solution of (22) in case , we get the Hawking temperature as and for the second solution of (22) in case of , we get In the case of we get We give a brief discussion of the entropy of black hole in gravity. For this purpose, we use the arguments presented in . From the Noether method used to calculate the entropy associated with black holes in theory that have constant Ricci scalar, one finds  where is the area of the event horizon. Using solution (17) we get for and for we get Using (17) and (27) in (59) we get the entropy in the form
Utilizing (52) and (58), the Smarr relation in the extended phase space can be obtained in the case of solution (17) as For the first solution of (22) we get the the Smarr relation in the extended phase as where we have put . Finally, for the second solution of (22) we get the Smarr relation as where we have put .
In the case , we introduce the extended phase space where the cosmological constant is identified as the thermodynamic pressure while the conjugate quantity is regarded as the thermodynamic volume. We adopt the following definition of pressure that is commonly used in the literatures of extended phase space