Corotating binary systems of identical Kerr-Newman black holes

In the present paper binary configurations of identical corotating Kerr-Newman black holes separated by a massless strut are derived and studied. After solving the axis conditions and establishing the absence of magnetic charges in the solution, one gets two 4-parametric corotating binary black hole models endowed with electric charge, where each source contains equal/opposite electric charge in the first/second configuration. Since the black hole horizons are given by concise expressions in terms of physical parameters, all their thermodynamical properties satisfying the Smarr relation for the mass are also obtained. We discuss the physical limits of both models.

Since the appearance of the electromagnetic field Φ increases enormously the complexity of finding exact solutions in Eq. (2), one must resort to applying soliton techniques like Sibgatullin's method (SM) [12] which provides the full metric in the entire spacetime once we pick a particular form of the axis data. For such a case, as a starting point to develop the SM with the main objective to treat binary configurations of massive rotating charged sources, we adopt the following suitable form of the Ernst potentials on the symmetry axis: In order to gain more insight on the spacetime structure for this particular choice of the axis data Eq. (4), the first Simon multipolar moments [13] can be computed by means of the Hoenselaers-Perjés procedure [14,15], where 2M and 2(Q + iB) represent the total mass and total electromagnetic charge of the system, respectively. In addition, the electric/magnetic dipole moment Q o /B o is given by where J 0 is the NUT singularity [16] expressed as Finally it is not difficult to show that the total angular momentum J T of the system is expressed in the simple form The sources are two thin rods separated by a coordinate distance R and their location on the symmetry axis depends on the values α 1 = −α 4 = R/2 + σ, α 2 = −α 3 = R/2 − σ, where the condition σ 2 ≥ 0 defines BHs whereas σ 2 < 0 might be useful to describe hyperextreme sources (relativistic disks or naked singularities) as shown in Fig. 1. As it is well-known, an asymptotically flat spacetime might be considered after killing the NUT charge J 0 , where in this case such a condition is satisfied by means of where we have first eliminated the total magnetic charge from the binary system; i.e., B = 0. At this point, it is worth noting that each source contains identical magnetic charge with opposite sign. Following the approach of Ref. [17], after the application of straight but non-trivial calculations, eventually one gets the following representation for the Ernst potentials and metric functions where Eq. (9) is depicted by a total of seven parameters {M, Q, q, q o , b o , δ, R}. Notice that Eq. (9) also shows the Kinnersley potential Φ 2 [18] in order to get directly the magnetic potential A 3 through

III. COROTATING KERR-NEWMAN BINARY BHS
The above solution Eq. (9) cannot be considered to describe a pair of BHs unless we have been able to solve the axis condition in the middle region among the sources, namely, where it ensures that both BHs will be apart by a massless strut. The substitution of Eq. (8) into Eq. (11) will leads us to a quadratic equation for any of the variables q o , b o or δ, namely and since this algebraic equation can be readily solved, it is possible to assume for one instant that we know explicitly its solution, thus having two corotating charged sources separated by a massless strut [2]. Later on, we will combine Eq. (12) together with the condition that annihilates to each individual magnetic charge with the aim to define corotating Kerr-Newman binary BHs. On the other hand, in stationary axisymmetric spacetimes Tomimatsu formulas [19] provide us an easy way to calculate straightforwardly Komar conserved quantities in a two-body system once we know a specific metric. Nevertheless, Clément and Gal'tsov [20] recently have shown that Tomimatsu formulas are not correct in the presence of magnetic charges, because the mass of the horizon M H suffers contributions coming from a Dirac string joined to the BHs; i.e., each BH is carrying a magnetic flux. Therefore, the corrected Tomimatsu formulas acquire the form [19,20] where Ψ = Im(E) andÃ 3 = A 3 + ωA 4 . Also, ω H is the value of the metric function ω over the horizon, while M S A is an extra term related to the presence of the Dirac string, which is given by For such a case, the Smarr formula [21] for each BH still holds and reads [20] where κ and S are the surface gravity and the area of the horizon, respectively; both are related to σ. Furthermore, Ω = 1/ω H is the angular velocity and Φ H E = −A H 4 − ΩA H 3 defines the electric potential in the corotating frame of the BH. The aforementioned integrals Eqs. (13) are evaluated on the corresponding region that each rod represents the BH horizon [see Fig. 1(a)]. In this case, one may consider the values of the upper BH horizon: R/2 − σ ≤ z ≤ R/2 + σ, ρ → 0, and 0 ≤ ϕ ≤ 2π, due that the length of both sources are equal. Replacing Eq. (9) into Eq. (13), it can be proven that the mass M H and electromagnetic charge Q H + iB H for the upper BH assume the form It is important to note that the individual masses and electric charges will not be necessarily half of 2M and 2Q, respectively, unless the magnetic charges vanish. Moreover, the lower BH will contain a mass 2M − M H and electromagnetic charge 2Q − Q H − iB H . In the following, we are going to consider two electrically charged models where an absence of magnetic charges is taken into account.

A. Identically charged Kerr-Newman BHs
The first case that is considered here emerges if q o = 0 and Q = Q H , in the formulas of the current section. We find from Eqs. (12) and (16) that an absence of magnetic charges (B H = 0) is achieved when and therefore M H = M since the term M S A vanishes. For such a situation, the expression for σ in the identically charged case is obtainable from Eq. (8); it reads whereas the angular momentum of each BH is given by Regarding now the thermodynamical properties contained within the Smarr formula Eq. (15) we have that Ω and Φ H and the area of the horizon S as well as the surface gravity κ may be obtained via the formulas [19,22] where γ H defines the value that metric function γ takes at the horizon. A straightforward calculation yields us to On the other hand, in order to interpret the pressure exerted on each source, it is necessary to compute the interaction force associated to the strut, which can be obtained in a simple manner with the formula F = (e −γs − 1)/4 [4,23], where γ s denotes now the constant value that metric function γ acquires on the middle region among the sources, thus getting Contrary to the vacuum scenario [7], Eq. (19) cannot be solved analytically because it is leading us to a quintic algebraic equation in the variable q. So, whenever the numerical analysis should be performed, it is necessary to bear in mind, the physical limits of the solution. In this sense, the minimal distance at which the BH horizons are touching each other (the merger limit) is given by R min = 2 M 2 − Q 2 H − q 2 , while the force F → ∞. After taking into account this distance value, we notice from Eqs. (18) and (19) that σ = M 2 − Q 2 H − (J/2M ) 2 and q = J/2M , respectively, having the following result which is leading us to very simple expressions for Ω, Φ H E , κ, and S: where we have employed the fact that the parameters b o and δ are equal to zero during the merger limit. Another physical limit to be considered is when R → ∞, representing the physical case in which the sources move far away from each other and the interaction force F → 0. It is quite easy to show from Eqs.
where one confirms that the size of the BH horizon 2σ during the merger limit is bigger in comparison to the isolated case, but its thermodynamical properties decrease their corresponding values. This statement is in agreement with the Smarr formula. At large distances the interaction force acquires the approximate value

B. Oppositely charged Kerr-Newman BHs
The second electrically charged model comes to light immediately when the total electric charge is eliminated from the binary system by doing now Q = 0. Then, the magnetic charges can be removed from the solution only if b o = 0 and where Eqs. (12) and (16) are identically fulfilled with these expressions. Notice that once again M H = M and M S A = 0. In the oppositely charged case σ takes the form and the angular momentum is expressed by means of another quintic algebraic equation It follows that we have obtained a corotating black dihole model which in the lack of rotation (q = 0) specifies a non-extreme Emparan's dihole [25]. With regard to the thermodynamical characteristics we have now that Ω, Φ H E , κ, and S are simplified as while the formula of the force reads Similar to the situation with identical electric charges, it follows that the merger limit now results to be where q = J/2M . In this limit value of the distance the properties on the horizon are written down as follows from which it is shown that the electric potential vanishes, and therefore, the event horizon 2σ contains the same length as in the vacuum case [6,7]. Furthermore, in the other limit R → ∞, we have that q = J/M and the electric dipole behaves as q o ∼ Q H R/2, where it is possible to recover the description of one isolated Kerr-Newman BH when deriving exactly the same formulas described above in Eq. (26). Finally, the force at infinite separation distance contains the next behavior The physical values for the variable q earlier discussed for the non-extreme case are shown in Fig. 2.

C. The extreme case of BHs
The extreme limit of Eq. (9) can be achieved once is established σ = 0, where such a metric may be expressed in a quite simple form by using the Perjes factor structure [26], thus having where (x, y) are prolate spheroidal coordinate denoted as Therefore, this metric allows to treat identically charged extreme Kerr-Newman BHs once is established first q o = 0, and Q = Q H on it. The explicit values of the angular momentum will depend on the parameter q after solving Eq. Let us now consider the extreme case of BHs during the merger limit, with the goal to derive non-trivial expressions for the force and area of the horizon at this particular distance value. We notice from Eq. (24) that the minimal distance is given by R min = 0 from which it is possible to get where q = M 2 − Q 2 H . For such a situation, the merger process will start to form a single extreme Kerr-Newman BH of total mass M T = 2M , total electric charge Q T = 2Q H , and total angular momentum J T = 2J, fulfilling a well-known relation, namely So, in order to compute the final values that the force and horizon area acquire in the physical scenario where both extreme BHs touch to each other (but they do not merge yet!), we are going to establish initially σ = 0 in Eq. (18), and later on, we will apply Taylor's expansion around R = 0 by using q = M 2 − Q 2 H + C 0 R, with the aim to calculate a first order contribution in the whole set of physical and thermodynamical properties. With this procedure, the result turns out to be whereas the force and area of the horizon are given explicitly by The angular velocity and electric potential are obtained from Eq. (25) by doing simply σ = 0 and these properties do not require to much special attention from us. It should be pointed out that there exist two states during the merger limit depending on the sign of ε. In the first/second case; i.e., when ε = +1/ − 1, the force is positive/negative and therefore attractive/repulsive, where the area of the horizon is smaller in the attractive case compared to the repulsive case. In addition, in the absence of electric charge Q H = 0 one gets the following formulas where it can be seen that only the attractive case has been considered earlier in [9]. [28] When is now placed Q = 0 and b o = 0 in Eq. (36) one may treat oppositely charged extreme Kerr-Newman BHs. In the same manner as in the identically charged situation, the angular momentum adopts the values from the parameter q that fulfill the condition σ = 0 [see Figs. 3 and 4]. Moreover, in the merger limit of extreme oppositely charged BHs, the same limiting procedure described before might be also included here. In this respect, we have from Eq. (33) that R min = 0 is valid only if the angular momentum and mass are related by means of J = 2M 2 , while now q = M . After carrying out once again Taylor's expansion around R = 0, eventually the interaction force and area of the horizon are expressed as [28] By applying the same limiting procedure in the unequal vacuum case [5,24], one might be able to obtain the non-identical version of Eq. (42), namely where M 1 = M 2 = M , recovers it.
where according to the sign taken by ε there exist two states that also behave exactly in the same manner as in the identically charged case. These expressions are reduced to the aforementioned Eq. (42) if the electric charges are not present.

IV. CONCLUSION
In this paper, we have worked out a physical metric that permits us the study of identical corotating Kerr-Newman binary BH models in which the sources contain equal (or opposite) electric charges. These clearly extend the earlier results provided in Refs. [6,7]. In order to derive this 4-parametric physical models, the axis condition in between sources has been combined with the one eliminating the magnetic charges; therefore, the sources are two Kerr-Newman BHs supported by a conical singularity. Our suitable parametrization allowed us to get concise formulas for the physical and thermodynamical features of the BHs, if after getting first the half-length parameter σ which represents the BH horizon in cylindrical coordinates. On one hand, in both models the horizon mass M H = M due to the fact that magnetic charges are not involved. On the other hand, it is quite clear that M H = M if there exists a Dirac string joined to the BHs [20,27]. The contribution of the string mass M S A clearly deserves further research. In the extreme limit of BHs we have introduced a binary metric with a quite simple aspect. Remarkably, our physical treatment led us to derive in both models two final states during the merger limit, in which the force is attractive/repulsive while the horizon area is smaller in the attractive case in contrast to the repulsive scenario. It is presumable to assume that such an extreme metric will be helpful to extend the previous results included in Ref. [9] on a near extreme binary BH geometry, with a more physical aspect. To conclude, we would like to mention that the most satisfactory exact solution as describing unequal configurations is extremely complicated, but we do not exclude that after some efforts this problem might be solved by following the approach considered within this work.