Opposite charged two-body system of identical counter-rotating black holes
I. Cabrera-Munguia, Claus Lämmerzahl, L. A. López, Alfredo Macías
aa r X i v : . [ g r- q c ] M a y Opposite charged two-body system of identical counter-rotating black holes
I. Cabrera-Munguia , ∗ , Claus L¨ammerzahl , † , L. A. L´opez , ‡ and Alfredo Mac´ıas , § Departamento de F´ısica, Universidad Aut´onoma Metropolitana-Iztapalapa A.P. 55-534, M´exico D.F. 09340, M´exico ZARM, Universit¨at Bremen, Am Fallturm, D-28359 Bremen, Germany ´Area Acad´emica de Matem´aticas y F´ısica., UAEH,carretera Pachuca-Tulancingo km 4.5, C.P. 42184, Pachuca, Hidalgo, M´exico A 4-parametric exact solution describing a two-body system of identical Kerr-Newman counter-rotating black holes endowed with opposite electric/magnetic charges is presented. The axis condi-tions are solved in order to really describe two black holes separated by a massless strut. Moreover,the explicit form of the horizon half length parameter σ in terms of physical Komar parameters, i.e.,Komar’s mass M , electric charge Q E , angular momentum J , and a coordinate distance R is derived.Additionally, magnetic charges Q B arise from the rotation of electrically charged black holes. As aconsequence, in order to account for the contribution to the mass of the magnetic charge, the usualSmarr mass formula should be generalized, as it is proposed by A. Tomimatsu, Prog. Theor. Phys. , 73 (1984). PACS numbers: 04.20.Jb,04.70.Bw,97.60.Lf
I. INTRODUCTION
Binary black hole systems in equilibrium, without asupport strut in between, have been extensively studiedin vacuum since the famous double-Kerr-NUT solutionwas presented by Kramer et al. in 1980 [1]. These type ofsolutions are nonregular [2], due to the fact that at leastone of the Komar masses results to be negative [3, 4], ap-pearing ring singularities off the axis. On the other hand,the electrovacuum sector has received less attention be-cause the electromagnetic field increases considerably thedifficulty of finding exact solutions in these type of two-body systems.A binary system of identical Kerr-Newman (KN)sources separated by a massless strut (conical singular-ity) [5] in between has been recently studied by Manko et al. [6]. The strut prevents the sources from fallingonto each other and provides an interaction force which,nevertheless in this case, does not contain any spin-spininteraction. Furthermore, the equilibrium condition ofthe two-body system is reached after removing the strutand it reveals that the system is composed by identicalcounter-rotating relativistic disks, lying on the equatorialplane, whose individual electric charges, equal to their re-spective masses, result to have the same sign [7, 8]. Allof these aforementioned two-body systems do not con-tain individual magnetic charges; hence, they fulfill thestandard Smarr formula for the mass [9].Additionally, following the ideas of Varzugin [10], wesolved, in Ref. [11] the axis conditions in order to definea 4-parametric asymptotically flat exact solution, whichdescribes two unequal counter-rotating black holes sepa- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] rated by a massless strut. We established a straightfor-ward procedure to obtain explicitly the functional formof the horizon of length 2 σ in terms of Komar physicalparameters [12]. It determines the structure of the wholespacetime and its geometrical properties in a more phys-ical way.The main purpose of the present paper is to solve theaxis conditions in order to describe a binary system oftwo identical counter-rotating black holes endowed withopposite electric/magnetic charges. We will show thatmagnetic charges arise as a result of the rotation of elec-trically charged black holes. In this description, in orderto account for the contribution of the magnetic charge Q B to the mass, the Smarr mass formula [9] is gener-alized; it becomes a cubic equation. This modificationis already proposed by Tomimatsu [13], so that we onlyprovide its physical form.The interaction force related with the strut containsnow, due to the rotation, a spin-spin interaction. Differ-ent limits of our solution are also discussed. Since theidentical KN black holes are counter-rotating and haveopposite electric charges, the full metric exhibits an equa-torial antisymmetry property in the sense proposed byErnst et al. in [14]. The upper black hole is character-ized by having J > Q E <
0, while the lower onehas
J < Q E > σ in terms of Komar physi-cal parameters, are derived. In Sec. IV the full metricfor the extreme limit case is obtained. The concludingremarks are presented in Sec. V. II. OPPOSITE CHARGED TWO-BODYSYSTEM OF IDENTICAL COUNTER-ROTATINGBLACK HOLES
According to Ernst’s formalism [15] the Einstein-Maxwell equations describing the stationary axisymmet-ric electrovacuum spacetimes can be reduced to the fol-lowing system of equations: (cid:0) Re E + | Φ | (cid:1) ∆ E = ( ∇ E + 2 ¯Φ ∇ Φ) ∇ E , (cid:0) Re E + | Φ | (cid:1) ∆Φ = ( ∇ E + 2 ¯Φ ∇ Φ) ∇ Φ , (1)where ∇ and ∆ are the gradient and Laplace operators defined in Weyl-Papapetrou cylindrical coordinates ( ρ, z )and acting over the complex potentials E = f − | Φ | + i Ψand Φ = − A + iA ′ . Here, A is the electric poten-tial and A ′ is associated with the magnetic potential A , both components of the electromagnetic 4-potential A i = (0 , , A , A ). Any solution of Eq.(1) determinesthe metric functions γ and ω of the line element [16] ds = f − (cid:2) e γ ( dρ + dz ) + ρ dϕ (cid:3) − f ( dt − ωdϕ ) , (2)by means of the following set of differential equations:4 γ ρ = ρf − (cid:2) |E ρ + 2 ¯ΦΦ ρ | − |E z + 2 ¯ΦΦ z | (cid:3) − ρf − ( | Φ ρ | − | Φ z | ) , γ z = ρf − Re (cid:2) ( E ρ + 2 ¯ΦΦ ρ )( ¯ E z + 2 ¯ΦΦ z ) (cid:3) − ρf − Re( ¯Φ ρ Φ z ) ,ω ρ = − ρf − Im( E z + 2Φ ¯Φ z ) , ω z = ρf − Im( E ρ + 2Φ ¯Φ ρ ) , (3)where the subindices ρ and z denote partial differentiation, the bar over a symbol represents complex conjugationand | x | = x ¯ x . An electrovacuum exact solution of Eq.(1), describing a binary system composed by KN sources, canbe obtained with the aid of the Sibgatullin Method (SM) [17, 18]. Following this approach, the Ernst potentials E , Φand the full metric read [18] E = E + E − , Φ = FE − , f = D | E − | , ω = 2Im (cid:2) E − ( ¯ G o + ¯ H o ) − F ¯ I (cid:3) D , e γ = D | a + | Q n =1 r n ,D = E + ¯ E − + ¯ E + E − + 2 F ¯ F , E ± = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ± ± C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , F = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( α ) f ( α ) f ( α ) f ( α )11 C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,G o = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p p p C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , H o = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − β − β C ¯ e ¯ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f + f f ( α ) f ( α ) f ( α ) f ( α ) z − β − − β − e C ¯ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,C = γ r γ r γ r γ r γ r γ r γ r γ r M M M M M M M M , a + = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ γ γ γ γ γ γ γ M M M M M M M M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , p n = 2 z − α n − r n ,M jn = (cid:2) ¯ e j + 2 ¯ f j f ( α n ) (cid:3) ( α n − ¯ β j ) − , f ( α n ) = X j =1 f j γ jn , γ jn = ( α n − β j ) − , r n = p ρ + ( z − α n ) . (4)where the parameters e j are functions of α n , f j , and β j , they read: e = 2 Q n =1 ( β − α n )( β − β )( β − ¯ β )( β − ¯ β ) − X k =1 f ¯ f k β − ¯ β k , e = 2 Q n =1 ( β − α n )( β − β )( β − ¯ β )( β − ¯ β ) − X k =1 f ¯ f k β − ¯ β k . (5)Equation (4) contains a set of twelve algebraic param-eters { α n , f j , β j } , where the real or complex values of α n define subextreme objects (black holes) or hyperex-treme objects (relativistic disks). It is important to notethat the metric (4) is not asymptotically flat at spatialinfinity, since NUT sources [19] as well as the total mag-netic charge are present. Therefore, in order to get rid ofsuch monopolar terms, which break the asymptotic flat-ness of the solution, it is necessary to impose and solvethe corresponding conditions on the symmetry axis (axisconditions). By construction, Eq.(4) satisfies an elemen-tary flatness condition on the upper part of the symmetryaxis: ω ( α < z < ∞ ) = 0 and γ ( α < z < ∞ ) = 0. Be-sides, the metric function γ ensures the fulfillment of thebalance condition on the lower part of the symmetry axis: γ ( −∞ < z < α ) = 0. The remaining conditions on thesymmetry axis read ω ( ρ = 0 , α < z < α ) = 0 ,ω ( ρ = 0 , −∞ < z < α ) = 0 . (6)A straightforward simplification leads us to the follow-ing algebraic system of equations:Im[¯ a − ( g − + h − )] = 0 , Im[¯ a + ( g + + h + )] = 0 , g ± = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ± ±
111 ( a ± )00 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , h ± = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ± )¯ e ¯ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , a ± = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ± γ ± γ γ γ ± γ ± γ γ γ M M M M M M M M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7)The total mass M , total electric charge Q , and totalmagnetic charge B of our binary system can be calculatedasymptotically from the Ernst potentials on the symme-try axis, which lead toRe[ e + e ] = − M , f + f = Q + i B . (8)Replacing Eq.(5) into the first equation of (8) yieldsthe relation β + β + ¯ β + ¯ β = − M . (9)By choosing β + β = −M := − M , Q := 0, and B := 0, we are describing a system of two identicalcounter-rotating KN black holes of mass M , endowedwith opposite electric/magnetic charge Q E /Q B with a well-known conical line singularity [5] in between. Thecorresponding horizons of the black holes are defined bythe real values of the constant parameters α n , they fulfillthe relation α + α = α + α = 0, as shown in Fig. 1.The parameters α n can be written in terms of the rela-tive coordinate distance R and the half length σ of eachrod describing the black holes; they read α = − α = R σ, α = − α = R − σ. (10)An explicit solution to the algebraic equations (7) isgiven by f , = ∓ q o √ p + iδ , β , = − M ± p p + iδ,p = R / − M + σ ,δ = p ( R − M )( M − σ − µQ o ) ,q o := Q o ( R/ − M ) , µ := R − MR + 2 M . (11)where the constant parameter Q o is the value of the elec-tric charge Q E in absence of magnetic charge Q B . Itis important to note the fact that M , J , − Q E are theparameters characterizing the upper constituent of thesystem, while M , − J , + Q E characterize the lower partof the system. The black holes are separated by a co-ordinate distance R . In order to guarantee equatorialantisymmetry of the exact solution [14], the electric andmagnetic charges of the constituents should have oppo-site sign. By using Eq.(11), one is able to prove thatEq.(4) reduce to R z a a a a FIG. 1: Two identical KN black holes on the symmetry axis,with the values α = − α = R/ σ , α = − α = R/ − σ ,and R > σ . E = Λ − ΓΛ + Γ , Φ = χ Λ + Γ , f = | Λ | − | Γ | + | χ | | Λ + Γ | , ω = Im (cid:2) (Λ + Γ) ¯ G − χ ¯ I (cid:3) | Λ | − | Γ | + | χ | , e γ = | Λ | − | Γ | + | χ | κ o r r r r , Λ = 4 σ ( κ + + 2 q o )( r − r )( r − r ) + R ( κ − − q o )( r − r )( r − r )+ 2 σR ( R − σ ) [ σR ( r r + r r ) + iδ ( r r − r r )] , Γ = 2
M σR ( R − σ )[ σR ( r + r + r + r ) − (2 M − iδ )( r − r − r + r )] ,χ = 4 q o σR [( R − σ )( ǫ + + 4 M )( r − r ) + ( R + 2 σ )( ǫ − − M )( r − r )] , G = − z Γ + 2 σR [4 σκ + ( r r − r r ) + 2 Rκ − ( r r − r r ) − M ( R − σ ) ν + ( r − r ) − M ( R + 2 σ ) ν − ( r − r )] , I = 4 M q o [2 σ ( R − M − iδ )( r r + r r ) + R (2 M − σ + iδ )( r r + r r )] − q o ( R − σ ) × (cid:8) M (cid:2) ( ǫ + + 4 M ) r r − ( ǫ − − M ) r r (cid:3) + σR (cid:2) ( ǫ + + 8 M )( r + r ) + ( ǫ − − M )( r + r ) + 8 σM R (cid:3)(cid:9) ,κ o := 4 σ R ( R − σ ) , κ ± := M ( R − σ ) ± q o , ν ± := ǫ ± ( R ± σ ) ± q o , ǫ ± := σR ∓ (2 M − iδ ) , (12)where r n have the following reparametrized form: r , = q ρ + ( z − R/ ∓ σ ) ,r , = q ρ + ( z + R/ ∓ σ ) . (13)The corresponding Ernst potentials on the symmetryaxis are given by e ( z ) = e + e − , f ( z ) = ∓ | q o | e − ,e ± = z ∓ M z + 2 M − R / − σ − iδ. (14)The above solution [Eq.(12)] possesses the equatorialantisymmetry property in the sense of [14], according towhich after making the change z → − z , the metric func-tion ω changes its global sign and the Ernst potentialson the symmetry axis satisfy the relations e ( z ) e ( − z ) = 1and f ( z ) = ∓ f ( − z ) e ( z ). From Eq.(12), Q o = 0 defines atwo-body system of equal counter-rotating black holes[10]. It is a particular case of the solution presentedin Ref. [11]. Furthermore, if σ = p M − µQ o with Q o = − Q E , Eq.(12) represents a binary system of identi-cal opposite charged Reissner-Nordstr¨om black holes [20].On the other hand, the analysis of the energy-momentumtensor of the strut in between, leads us to the expressionfor the interaction force [5, 21]: F = 14 ( e − γ −
1) = M ( R − σ ) + Q o ( R − M ) ( R − M )( R − σ − µQ o ) , (15)where γ is the value of the metric function γ on theregion of the strut. III. ANALYTIC FORM OF σ ANDGEOMETRICAL PROPERTIES OF THESOLUTION
In this case, the half length of the horizon σ can bewritten as a function of physical Komar parameters [12]: M , Q E , J , and a coordinate distance R . Nevertheless,the individual magnetic charge Q B is not vanishing inthis approach. In order to calculate σ we are going touse the Tomimatsu’s formulas [22], M = − π Z H ω Ψ z dϕdz,Q E = 14 π Z H ωA ′ z dϕdz, Q B = 14 π Z H ωA z dϕdz,J = − π Z H ω (cid:20) ω Ψ z − ˜ A A ′ z − ( A ′ A ) z (cid:21) dϕdz, (16)with ˜ A := A + ωA and the magnetic potential A isdefined as the real part of Kinnersley’s potential Φ [23].Using the SM [18], one finds thatΦ = − i IE − = i (cid:18) z Φ − I Λ + Γ (cid:19) . (17)Since the black holes are identical, the horizon for theupper object is defined as a null hypersurface H = {− σ ≤ z − R ≤ σ, ≤ ϕ ≤ π, ρ → } . From Eq.(16), one canshow that M represents the individual mass of each of theblack holes. On the other hand, the electric and magneticcharges read Q E = − Q o ( R − M ) R − σ − µQ o ,Q B = 2 Q o p ( R − M )( M − σ − µQ o ) R − σ − µQ o . (18)Notice that the electric and magnetic charges possess op-posite sign. After combining both equations in (18), onegets Q E + Q B = − Q E Q o , ( Q E < , Q B > , (19)where the parameter Q o allows us to define a new auxil-iary variable as follows: Q o = − Q E X, X := 1 + Q B Q E , (20)and σ can be written as a function of X in the form σ = r X ( M − Q E µX ) + R − X ) . (21)The last integral in Eq.(16), which defines the angu-lar momentum J , is not vanishing and the usual Smarrformula for the mass [9] is not anymore fulfilled. AsTomimatsu proposed [13], it should be enhanced in orderto include the contribution to the mass of the magneticcharge; it acquires the corresponding additional term: M = κS π + 2Ω J + Φ HE Q E + M SA = σ + 2Ω J + Φ HE Q E + M SA (22)where Φ HE = − A H − Ω A H is the electric potential in theframe rotating with the black hole and M SA is an extraboundary term associated to the magnetic charge [13],given by M SA = − π Z H (cid:16) A A ′ (cid:17) z dϕdz = 2 Q o ( R − M ) (cid:20) M − σ − µQ o ( R − σ − µQ o ) (cid:21) × (cid:20) ( R + 2 σ )( R + 4 M + 2 σ ) + 4 µQ o M ( M + σ )( R + 2 σ ) − µQ o ( R − M ) (cid:21) . (23)Let us call ω H the constant value of the metric function ω at the horizon. Ω := 1 /ω H is the angular velocity. Asimple calculation leads us to the following expressionsfor Ω and Φ HE :Ω = ( R + 2 σ ) p µ ( M − σ − µQ o )2 M ( M + σ )( R + 2 σ ) − µQ o ( R − M ) , Φ HE = − Q o ( R − M )( M + σ )2 M ( M + σ )( R + 2 σ ) − µQ o ( R − M ) . (24)Replacing Q o from Eq.(20) and σ from Eq.(21) intoEqs.(23) and (24), one gets M SA = µQ E ( X − R + 2 σ ) − ( R − M ) X ]2 ( M [ R + 2 σ − ( R − M ) X ] − µQ E X ) , Ω = µ R + 2 σ ) √ X − M [ R + 2 σ − ( R − M ) X ] − µQ E X , Φ HE = Q E µ ( M + σ ) XM [ R + 2 σ − ( R − M ) X ] − µQ E X . (25)Combining Eq.(25) with each other, it is easy to finda kind of enhanced form for the Smarr formula, whichincludes the contribution to the mass of the boundaryterm associated with the magnetic charge [13], M = σ +Ω (cid:20) J − Q E Q B (cid:18) − Q B Q E (cid:19)(cid:21) +Φ HE (cid:18) Q B Q E (cid:19) Q E . (26)The substitution of σ from Eq.(21) into Eq.(26) leadsus to the following cubic equation in terms of the auxil-iary variable X :( X − (cid:20) X − (cid:18) − M Q E (1 − µ ) (cid:19)(cid:21) − J Q E = 0 . (27) The explicit real root solution of Eq.(27) is given by X = 1 + [ a + [ b − a + p b ( b − a )] / ] [ b − a + p b ( b − a )] / ,a := 13 (cid:18) − M Q E (1 − µ ) (cid:19) , b := 2 J Q E , b ≥ a . (28)From the second Eq.(20) the individual magneticcharge reads Q B = − Q E √ X − . (29)In the electrostatic limit, i.e., J = 0, X = 1, σ reducesto σ E = q M − Q E µ. (30)which is one special case of the corresponding relationgiven in [20]. On the other hand, in the vacuum limit,i.e., Q E = 0, X = 1 + (1 − µ ) J / M , σ reads [10] σ V = r M − J M µ. (31)In both limits the magnetic charge, Eq.(29), vanishes.However, this is not true in the electrovacuum case as weshall show later. The interaction force due to the strutin between reads F = M R − M + ( Q E + Q B ) µ (cid:18) RR − M (cid:19) . (32)Notice the explicit appearance of the magnetic charge.It is worthwhile to mention that Eq.(32) has the sameform in the non-extreme case, as well as in the extremecase. The difference is that in the extreme case the con-dition σ = 0 relates the parameters appearing in Eqs.(21)and (27). A. Analytic calculation of σ A counter-rotating opposite charged two-body systemclearly reveals the appearance of magnetic charges as aconsequence of the rotation of electrically charged blackholes. In order to derive an explicit form of σ we willconsider first a system of two identical counter-rotatingblack holes in a weak electromagnetic field. Therefore,the corresponding value of X is X ≃ J (1 − µ ) M . (33)Hence, the magnetic charge reads Q B ≃ − Q E J (1 − µ )2 M , (34)and σ reduces to σ ≃ s M − J M µ − Q E (cid:18) J (1 − µ ) M (cid:19) µ. (35)The interaction force can now be written as F ≃ M R − M + Q E µ (cid:18) RR − M (cid:19) (cid:18) J (1 − µ ) M (cid:19) . (36)Let us now consider a system of opposite electriccharged black holes with slow rotation. The correspond-ing value of X is now given by X ≃ J (1 − µ ) [4 M − Q E (1 − µ )] , (37)the magnetic charge reads Q B ≃ − Q E J (1 − µ )4 M − Q E (1 − µ ) , (38)and σ can be written as σ ≃ s M − Q E µ − J (cid:18) M + Q E (1 − µ ) [4 M − Q E (1 − µ )] (cid:19) µ. (39)The corresponding interaction force is given by F ≃ M R − M + Q E µ (cid:18) RR − M (cid:19) × (cid:18) J (1 − µ ) [4 M − Q E (1 − µ )] (cid:19) . (40)It is important to note that the interaction force cor-responding to the two examples given above presents aspin-spin interaction and consequently magnetic chargesappear in the systems under consideration. B. Geometrical properties
The surface gravity κ and area of the horizon S can beobtained directly from Eq.(12) and without any previousknowledge of the explicit form of σ . In order to calculate κ , one can use the formula [13] κ = p − Ω e − γ H , (41)where γ H is the metric function γ evaluated at the hori-zon. A straightforward calculation leads us to the follow-ing expressions for the surface gravity and the area of thehorizon: κ = Rσ ( R + 2 σ )2 M ( M + σ )( R + 2 σ )( R + 2 M ) − Q o ( R − M ) S = 4 π (cid:20) M ( M + σ ) (cid:18) MR (cid:19) − Q o ( R − M ) R ( R + 2 σ ) (cid:21) , (42)where Q o was already defined in Eq.(20). One shouldnote that the strut between the KN black holes disap-pears in the limit R → ∞ and the bodies are isolated.In this limit both magnetic charges vanish as well asthe extra boundary term given by Eq.(23) and thereforeEqs.(35) and (39) reduce to σ = p M − Q E − J /M .In addition, if R → M , the two horizons can touch eachother, the angular velocities are stopped and the two-body system evolves into one single Schwarzschild blackhole. IV. THE EXTREME LIMIT OF THE SOLUTION
The extreme limit can be obtained by setting σ = 0in Eq.(12), and after a careful use of the l’Hˆopital’s rule,one gets E = Λ − αM x Γ + Λ + 2 αM x Γ + , Φ = 2 q o y Γ − Λ + 2 αM x Γ + , f = DN , ω = 4 α δ o y ( x − y − WD , e γ = Dα ( x − y ) , Λ = α ( α − M )( x − y ) + α M ( x −
1) + q o (1 − y ) + 2 iα δ o ( x + y − x y ) , Γ ± = ± n(cid:16)p M − µQ o ∓ i p α − M (cid:17) hp M − µQ o ( x − ± i p α − M ( x − y ) i + µQ o ( x − o ,D = [ α ( α − M )( x − y ) + α M ( x − − q o ( y − ] − α δ o x y ( x − − y ) ,N = { α ( α − M )( x − y ) + α M ( x −
1) + q o (1 − y ) + 2 αM x [( α − M )( x − y ) + M ( x − } + 4 α δ o (cid:2) α ( x + y − x y ) + M x (1 − y ) (cid:3) ,W = M α [( α − M )( x − y )(3 x + y ) + M (3 x + 6 x −
1) + 8 αM x ] + q o [ M ( y − − αxy ] ,δ o := p ( α − M )( M − µQ o ) , α := R , (43)where ( x, y ) are prolate spheroidal coordinates x = r + + r − α , y = r + − r − α , r ± = p ρ + ( z ± α ) , (44)related to the cylindrical coordinates ( ρ, z ) through the following transformation formulas: ρ = α p ( x − − y ) , z = αxy. (45)The extreme limit case given by Eq.(43) is a 3-parametric exact solution where the physical parameters are relatedby Eq.(27) and by the following equation: X + ( α + M ) Q E X − α Q E (cid:18) α + Mα − M (cid:19) = 0 . (46)Nevertheless, it is quite complicated to derive an analytic expression of one of the parameters in terms of the otherthree. After combining Eqs.(27) and (46), it is possible to get the following relation: | J | = h µ ( M (1 − µ ) − Q E (1 − µ ) ) + M p µ [4 M µ + Q E (1 − µ ) ] i | Q E | µ / (1 − µ ) × r M q µ [4 M µ + Q E (1 − µ ) ] − µ (2 M + Q E (1 − µ ) ) , (47)whose asymptotic expansion lead us to | J | M p M − Q E ≃ M − Q E ( M − Q E ) M ( M − Q E ) (cid:18) R (cid:19) > , (48)and it implies that the inequality J /M > M − Q E > R ≫ M , forwhich 0 < µ <
1. The equality J /M = M − Q E isreached if the distance becomes large enough and tendsto infinity (i.e, µ = 1), where the black holes are isolated.It is important to stress the fact that magnetic chargesdepend also on the coordinate distance R as shown inEqs.(34) and (38), but they vanish if the distance tendsto infinity.It should be pointed out that the metric Eq.(43) ful-fills the axis condition for all the regions on the sym-metry axis: ω ( y = ±
1) = 0 for | z | > α and ω ( x =1) = 0 for | z | < α . It reduces to well-known lim-its: by setting Q E = 0 it results to be one particularcase of the Kinnersley-Chitre solution [24]. The extremedouble-Reissner-Nordstr¨om (DRN) solution is obtainedif | Q E | = µ − / M > M and J = 0 (black dihole solution[25]). The extreme DRN solution was already consid-ered in Ref. [26] for unequal constituents. However, theexpression for the interaction force between identical ex-treme Reissner–Nordstr¨om components is not written, itexplicitly reads F = 2 M R − M (cid:20) M R − M (cid:21) , R > M. (49) V. CONCLUDING REMARKS
This paper deals with a complementary asymptoticallyflat exact solution related to identical counter-rotatingblack hole sources, in the presence of electromagneticfield. Particularly, the black holes are endowed withopposite electric/magnetic charges. Our descriptionprovides an analytical way to derive the expression for σ in terms of the physical Komar parameters and the coordinate distance. This new exact solution gives aphysical explanation of the appearance of magneticcharges in the solution as a consequence of the rotationof electrically charged black holes. Moreover, thepresence of magnetic charges violates the usual Smarrformula for the mass; it should be enhanced in order totake into account the contribution to the mass of themagnetic charge.On the other hand, it is worthwhile to mention that ac-cording to the positive mass theorem [27, 28] a regularsolution of Eqs.(12) and (43) should fulfill the mass for-mula Eq.(26): M ≥ Ω[2 J − Q E Q B (1 − Q B /Q E )]+Φ HE (1+ Q B /Q E ) Q E >
0. Nevertheless, the theorem does notimply that the condition
M > αM x Γ + = F R + iF I = 0 , (50)where F R = α ( α − M )( x − y ) + α M ( x − αM x (cid:2) ( α − M )( x − y ) + M ( x − (cid:3) + q o (1 − y ) = 0 ,F I = 2 αδ o (cid:2) α ( x + y − x y ) + M x (1 − y ) (cid:3) = 0 . (51)In Weyl-Papapetrou cylindrical coordinates ( ρ, z ) theinterior naked singularity of a black hole lies on the sym-metry axis. In the plane ( x, y ) the region x ≥ | y | ≥ M >
M <
0, in the solution Eq.(43) ringsingularities off the axis arise due to the intersections ofthe curves of Eq.(51) in the region x > | y | <
1, (seeFig. 3). - - - - x y a - - - Ρ z b FIG. 2: (a) No zeros in the denominator of the Ernst poten-tials in the ( x, y ) plane, for the values M = 1, Q E = − . Q B = 0 . J = 3 .
31, and α = 1 . F R and F I are repre-sented by the continuous and dashed lines, respectively. (b)The stationary limit surfaces of two identical counter-rotatingextreme KN black holes with M > - - - - x y a - - - Ρ z b FIG. 3: (a) Singularities located at x = 1 . y = ± .
7, for thevalues M = − Q E = − Q B = − . J = 0 .
3, and α ≃ . M < ρ ≃ . z ≃ ± .
23. The small displacement of the ringsingularities with respect to their corresponding ergosurfaceis due to the presence of the electromagnetic charge.
Additionally, the easiest analytical proof on the regu-larity of the solution can be made in the extreme DRNsector, since the curves defined by Eq.(51) are now re-duced to a geometric locus of two straight lines whoseintersection forms an angle of θ = arctan[ α/ √ α − M ], F R ≡ F DRN = (cid:18) x + Mα (cid:19) − (cid:18) − M α (cid:19) y = 0 , (52)where the straight lines are given by y = ± p − M /α (cid:18) x + Mα (cid:19) . (53) The conditions x = 1 and | y | < x > | y | <
1, forming singular surfaces off the axis (see Fig.4), 1 p − M /α (cid:18) Mα (cid:19) < , ⇒ M < , − p − M /α (cid:18) Mα (cid:19) > − , ⇒ M < . (54)Moreover, the conditions x = 1 and | y | > p − M /α (cid:18) Mα (cid:19) > , ⇒ M > , − p − M /α (cid:18) Mα (cid:19) < − , ⇒ M > . (55) - - - - - x y a - - Ρ z b FIG. 4: (a) Crossing inside the region x > | y | < M <
0, for the values α = 1, M = − .
4. (b) Emergence ofsingular surfaces if
M <
To conclude, we should mention that our descriptioncan be reduced to the well-known limits as the vacuumand electrostatic ones [10, 20]. In the extreme limit pre-sented, it is not trivial to derive relations between theparameters and it remains as a future work to be ana-lyzed. Particularly it would be also quite interesting toaccomplish a deeper analysis of the inequalities betweenstruts and Komar physical quantities, provided and dis-cussed by Gabach Clement [29].
ACKNOWLEDGEMENTS
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