Stationary two-black-hole configurations: A non-existence proof
aa r X i v : . [ g r- q c ] M a y Stationary two-black-hole configurations:A non-existence proof
Gernot Neugebauer ∗ Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at,Max-Wien-Platz 1, 07743 Jena, Germany
J¨org Hennig † Department of Mathematics and Statistics, University of Otago,P.O. Box 56, Dunedin 9054, New Zealand
Based on the solution of a boundary problem for disconnected (Killing) horizonsand the resulting violation of characteristic black hole properties, we present a non-existence proof for equilibrium configurations consisting of two aligned rotating blackholes. Our discussion is principally aimed at developing the ideas of the proof andsummarizing the results of two preceding papers (Neugebauer and Hennig, 2009[29], Hennig and Neugebauer, 2011 [12]). From a mathematical point of view, thispaper is a further example (Meinel et al., 2008 [22]) for the application of the inverse(“scattering”) method to a non-linear elliptic differential equation.
I. INTRODUCTION
The investigation of gravitational interactions in static two-body systems dates back tothe early days of General Relativity and was initiated by Hermann Weyl and Rudolf Bach. Ina joint paper [2] they discussed, as a characteristic example, an axisymmetric configurationconsisting of two “sphere-like” bodies at rest. Bach, who constructed a corresponding solu-tion for the vacuum region outside the bodies by superposition of two exterior Schwarzschildsolutions, noted that this solution becomes singular on the portion of the symmetry axisbetween the two bodies as expected. In a supplement to Bach’s contribution, Weyl fo-cused on the interpretation of this type of singularity and used stress components of theenergy-momentum tensor to define a non-gravitational repulsion between the bodies whichcompensates the gravitational attraction. Weyl’s result is based on some artificial assump-tions but implies an interesting question: Are there repulsive effects of gravitational originwhich could counterbalance the omnipresent mass attraction?Newtonian approximations tell us that the interaction of the angular momenta of rotatingbodies (“spin-spin interaction”) could indeed generate repulsive effects. This is a goodmotivation to study, in a rigorous way, the equilibrium between two (aligned) rotating blackholes with parallel (or anti-parallel) spins as a characteristic example for a stationary two-body problem. In preceding papers [12, 29], which involved degenerate (“extreme”) blackholes we came to a negative conclusion. This paper is meant to summarize the steps of thisnon-existence proof, to point out the main points of the matter and to refer non-specialists ∗ Electronic address: [email protected] † Electronic address: [email protected]
PSfrag replacements ̺ z K K K K K = K K = K A + A A − H ( ) H ( ) C (a)(b)(c)(d) FIG. 1: Illustration of a two-black-hole configuration with one degenerate (point-like) horizon H (1) ( ̺ = 0 , ζ = K ) and one sub-extremal horizon H (2) ( ̺ = 0 , K ≥ ζ ≥ K ) in Weyl-Lewis-Papapetrou coordinates. A + , A and A − denote the three parts of the axis of symmetry. to papers dealing with the technical details. It should be noted that a non-existence prooffor special symmetric equilibrium configurations by Beig et al. [3, 4] is essentially basedon symmetry arguments and does not apply to black holes with different horizon areas andangular momenta.Another aspect with some relevance for the two-black-hole configurations in question isthe interpretation of the so-called double-Kerr-NUT solution [17, 24], a seven parametersolution constructed by a two-fold B¨acklund transformation of Minkowski space. Since asingle B¨acklund transformation generates the Kerr-NUT solution that contains, by a specialchoice of its three parameters, the stationary black hole solution (Kerr solution) and sinceB¨acklund transformations act as a non-linear superposition principle, the double-Kerr-NUTsolution was considered to be a good candidate for the solution of the two-horizon problem inquestion and extensively discussed in the literature [9, 13–15, 17–21, 30, 33]. However, therewas no argument that this particular solution is the only candidate. Therefore, defects ofthis special solution would not a priori imply a general non-existence proof for our stationarytwo-black-hole problem.In papers [12, 27–29] we could remove this objection and show that the discussion ofa boundary value problem for the Ernst equation which represents a part of the vacuumEinstein equations, necessarily leads to a subclass of the double-Kerr-NUT solution. Thisresult is in line with a theorem of Varzugin [31, 32] which says that the 2 N -soliton solutionby Belinski and Zakharov contains all possible solutions (if any exist) corresponding to anequilibrium configuration of black holes.The subclass is characterized by a set of restrictions for the parameters of the generaldouble-Kerr-NUT solution. These restrictions ensure the “correct” behavior of the double-Kerr-NUT solution along the axis of symmetry and the horizons. For solving the restrictionswe could go back to the already mentioned discussions of the equilibrium conditions for thedouble-Kerr-NUT solution. After a too restrictive ansatz in [17], Tomimatsu and Kihara [15,30] derived and discussed a complete set of equilibrium conditions on the axis of symmetry.Reformulations and numerical studies by Hoenselaers [14] made plausible that the double-Kerr-NUT solution cannot describe a two-black-hole equilibrium if the individual Komarmasses of its two sources (“horizons”) are assumed to be positive. Manko et al. [20] andfinally Manko and Ruiz [21] were able to prove the conjecture. The critical point of this proofin view of a non-existence theorem is, however, the presumed positiveness of the two Komarmasses. To the best of our knowledge there is no argument in favor of this assumption (onthe contrary, Ansorg and Petroff [1] gave convincing counterexamples). Instead, we replacethe Komar mass inequality M i > i = 1 ,
2) by inequalities connectingangular momenta J i and surface areas A i (8 π | J i | < A i , i = 1 ,
2) [11]. These relations arebased on the causal structure of trapped surfaces in the interior vicinity of any event horizon[6]. In the case of two non-degenerate black holes it turns out that one of the two inequalitiesis always violated, i.e. one of the sources cannot be a black hole.This type of non-existence proof avoids more laborious investigations of the domain offthe axis of symmetry. In some degenerate cases, see Fig. 1, we need additional eliminatingcriteria such as the positiveness of the total (ADM) mass or the absence of singular ringsfor proving the non-existence. We summarize the results of the discussions of all subcasesin Sec. VI.
II. A BOUNDARY PROBLEM FOR DISCONNECTED HORIZONS
The exterior vacuum gravitational field of axially symmetric and stationary gravitationalsources can be described in cylindrical Weyl-Lewis-Papapetrou coordinates ( ̺, ζ , ϕ, t ) , inwhich the line element takes the formd s = e − U (cid:2) e k (d ̺ + d ζ ) + ̺ d ϕ (cid:3) − e U (d t + a d ϕ ) , (1)where the “Newtonian” gravitational potential U , the gravitomagnetic potential a and the“superpotential” k are functions of ̺ and ζ alone. At large distances r = | p ̺ + ζ | → ∞ from isolated sources located around the origin of the coordinate system, r = 0, the spacetimehas to be Minkowskian, r → ∞ : d s = d ̺ + d ζ + ̺ d ϕ − d t . (2)According to the objective of this paper (see Fig. 1) we will exclusively discuss gravitationalfields (1) under condition (2).Metric (1) admits an Abelian group of motions G with the generators (Killing vectors) ξ i = δ it (stationarity) η i = δ iϕ (axisymmetry) (3)where the Kronecker symbols δ it , δ iϕ indicate that ξ i has only a t -component whereas η i pointsin the azimuthal ( ϕ ) direction. η i has closed compact trajectories about the axis of symmetryand is therefore space-like off the axis (and the horizons). ξ i is time-like sufficiently far fromthe black holes but can become space-like inside ergoregions. Obviously,e U = − ξ i ξ i , a = − e − U η i ξ i (4)is a coordinate-free representation of the gravitational potentials U and a . In the following, we also use the complex coordinates z = ̺ + i ζ and ¯ z = ̺ − i ζ . t is the time coordinate. According to Carter’s theorems [8] we can assume that the event horizons of the twoblack holes under discussion are Killing horizons. Here a Killing horizon can be defined bya linear combination ξ ′ of the Killing vectors ξ and η , ξ ′ = ξ + Ωη (5)with the norm e V = − ( ξ ′ , ξ ′ ) = e U (cid:2) (1 + Ωa ) − ̺ Ω e − U (cid:3) (6)where Ω is the constant angular velocity of the horizon. A connected component of the setof points with e V = 0, which is a null hypersurface, (de V , de V ) = 0, is called a Killinghorizon H ( ξ ′ ), H ( ξ ′ ) : e V = − ( ξ ′ , ξ ′ ) = 0 , (de V , de V ) = 0 . (7)Since the Lie derivative L ξ ′ of e V vanishes, we have ( ξ ′ , de V ) = 0. Being null vectors on H ( ξ ′ ), ξ ′ and de V are proportional to each other, H ( ξ ′ ) : de V = − κξ ′ . (8)Using the field equations one can show that the surface gravity κ is a constant on H ( ξ ′ ).In the ̺ - ζ plane ( t = constant, ϕ = constant) of the Weyl-Lewis-Papapetrou coordinatesystem (1) horizons are located on the ζ -axis ( ̺ = 0) and cover a finite portion of the axis( H (2) in Fig. 1) or shrink to a single point ( H (1) in Fig. 1)[8]. It turns out that extendedhorizons (“sub-extremal horizons”) and point-like horizons (“degenerate horizons”) requiredifferent considerations. Note that a Killing horizon is always a two-surface in the time slice t = constant. The degeneracy to a line or a point is a peculiarity of the special coordinatesystem.The dashed line in Fig. 1 sketches the boundaries of the vacuum region: A + , A , A − are the regular parts of the ζ -axis (axis of symmetry), H (1) and H (2) denote the two Killinghorizons (Fig. 1 shows a point-like and an extended horizon), and C stands for spatial infinity.The gravitational fields a , k , U have to satisfy the following boundary conditions A ± , A : a = 0 , k = 0 , (9) H ( i ) : 1 + Ω i a = 0 , i = 1 , , (10) C : U → , a → , k → , (11)where Ω and Ω are the angular velocities of the two horizons. Equations (9) characterizethe axis of symmetry (rotation axis). The first relation originates from the second equationin (4), since the compact trajectories of η with the standard periodicity 2 π become infinites-imal circles with the consequence η →
0. The second relation is a necessary condition forelementary flatness (Lorentzian geometry in the vicinity of the rotation axis). Equation (10)is a reformulation of Eqs. (7) (e V = 0) and (6) since the horizons are located on the ζ -axis( ̺ = 0); see Fig. 1. Finally, Eq. (11) ensures the asymptotic flatness of the metric (1); see(2).In our discussion we will essentially use the Ernst formulation of the field equations [10].For this purpose, we introduce the complex Ernst potential f = e U + i b, (12)where the (real) twist potential b is defined by a ,̺ = ̺ e − U b ,ζ , a ,ζ = − ̺ e − U b ,̺ . (13)In this formulation, a part of the Einstein vacuum equations is equivalent to the complex Ernst equation ( ℜ f ) (cid:16) f ,̺̺ + f ,ζζ + 1 ̺ f ,̺ (cid:17) = f ,̺ + f ,ζ . (14)Obviously, the imaginary part of the Ernst equation is nothing but the integrability condition a ,̺ζ = a ,ζ̺ . On the other hand, the condition b ,̺ζ = b ,ζ̺ leads back to the field equation for a as an element of the original Einstein equations.The metric potential k can be calculated from f via a line integral, k ,̺ = ̺ h U ,̺ − U ,ζ + 14 e − U ( b ,̺ − b ,ζ ) i , k ,ζ = 2 ̺ h U ,̺ U ,ζ + 14 e − U b ,̺ b ,ζ i . (15)The result of the integration does not depend on the path of integration, since (14) implies( k ,̺ ) ,ζ = ( k ,ζ ) ,̺ .Equations (14) and (15) are completely equivalent to the Einstein vacuum equations.Thus one can first integrate the Ernst equation to obtain e U and b or, alternatively, e U and a (see (13)), and determine the remaining metric potential k (see (1)) by line integration afterward . Taking advantage of this circumstance we will first analyze the boundary problem A ± , A : a = 0 , (16) H ( i ) : 1 + Ω i a = 0 , i = 1 , , (17) C : U → , a → , (18)for the Ernst equation (14). Note that the connection between b = ℑ f and a is non-local;see (13). III. THE INVERSE METHOD
The inverse (scattering) method, first applied for solving initial (value) problems of specialclasses of non -linear partial differential equations in many areas of physics (such as Korteweg-de Vries equation in hydrodynamics, non-linear Schr¨odinger equation in non-linear opticsetc.), is based on the existence of a linear problem (LP), whose integrability condition isequivalent to the non-linear differential equation. Surprisingly, the Ernst equation has an LP.This fact is the background for the rich gain of exact solutions with interesting mathematicalproperties; see [5, 16, 17, 22, 24, 28] and the references therein. However, we cannot expect a priori one of these solutions to solve our physical question, and are therefore referred tomethods applicable to boundary (value) problems. In a sense, we can try to borrow ideasfrom the above mentioned analysis of initial (value) problems.We use the LP [23, 25] Φ ,z = (cid:20)(cid:18) N M (cid:19) + λ (cid:18) NM (cid:19)(cid:21) Φ , Φ , ¯ z = (cid:20)(cid:18) ¯ M
00 ¯ N (cid:19) + 1 λ (cid:18) M ¯ N (cid:19)(cid:21) Φ , (19)where the pseudopotential Φ ( z, ¯ z, λ ) is a 2 × λ = r K − i¯ zK + i z , K ∈ C , (20)as well as on the complex coordinates z = ̺ + i ζ , ¯ z = ̺ − i ζ , (21)whereas M , N and the complex conjugate quantities ¯ M , ¯ N are functions of z , ¯ z (or ̺ , ζ )alone and do not depend on the constant parameter K . Since the integrability conditions Φ ,z ¯ z = Φ , ¯ zz must hold identical in K (or λ ) they yield the first order equations M , ¯ z = M ( ¯ N − ¯ M ) − ̺ ( M + ¯ N ) , N , ¯ z = N ( ¯ M − ¯ N ) − ̺ ( N + ¯ M ) (22)with the “first integrals” M = f ,z f + ¯ f , N = ¯ f ,z f + ¯ f , (23)where f is any complex function of z , ¯ z . Resubstituting M and N in Eqs. (22) one obtainsthe Ernst equation (14) for f ( z, ¯ z ). Thus the Ernst equation is the integrability condition ofthe LP (19). Vice versa, the matrix Φ calculated from M , N does not depend on the pathof integration if f is a solution of the Ernst equation.Without loss of generality the matrix Φ may be assumed to have the form Φ = (cid:18) ψ ( ̺, ζ , λ ) ψ ( ̺, ζ , − λ ) χ ( ̺, ζ , λ ) − χ ( ̺, ζ , − λ ) (cid:19) . (24)Note that both columns are independent solutions of the LP. The particular form of (24) isequivalent to Φ ( − λ ) = (cid:18) − (cid:19) Φ ( λ ) (cid:18) (cid:19) . (25)Furthermore ¯ ψ (cid:18) ̺, ζ , λ (cid:19) = χ ( ̺, ζ , λ ) (26)due to the special structure of the coefficient matrices of the LP.For K → ∞ and λ → − ψ , χ may be normalized by ψ ( ̺, ζ , −
1) = χ ( ̺, ζ , −
1) = 1 . (27)For K → ∞ and λ → f ( ̺, ζ ) = χ ( ̺, ζ , (cid:0) ¯ f ( ̺, ζ ) = ψ ( ̺, ζ , (cid:1) (28)as a consequence of the LP (19). Thus one obtains the solution f of the Ernst equation fromthe pseudopotential Φ ( χ = Φ , ψ = Φ ) for a particular choice of the spectral parameter λ . Similarly, the gravitomagnetic potential a can be rediscovered at K → ∞ and λ = 1 inthe first derivatives of the pseudopotential Φ [27], a ( ̺, ζ ) = − ̺ e − U (cid:18) ∂∂λ [ χ ( − λ ) − ψ ( − λ )] (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = 1 K → ∞ − C. (29)To prove this one has to make use of the LP (19) and Eqs. (23) and (13). The arbitraryreal constant C may be fixed by setting a = 0 for r = p ̺ + ζ → ∞ . An alternative formof (29) is a ( ̺, ζ ) = ie − U (cid:18) K ∂∂K [ χ ( − λ ) − ψ ( − λ )] (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = 1 K → ∞ − C. (30)(Note that ∂/∂K = λ ,K ∂/∂λ with λ from (20).) The idea of the inverse (scattering) methodis to discuss Φ , for fixed but arbitrary values of ̺ , ζ ( z , ¯ z ) as a holomorphic function of λ . We will show that the boundary values (9)-(11) together with the LP (19) yield thenecessary information that enables us to construct Φ ( ̺, ζ , λ ). (The Ernst potential andthe gravitomagnetic potential can then be determined in a simple way; see (28), (29)). Torealize the program we will integrate the LP along the dashed line in Fig. 1 starting fromand returning to any point ̺ = 0, ζ ∈ A + . Obviously, λ degenerates at ̺ = 0 to λ = ± K -surfaceconnected with λ according to (20). Note that the mapping of the Riemann surface of K onto the λ -plane depends on the parameters ̺ , ζ . Thus one has movable branch points K B = i¯ z , ¯ K B = − i z and the branch cut between them changes with the coordinates. IV. SOLUTION OF THE BOUNDARY PROBLEMA. Integration along the boundary
Integrating the LP (19) along A + , H (1) , A , H (2) , A − and using (23) one finds for thevalues of Φ on each interval I , I = A + , H (1) , A , H (2) , A − , the representation I : Φ = (cid:18) ¯ f I f I − (cid:19) L I , L I = (cid:18) A I ( K ) B I ( K ) C I ( K ) D I ( K ) (cid:19) , (31)where f I is the value of the Ernst potential in the interval I and A I ( K ), B I ( K ), C I ( K ), D I ( K ) are integration “constants” depending on K alone. On the axis of symmetry ( I = A ± , A ) f is a function of ζ alone ( ̺ = 0, ζ ∈ A ± , A ), I = A ± , A : f I = f I ( ζ ) . (32)The same holds on extended horizons which can be characterized by ̺ = 0, too. Weyl-Lewis-Papapetrou coordinates are, however, inappropriate for integrating along degenerate (“point-like”) horizons. As was shown by Meinel, see [22], this defect can be repaired by theintroduction of suitable (local) coordinates in the vicinity of such a horizon. To demonstratethe procedure we consider the point-like horizon in Fig. 1 at ζ = K and replace ̺ and ζ by polar coordinates ( R, θ ), ̺ = R sin θ , ζ = K + R cos θ , in which the horizon is describedby R → θ ∈ [0 , π ], i.e. by a line in an R - θ diagram. Performing the integration alongthe degenerate horizon in these coordinates one indeed obtains the structure (31) with f I = f I ( θ ). Thus we have I = H (1) , H (2) : f I = ( f I ( ζ ) for extended horizons f I ( θ ) for point-like horizons . (33)The definition of our Killing horizons is based on the Killing vectors ξ ′ = ξ + Ωη , see (5), (6),(7), where Ω = Ω for H (1) and Ω = Ω for H (2) . To exploit the characteristic propertiesof the horizons such as (7) we discuss metric potentials e U ′ , a ′ constructed from ξ ′ , η ′ = η according to (4). Thus we obtain the transformationse U ′ ≡ e V = e U [(1 + Ωa ) − Ω ̺ e − U ] , (34)(1 − Ωa ′ )e U ′ = (1 + Ωa )e U , (35)where Ω = Ω for H (1) and Ω = Ω for H (2) . It can easily be verified that the line elementretains its form (1) after the coordinate transformation ̺ ′ = ̺, ζ ′ = ζ , ϕ ′ = ϕ − Ωt, t ′ = t, (36)where Ω = Ω , Ω , if one simultaneously replaces e U and a by e U ′ and a ′ . Because of(36) we may call the primed quantities e U ′ , a ′ , etc. “corotating potentials”. Applying (13)to the primed potentials a ′ , b ′ and using (34), (35), (13) one obtains the corotating Ernstpotential f ′ from f and finally the corotating quantities M ′ , N ′ via (23). This procedureensures from the outset that f ′ satisfies the Ernst equation and guarantees the existence ofan LP (19) in the corotating system. The Φ -matrices of the two systems of reference areconnected by the relation Φ ′ = T Ω Φ , (37)where T Ω = (cid:18) Ωa − Ω̺ e − U
00 1 + Ωa + Ω̺ e − U (cid:19) + i( K + i z ) Ω e − U (cid:18) − − λλ (cid:19) (38)with Ω = Ω , Ω . This can be checked up by a straightforward calculation.The matrix Φ as defined in (24) can be considered as a unique function of λ , whichis therefore defined on both sheets of the K -surface. From this point of view, Eqs. (31)determine Φ on one sheet only, say, on the upper sheet with λ = 1. Its values on theother (lower) sheet with λ = − Φ along C . It does notdepend on ̺ , ζ , i.e., Φ [ C ] = Φ ( K ), since M and N vanish (see (23), (11)). Along C , thespectral parameter λ (20) changes from λ = 1 at ̺ = 0, ζ → ∞ to λ = − ̺ = 0, ζ → −∞ , i.e., starting in the upper sheet at the points of intersection A + / C , one arrives inthe lower sheet at A − / C . (Using polar coordinates ζ = r cos θ , ̺ = r sin θ , θ ∈ [0 , π ], onehas λ = p ( K − r e i θ ) / ( K − r e − i θ ) → λ = e i θ as r → ∞ and therefore λ = 1 for θ = 0 and λ = − θ = π .) Hence, to return to the upper sheet ( λ = 1), one has to apply (25) to Φ [ C ] = Φ [ A + / C ] = lim ζ →∞ Φ [ A + ], where we have used that Φ has to be continuous at thepoint of intersection A + / C . Thus we obtain Φ [ C / A − ] = lim ζ →∞ Φ [ A − ] = (cid:18) − (cid:19) (cid:26) lim ζ →∞ Φ [ A + ] (cid:27) (cid:18) (cid:19) , (39)and together with (31) (cid:18) A − ( K ) B − ( K ) C − ( K ) D − ( K ) (cid:19) = (cid:18) (cid:19) (cid:18) A + ( K ) B + ( K ) C + ( K ) D + ( K ) (cid:19) (cid:18) (cid:19) , (40) λ = 1 is arbitrarily ascribed to the upper sheet. where A − means A I for I = A − , etc.In a similar way one can use continuity arguments to interlink the ABCD -matrices atthe other points of intersection. Since the Ernst equation has to hold at all these points, f must be continuous and unique there. Note that the intersection values f = f [ A + / H (1) ] , f = f [ H (1) / A ] , f = f [ A / H (2) ] , f = f [ H (2) , A − ] (41)are purely imaginary: As a metric coefficient, e V = − g ′ tt is continuous at the points ofintersection. According to (34) and (9) e V = e U on the regular parts of the ζ -axis. Sincee V = 0 on the horizons, e U has to vanish at these points.Furthermore, the validity of the Ernst equations of the non-rotating and corotating systemat the points of intersection implies that Φ and Φ ′ must be continuous there as well. Byway of example let us consider the point of intersection A − / H (2) ( ̺ = 0, ζ = K ) of Fig. 1,i.e., the transition from a regular axis ( A − ) to an extended horizon ( H (2) ). According to(31) and (41) one has Φ [ A − / H (2) ] = (cid:18) − f f − (cid:19) L − = (cid:18) − f f − (cid:19) L (2) , (42)where L − = L I for I = A − and L (2) = L I for I = H (2) . Note that the determinant of thefirst factor of the matrix products vanishes such that one may cancel one line of the matrixequation. The corresponding equation for Φ ′ at z = i K follows from Eqs. (31), (37) and(38) under conditions (9) ( a = 0 on A − ) and (10) (1 + Ω a = 0 on H (2) ) Φ [ A − / H (2) ] = (cid:18) − f − Ω ( K − K ) 1 f + 2i Ω ( K − K ) − (cid:19) L − = 2i Ω ( K − K ) (cid:18) − (cid:19) L (2) . (43)Again, the matrix equation consists of two identical lines. Combining a line of (42) with aline of (43) one obtains (cid:18) f − f + 2i Ω ( K − K ) − (cid:19) L − = (cid:18) f − Ω ( K − K ) 0 (cid:19) L (2) . (44)Consider now two extended horizons: H (1) : ζ ∈ [ K , K ], H (2) : ζ ∈ [ K , K ], K > K >K > K . Following the idea that led to equation (44) one can continue the connection ofadjoining L -( ABCD -) matrices. Involving (40) and (44) and defining F i := (cid:18) − f i − f i f i (cid:19) , i = 1 , , , , (45)one arrives at the following chain of equations that reflects the integration of the LP alongthe closed contour A + H (1) A H (2) A − CA + , L + = (cid:18) + F Ω (1) ( K − K ) (cid:19) L (1) , L (1) = (cid:18) − F Ω (2) ( K − K ) (cid:19) L L = (cid:18) + F Ω (3) ( K − K ) (cid:19) L (2) , L (2) = (cid:18) − F Ω (4) ( K − K ) (cid:19) L − , L − = (cid:18) (cid:19) L + (cid:18) (cid:19) , (46)0where Ω (1) = Ω (2) = Ω , Ω (3) = Ω (4) = Ω . Point-like horizons can be involved without any difficulty by setting K = K or/and K = K in relations (46). Consider, for example, the point-like horizon in Fig. 1. Thoughthe horizon is placed on a point (as a peculiarity of the Weyl-Lewis-Papapetrou coordinatesystem), the Ernst potential has different values at A + / H (1) and H (1) / A : f = f ; see (41).Setting ζ = K = K does not require special considerations such that one can adopt thediscussion of the “extended” case step by step.Eliminating L (1) , L , L (2) , L − by means of (46) and defining R + := Y i =1 (cid:18) − ( − i F i Ω ( i ) ( K − K i ) (cid:19) (cid:18) (cid:19) , (47)where Ω = Ω (1) = Ω (2) , Ω = Ω (3) = Ω (4) , one arrives at the final result of the integrationalong the boundaries L + (cid:18) (cid:19) ( L + ) − = R + , (48)which specifies the holomorphic structure of the “integration constants” L as functions ofthe complex spectral parameter K . Since the trace of the left hand side of (48) vanishes, R + has to be trace free, tr R + = 0 . (49)As a condition identical in K , equation (49) yields four constraints among Ω , Ω ; K − K , K − K , K − K ; f , . . . , f . By way of example, Ω Ω = f − f f − f (50)as a consequence of lim K →∞ tr( R + ,K K ) = 0. The asymptotic behavior of L + is prescribedby (24), (27), (28) and (31), where one has to choose I = A + , L + = (cid:18) A + ( K ) B + ( K ) C + ( K ) D + ( K ) (cid:19) → (cid:18) (cid:19) as K → ∞ . (51)Nevertheless, equation (48) is not sufficient for determining L + uniquely. To illustrate thedegree of freedom, we factorize L + , L + = (cid:18) F ( K ) 0 G ( K ) 1 (cid:19) (cid:18) α ( K ) β ( K ) β ( K ) α ( K ) (cid:19) (52)which is always possible for det L + = 0 and A + = ± B + ; see (51). Inserting (52) into (48),one obtains (cid:18) − G F − G F G (cid:19) = R + , (53)i.e., F ( K ) = ¯ F ( ¯ K ) and G ( K ) = − ¯ G ( ¯ K ) are uniquely determined. Both functions areregular everywhere in the complex K -plane with the exception of simple poles (two extendedhorizons) or/and confluent poles of second order at most (one or two point-like horizons).In the next section, we shall determine the axis values of the Ernst potential f + from R + and show that α and β do not affect these values, i.e., (cid:16) α ββ α (cid:17) is a gauge matrix.1 B. Axis values of the Ernst potential
The elements of the matrix Φ = ( Φ ik ) (24) are unique functions of λ in the vacuum regionoutside the horizons. That implies that its elements ψ and χ must be unique ( Φ = Φ , Φ = − Φ ) at the confluent branch points K B = ¯ K B = ζ of the Riemann K -surfacesbelonging to axis values ̺ = 0, ζ ∈ A ± , A , i.e., according to (31), (32) I = A ± , A ; K = K B = ζ : Φ = (cid:18) ψ ψχ − χ (cid:19) = (cid:18) ¯ f I ( ζ ) 1 f I ( ζ ) − (cid:19) (cid:18) A I ( ζ ) B I ( ζ ) C I ( ζ ) D I ( ζ ) (cid:19) , Φ (cid:18) (cid:19) Φ − = (cid:18) − (cid:19) . (54)Then I = A ± , A : f I ( ζ ) = D I ( ζ ) + C I ( ζ ) A I ( ζ ) + B I ( ζ ) , ¯ f I ( ζ ) = D I ( ζ ) − C I ( ζ ) A I ( ζ ) − B I ( ζ ) , (55)i.e. one can express the values of the Ernst potential on the regular portions of the axis bythe “integration constants” A , B , C , D . According to (52), this means for f I ( ζ ) = f + ( ζ ), I = A + , f + ( ζ ) = 1 + G ( ζ ) F ( ζ ) , ¯ f + ( ζ ) = 1 − G ( ζ ) F ( ζ ) , (56)or F ( ζ ) = 2 f + ( ζ ) + ¯ f + ( ζ ) , G ( ζ ) = f + ( ζ ) − ¯ f + ( ζ ) f + ( ζ ) + ¯ f + ( ζ ) , (57)i.e., the matrix (cid:16) α ( ζ ) β ( ζ ) β ( ζ ) α ( ζ ) (cid:17) does not affect the axis values of the Ernst potential on A + .Because of the successive transformation (46) this holds for f − ( ζ ) and f ( ζ ), too. (Notethat (55) and (40) imply f + ( ζ ) = 1 /f − ( ζ ), ζ ∈ A + , A − , where f + and f − are continuationsof f + ( ζ ), ζ ∈ A + , f − ( ζ ), ζ ∈ A − via A ( K ), B ( K ), C ( K ), D ( K ).) Thus we obtain the axisvalues f ± ( ζ ), f ( ζ ) via (56) and (53) from R + ( K = ζ ), see (47), as well-defined functionsof ζ . The parameters entering these functions are restricted by the constraints tr R + = 0.Since the matrix (cid:16) α ( ζ ) β ( ζ ) β ( ζ ) α ( ζ ) (cid:17) does not affect f ± , f we may set α ( ζ ) = 1, β ( ζ ) = 0 withthe consequence (cid:18) α ( K ) β ( K ) β ( K ) α ( K ) (cid:19) = (cid:18) (cid:19) . (58)(Note that α ( K ), β ( K ) are analytic continuations of α ( ζ ), β ( ζ ).) The particular choice (58)is closely connected with a gauge transformation of the matrix Φ : Any transformation Φ new = Φ old (cid:18) a ( K ) b ( K ) b ( K ) a ( K ) (cid:19) (59)with a ( K ) = a ( ¯ K ) , b ( K ) = − b ( ¯ K ); a ( K ) → , b ( K ) → K → ∞ (60)leaves the normalizations (24)-(27) unaffected and can be used to remove the α - β -matrixfrom (52). Hence (59) is a gauge transformation and one can adjust the gauge so that L + = (cid:18) F ( K ) 0 G ( K ) 1 (cid:19) , (cid:18) α ( K ) β ( K ) β ( K ) α ( K ) (cid:19) = (61) K B = i¯ z → K B = − i z → ζ as ̺ → I = A + : Φ = ψ ( −
1) = 1 , Φ = − χ ( −
1) = 1 , (62)i.e., the gauge (61) is equivalent to the formulation of special initial conditions for ψ and χ at some starting point ̺ = 0, ζ = ζ ∈ A + , λ = − K in the lower sheet) of the integrationalong the closed dashed line in Fig. 1 (which we performed with unspecified integrationconstants A + , B + , C + , D + ).Let us summarize the results of the integration of the LP along the boundary and thedetermination of the axis values f ± ( ζ ), f ( ζ ): With the standard gauge (61) and the repre-sentation (31) we obtain for Φ on A + I = A + : Φ = (cid:18) ¯ f + ( ζ ) 1 f + ( ζ ) − (cid:19) L + , L + = (cid:18) F ( K ) 0 G ( K ) 1 (cid:19) , (63)where F ( K ), G ( K ) are elements of the matrix R + ; see (53) and (47). R + must be trace free, tr R + = 0 , (64)see (49), which affects via F ( K ) and B ( K ) the constant parameters entering the Ernstpotential f + ( ζ ) on the axis A + I = A + : f + ( ζ ) = 1 + G ( ζ ) F ( ζ ) , (65)see (56). The particular form of the pseudopotential Φ on the intervals H (1) , A , H (2) , A − and the axis values f ( ζ ), f − ( ζ ) result from (46) with the “starting matrix” L + as chosenin (61) and expressions (55). f + ( ζ ) seems to be a quotient of two normalized polynomials of fourth degree; see (56),(53), (47). However, the constraints (49) take care that the numerator as well as the denom-inator are of second degree. Inserting the first equation in (54) ( I = A + ) into the secondone and using (48) one obtains R + ( ζ ) = (cid:18) ¯ f ( ζ ) 1 f ( ζ ) − (cid:19) − (cid:18) − (cid:19) (cid:18) ¯ f ( ζ ) 1 f ( ζ ) − (cid:19) , (66)with the consequence[ R + ( ζ ) − ] (cid:18) f + ( ζ ) (cid:19) = 0 , [ R + ( ζ ) + ] (cid:18) − ¯ f + ( ζ ) (cid:19) = 0 . (67)By definition (47), the elements R + ik of the matrix R + obey the conditions¯ R +11 = − R +11 , ¯ R +22 = − R +22 , ¯ R +12 = R +12 , ¯ R +21 = R +21 . (68)Hence the two equations (67) are complex conjugate. f + ( ζ ) can now be calculated from (67)provided that det( R + −
1) = 0 holds. By definition, det R + = −
1, such that det( R + −
1) = − tr R + . Hence, det( R + − ) = 0 ⇔ tr R + = 0 (69) The fourth order coefficient is equal to one. r + = ( r + ik ), r + = R + 4 Y l =1 ( ζ − K l ) , (70) f + ( ζ ) takes the form f + ( ζ ) = Q l =1 ( ζ − K l ) − r +11 r +12 , r +11 + r +22 = 0 . (71)According to (69), the constraints r +11 + r +22 = 0 can be reformulated to give " Y l =1 ( ζ − K l ) − r +11 Y l =1 ( ζ − K l ) + r +11 = r +12 r +21 , (72)where by definition the two factors (“brackets”) on the left hand side are normalized complexconjugate polynomials of fourth degree (¯ r +11 = − r +11 ) whereas r +12 and r +21 are normalized real polynomials of fourth degree.Identifying the zeros on both sides of (72) we see see that each factor on the left handside has to have two zeros of r +12 as well as of r +21 . (Note that the brackets are complexconjugate.) Hence the numerator and the denominator of f + as factors in (72) have twocommon zeros such that f + has to be a quotient of two polynomials of second degree, f + ( ζ ) = n ( ζ ) d ( ζ ) = ζ + qζ + rζ + sζ + t , (73)where the complex constants q , r , s , t are restricted by constraints (49). Axis values of theform (73) are characteristic for the Ernst potential of the double-Kerr-NUT solution. Beforecontinuing f + ( ζ ) to all space, we will discuss the gravitomagnetic potential a on A ± , A , H (1 / . For this purpose we express χ ( − λ ), ψ ( − λ ) in (30) by the elements of the last columnof Φ in (31). The result a I = − K ∂∂K B I ( K ) (cid:12)(cid:12) K →∞ − C (74)tells us that the gravitomagnetic potential has constant values on all intervals A ± , A , H (1 / . Since a ( ̺, ζ ) is only determined up to an arbitrary constant, we adjust C so that a + = 0 . (75)To connect the values of a on adjoining intervals we make use of the successive transfor-mations (46) but forgo for the moment the special gauge (61). Considering the transition A + / H (1) , we assume that the asymptotics of L + can be described by a power series in 1 /K ,with D + ( K ) = 1 + O (cid:18) K (cid:19) , B + ( K ) = O (cid:18) K (cid:19) . (76)Obviously, our standard gauge (61) satisfies this assumption. Applying (74) to the 1-2element of the matrix equation L (1) = (cid:18) − F Ω (1) ( K − K ) (cid:19) L + , (77)4see (46), one obtains a (1) = a + − Ω (1) . (78)Continuing the procedure according to (46) one arrives at a = a (1) + 1 Ω (2) , a (2) = a − Ω (3) , a − = a (2) + 1 Ω (4) . (79)In our context, Ω (1) = Ω (2) = Ω , Ω (3) = Ω (4) = Ω . However, the form of Eqs. (78), (79)remains unchanged repeating their derivation with Ω (1) = Ω (2) , Ω (3) = Ω (4) in (46). Weshall make use of this later on; see (93).We can now make sure that our standard representation (63) satisfies the boundaryconditions (16) and (17). Indeed, L +12 = B + ( K ) = 0 and L +22 = D ( K ) = 1 obey assumption(76) such that Eqs. (78), (79) hold. With a + = 0 (see (75)) and Ω (1) = Ω (2) = Ω , Ω (3) = Ω (4) = Ω they take the form A ± , A : a ± = 0 , a = 0 H ( i ) : a ( i ) = − Ω i , i = 1 , C. Equilibrium conditions
We have shown that the Ernst potential (71) can be reduced to a quotient of two poly-nomials of second degree by means of the constraints r + r = 0. However, the explicitdetermination of the coefficients q , r , s , t in (73) turns out to be a subtle point. It dependson a suitable reparametrization of the constants Ω , Ω , K − K , K − K , K − K , f ,. . . , f , which permits an easier handling of the constraints. For this reason we introducethe functions α ( ζ ) = ¯ d ( ζ ) d ( ζ ) , α ¯ α = 1 , β ( ζ ) = ¯ n ( ζ ) n ( ζ ) , β ¯ β = 1 (81)and discuss their behavior at the points ζ = K i , ( i = 1 , . . . ,
4) which fix the positions of thehorizons. To begin with, we examine the configuration of two extended horizons H (1) : K ≥ ζ ≥ K , H (2) : K ≥ ζ ≥ K , K > K > K > K . (82)Introducing the parameters α i = α ( K i ) , α i ¯ α i = 1 , i = 1 , . . . , ,β i = β ( K i ) , β i ¯ β i = 1 , i = 1 , . . . , , (83)we obtain from (81) the two linear algebraic systems of equations¯ d ( K i ) − α i d ( K i ) = 0 , ¯ n ( K i ) − β i n ( K i ) = 0 , i = 1 , . . . , s , t (¯ s , ¯ t ); q , r (¯ q , ¯ r ) with K i , α i , β i as coefficients. According to (57), (53) and (47) andusing ¯ r +11 = − r +11 we havee U + = ( ζ − K )( ζ − K )( ζ − K )( ζ − K ) r +12 , (85)5where r +12 is a real normalized polynomial of fourth degree in ζ . From e U + ( K i ) = 0 ( r +12 ( K i ) =0, i = 1 , . . . ,
4) we get f + ( K i ) = f i = − ¯ f + ( K i ) = − ¯ f i , (86)with the consequence β i = − α i . (87)Hence, f + can be expressed in terms of α i (and K i ) alone. Solving the linear equations (84)for q , r , s , t and plugging the result into (73) we arrive at a determinant representation ofthe axis potential f + on A + , f + ( ζ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K K K α K ( ζ − K ) α K ( ζ − K ) α K ( ζ − K ) α K ( ζ − K )0 K K K K α ( ζ − K ) α ( ζ − K ) α ( ζ − K ) α ( ζ − K )0 1 1 1 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K K K − α K ( ζ − K ) α K ( ζ − K ) α K ( ζ − K ) α K ( ζ − K )0 K K K K α ( ζ − K ) α ( ζ − K ) α ( ζ − K ) α ( ζ − K )0 1 1 1 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (88)It can easily be seen that f ( ̺, ζ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K K K α K r α K r α K r α K r K K K K α r α r α r α r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K K K − α K r α K r α K r α K r K K K K α r α r α r α r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (89)where r i := p ( ζ − K i ) + ̺ ≥ , i = 1 , . . . , , (90)is a continuation of f + ( ζ ) to all space. (Replace ζ − K i = | ζ − K i | , i = 1 , . . . , r i , i = 1 , . . . , f ( ̺, ζ ) is a solution of the Ernst equation [17]. As we have already mentioned, the othergravitational potentials k , a (e U = ℜ f !) can be calculated from f ( ̺, ζ ) via line integrals.This solution of the Einstein vacuum equations is known under the name of double-Kerr-NUT solution. Since it can be shown by the inverse scattering methods that the axis values6 f + ( ζ ) uniquely determine the Ernst potential everywhere in the ̺ - ζ plane, f ( ̺, ζ ) as definedin (89) is the only solution of the Ernst equation to the boundary values (88). Hence, thesolution of the two-horizon problem must be a (particular) double-Kerr-NUT solution .The double-Kerr-NUT solution itself is a particular case ( N = 2, Minkowski seed) of aclass of solutions generated by an N -fold B¨acklund transformation from an arbitrary seedsolution of the vacuum field equations [23–25] . The interrelationship with a class of solitonicsolutions discovered by Belinski and Zakharov [5] is discussed in [16].Many attempts have been made to establish a connection between the double-Kerr-NUTsolution and two-black-hole equilibrium configurations [9, 14, 15, 17, 20, 21, 30]. Applyingthe boundary conditions A + , A : a = 0 , k = 0 C : f → , k → f ( ̺, ζ ), Tomimatsu and Kihara [15, 30] derived a complete set of algebraic equilibriumconditions on the axis of symmetry connecting the parameters α i , K i ( i = 1 , . . . ,
4) be-tween each other. Particular solutions of the algebraic system involving numerical resultswere discussed by Hoenselaers [14], who came to conjecture that the double-Kerr-NUT solu-tion cannot describe the equilibrium between two aligned rotating black holes with positiveKomar masses. Hoenselaers and Dietz [9, 13] and Krenzer [19] were able to prove this con-jecture for symmetric configurations K − K = K − K , Ω = Ω . The explicit solution ofthe Tomimatsu-Kihara equilibrium conditions was found by Manko et al. [20, 21]. Finally,Manko and Ruiz [21] were able to prove Hoenselaers’ conjecture. The results derived byManko and collaborators are important steps toward a non-existence proof. In particular,we will make use of their solution of the equilibrium conditions. Before using these results,we had to formulate and analyze a boundary problem for disconnected horizons, cf. [31],since a non-existence proof cannot be based on an arbitrarily chosen solution, even thoughthis solution (here: the double-Kerr-NUT solution) seems to be a promising candidate. For-tunately (or, as expected) the analysis of the boundary problem led to the double-Kerr-NUTsolution.There is another critical point in the argumentation of Hoenselaers and Dietz and Mankoet al. To the best of our knowledge there is no argument in favor of the positiveness of theindividual Komar masses of interacting bodies and black holes. On the contrary, Ansorg andPetroff [1] have given convincing counterexamples. We replace the Komar mass inequality(“positivity of the Komar mass of each black hole”) by an inequality connecting angularmomentum and horizon area [11]; see Sec. VI. This relation is based on the geometry oftrapped surfaces in the interior vicinity of the event horizon.Our next goal is the reformulation of the constraints (69) in terms of the new parameters α i ( i = 1 , . . . , Φ ( ̺, ζ , λ ) as a solution of the LP (19), whose coefficients N ( ̺, ζ ), It should be noted that these solutions can also be written as a quotient of two determinants in completeanalogy to (89); see [17] or [16]. M ( ̺, ζ ) are explicitly known by (89) and (23). The integration of the LP yields (cf. [26]) χ ( ̺, ζ , λ ) = 1 K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K K K K λK ( K + i z ) α K r α K r α K r α K r K K K K K λ ( K + i z ) α r α r α r α r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K K K − α K r α K r α K r α K r K K K K α r α r α r α r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (92)where r i = p ( ζ − K i ) + ̺ ≥ i = 1 , . . . ,
4. Note that the remaining elements of Φ can easily be constructed from χ ( ̺, ζ , λ ) = Φ : Φ = ψ ( ̺, ζ , λ ) = ¯ χ ( ̺, ζ , ¯ λ − ) , see(26), Φ = ψ ( ̺, ζ , − λ ), Φ = − χ ( ̺, ζ , − λ ). The straightforward verification of (92) byinserting Φ ( ̺, ζ , λ ) into the LP (19) with coefficients (23) and (89) is laborious. Instead[26], one realizes that Φ ,z Φ − , for fixed values of ̺ , ζ is regular in the λ -plane with theexception of a simple pole of first order at λ = ∞ . From Liouville’s theorem one mayconclude that Φ ,z Φ − = P + λ Q , where the 2 × P ( ̺, ζ ), Q ( ̺, ζ ) do not dependon λ . As a consequence of (25), P becomes diagonal and Q off-diagonal with Q = P and Q = P such that one arrives at the first equation (19). From this equation one obtainsthe second equation (19) by using (26). Note that the Ernst potential (89) follows from(92), f ( ̺, ζ ) = χ ( ̺, ζ ,
1) ( K → ∞ ) as expected; see (28).Having discussed some implications of the reparametrization for the representation ofthe Ernst potential and the pseudopotential Φ , we will now formulate the constraints interms of α i and K i , i = 1 , . . . ,
4. In particular, we will show that representation (88) of f + ( ζ ) together with the boundary conditions (16), (17) for the gravitomagnetic potential a is equivalent to representation (71) with tr r + = 0 ( ⇔ tr R + = 0).Denote the pseudopotential Φ constructed from (92) by Φ α and equip all quantitiesderived from Φ α with an index α . Since Φ α is an integral of the LP (19), its values onthe axis and horizons have the form (31). They are continuous at the points of intersection A + / H (1) , H (1) / A , A / H (2) , H (2) / A − . This may directly be verified in (92) and for the otherelements of the matrix Φ α . The introduction of rotating systems of reference in IV A andcorotating quantities such as the corotating pseudopotential Φ ′ are closely connected withthe definition of the Killing horizon, which has a well-defined angular velocity in contrastto the intervals [ K , K ] and [ K , K ] as “potential” horizons. However, one can introducedifferent corotating systems of reference at the ends of each “potential” horizon with angularvelocities Ω ( i ) , i = 1 , . . . ,
4, as defined in (78) and (79), Ω (1) = (cid:2) a + α − a (1) α (cid:3) − , Ω (2) = (cid:2) a α − a (1) α (cid:3) − , Ω (3) = (cid:2) a α − a (2) α (cid:3) − , Ω (4) = (cid:2) a − α − a (2) α (cid:3) − , (93) Due to (20), K is a rational function of λ and can be replaced in all elements of Φ to obtain Φ ( ̺, ζ, λ ) asa polynomial of fourth degree in λ . λ → / ¯ λ implies K → ¯ K and vice versa. a + α = a α = a − α . Thus one obtains L + α (cid:18) (cid:19) ( L + α ) − = R + α := Y i =1 (cid:18) − ( − i F i Ω ( i ) ( K − K i ) (cid:19) (cid:18) (cid:19) , (94)where f i = f ( ̺ = 0 , ζ = K i ), i = 1 , . . . ,
4, entering F i according to (45), can be taken from(89). A direct consequence of these equations istr R + α = 0 (95)which may be compared with the constraints (49). Obviously, the constraints tr R + = 0 aresatisfied if Ω (1) = Ω (2) , Ω (3) = Ω (4) , i.e. if a + α = a α , a − α = a α ,Ω = Ω (1) = Ω (2) = (cid:2) a + α − a (1) α (cid:3) − , Ω = Ω (3) = Ω (4) = (cid:2) a − α − a (2) α (cid:3) − . (96)Conversely, we have shown that the representation (71) of f + ( ζ ) together with the con-straints tr r + = 0 = tr R + implies conditions (80). Thus we may conclude that the (four)conditions (96) are a reformulation of the (four) constraints (64). According to (16), (17)the conditions (96) are necessary conditions for the equilibrium of the two-black-hole con-figuration. Hence, constraints (64) and restrictions (96) are equivalent formulations of theequilibrium conditions to the respective Ernst potentials .The explicit form of the axis equilibrium conditions (96) can be evaluated by using (29)or (74). In both cases, one may start with expression (92) that determines all elements of Φ α (e U = ℜ f in (29) can be taken from (89)). Equation (29) yields the gravitomagneticpotential a ( ̺, ζ ) everywhere including the axis intervals I whereas equation (74) directlyleads to the axis values of a I , I = A ± , A , H (1 / ( B I ( K ) may be taken from representation(31)). Straightforward calculations result in a I + C = − i H I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K K K K K K K π I π I π I π I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (97)where H I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ I α K λ I α K λ I α K λ I α K K K K K λ I α λ I α λ I α λ I α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (98)with π Ik = Q n =1 (1 + λ In α n )1 + λ Ik α k , λ Ik = | K k − ζ | K k − ζ , I = A ± , A , H (1 , , i = 1 , . . . , . (99) Henceforth we omit the index α . λ Ik = ± I , e.g., I = A , K ≤ K ≤ ζ ≤ K ≤ K :( λ k ) = (1 , , − , − one obtainsthe angular velocities Ω , Ω of the horizons H (1 / in terms of the parameters α i , K i , i =1 , . . . , a + = a , a − = a ) that restrict the choice of these parameters.These restrictions must be taken into account when examining the Ernst potential (89). D. Discussion of the solution
We have shown that the solution of the boundary problem (16)-(18) for the Ernst equation(14) is given by the Ernst potential (89), whose constant parameters α i , K i , i = 1 , . . . , a + = a , a − = a with a ± , a from (97). ( Ω , Ω can be calculated straightforwardly.) The appearance of the equilibriumcondition is not promising and makes a comprehensive discussion of the solution difficult.We will only list a few aspects of the interpretation.
1. Number of parameters
Written in dimensionless coordinates such as˜ ̺ = ̺K , ˜ ζ = ζ − K K , K = K − K , (100)the Ernst potential contains four free parameters. Since the quotient of determinants (89)remains unchanged under a translation of the two-black-hole configuration along the ζ -axis and a multiplication of ̺ , ζ and K i by a common (real) factor, f depends on thecoordinates ˜ ̺ , ˜ ζ and the six parameters α , . . . , α ; K /K , K /K , where we have usedthe abbreviation K ij = K i − K j , i, j = 1 , . . . , . (101)It can easily be seen that the two conditions a + = a , a − = a can be rescaled to be writtenin terms of the six parameters alone. Hence f = f ( ˜ ̺, ˜ ζ ) is a four parameter solution.
2. Singularities outside the horizons
The solution is a necessary consequence of the integration of the LP along the boundary(closed dashed line in Fig. 1). We have (implicitly) assumed the validity of the Ernstequation everywhere in the enclosed ̺ - ζ domain. It is, however, not clear, whether theErnst potential is really free of singularities there. As matters stand at present the questionmust remain undecided. Interestingly, the solution of the static two-horizon problem on thelevel of the Laplace equation ∆U = 0 (which is the static form of the Ernst equation) hasno singularities outside the horizons. (Applying the inverse formalism one simply obtains asuperposition of two Schwarzschild solutions in Weyl-Lewis-Papapetrou coordinates). The We omit α as arranged. A that forbids the existence of static two-black-hole configurations only appears when the metric coefficient e k is involved. Calculatedfrom a regular solution of the Laplace equation (the “double-Schwarzschild” solution), e k violates the regularity condition e k = 1 on A ; see (9). We will discuss this condition (whichensures elementary flatness on the axis of symmetry) in the next section.
3. Generalizations
Obviously, our analysis of the two-horizon problem can easily be extended to an arbitrarynumber n , n >
2, of aligned disconnected horizons. Integrating the LP (19) along the 2 n + 1intervals I ( n horizons H and n + 1 “regular” intervals A ) one arrives at a representationof the form (31) for each of the 2 n + 1 intervals. Consequently, one has 2 n + 1 matrices L I and 2 n matrices F i ; cf. (45). Replacing the symbol Q i =1 in (47) by Q ni =1 one obtains,via (53) and by (56), a representation for the Ernst potential f + ( ζ ) on A + . The rationalstructure of this potential ( f + is a quotient of two normalized polynomials of equal degreein ζ ) is a characteristic feature of solutions to the Einstein equations derived by iterativeB¨acklund transformations of the metric of the Minkowski space [16], [24], or, equivalently,by the Belinski-Zakharov approach [5]. This confirms Varzugin’s result [31] which says thatany equilibrium configuration of aligned black holes can be described by a Belinski-Zakharovsolution [5]. After a reparametrization in full analogy to (81), f ( ̺, ζ ) again turns out to be aquotient of (2 n +1) × (2 n +1) determinants whose structure is an obvious generalization of thedeterminants in (89). Finally, the equilibrium conditions can be derived straightforwardly. V. ELEMENTARY FLATNESS AND EQUILIBRIUM CONDITIONS
We know from the Bach-Weyl paper [2] that the metric coefficient e k is a measure forthe interaction of the two black holes. To guarantee equilibrium, k has to vanish on theportion of the axis of symmetry between the two black holes. From a geometrical point ofview, the condition e k = 1 on A is a necessary condition for elementary flatness (Lorentziangeometry) of spacetime in the vicinity of the rotation axis A . Our discussion of the metricpotential k = k ( ̺, ζ ) is based on Kramer’s representation [18], which is a result of theintegration of the defining relation (15) with f ( ̺, ζ ) from (89). It turns out that k ( ̺ = 0 , ζ )is a step function with constant values on the intervals A ± , A . In particular, one hase k + = e k − . Equation (15) determines k up to an additive constant, which can be chosensuch that k + = 0 with the consequence e k − = 1. For this choice, e k takes the forme k = 1 + (cid:18) H H + − (cid:19) (cid:18) H H + + 1 (cid:19) (102)with H I , I = A , A + as in (98). Obviously, the equilibrium condition e k = 1 has twosolution branches, H = ± H + . Tomimatsu and Kihara [30] ruled out the condition H = H + ( H = H + together with a + = a − , a − = a leads to overlapping horizons). H = − H + yields α α + α α = 0 (103)1and effects a considerable simplification of the equilibrium conditions (96). Putting (97)-(99)in a + = a , a − = a and observing H + = − H one finds α (1 − α ) ( K + K ) − (1 − α α ) α K K = 0 α (1 + α ) ( K + K ) + (1 − α α ) α K K = 0 , (104)where K ij = K i − K j , α ij = α i − α j K i − K j . (105)For point-like horizons K = K or/and K = K , the parametrization (83) does not ap-ply, since the mapping of the four coefficients q , r , s , t in (73) onto less than four parameters α i ( α = α or/and α = α ) is not invertible. However, the invertibility can be restored byintroducing the derivatives α ′ ( ζ ), β ′ ( ζ ) in the confluent points K = K or/and K = K .As was shown in [12] this concept makes it possible to consider the equilibrium conditionsfor configurations with degenerate horizons as particular cases of (103), (104): To describepoint-like configurations one has to set K = K , α = α , α = α ′ ( K ) = − i α γ or/and K = K , α = α , α = α ′ ( K ) = − i α γ , (106)where γ and γ are real constants.To introduce α and α in the Ernst potential (89), one has to subtract the secondcolumn from the third one and the fourth column from the fifth one and to decompose the α -parameters into symmetric and antisymmetric parts (e.g. α = ( α + α ) + ( α − α ), α = ( α + α ) − ( α − α )).We are now prepared to discuss all constellations (extended/extended, extended/point-like, point-like/point-like) by using (89), (103) and (104). VI. BLACK HOLE INEQUALITIES AND SINGULARITIESA. Two sub-extremal black holes
We start the discussion of the different types of possible two-black-hole equilibrium con-figurations by considering spacetimes containing two black holes with extended horizons.Following Both and Fairhurst [6], we will assume that a physically reasonable non-degenerateblack hole should be sub-extremal , i.e. characterized through the existence of trapped sur-faces (surfaces with a negative expansion of outgoing null geodesics) in every sufficientlysmall interior neighborhood of the event horizon. As shown in [11], the presence of trappedsurfaces implies the inequality 8 π | J | < A between angular momentum J and horizon area A of the black hole. In a regular spacetime with two sub-extremal black holes, both blackholes have to satisfy this inequality individually,8 π | J i | < A i , i = 1 , . (107)This is the key ingredient for the non-existence proof of two-black-hole equilibrium config-urations, as we will see below.In order to test these inequalities for that subclass of the double-Kerr-NUT family ofsolutions which describes two gravitating objects with extended horizons, we have to solve2the general axis regularity conditions (103), (104). For K > K > K > K these conditionscan be written as α α + α α = 0 (108)and (1 − α ) α w = (1 − α ) α , w := r K K K K ∈ [1 , ∞ ) , (1 + α ) α w ′ = (1 + α ) α , w ′ := r K K K K ∈ (0 , . (109)As shown by Manko et al. [21], these equations can be explicitly solved for α , . . . , α , α = w ′ α + i εαw ′ − i εα , α = α + i w ′ εα − i w ′ εαα = wα − αw − α , α = α − wα − wα , (110)where α := √− α α = √ α α =: e i φ , α ¯ α = 1 and ε = ± p i := 8 πJ i A i , i = 1 , , (111)in terms of the parameters w , w ′ and φ ∈ [0 , π ). The result is p = ε Φw ′ w ′ ( Φ + w ′ ) , p = ε w ( w − Φ )1 − wΦ (112)with Φ := cos φ + ε sin φ, ε = ± . (113)Rewriting the inequalities in (107) as p i < i = 1 ,
2, we find with the previous formulaethe two conditions w ′ + 2 Φw ′ + 1 < w − Φw + 1 < . (114)Since the latter inequalities imply Φw ′ < Φw >
0, we arrive at a contradiction, because w and w ′ are positive. In this way, we have shown that two sub-extremal black holes cannotbe in equilibrium. B. One sub-extremal and one degenerate black hole
Without loss of generality, we may assume that the upper horizon is point-like ( K = K )and the lower one is extended ( K > K > K ) as sketched in Fig. 1. Then, the equilibriumconditions (103), (104) can be written as α + α α = 0 , (115)with the solution α = i εα, α α = α , (116)3where ε = ± α ∈ C , α ¯ α = 1, and(1 − α ) w = α − α α , ( α − i ε ) [ γ K w ( εα + i) + i( w + 1)( εα − i)] = 0 . (117)As solution to these equations we obtain two sets of parameters. In the first solution branch,the parameters α , γ , α and α have to be chosen according to α = i εα, γ = i( w + 1) wK i − εα i + εα , α = wα − αw − α , α = α − wα − wα , (118)which is the limit K → K ( ⇔ w ′ →
1) of solution (110) for extended horizons. Thisfamily of solutions depends on the two parameters α and w (and on two additional scalingparameters, e.g. K and K ).The second solution branch of the equilibrium conditions is given by α = − , γ ∈ R , α = 1 − i εw εw , α = − εw − i εw , (119)i.e. the corresponding Ernst potential depends on the two parameters γ and w (plus twoscaling parameters). Interestingly, this solution has no counterpart in the case of extendedhorizons.The desired non-existence proof follows the same idea for both families: The ADM mass M of the spacetime can be expressed in terms of the two parameters (and the additionalscaling parameter K ). The resulting expressions can be estimated, using the inequality8 π | J | < A for the sub-extremal object. We obtain M <
C. Two degenerate black holes
In the case of possible equilibrium configurations with two degenerate black holes , wefind three one-parametric families of candidate solutions: The equilibrium conditions arenow α + α = 0 , (121)which is solved by α = i εα, α = − α, (122)with ε = ± α ∈ C , α ¯ α = 1, and( α + 1) [( α − γ K − α + 1)] = 0 , ( α − i ε ) [ γ K ( εα + i) + 2i( εα − i)] = 0 . (123) See also [7] for a discussion of properties of spacetimes with two degenerate objects. α = i εα, α = − α, γ = 2i K · εα − i εα , γ = 2i K · α + 1 α − α . This solution can be obtained in the limit K → K , K → K ( ⇔ w → , w ′ →
1) from (110).The second and third solution branches are α = − i ε, α = 1 , γ = 2 εK , γ ∈ R , (125)and α = − , α = − i ε, γ ∈ R , γ = 2 εK , (126)where now γ or γ are free parameters.In order to show that the above solutions do not lead to regular two-black-hole configu-rations, we calculate the ADM mass M . For the first solution branch, the result is M = − K √ (cid:0) εφ + π (cid:1) . (127)Obviously, M is always negative and we arrive again at a contradiction to the positive masstheorem.For the second and third solution branches, M has the form M = − K · ˜ γ − γ − , ˜ γ = εγ K . (128)Hence, the mass is negative for ˜ γ < γ > M > γ ∈ [1 , M > p ( A + A ) / π . Since this inequality isrelated to cosmic censorship, this might indicate that these configurations are not regularoutside the two gravitational sources. However, since so far no rigorous proof of the Penroseinequality for axisymmetric configurations was given, this does not yet exclude the possibilitythat two sub-extremal black holes can be in equilibrium. Instead, one can directly show thatthese solutions always suffer from the presence of singular rings; see Appendix A in [12].Therefore, this solution branch can be excluded, too. VII. SUMMARY
As a characteristic example for the present discussion about existence or non-existence ofstationary equilibrium configurations within the theory of general relativity, we have studiedthe question whether two aligned black holes can be in equilibrium. We have shown how thisquestion can be reformulated in terms of a boundary value problem for two disconnectedKilling horizons with specific boundary conditions at the horizons, on the axis of symmetryand at infinity. Using the Ernst formulation of the Einstein equations, it was possible toapply methods from soliton theory which allow us to study an associated linear problem(LP) instead of the non-linear field equations themselves.5By integrating this LP along the boundaries of the vacuum region (along the axis, thehorizons and at infinity) we found an explicit expression for the Ernst potential f + on theupper part of the symmetry axis A + (see Fig. 1). In particular, it turned out that f + is aquotient of two normalized polynomials in ζ (the “axis coordinate”), which depends on anumber of parameters. In addition, the solution of the LP led to constraints between theseparameters which we have shown to be equivalent to the “equilibrium condition” (regularitycondition) a = 0 ( a : gravitomagnetic potential) on the axis of symmetry. By analyzing theconstraints, we found that the Ernst potential f + is a quotient of two polynomials of seconddegree in ζ and hence all possible equilibrium configurations with two black holes wouldnecessarily belong to the double-Kerr-NUT family of solutions.Discussing the three possible configurations(i) two sub-extremal black holes(ii) one sub-extremal and one degenerate black hole(iii) two degenerate black holeswhich are characterized by specific restrictions of the parameters (“constraints”) of thedouble-Kerr-NUT solutions, we could show that none of them does correspond to a physicallyreasonable black hole solution: All solution families (i), (ii) and (iii) suffer from the presenceof naked singularities, which manifests in the violation of physical black hole inequalities (thepositivity of the ADM mass or inequality (107) between angular momentum and horizonareas for sub-extremal black holes). Hence we can conclude that two-black hole equilibriumconfigurations do not exist . Acknowledgments
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