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Black Hole-Naked Singularity Dualism and the Repulsion of Two Kerr Black Holes Due to Spin-Spin Interaction ∗ V.S

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Black Hole-Naked Singularity Dualism and the Repulsion of Two Kerr Black Holes Due to Spin-Spin Interaction ∗ V.S Physics Letters B 791 (2019) 26–29 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Black hole-naked singularity dualism and the repulsion of two Kerr black holes due to spin-spin interaction ∗ V.S. Manko a, , E. Ruiz b a Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, A.P. 14-740, 07000 Ciudad de México, Mexico b Instituto Universitario de Física Fundamental y Matemáticas, Universidad de Salamanca, 37008 Salamanca, Spain a r t i c l e i n f o a b s t r a c t Article history: We report about the possibility for interacting Kerr sources to exist in two different states – black holes Received 22 October 2018 or naked singularities – both states characterized by the same masses and angular momenta. Another Received in revised form 14 February 2019 surprising discovery reported by us is that in spite of the absence of balance between two Kerr black Accepted 18 February 2019 holes, the latter nevertheless can repel each other, which provides a good opportunity for experimental Available online 20 February 2019 detection of the spin-spin repulsive force through the observation of astrophysical black-hole binaries. Editor: M. Cveticˇ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 Keywords: (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . Double-Kerr solution Spin-spin interaction Binary black hole Naked singularity 1. Introduction and is based on the recent results [9] obtained for the configu- rations of two equal Kerr sources. We remind that in Ref. [9]it The recent direct detection of gravitational waves by the LIGO has been established that in a highly nonlinear regime the masses and Virgo teams [1] has aroused much interest in the far-reaching and angular momenta may cease to define a binary configura- study of binary black hole systems. Though the numerical simula- tion uniquely, but the only example given in [9]treated the case tions are obviously the main tool for modeling the generic mergers of hyperextreme sources, not black holes. Lately nonetheless we of two black holes, the analytical results have also been obtained have eventually succeeded in finding physically meaningful exam- under some simplifying assumptions [2]. It appears that an im- ples of nonuniqueness for the black hole constituents too, and the portant physical information pertinent to the case of the head- remarkable feature of the newly found binary black-hole configu- on collisions of black holes was also supplied by the well-known rations is that the black holes in them can repel each other due to double-Kerr solution [3]– the two black holes cannot be in sta- prevailing of the spin-spin repulsion over gravitational attraction, tionary equilibrium [4,5], the usual interpretation for which (see which prevents the two black holes from merging into a single the conjecture [6]on the positiveness of the interaction force) is black hole. Moreover, we have discovered that the nonuniqueness that the spin-spin repulsion in a binary black hole system is in- also includes a surprising phenomenon when the same mass and sufficient to counterbalance the gravitational attraction. This could angular momentum determine simultaneously a black-hole and a probably make one think that the end state of a pair of collid- hyperextreme configuration of two Kerr sources, which allows to ing Kerr black holes must inevitably be a single rotating black speak about the black hole-naked singularity (BH/NS) dualism as a hole [7] whose mass and angular momentum would be equal to new highly nonlinear effect occurring in the binary systems sub- the aggregate masses and angular momenta of the colliding con- ject to a very strong gravity. We now turn to the details. stituents, as was surmised by Hawking [8]long ago. However, as will be shown in the present letter, a different scenario of the dy- 2. The configurations of two equal Kerr sources namical evolution of a binary black hole system is still possible The family of two equal Kerr sources kept apart by a massless strut is described by the metric [9] Corresponding author. * 2 −1 2γ 2 2 2 2 2 E-mail address: vsmanko@fis.cinvestav.mx (V.S. Manko). ds = f [e (dρ + dz ) + ρ dϕ ]− f (dt − ωdϕ) , https://doi.org/10.1016/j.physletb.2019.02.025 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. V.S. Manko, E. Ruiz / Physics Letters B 791 (2019) 26–29 27 Fig. 1. Two possible configurations of a pair of equal nonextreme Kerr sources: (a)ablack-hole binary, (b) a system of two naked singularities. A A¯ − B B¯ A A¯ − B B¯ = 2γ = f ¯ ¯ , e ¯ , (A + B)(A + B) 16λ0λ0 R1 R2 R3 R4 2Im[G(A¯ + B¯ )] Fig. 2. The curved triangle separating the unique configurations with particular m and a from those for which the same particular values of mass and angular momen- ω = ω0 − , A A¯ − B B¯ tum are shared by three different configurations (the latter lie inside the triangle). 2 The dashed line separates the configurations with positive and negative mass. A = (R1 − R2)(R3 − R4) − 4σ (R1 − R3)(R2 − R4), 3 − 2 + ¯ − = ≡ 2 B = 2sσ [(1 − 2σ )(R1 − R4) − (1 + 2σ )(R2 − R3)], Z pZ pZ 1 0, Z φ , + + 2 G =−zB + sσ [2R R − 2R R − 4σ (R R − R R ) m 2 i(2m 1) 1 3 2 4 1 2 3 4 p ≡ px + ip y =− − , (4) m ma −s(1 − 4σ 2)(R − R − R + R )], (1) 1 2 3 4 while the dependence ν(m, a) is determined by the equation where the functions R are defined by the expressions i i[m(φ2 + 1)2 + 2φ2] ν =− , (5) 2 − 2 2 mφ(φ 1) Ri = Xi ρ + (z − αi) , into which the explicit form of φ should be inserted after the res- 2 X1 =−1/X4 = φ(μ + μ − 1), olution of equation√ (4), taking into account that the correct choice of sign in φ =± Z must ensure ν > 0. Obviously, for m and a the 2 X2 =−1/X3 = φ(μ − μ − 1), following relations hold: − 2 1 1 =− 2 = (px 3) α1 =−α4 = s + σ , α2 =−α3 = s − σ , m , a . (6) 2 2 p + 1 2p (p + 1) x y x i μ2 − 1 As has been established in [9], equation (4)may have, for given σ =− , ν values of m and a, up to three unitary roots, and the nonunique- ness zone is defined by the “tanga curve” depicted in Fig. 2, inside (φ2 + 1)[φ(ν2 − 4) + iν(φ2 − 1)] μ = , (2) of which (and on the curve itself) the values of m and a do not 2[(φ2 + 1)2 − iνφ(φ2 − 1)] lead to unique configurations of the Kerr constituents. The only and the constants λ0 and ω0 have the form example of nonuniqueness considered in [9]involved the hyper- extreme Kerr sources because at that time we did not yet know 1 2 2 2 2 2 how to extend our analysis to the black-hole case. The difficulty λ0 = (1 − μ )[(1 + φ ) − ν φ ], ν2 that exhibits the latter case is the need, in addition to finding 2is(1 − φ4 − 2iμνφ2) configurations with real positive m and σ , to satisfy also the con- = . (3) 1 ω0 2 2 2 2 2 (1 + φ ) − ν φ dition σ < 4 ensuring the separation of the black holes on the symmetry axis. The strategy that eventually allowed us to get the The above formulas (1)-(3)involve three arbitrary parameters, desired two different binary black-hole configurations defined by which are the coordinate distance s between the centers of the the same values of m and a consisted in first identifying one Kerr constituents (see Fig. 1), the real positive constant ν, and physically meaningful black-hole configuration on the tanga curve ¯ = the unitary complex parameter φ, so that φφ 1; the black-hole (where only two unitary Z are possible) and then searching the configurations correspond to real positive σ , while the pure imag- roots of the cubic equation (4)inside the curved triangle which inary σ describe the binary systems of hyperextreme Kerr objects would be slightly different from the boundary values. Below we (naked singularities). Instead of ν and φ, however, one may use give the numerical value of p found in this way which determines the dimensionless individual mass m and angular momentum per two different black-hole configurations and one hyperextreme con- unit mass a of each constituent (these are related to the total di- figuration of two Kerr constituents, all three configurations sharing = = 2 mensional quantities by the relations MT 2ms, J T 2mas ), and the same mass and angular momentum: finding φ in terms of m and a then reduces to solving the cubic equation p =−1.146560133416505 + 2.152383532475482i. (7) 28 V.S. Manko, E. Ruiz / Physics Letters B 791 (2019) 26–29 The corresponding values of m and a obtainable from (6)are the black hole or naked singularity – does not really have any seri- following: ous effect on the interaction force in a binary configuration if the subextreme and hyperextreme constituents have the same masses m = 13.64628, a =−27.25276, (8) and angular momenta and are separated by the same coordinate distance. and these are given up to five decimal places (like all the re- maining quantities hereafter).
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