PHYSICAL REVIEW D 100, 044033 (2019)

Generalized near horizon extreme binary geometry

† ‡ Jacob Ciafre,1,* Shahar Hadar,2,4, Erin Rickenbach,1, and Maria J. Rodriguez1,2,3,§ 1Department of Physics, Utah State University, 4415 Old Main Hill Road, Utah 84322, USA 2Max Planck for Gravitational Physics-Albert Einstein Institute, Am Muhlenberg 1, Potsdam 14476, Germany 3Instituto de Física Teórica IFT-UAM/CSIC, C/ Nicolás Cabrera 13-15, Universidad Autónoma de Madrid, 28049 Madrid, Spain 4Center for the Fundamental Laws of Nature, Harvard University, Cambridge, Massachusetts 02138, USA

(Received 27 June 2019; published 15 August 2019)

We present a new vacuum solution of Einstein’s equations describing the near horizon region of two neutral, extreme (zero-temperature), corotating, nonidentical Kerr black holes. The metric is stationary, asymptotically near horizon extremal Kerr (NHEK), and contains a localized massless strut along the symmetry axis between the black holes. In the deep infrared, it flows to two separate throats which we call “pierced-NHEK” geometries: each throat is NHEK pierced by a conical singularity. We find that in spite of the presence of the strut for the pierced-NHEK geometries the isometry group SLð2; RÞ ×Uð1Þ is restored. We find the physical parameters and entropy.

DOI: 10.1103/PhysRevD.100.044033

I. INTRODUCTION hypothesizes that the Kerr BH is dual to a (1 þ 1 dimen- sional) conformal field theory (CFT) living on the boun- Rapidly rotating, (near-)extreme Kerr black holes (BHs) dary of this near horizon geometry. This boundary can be constitute a unique arena which offers both observational thought of as the region in which the NHEK relevance and enhanced theoretical control. Several high- geometry is glued to the external, asymptotically flat, Kerr spin candidates (cf. [1–4]) have been observed, and such spacetime. BHs could produce characteristic signatures for various The NHEK geometry has a simpler cousin—the current and future experiments, including gravitational- 2 Robinson-Bertotti universe or AdS2 ×S. This spacetime wave detectors such as LIGO/Virgo, and optical observa- arises as an analogous near-horizon limit of maximally tories such as the recently triumphant [5] Telescope. Theoretically, (near-)extreme BHs are espe- charged Reissner-Nordström BHs. This type of BHs can be cially tractable since they develop an emergent conformal used to construct, remarkably simply, multi-BH configu- symmetry. More precisely, they admit a nondegenerate rations [8]. Those are static solutions to Einstein-Maxwell near-horizon geometry, the so-called near-horizon extreme theory with an arbitrary number of maximally charged (all Kerr (NHEK) geometry [6]. This geometry is interesting: with the same sign), nonrotating BHs of any mass. The every fixed polar angle slice of it can be thought of either as time-independence of these solutions is possible since the BHs’ gravitational attraction and electric repulsion cancel 2-dimensional anti–de Sitter space (AdS2) with a circle — — nontrivially fibered upon it or (equivalently) as a quotient of each other precisely in the full nonlinear theory for arbitrary BH positions. A neat observation regarding these the so-called warped AdS3 spacetime. Consequently it enhances the isometry group of Kerr, R × Uð1Þ (corre- solutions was made in [9]. Consider a system of two such sponding to stationarity and axisymmetry), to SLð2; RÞ× maximally charged BHs. When they are widely separated, 1 there exist also well-separated near-horizon (approximately Uð Þ. This motivated the Kerr/CFT conjecture [7], which 2 AdS2 ×S ) throats surrounding each one of the BHs. When the BHs are close to each other (relative to a length scale *[email protected] † defined by a characteristic mass), however, there exists a [email protected] ‡ 2 [email protected] region around them which is approximately an AdS2 ×S §[email protected]; [email protected] throat which surrounds both horizons, and only when moving further towards either one of the horizons does Published by the American Physical Society under the terms of one recover the two separate throats. This phenomenon was the Creative Commons Attribution 4.0 International license. “ ” Further distribution of this work must maintain attribution to coined in [9] AdS fragmentation : the joint throat frag- the author(s) and the published article’s title, journal citation, ments into two smaller ones, when moving deeper into the and DOI. Funded by SCOAP3. infrared. This generalizes to an arbitrary number of throats:

2470-0010=2019=100(4)=044033(9) 044033-1 Published by the American Physical Society CIAFRE, HADAR, RICKENBACH, and RODRIGUEZ PHYS. REV. D 100, 044033 (2019) one “trunk” throat can fragment into several branches [14]. It should be possible to generalize our construction which can then branch again, and so forth. to an arbitrary number of BHs with arbitrary masses. This compelling picture depends strongly on the proper- The workhorses of this paper are the binary BH solutions ties of the special system of choice. The fact that it can be first found in [15] and further studied, including their embedded in a supersymmetric theory, as a solution which construction via various solution generating techniques in preserves some supersymmetry [10], guarantees this type of [16–21]. These exact solutions are stationary, axisymmet- behavior. In this paper, we propose the closest possible ric, asymptotically flat solutions which describe two rotat- analogue, presumably, to fragmentation in the case of ing BHs held apart by a strut along the symmetry axis. The maximally rotating, uncharged BHs. Since these are not BHs of these solutions can have arbitrary masses and spins supersymmetric anymore and there is no known smooth and in particular can be either co- or counterrotating. We stationary solution involving such BHs, we allow for a are interested in the case in which the BHs are maximally conical singularity between the BHs which balances the corotating, with arbitrary masses. In particular, we start gravitational attraction and keeps the system stationary. We from the corotating solution described in [22], and for the study a 1-parameter family of exact axis-symetric solutions convenience of the interested reader we describe it explic- describing two corotating extreme Kerr BHs of arbitrary itly in the so-called Weyl-coordinates in Appendix A. This masses which are held apart by a conical singularity with coordinate choice serves best to describe classes of sta- effective pressure, usually called a strut and as we rescale tionary and axisymmetric solutions of Einstein’s theory of coordinates to zoom-in on the near-horizon region, we also in vacuum. shorten the strut separating the BHs. In this way we construct The rest of this paper is organized as follows. We first the exact solution corresponding to the region where NHEK construct the new generalized near horizon geometry of the fragments into two NHEK-like throats which are held apart stationary binary extreme-Kerr BH solution in Sec. II and by the strut. We call these “NHEK2” geometries. The analyze its physical properties. In particular, we show how it solution presented here generalizes [11], which studied a admits a localized strut along the symmetry axis between the similar construction for the equal-mass case. These infrared black holes but is asymptotically NHEK. In Sec. III we zoom- near-horizon geometries which the strut pierces on its way to in further to the infrared of each throat, and find the near- the horizons are analogues of NHEK which include a conical horizon geometries in which the strut pierces the horizons, singularity at one of the poles, extending from the horizon all extending from thehorizon all thewayto theNHEKboundary. the way to the NHEK boundary. We verify that this does not We show that in spite of the strut, the pierced-NHEK geo- “ ” ruin the symmetry structure: the pierced-NHEK geometry metries have an SLð2; RÞ ×Uð1Þ isometry group. Finally, we 2 1 still has an SLð ; RÞ ×Uð Þ isometry group. So while the will summarize the key results of the paper in Sec. IV. full NHEK2 does not have SLð2; RÞ ×Uð1Þ, it interpolates from a geometry that does have conformal symmetry in the II. GENERALIZED-NHEK2: GENERALIZED NEAR ultraviolet to two throats that are also conformally sym- HORIZON GEOMETRY OF EXTREME BINARY metric, in the infrared. KERR BLACK HOLES SOLUTION Introducing conical singularities has caveats which are important to stress. First, the stability, both classical and In this section, we construct the generalized near horizon quantum mechanical, of these solutions is questionable. geometry of extreme binary Kerr (Generalized-NHEK2) A second point is that the type of conical singularities we black hole solution. Our starting point, is the stationary use here, the struts, are of excess angle type (rather than solution to Einstein equations in vacuum [22] that contains deficit angle); the effective stress-energy associated to two extremal (zero-temperature) corotating black holes. such objects has negative energy density. Keeping these For convenience and for fixing the notation, we reproduced caveats in mind, we still hope that this construction may the original results of [22] in Appendix A. We will only be useful in various contexts. First, such stationary BH consider the solutions characterized by positive values of binary solutions have been recently applied to study the mass that correspond to the parameter range astrophysically motivated problems involving dynamical binaries (see for example [12] for the use of quasistationary, 1 − pffiffiffi ≤ p<0;q>0;q

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A. Near-horizon limiting procedure   β0ðβ1 − K1Þ 2 π0 ¼ 1 − 4Δ0β0 þ β1 þ 4x In previous works [11] we developed the necessary tools K0 to inspect the extreme corotating binary Kerr black hole pffiffiffi − 2 1 2 − 1 2 solution. This section is nevertheless self-contained. We ðPxð þ x Þ Qyð þ y ÞÞ ð10Þ proceed to compute the near horizon geometry of extremal ffiffiffi ffiffiffi p p 2 2 nonidentical binary Kerr black holes solution, that we are þð 2ðβ0P þ α0QÞx − 2ðα0P þ β0QÞy − β3Þðx − y Þ going to refer to as “generalized-NHEK2”. ð11Þ The solution of extremal BBHs [22]—that we repro- duced in Appendix A—has a rather more compact repre-   α0ðβ1 − K1Þ sentation in Weyl coordinates. We therefore perform the þ 4Δ0α0 − α1 − 4xy; ð12Þ scaling computations in these coordinates. In this case, we K0   find that the appropriate near-horizon limiting procedure ffiffiffi p 2 Q 2 2 for the extremal BBHs is τ0 ¼ 2ðPx þ QyÞðx − 1Þþ α3 þ pffiffiffi x ðx − y Þ 2α   0 tˆ 1 tˆ ρ ¼ ρλˆ ;z¼ zˆλ;t¼ ; ϕ ¼ ϕˆ þ ; ð2Þ − α Qffiffiffi 1 − 2 λ 2M λ 3 þ p ð y Þ; ð13Þ 2α0 pffiffiffi 1 3 2 − 2 − ffiffiffi P λ κ λ where we use prolate spheroidal coordinates p ¼ p þ 4 ; ¼ M : ð3Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρˆ2 þðzˆ þ MÞ2 þ ρˆ2 þðzˆ − MÞ2 Taking λ → 0 and keeping ðt;ˆ ρˆ; z;ˆ ϕˆÞ fixed. As a result of x ¼ ; ð14Þ 2M this procedure, we find the generalized (nonidentical mass) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi generalized-NHEK2 geometry 2 2 2 2 ρˆ þðzˆ þ MÞ − ρˆ þðzˆ − MÞ y ¼ ; ð15Þ ρˆ2 2M ds2 ¼ − dtˆ2 þ fðdϕˆ þ ωdtˆÞ2 þ e2νðdρˆ2 þ dzˆ2Þ; ð4Þ f and introduce the notation pffiffiffi defined by the equations pffiffiffiffiffiffiffiffiffiffiffiffiffi 3 − 2 1 − 2 Δ P β − 1 2 Q ¼ P ; 0 ¼ 2 ;K0 ¼ 0 = ; 4 2μ μ 2σ2 − M 0ð 0 þ 0Þ f ¼ 2 2 ; ð16Þ μ0ðμ0 þ 2σ0 − 2σ1 þ π0Þþμ1π1 þð1 − y Þσ0τ0 2   π0σ0 þ π1σ1 − μ1 − 4σ0σ1 − ð1 − y Þτ0=2 1 4 2β ω − ; − Δ − 7 1 − 2β − 1K0 − 4Δ ¼ 2 μ 2σ2 K1 ¼ 0 þ ð 0Þ 0K0; Mð 0 þ 0Þ 4 Δ0 1 − 2β0 2 2 μ0ðμ0 þ 2σ − 2σ1 þ π0Þþμ1π1 þð1 − y Þσ0τ0 e2ν 0 ; ð17Þ ¼ 2 2 − 2 4 K0ðx y Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 2 ð5Þ α0 ¼ pffiffiffi ; β0 ¼ − α0 þ 1=2 ð18Þ 2 − 2P where 2 2 2Q β0 Δ0 2 α1 ¼ − pffiffiffi Qð1 þ 4β Þ; ρˆ2 2 − 2 α 2 0 μ − σ − x y β 2 2 − 2α 0 0 ¼ 2 ; 0 ¼ þ 0ðx þ y Þ 0xy; 2 2M 2 2PQβ0 Δ0 β1 ¼ − pffiffiffi ðP þ 4Qα0β0Þ; ð19Þ ð6Þ α0 2   π 4 β − α − 1 2β 2 − 2 5Δ 2 3 1 ¼ xð 0x 0yÞ ð þ 0Þðx y Þ; ð7Þ α − 0 − 3 ¼ 2 Δ þ 2 ; pffiffiffi 0 2 β2 2 2 2 2 Q 0 2 2 2 Δ0ð1 − 8β0Þþ2ðα0α1 þ β0β1Þ ð1 − 4β0ÞK1 μ1 ¼ −Δ0ð−1 þ x Þ þ ðx − y Þ ; ð8Þ β α 3 ¼ þ 2 ; 0 K0 2K0 2 2 2 2 ð20Þ σ1 ¼ Δ0ðx − y Þþð−2Δ0β0 þ β1Þðx þ y Þ 2 2Δ α − α pffiffiffi þ ð 0 0 1Þxy; ð9Þ for 1= 2

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B. Physical parameters C. Let us now consider the physical parameters of the The generalized-NHEK2 spacetime that we constructed generalized-NHEK2 solution. As we did at the level of the contains regions where the vector ∂t becomes null. We will geometry, the near-horizon limiting procedure can be refer to the boundary region as the , since they applied to the original physical parameters found in [22] are inherited from the presence of such regions in the (also reviewed here in Appendix A). Applying this original stationary extreme BBHs geometries. For NHEK2 technique yields the expressions for the masses M1, M2, these are defined by regions where gtt ¼ 0 and give rise to a and angular momenta J1, J2 in the generalized-NHEK2 set of disconnected regions as shown in Fig. 1. Different solution values of the parameter P are bounded by the extreme pffiffiffi     mass ratio solution when P ¼ 1= 2 and identical mass M Q M Q solution when P ¼ 1. The horizons of the black holes M1 ¼ 1 − pffiffiffi ;M2 ¼ 1 þ pffiffiffi ; ρˆ ˆ 2 2 − P 2 2 − P in generalized-NHEK2 are points in the ð ; zÞ-plane and have finite horizon areas. There is a self-similar behavior ð21Þ close to each black hole that resembles the ergospheres’     diagrams of isolated extremal Kerr black holes. −1 −1 2 Q 2 Q J1 ¼ 2M1 1 − pffiffiffi ;J2 ¼ 2M2 1 þ pffiffiffi ; 2 − P 2 − P D. Asymptotic behavior ð22Þ In the asymptotic limit, for ρˆ ¼ r sin θ, zˆ ¼ r cos θ and r → ∞, the generalized-NHEK2 geometry in Sec. II A has and the corresponding angular velocities a limiting metric that corresponds to the NHEK metric—in Weyl coordinates—(4) with functions 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ω1 ¼ Ω2 ¼ ; ð23Þ 4M2ρˆ2 zˆ2 þ ρˆ2 M2ð2zˆ2 þ ρˆ2Þ 2M f ¼ ; ω ¼ ;e2ν ¼ : 2zˆ2 þ ρˆ2 2M2 ðzˆ2 þ ρˆ2Þ2 2 Ω satisfying at the same time the Smarr relation M1 ¼ J1 1 ð25Þ and M2 ¼ 2J2Ω2. It is worth noticing that the new solution contains objects that are in thermal equilibrium. The black In other words, the generalized-NHEK2 solution is asymp- hole entropy is, as usual, the area of the event horizon totically NHEK. It is worthwhile to mention at this point divided by 4. This gives that in [23,24] it was shown that in the case of 4D Einstein ffiffiffi gravity the NHEK geometry is the unique (up to diffeo- 2 p S1 ¼ 4πM ð2 − 2ðP þ QÞÞ; morphisms) regular stationary and axisymmetric solution pffiffiffi asymptotic to NHEK with a smooth horizon. The NHEK2 4π 2 2 − 2 − S2 ¼ M ð ðP QÞÞ: ð24Þ geometry that we unveil is asymptotically NHEK, but is not

FIG. 1. Ergoregion (shaded orange region) of the generalized-NHEK2-black hole solution for M ¼ 1 and P ¼ 1, 0.9, 0.71 (from left to right). Each black hole is located at ρ ¼ ρˆH ¼ 0;z¼ zˆH ¼1. Magnified diagrams, close to the locations of the black holes appear in the corners. The dashed line corresponds to the boundary where ∂ ˆ is null. The different values of P in the diagrams correspond to t pffiffiffi solutions with BBHs of distinctive mass ratios. Note that in the generalized-NHEK2 solution the parameter 1= 2

044033-4 GENERALIZED NEAR HORIZON EXTREME BINARY … PHYS. REV. D 100, 044033 (2019) diffeomorphic to NHEK; this is not in contradiction with singularity balances the gravitational attraction of the com- the results of [23,24] since the NHEK2 geometry is not panion BH, thereby enabling stationarity. The cosmic string smooth on the strut which keeps the BHs apart. extends all the way from the horizon to infinity in this geometry which we call the “pierced-NHEK.” E. Conical singularity Our starting point is the solution given in [21] (that As we have shown in the previous subsection, the corresponds to the identical mass binary black hole 1 generalized-NHEK2 is exactly asymptotically NHEK with- metric in [22] for P ¼ which for convenience we out any conical defects. However, as in the original sta- reviewed in Appendix A). As the most general solution tionary, extremal BBHs geometry there is in the bulk, a is quite involved, we will start by fixing the parameters at a conical singularity on the ρˆ ¼ 0 axis localized between the specific, convenient value which will be enough to convey two black holes. In Weyl coordinates the conical singu- our point regarding the existence of a nonsingular near- larities can be easily computed horizon geometry. It could be nice to explicitly write down sffiffiffiffiffiffiffiffiffiffiffi! the full most general expression, for arbitrary value of P, but for the sake of simplicity we will only focus on the f Δϕˆ ¼ 2πlim 1 − ; −M

pffiffiffiffiffi pffiffiffiffiffi 2 3 33 − 13 Θ 15 − 33 3 2Θ Γ Θ ð Þ cos þð Þð þ cos Þ ð Þ¼ 16 ; 256sin2Θ ΓðΘÞΛðΘÞ2 ¼ pffiffiffiffiffi pffiffiffiffiffi : ð31Þ 4ð−59 þ 11 33Þ cos Θ þð93 − 13 33Þð3 þ cos 2ΘÞ

In this geometry, a priori, there could be a conical singularity either at Θ ¼ 0 or at Θ ¼ π. Using (27), however, shows explicitly that at Θ ¼ 0 there is no conical singularity whereas for θ ¼ π there is an angular excess of

pffiffiffiffiffi 33 − 1 ΔΦ 2π ¼ 8 : ð32Þ

Writing the pierced-NHEK geometry in the form (30) shows immediately that it enjoys the isometry group SLð2; RÞ ×Uð1Þ, just like NHEK: the strut on the symmetry axis does not spoil this symmetry.

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FIG. 2. The diagram represents a spatial cross section of the two extremal corotating BHs metric [22] reproduced in Appendix A. The geometry (in black and gray) has an asymptotically Minkowskian region and a single black hole throat of mass M1 þ M2 which divides into two throats of masses M1 and M2. The strut (conical singularity) in the solution is localized between the two black holes (dashed blue). In the infrared limit λ → 0, when zooming into the near horizon limit, the throat becomes infinitely long and the Minkowski region decouples. This is the new generalized NHEK2 solution that we constructed (represented in green) that is asymptotically NHEK (in orange). The splitting of the throat into two pieces survives this limit. In the deep infrared when zooming close to one of the horizons, we find new geometries that we call “pierced NHEK.” These correspond to the NHEK metric pierced by a conical singularity on the symmetry axis, which runs from one of the poles up to the boundary (in purple).

IV. DISCUSSION not diffeomorphic to NHEK. This is not in contradiction to the discussions in [25,26] since in these papers, smoothness The aim of this paper was to unveil and analyze the is assumed while here we allow for a conical singularity generalized-NHEK2 geometry. This geometry is obtained which balances the gravitational attraction between the via a limiting procedure that we developed: a zoom-in BHs. This paper generalizes the construction studied on the near-horizon region of a 1-parameter family of 2 recently in [11] for the equal mass case. corotating, double-extreme Kerr solutions of arbitrary masses where the two BHs are parametrically close to each other and are held apart by a conical singularity (strut). ACKNOWLEDGMENTS The distance between the BHs is scaled to zero at the same We would like to thank David Chow and Oscar Varela rate of the zoom-in on the near-horizon region. This gives a for helpful discussions. This work was supported by the relatively simple solution, which is asymptotically NHEK, NSF Grant No. PHY-1707571 at Utah State University and in the infrared flows to two separate throats which we and, partially, by the Max Planck Gesellschaft through the call “pierced-NHEK” geometries: each of them is, approx- Gravitation and Black Hole Theory Independent Research imately (when zooming further towards one of the hori- Group. S. H. gratefully acknowledges support by the Jacob zons), NHEK pierced by a conical singularity on the Goldfield Postdoctoral Support Fund. symmetry axis, which runs from one of the poles up to the boundary. We find that in the deep infrared where the geometry is approximately pierced-NHEK, the presence of APPENDIX: STATIONARY EXTREME KERR the strut does not break the isometry group SL 2; R × ð Þ BINARY WITH STRUT—FULL SOLUTION Uð1Þ—it is restored there. In Fig. 2, we illustrate the structure of the generalized-NHEK2 geometry. The gen- Here we record, for completeness, the full exact solution eralized-NHEK2 solution asymptotes to NHEK, yet it is corresponding to two extremal co-rotating BHs which are held apart by a strut lying on the joint rotation axis, between 2The counterrotating counterpart cannot be used to construct the BHs. The solution is axisymmetric, stationary and a similar solution since asymptotically it appears as a non- asymptotically flat. We follow the conventions of [22] in extreme BH. which this solution was presented.

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Define prolate spheroidal coordinates ðx; yÞ by the metric is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ2 ρ2 κ 2 ρ2 − κ 2 2 2 2 2ν 2 2 þðz þ Þ þ þðz Þ ds ¼ − dt þ fðdϕ þ ωdtÞ þ e ðdρ þ dz Þ; ðA3Þ x ¼ 2κ ; ðA1Þ f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ2 þðz þ κÞ2 − ρ2 þðz − κÞ2 y ¼ 2κ ; ðA2Þ where:

κ 2 − 1 κ 2 − 1 ðy ÞF 2ν D ω − ðy ÞFN f ¼ ;e¼ 2 2 2 4 ; ¼ 2 2 2 2 ; Dω K0ðx − y Þ ½ðκðy − 1ÞFÞ − ρ D N ¼ μ2 − ðx2 − 1Þð1 − y2Þσ2; D ¼ N þ μπ þð1 − y2Þστ; F ¼ðx2 − 1Þσπ þ μτ; μ ¼ p2ðp2ðx2 − 1Þ2 þ q2ð1 − y2Þ2 þðα2 − β2Þðx2 − y2Þ2Þ; σ ¼ p2ð2½pqðx2 − y2Þþβðx2 þ y2Þ − 2αxyÞ; 2 2 2 2 2 2 π ¼ p ðð4p =K0ÞfðK0=p Þ½pPsxðx þ 1Þþ2x þ qQyðy þ 1Þ þ 2ðpQ þ pPα þ qQβÞ½pqyðx2 − y2Þþβyðx2 þ y2Þ − 2αxy2 2 2 2 2 2 2 2 2 2 − ðK0=p Þðx − y Þ½ðpQα − qPβÞx þðqPα − pQβÞy − 2ðq α þ p β Þðx − y Þ þ 4ðpq þ βÞðβx2 − αxyÞgÞ; 2 2 2 2 2 2 τ ¼ p ðð4p =K0ÞfðK0=p Þx½ðqQα þ pPβÞðx − y Þ − qPð1 − y Þ þðpQ þ pPα þ qQβÞy½ðp2 − α2 þ β2Þðx2 − y2Þþy2 − 1 2 2 2 2 2 2 2 − pQðK0=p Þyðx − 1Þ − 2pðqα − qβ − pβÞðx − y Þ − ðpq þ βÞð1 − y ÞgÞ; 2 2 2 2 K0 ¼ p ðp þ α − β Þ; ðA4Þ and the parameters are constrained so that

p2 þ q2 ¼ 1;P2 þ Q2 ¼ 1: ðA5Þ

For the corotating solution in which we are interested in this paper,

Q½qΔ þ pq2 þ Pð1 þ p2Þ α ¼ − ; 2ðp2 − Q2Þ p½PΔ þ qð1 þ pP þ Q2Þ β ¼ ; 2ðp2 − Q2Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ ¼ 4p2ð1 þ pPÞþq2ðp þ PÞ2: ðA6Þ

1. Physical parameters

The asymptotic metric does not contain a conical singularity, then the mass M1, M2 and the angular momenta J1, J2 of the black holes can be easily calculated

κ½ðq þ pqP − p2QÞΔ − ð1 þ pPÞðp þ p3 þ q2P − pqQÞþpq3Q M1 ¼ ; 2pð1 þ pPÞðp2 − q2Þ κ½ðq þ pqP þ p2QÞΔ − ð1 þ pPÞðp þ p3 þ q2P þ pqQÞ − pq3Q M2 ¼ ; 2pð1 þ pPÞðp2 − q2Þ

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2 ð1 þ pP þ qQÞM1 2 2 J1 ¼ ½ð1 þ pP þ q ÞΔ − 4pq þ pqðp − PÞ ; 2ðp þ PÞ2 2 ð1 þ pP − qQÞM2 2 2 J2 ¼ ½ð1 þ pP þ q ÞΔ − 4pq þ pqðp − PÞ ; ðA7Þ 2ðp þ PÞ2 and, employing the Smarr relation, we can easily find the expressions for the angular velocities

M1 M2 Ω1 ¼ ; Ω2 ¼ : ðA8Þ 2J1 2J2

Additionally, the entropy for each black hole can be calculated to give:

2π 2κ2 p 1 − 2 2 α − β α 1 β S1 ¼ 2 ðK0ð þ pP þ qQÞ p ð ÞðpðQ þ P þ qÞþðqQ þ Þ ÞÞ; ðA9Þ K0 2π 2κ2 p 1 − − 2 2 α β α − − 1 β S2 ¼ 2 ðK0ð þ pP qQÞ p ð þ ÞðpðQ þ P qÞþðqQ Þ ÞÞ: ðA10Þ K0

[1] L. Gou, J. E. McClintock, M. J. Reid, J. A. Orosz, J. F. [11] J. Ciafre and M. J. Rodriguez, A near horizon extreme Steiner, R. Narayan, J. Xiang, R. A. Remillard, K. A. binary black hole geometry, arxiv:1804.06985. Arnaud, and S. W. Davis, The extreme spin of the black [12] J. Camps, S. Hadar, and N. S. Manton, Exact gravitational hole in Cygnus x-1, Astrophys. J. 742, 85 (2011). Wwave signatures from colliding extreme black holes, [2] G. Risaliti et al., A rapidly spinning supermassive black Phys. Rev. D 96, 061501 (2017). hole at the centre of NGC 1365, Nature (London) 494, 449 [13] P. V. P. Cunha, C. A. R. Herdeiro, and M. J. Rodriguez, Does (2013). the black hole shadow probe the event horizon geometry?, [3] A. C. Fabian, The innermost extremes of black hole accre- Phys. Rev. D 97, 084020 (2018). tion, Astron. Nachr. 337, 375 (2016). [14] A. Anabalon, M. Appels, R. Gregory, D. Kubiznk, [4] J. E. McClintock, R. Shafee, R. Narayan, R. A. Remillard, R. B. Mann, and A. Ovgun, Holographic thermodynamics S. W. Davis, and L. x. Li, The spin of the near-extreme of accelerating black holes, Phys. Rev. D 98, 104038 Kerr black hole GRS 1915 þ 105, Astrophys. J. 652, 518 (2018). (2006). [15] W. Kinnersley and D. M. Chitre, Symmetries of the sta- [5] K. Akiyama et al. (Event Horizon Telescope Collaboration), tionary EinsteinMaxwell equations. IV. Transformations First M87 event horizon telescope results. I. The shadow of which preserve asymptotic flatness, J. Math. Phys. (N.Y.) the , Astrophys. J. 875, L1 (2019). 19, 2037 (1978). [6] J. M. Bardeen and G. T. Horowitz, The extreme Kerr throat [16] D. Kramer and G. Neugebauer, The superposition of two 2 geometry: A vacuum analog of AdS2 ×S , Phys. Rev. D 60, Kerr solutions, Phys. Lett. 75A, 259 (1980). 104030 (1999). [17] C. A. R. Herdeiro and C. Rebelo, On the interaction [7] M. Guica, T. Hartman, W. Song, and A. Strominger, between two Kerr black holes, J. High Energy Phys. 10 The Kerr/CFT correspondence, Phys. Rev. D 80, 124008 (2008) 017. (2009). [18] V. S. Manko, E. D. Rodchenko, E. Ruiz, and B. I. [8] J. B. Hartle and S. W. Hawking, Solutions of the Einstein- Sadovnikov, Exact solutions for a system of two Maxwell equations with many black holes, Commun. Math. counter-rotating black holes, Phys. Rev. D 78, 124014 Phys. 26, 87 (1972). (2008). [9] J. M. Maldacena, J. Michelson, and A. Strominger, [19] M. S. Costa, C. A. R. Herdeiro, and C. Rebelo, Dynamical Anti-de Sitter fragmentation, J. High Energy Phys. 02 and thermodynamical aspects of interacting Kerr black (1999) 011. holes, Phys. Rev. D 79, 123508 (2009). [10] G. W. Gibbons and C. M. Hull, A bogomolny bound for [20] V. S. Manko, E. D. Rodchenko, E. Ruiz, and B. I. general relativity and solitons in N ¼ 2 supergravity, Phys. Sadovnikov, On the simplest binary system of rotating Lett. 109B, 190 (1982). black holes, AIP Conf. Proc. 1122, 332 (2009).

044033-8 GENERALIZED NEAR HORIZON EXTREME BINARY … PHYS. REV. D 100, 044033 (2019)

[21] V. S. Manko and E. Ruiz, On a simple representation of the [24] O. J. C. Dias, H. S. Reall, and J. E. Santos, Kerr-CFT Kinnersley-Chitre metric, Prog. Theor. Phys.125, 1241 (2011). and gravitational perturbations, J. High Energy Phys. 08 [22] V. S. Manko, E. Ruiz, and M. B. Sadovnikova, Stationary (2009) 101. configurations of two extreme black holes obtainable from [25] A. J. Amsel, G. T. Horowitz, D. Marolf, and M. M. Roberts, the Kinnersley-Chitre solution, Phys. Rev. D 84, 064005 Uniqueness of extremal Kerr and Kerr-Newman black holes, (2011). Phys. Rev. D 81, 024033 (2010). [23] A. J. Amsel, G. T. Horowitz, D. Marolf, and M. M. Roberts, [26] H. K. Kunduri and J. Lucietti, Classification of near-horizon No dynamics in the extremal Kerr throat, J. High Energy geometries of extremal black holes, Living Rev. Relativity Phys. 09 (2009) 044. 16, 8 (2013).

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