View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DigitalCommons@USU Eur. Phys. J. C (2019) 79:754 https://doi.org/10.1140/epjc/s10052-019-7188-3 Regular Article - Theoretical Physics A near horizon extreme binary black hole geometry Jacob Ciafre1,a, Maria J. Rodriguez1,2,b 1 Department of Physics, Utah State University, 4415 Old Main Hill Road, Logan, UT 84322, USA 2 Max Planck for Gravitational Physics-Albert Einstein Institute, Am Muhlenberg 1, 14476 Potsdam, Germany Received: 21 August 2018 / Accepted: 1 August 2019 © The Author(s) 2019 Abstract A new solution of four-dimensional vacuum From a theoretical perspective, it is conjectured that a General Relativity is presented. It describes the near horizon rotating Kerr black hole obeys cosmic censorship: any sin- region of the extreme (maximally spinning) binary black hole gularities must lay behind the black hole’s event horizon. system with two identical extreme Kerr black holes held in As the black hole spin increases, for a fixed mass, the event equilibrium by a massless strut. This is the first example of a horizon area protecting the singularity decreases. The maxi- non-supersymmetric, near horizon extreme binary black hole mum allowed spin is reached before exposing the singularity. geometry of two uncharged black holes. The black holes are These maximally rotating black holes are called extreme and co-rotating, their relative distance is fixed, and the solution are the central objects of study of this work. is uniquely specified by the mass. Asymptotically, the geom- At extremality, the black hole’s temperature drops to zero. etry corresponds to the near horizon extreme Kerr (NHEK) Interestingly, at this point the geometry close to the event black hole. The binary extreme system has finite entropy. horizon, the so called Near-Horizon Extreme Kerr (NHEK) geometry [14], displays a very special feature: the isom- etry group is enhanced to an SL(2, R), conformal sym- Contents metry and develops a warped version of AdS3. NHEK is obtained through scalings from the extreme Kerr black hole, 1 Introduction ...................... has finite size, namely, finite event horizon area, and retains 2 Near-horizon extreme black hole binary geometry .. all the relevant aspects of black holes: a horizon and an ergo- 3 Discussion ....................... sphere which may be associated with those corresponding to A Extreme co-rotating binary black hole review ..... extreme Kerr. The enhanced conformal symmetry in NHEK, References ......................... which does not extend to the full Kerr geometry, motivated works such as the Kerr/CFT conjecture concerning the quan- tum structure of black holes [15] and studies on the dynamics 1 Introduction of stars [16] and energy extraction [17] in this region. One then wonders, could NHEK be formed from a merger Several black holes have been observed to rotate at nearly the of the near horizon geometries of extreme BBHs? And, in this speed of light [1–5]. Although there is considerable uncer- case, is the conformal symmetry also present in the geom- tainty in how black holes attain nearly extremal spins, several etry before merger giving rise to NHEK? To answer these mechanisms in Nature can be envisioned. These include the questions, in this paper, we propose to focus on the sta- merging of binary black holes (BBHs) possibly spinning at tionary – rather than dynamical – BBHs and find explicitly nearly extremal spins, and prolonged disk accretion. Numer- the Near-Horizon Extreme Kerr Binary Black Hole geom- ical simulations, that are key in developing the most precise etry, “NHEK2”. We will show that this new metric retains predictions of dynamical BBHs mergers [6–8] and black hole only partially the symmetry of NHEK, and are “tree”-like disk accretion [9–13], have confirmed the feasibility of these geometries: they are solutions with NHEK asymptotics, but mechanisms giving rise to nearly extremal spinning black as one moves inward the geometry branches into two smaller holes. warped AdS3 regions. Fragmentation of the AdS near hori- zon extremal charged static black hole geometries was previ- ously investigated in [18]. Further, a description in the dual a e-mail: [email protected] CFT was presented where the AdS trees in the geometry b e-mails: [email protected]; [email protected] 0123456789().: V,-vol 123 754 Page 2 of 6 Eur. Phys. J. C (2019) 79:754 correspond to different classical vacua of a quantum mechan- 2 Near-horizon extreme black hole binary geometry ical theory. Luckily, stationary (non-dynamical) BBHs solutions of The solution of extremal co-rotating identical BBHs [28] Einstein’s equations of General Relativity in vacuum with (that for completeness we review in Appendix A) from which two-(neutral)Kerr black holes are also known analytically. we derive the new near-horizon geometry of extreme BBHs, These are exact asymptotically flat vacuum solutions with takes a simpler representation in Weyl coordinates. We there- two spinning Kerr black holes supported by a conical singu- fore perform the scaling computations in this frame. In this larity along the line separating the black holes. They repre- case, we find that the appropriate near-horizon limiting pro- sent special subfamilies of vacuum spacetimes constructed cedure for the extremal BBHs is via a variety of solution-generating techniques in e.g. [19– ρ =ˆρλ, z =ˆz λ, (2) 28]. The black holes in these solutions can co-rotate or spin tˆ tˆ in opposite directions. They become, for certain range of t = ,φ= φˆ + , (3) the parameters, extremal zero-temperature stationary BBHs λ √ 2Mλ solutions [25–28]. Remarkably, as observed in [22,26,27], 1 (3 2 − 2) p =−√ + λκ= M λ, (4) the extreme BBHs overtake the extremal Kerr bound. 2 4 The fact that there is a conical singularity in all these solu- λ → (ˆ, ρ,ˆ ˆ, φ)ˆ κ tions may seem discouraging, however the solution, and the taking 0 and keeping t z fixed. In [28], p and contribution from the conical singularities have a well defined are free parameters related to the definitions of positive mass gravitational action [30]. Thus, standard Euclidean Gravity and the separation between the black holes respectively. It thermodynamical arguments can be employed, and one can is worth emphasizing that at the same rate of the zoom-in extract new results from an apparently ill defined geometry. on the near-horizon region, the coordinate distance between We interpret the conical singularities as boundary conditions the black holes is scaled to zero. The Newtonian distance = necessary to prevent the collapse of the array and maintain between the black holes is z 2M. Therefore, after the the holes at fixed distance. Note also that the total energy scaling procedure, the distance separating the black holes is associated with the strut as seen by this observer at infinity finite, and yet the two black holes are held apart by a con- is negative. For large separation z respect to the mass of ical singularity (massless strut). As a result of this process, the black holes M, the energy is E =−M2G/z, that cor- we find the near-horizon extreme black hole binary NHEK2 responds to the Newtonian potential between two particles. geometry in Weyl coordinates (1). This is defined by the All of the above geometries were originally written in equations 2 2 Weyl coordinates, where many expressions simplify. In these 4M μ0 (μ0 + 2σ ) f =− 0 , coordinates, the metric takes the form μ (μ + σ 2 − σ + π ) + μ π + ( − 2)σ τ 0 0 2 0 2 1 0 1 1 1 y 0 0 2 ρˆ2 π0 σ0 + π1 σ1 − μ1 − 4σ0 σ1 − (1 − y )τ0/2 2 =− ˆ2 + ( φˆ + ω ˆ)2 + 2ν( ρˆ2 + ˆ2), ω =− , (5) ds dt f d dt e d dz (1) 2M(μ + 2 σ 2) f 0 0 2 2 μ0 (μ0 + 2σ − 2σ1 + π0) + μ1 π1 + (1 − y )σ0 τ0 e2ν = 0 , where f,ν,ω are functions only of the (ρ,ˆ zˆ)-coordinates, 2 ( 2 − 2)4 K0 x y and tˆ ∈ (−∞, ∞), ρ,ˆ zˆ ∈ (−∞, ∞) and φˆ ∼ φˆ + 2π.Note that all stationary axis-symmetric black hole solutions can where the functions yield be written in Weyl coordinates (1), including the stationary ρˆ2 x2 − y2 x2 + y2 BBHs central to our discussion, and also the Kerr and NHEK μ =− ,σ=− − √ , (6) 0 2M2 0 2 geometries. Our starting point to find the NHEK2 geometry √ √ 2 π =− 2 − ( − )( 2 − 2), are the co-rotating extreme double-Kerr black hole solutions 1 2 2 x√ 1 2 x y (7) in [28] (see Appendix A for a review). We develop an appro- ( − ) √ μ =− 3 2 (− + 2)2 + ( − )( 2 − 2)2, priate near-horizon limiting procedure in Weyl coordinates1, 1 1 x 1 2 x y (8) √2 and apply it to the extreme BBHs geometry. As a result we ( − ) 3 2 2 2 obtain the new NHEK2 geometries, which becomes a distinct σ1 = (x − y ) 2 object with finite entropy. Our results may be of considerable √ √ (1 − 2)(3 − 2 2) interest in dual CFT studies as in [15,18], entropy calcula- + √ (x2 + y2), (9) 2 2 tions, and gravitational signatures of near exremal spinning √ binary black holes. −74 + 85 2 √ π = x2 − 2x(1 + x2) 0 14 √ 1 The procedure to describe the near horizon geometries of extremal 145 − 108 2 black holes has most frequently been developed in Boyer–Lindquist + − x (x2 − y2), (10) coordinates 28 123 Eur. Phys. J. C (2019) 79:754 Page 3 of 6 754 √ √ 99 − 51 2 √ with τ = 2x(x2 − 1) − − (1 − 2)x 0 28 √ √ √ (22 − 15 2)(3 + cos 2 ) ± (72 − 52 2) cos 71 − 23 2 ( ) = , (x2 − y2) + (1 − y2), (11) √ 8 28 16(2 − 2)2 sin2 ( )( ) = √ √ , ( − )( + ) ± ( − ) and 6 2 3 cos 2 8 12 2cos ρˆ2 + (zˆ + M)2 + ρˆ2 + (zˆ − M)2 when the leading terms in each metric component are x = , 2M retained.