Rings, ripples, and rotation: connecting black holes to black rings

Abstract

Singly-spinning Myers-Perry black holes in d ≥ 6 spacetime dimensions are unstable for sufficiently large angular momentum. We numerically construct (in d = 6 and d = 7) two new stationary branches of lumpy (rippled) black hole solutions which bifurcate from the onset of this ultraspinning instability. We give evidence that one of these branches connects through a topology-changing merger to black ring solutions which we also construct numerically. The other branch approaches a solution with large curvature invariants. We are also able to compare the d = 7 ring solutions with results from finite-size corrections to the blackfold approach, finding excellent agreement. JHEP07(2014)045Introduction. As expressed by John Wheeler's statement, "Black holes have no hair" [1], black holes (BHs) in four spacetime dimensions are remarkably simple objects. The topology, rigidity, uniqueness, and no-hair theorems ensure that Kerr BHs are the only stationary, vacuum, and asymptotically flat solutions to general relativity, and that they are uniquely specified by their mass M and angular momentum J [2]. Because Kerr BHs are also linear mode stable [3], any stellar object undergoing gravitational collapse towards a BH is expected to settle to the Kerr solution.Yet, the uniqueness of the Kerr BH may seem surprising given the close connection between gravity and fluids. According to the membrane paradigm [4], external observers will find that BH horizons behave like fluid membranes, endowed with a viscosity, conductivity, temperature, entropy, etc. Moreover, in certain circumstances there are formal mappings between solutions of general relativity and solutions of Navier-Stokes [5,6]. Fluids, however, lack strong uniqueness theorems and admit a rich structure of solutions. Indeed, rotational instabilities appear in fluid droplets and non-spherical solutions develop [7]. In particular, a ring configuration is preferred for high spin. Therefore, it may be natural to suspect that BHs behave like fluid droplets and have a greater diversity of solutions. This is the emerging picture in d > 4 spacetime dimensions.In higher dimensions, gravity becomes weaker as it spreads out over extra dimensions. Horizons are therefore more flexible, creating new gravitational phenomena with no fourdimensional counterparts. For example, in d ≥ 5 black strings exist and suffer from the Gregory-Laflamme instability [8]. This instability leads to a fractal-like array of spherical BHs and cosmic censorship violation [9,10]. This behaviour is similar to the RayleighPlateau instability, where a fluid jet breaks into an array of spherical droplets [11].In addition, there are black rings with horizon topology S 1 × S d−3 . These have been constructed in closed analytic form in d = 5 [12] and numerically in d = 6 [13]. (The list of other BH horizon topologies that are allowed in d > 4 were discussed in [14]). In any d ≥ 5, black rings can be found perturbatively using the blackfold approach if the S 1 radius is much larger than the S d−3 radius [15]...

Blackfolds, plane waves and minimal surfaces

“…One of these effective theories, known as the blackfold approach [3,4], describing the long wavelength dynamics of black branes in a derivative expansion including hydrodynamic and elastic degrees of freedom [5][6][7][8][9], has allowed to scan for non-trivial black hole horizon topologies in asymptotically flat and (Anti)-de Sitter space-times [10][11][12][13]. These new black hole topologies include black rings in higher-dimensions, black odd-spheres and black cylinders, some of which have been constructed numerically [14][15][16][17]. However, these works have only scratched the surface of the entire set of possible horizon topologies.…”

“…where η b is a unit normalised normal vector to the worldvolume boundary ∂W p+1 and the effective stress-energy tensor T ab takes the perfect fluid form 15) where the prime denotes a derivative with respect to k and the fluid velocity u a is aligned with the worldvolume Killing vector field k a . Because of worldvolume general covariance, the stress tensor (2.15) automatically solves the conservation equation in (2.14).…”

“…We shall refer to these interpolating surfaces simply as Scherk surfaces in this paper. 15 We will analyse in detail this general solution in section 3.4. The metric (2.63), using (2.73) and after rescaling back φ → aφ, is diagonal and takes the form …”

“…It would be interesting to study this for more non-trivial embeddings of the black ring as in appendix D.1 using the methods of [27,41] and to use numerical methods as in [14][15][16][17] in order to construct the full solution.…”

Lattice black branes: sphere packing in general relativity

“…In particular, they can be unstable [1][2][3][4][5][6], violate black hole uniqueness [2,3,7,8] and also lead to a violation of weak cosmic censorship [9][10][11].…”