Rings, Ripples, and Rotation: Connecting Black Holes to Black Rings

Singly-spinning Myers-Perry black holes in d>5 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.

Introduction and Summary. 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 solutions to general relativity, and that they are uniquely specified by their mass M and angular momentum J (see [2] for a review). Because Kerr BHs are also linearly stable [3], it is conjectured that any stellar object undergoing gravitational collapse towards a BH will settle down to the Kerr solution.
Yet, the uniqueness of the Kerr BH is perhaps surprising in light of the 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 sufficiently high spin. Therefore, it may be natural to suspect that BHs behave like fluid droplets and have a greater diversity of solutions. As we shall see, this is the emerging picture in d > 4 spacetime dimensions.
In higher dimensions, gravity becomes weaker as it spreads out over extra dimensions. As a result, horizons are more flexible, giving rise to a plethora of gravitational phenomena that have no four-dimensional counterpart. For example, in d ≥ 5 black strings (direct products of Kerr BHs and flat extra direction(s)) are unstable to the Gregory-Laflamme instability [8] and evolve towards a fractal-like array of spherical BHs [9,10]. This behaviour is similar to the Rayleigh-Plateau instability, where a fluid jet breaks into an array of spherical droplets [11].
In addition, in d ≥ 5 there are asymptotically flat black rings with non-spherical 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]. In any d ≥ 5, they can be found perturbatively using the blackfold approach if the S 1 radius is much larger than the S d−3 radius [14]. In addition to new topologies, these rings also introduce non-uniqueness since there are different ring solutions with the same M and J. Given the loss of uniqueness, understanding the space of BH solutions in higher dimensions is an open and difficult task. Though, it is also an important task in scenarios where higher dimensions are unavoidable, such as those arising from string theory or holography.
Let us, therefore, summarise of the state of the art, focusing on asymptotically flat stationary solutions. In d = 4, the Kerr family forms the only stationary BH solution. For a fixed mass M , increasing the angular momentum J decreases the BH temperature T H and the horizon area A H , until the BH reaches extremality at J = GM 2 , where the BH will have vanishing temperature but non-vanishing horizon area [15].
In d ≥ 5, the higher dimensional analogues to the Kerr BH are the Myers-Perry (MP) solutions [16]. Also in higher dimensions, BHs can have d−1 2 independent angular momenta. For simplicity, we only consider the singly-spinning case with one non-zero J.
Like Kerr BHs, singly-spinning MP BHs in d = 5 at fixed M will have decreasing T H and A H with increasing J. Unlike Kerr BHs however, both T H and A H vanish at extremality [15], giving a naked singularity. This singularity is the merger point between the MP solutions and the S 1 × S 2 black rings [12]. As mentioned earlier, these rings are not uniquely specified by M and J. There are two branches: the "fat" branch connected to the naked singularity, and a "thin" branch which for large J resembles bent black strings. There is also an infinite family of solutions with disconnected horizons, such as black saturns [17] and black di-rings [18], which all connect to the black ring and MP BH at the naked singularity [31].
In d ≥ 6, singly-spinning MP BHs can have arbitrarily large angular momentum. Consequently, there are situations where the horizon spreads along the rotation plane to a radius much larger than the thickness in directions transverse to this plane. Above a critical rotation, such BHs become unstable to the ultraspinning instability, which is of Gregory-Laflamme type [19][20][21]. It was conjectured that the threshold mode of this instability signals a bifurcation to a new branch of axisymmetric rotating BHs with lumpy or rippled S d−2 horizons. These "lumpy" BHs are conjectured to connect to the fat black ring branch [14,19]. Further lumpy BH solutions may appear from the threshold of higher harmonic modes and connect to black saturns and di-rings, etc.
In this Letter, we will take a firm step towards the completion of the phase diagram of stationary asymptotically flat solutions of general relativity with a single angular momentum in d ≥ 6. We confirm the existence of lumpy BHs by explicit numerical construction in d = 6 and d = 7. We also numerically construct black rings in d = 6 (reproducing [13]) and in d = 7. We find that there are not one but two families of lumpy BHs that bifurcate from the same MP solution. Our results give robust evidence that one of these lumpy BH branches connects to the fat black ring via a topology changing merger. The existence of the complementary branch of lumpy BHs was not anticipated in previous studies [14], and we find that they approach a solution containing large curvature. As a byproduct of our analysis, we are also able to check the finite-size corrections to the blackfold approach for the thin black rings in d = 7. We find excellent agreement between these analytic results and our numerics, even for an adimensional angular momentum of order one.
Method. The singly-spinning MP BHs, the lumpy BHs, and the black rings all have an asymptotic timelike Killing vector ∂ t and rotate with angular velocity Ω H along the orbit of the rotational Killing vector ∂ ψ such that the linear combination K = ∂ t + Ω H ∂ ψ generates a Killing horizon. These geometries can be written in the form where dΩ 2 d−4 is the line element of a unit (d − 4)-sphere, and A, B, C, S 1 , S 2 , W, F are functions of the coordinates y and x. For the MP and lumpy BHs, y is a radial coordinate and x is an angular coordinate. For the black ring, x and y resemble the bipolar ring coordinates given in [22]. We choose these functions to vanish smoothly in the appropriate places to yield horizons with the correct topology, and demand that they have flat asymptotics.
To find A, B, C, S 1 , S 2 , W, F , we use the DeTurck method, first introduced in [23,24]. This method requires noà priori gauge fixing and yields equations with a well defined character. For our choice of boundary con-ditions, our problem is elliptic [25]. We use pseudospectral collocation on a Gauss-Chebyshev-Lobbato grid (a patched grid for the rings) to discretise our PDE system. The resulting system of nonlinear algebraic equations is solved using a standard Newton-Raphson method. In our case, solutions to the Einstein-DeTurck equations are also Ricci-flat, meaning the so-called DeTurck vector ξ must vanish. We therefore check that |ξ| = 0 with an error smaller than 10 −8 , and that |ξ| vanishes exponentially with increasing grid size, as predicted by pseudospectral methods. For the black rings, we also confirm that our results do not depend on our specific choice of reference metric or patch boundary.
We compute the mass M and angular momentum J from a Komar Integral at infinity, and obtain the angular velocity Ω H , entropy S H = A H /4 and temperature T H . We verify that the Smarr relation d−3 d−2 M = T H S H +Ω H J and the first law dM = T H dS H + Ω H dJ are satisfied to less than 5% error [32]. The lumpy BHs are parametrised by a rotation parameter in horizon radius units a/r + . The black rings are parametrised by Ω H /T H .
Results. The phase diagram of stationary solutions and the connections between different BH families are best described by thermodynamic quantities. A meaningful comparison can be made if one fixes the mass M and introduces the dimensionless spin j, area a H , angular velocity ω H , and temperature t H via [14] where the constants c j , c a , c ω , c t can be found in [14]. We can now present several phase diagrams of stationary BH solutions with a single angular momentum. Fig. 1 shows the horizon area a H as a function of the angular momentum j in d = 6. The solid green line describes the singly spinning MP BH [16]. The large red square indicates the zero mode of the ultraspinning instability computed in [20,21]. The zero mode gives a bifurcation point to two branches of the lumpy BHs (the black squares and brown dots). Note, however, that the lumpy BHs follow a curve that is close to the MP curve. To better understand their relationship, Fig. 2 shows ∆a H as a function of j, where ∆a H is the difference in a H between a given solution and the MP BH with the same j. In this plot, it is clear that the lumpy BHs meet with the MP BHs in a second-order phase transition.
Note that the brown dots move towards a higher j with a higher entropy than MP, hits a cusp, and then decreases in a H and j, eventually having a lower entropy than the MP BH. This implies that the lumpy BHs are non-unique even among their own family, and provide the first example of non-uniqueness among an asymptotically flat spherical family. Continuing along this curve, we find that this branch of lumpy BHs approaches the black ring solutions given by the blue circles. (This curve for the black ring extends and agrees with the curve in [13]). A topology-changing merger between the lumpy BHs and the black rings is expected to occur through a local conical geometry as discussed in [26]. Indeed, we find that the Ricci scalar of the induced horizon geometry grows as we approach the merger. As further evidence for a merger, we plot the other thermodynamic variables as a function of j in Fig. 3 and Fig. 4. Again, we find the black ring and lumpy BHs approaching each other. In these plots, the cusps are described by smooth turning points.
The second family of lumpy BHs (the black squares) was not anticipated. To understand their existence, consider a perturbative expansion of the lumpy BHs around the ultraspinning merger point. At leading order, the amplitude of the lumpy BHs can be positive or negative and their entropy is linear in the amplitude; hence we have two branches of solutions. Note that previous studies [14] drew intuition from the Kaluza-Klein (non)uniform black string system [8,9] where the entropy is instead quadratic in the linear amplitude because periodicity ensures that both signs yield the same solution.
Predicting where these lumpy BHs lead is difficult; we can only conjecture a few possibilities. As we move along this family away from the merger with the MP BH (red square), we find that curvature invariants of these lumpy BHs grow large [33]. The curvature is largest at a location where the function S 2 in (1) vanishes. This suggests (but by no means implies) a possible topological transition to a S 2 × S d−4 solution with rotation on the S 2 . Such a solution would be supported by spin-spin interaction and hence does not have a blackfold approximation at lowest order [27]. We admit that there are other possibilities such as a double MP BH (also supported by spin-spin repulsion). This lumpy BH branch might also simply end in a nakedly singular configuration. A zero-temperature limit is also possible, but Fig. 3 suggests we are still far from zero temperature. So far, we have described the phase diagram for d = 6, but we expect similar behaviour in d > 6. Indeed, this is what happens in the d = 7 phase diagrams that are displayed in Fig. 5 and Fig. 6 [34].
We can compare our numerical black ring results with analytical results obtained from the leading-order blackfold approximation, which in this case is valid for large j. In d = 6, this is given by a H 1 j , t H 4j, ω H 1 4j 2 , and is shown by the dashed red line in Figs 1, 3 and 4. These results agree with our numerics for large j, as they should. In d > 6, one can also include finite-size corrections to the blackfold approximation (which only become dominant over self-gravitational effects in d > 6 [28]). For d = 7, these are given by [29] ξ 0 = 13Γ( 4 3 ) 2 (120) 2 This is illustrated for d = 7 in Fig. 5 and Fig. 6, where the dashed red line is the leading order blackfold result and the solid red line includes finite-size corrections. With the corrections, the agreement is impressive, even giving a H to ∼ 1% when j ∼ O(1). This is the first time these corrections have been compared to a numerical solution.
Finally, note that in Figs 1, 3, 4, 5, and 6, the isolated magenta squares describe the zero modes of the ultraspinning higher harmonics ( > 2) [20,21]. Given our findings for the first harmonic = 2, we conjecture that two families of lumpy solutions will bifurcate from each of these points, one of which will connect to (possi- The main plot is the horizon area aH as a function of the spin j. The inset is the difference in area ∆aH vs the spin j. The colour coding is the same as in Fig. 1 with the added solid red line representing the blackfold curve with finite-size corrections. bly lumpy) black saturns or black di-rings etc.
BW was supported by European Research Council grant no. ERC-2011-StG 279363-HiDGR. The authors thankfully acknowledge the computer resources, technical expertise, and assistance provided by CENTRA/IST. Some of the computations were performed at the cluster "Baltasar-Sete-Sóis" and supported by the DyBHo-256667 ERC Starting Grant.