List of Figures

View Image Figure 1:
Horizon area vs. angular momentum for Myers–Perry black holes with a single spin in d = 5 (black), d = 6 (dark gray), and d = 10 (light gray).
View Image Figure 2:
Phase space of (a) five-dimensional and (b) six-dimensional MP rotating black holes: black holes exist for parameters within the shaded regions. The boundaries of the phase space correspond to extremal black holes with regular horizons, except at the corners of the square in five dimensions, where they become naked singularities. The six-dimensional phase space extends along the axes to arbitrarily large values of each of the two angular momenta (ultraspinning regimes).
View Image Figure 3:
Phase space of (a) seven-dimensional, and (b) eight-dimensional MP rotating black holes (in a representative quadrant ji ≥ 0). The surfaces for extremal black holes are represented: black holes exist in the region bounded by these surfaces. (a) d = 7: the hyperbolas at which the surface intersects the planes ji = 0 (which are √ -- jkjl = 1∕ 6, i.e., √ -- akal = μ and r0 = 0) correspond to naked singularities with zero area; otherwise, the extremal solutions are nonsingular. The three prongs extend to infinity; these are the ultraspinning regimes in which one spin is much larger than the other two. The prong along ji becomes asymptotically of the form |jk| + |jl| ≤ f(ji), i.e., the same shape as the five-dimensional diagram in Figure 2(a). (b) d = 8: ultraspinning regimes exist in which two spins are much larger than the third one. The sections at large constant ji asymptotically approach the same shape as the six-dimensional phase space Figure 2(b).
View Image Figure 4:
Horizon area a (j ,j ) H 1 2 of five-dimensional MP black holes. We only display a representative quadrant j1,j2 ≥ 0 of the full phase space of Figure 2(a), the rest of the surface being obtained by reflection along the planes j1 = 0 and j2 = 0.
View Image Figure 5:
Curve aH (j) of horizon area vs. spin for five-dimensional black rings rotating along their 1 S (solid). The dashed curve corresponds to five-dimensional MP black holes (see Figure 1). The solid curve for black rings has two branches that meet at a regular, nonextremal minimally-rotating black ring at ∘ ------ j = 27∕32: an upper branch of thin black rings, and a lower branch of fat black rings. Fat black rings always have a smaller area than MP black holes. Their curves meet at the same zero-area naked singularity at j = 1.
View Image Figure 6:
Phase space of doubly-spinning black rings (j1 ≡ jψ, j2 ≡ jφ), restricted to the representative region j1 > j2 ≥ 0. The dashed line j1 + j2 = 1 corresponds to extremal MP black holes (see Figure 2(a)). The (upper) thin black curve corresponds to regular extremal black rings with degenerate horizons at maximal S2 spin j 2, for given S1 rotation j 1. It ends on the extremal MP curve at (3∕4,1∕4). The (lower) thick black curve corresponds to regular nonextremal black rings with minimal spin j1 along 1 S for given j2 on 2 S. It ends on the extremal MP curve at (4∕5,1∕5). Black rings exist in the gray-shaded parameter regions bounded by the black curves, the segment j1 ∈ [3∕4,4∕5] of the extremal MP dashed line, and the j1 axis with j ≥ ∘27--∕32 1. In the light-gray region there exist only thin black rings. In the dark-gray spandrel between the dashed MP line, the thick black curve, and the axis ∘ ------ 27∕32 < j1 < 1, there exist thin and fat black rings and MP black holes: there is discrete three-fold nonuniqueness.
View Image Figure 7:
The phase space (j1,j2) covered by doubly-spinning MP black holes and black rings, obtained by replicating Figure 6 taking ±j1 ↔ ±j2. The square |j1| + |j2| ≤ 1 corresponds to MP black holes (see Figure 2(a)). The light-gray zones contain thin black rings only, and the medium-gray zone contains MP black holes only. At each point in the dark-gray spandrels near the corners of the square there exist one thin and one fat black ring, and one MP black hole.
View Image Figure 8:
Rod structures for the (a) 4D Schwarzschild and (b) 5D Tangherlini black holes. From top to bottom, the lines represent the sources for the time, φ and ψ (in 5D) potentials Ut, U φ, U ψ.
View Image Figure 9:
Rod structures for (a) the seed used to generate (b) the rotating black ring. The seed metric is diagonal, and the dotted rod has negative density − 1∕2. In the final solution the parameters can be adjusted so that the metric at z = a1 on the axis is completely smooth. The (upper) horizon rod in the final solution has mixed direction (1,0,Ω ψ), while the other rods are aligned purely along the φ or ψ directions.
View Image Figure 10:
Curves aH (j) for phases of five-dimensional black holes with a single angular momentum: MP black hole (black), black ring (dark gray), black Saturn (light gray). We only include those black Saturns, where the central black hole and the black ring have equal surface gravities and angular velocities. The three curves meet tangentially at a naked singularity at j = 1, aH = 0. The cusp of the black-ring curve occurs at ∘ ------ j = 27∕32 ≈ 0.9186, aH = 1. The cusp of the black-Saturn curve is at j ≈ 0.9245, with area aH ≈ 0.81.
View Image Figure 11:
Correspondence between phases of black membranes wrapped on a two-torus of side L (left) and quickly-rotating MP black holes with rotation parameter a ∼ L ≥ r0 (right: must be rotated along a vertical axis): (i) uniform black membrane and MP black hole; (ii) nonuniform black membrane and pinched black hole; (iii) pinched-off membrane and black hole; (iv) localized black string and black ring (reproduced from [80]).
View Image Figure 12:
Proposal of [80] for the phase curves aH(j) of thermal equilibrium phases in d ≥ 6. The solid lines and figures have significant arguments in their favor, while the dashed lines and figures might not exist and admit conceivable, but more complicated, alternatives. Some features have been drawn arbitrarily; at any given bifurcation, and in any dimension, smooth connections are possible instead of swallowtails with cusps; also, the bifurcation into two black-Saturn phases may happen before, after, or right at the merger with the pinched black hole. Mergers to di-rings or multiple-ring configurations that extend to asymptotically large j seem unlikely. If thermal equilibrium is not imposed, the whole semi-infinite strip 0 < aH < aH(j = 0), 0 ≤ j < ∞ is covered, and multiple rings are possible.
View Image Figure 13:
GM ∕ℓ against G |J |∕ ℓ2 for d = 4 Kerr-AdS black holes. The thick curve corresponds to extremal black holes. Black-hole solutions lie on, or above, this curve (which was determined using results in [20]). The thin line is the BPS bound M = |J|∕ℓ.
View Image Figure 14:
GM ∕ ℓ2 (vertical) against G|Ji|∕ℓ3 for d = 5 Myers–Perry-AdS black holes. Nonextremal black holes fill the region above the surface. The surface corresponds to extremal black holes, except when one of the angular momenta vanishes (in which case there is not a regular horizon, just as in the asymptotically-flat case). This “extremal surface” lies inside the square-based pyramid (with vertex at the origin) defined by the BPS relation M = |J1|∕ℓ + |J2|∕ℓ, so none of the black holes are BPS.