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Figure 1:
Horizon area vs. angular momentum for Myers–Perry black holes with a single spin in
(black), (dark gray), and (light gray). |
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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). |
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Figure 3:
Phase space of (a) seven-dimensional, and (b) eight-dimensional MP rotating black holes
(in a representative quadrant ). The surfaces for extremal black holes are represented: black
holes exist in the region bounded by these surfaces. (a) : the hyperbolas at which the surface
intersects the planes (which are , i.e., and ) 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 becomes asymptotically of the form , i.e.,
the same shape as the five-dimensional diagram in Figure 2(a). (b) : ultraspinning regimes
exist in which two spins are much larger than the third one. The sections at large constant
asymptotically approach the same shape as the six-dimensional phase space Figure 2(b). |
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Figure 4:
Horizon area of five-dimensional MP black holes. We only display a
representative quadrant of the full phase space of Figure 2(a), the rest of the surface
being obtained by reflection along the planes and . |
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Figure 5:
Curve of horizon area vs. spin for five-dimensional black rings rotating along their
(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 : 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 . |
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Figure 6:
Phase space of doubly-spinning black rings ( , ), restricted to the
representative region . The dashed line 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 spin , for given rotation . It ends on the
extremal MP curve at . The (lower) thick black curve corresponds to regular nonextremal
black rings with minimal spin along for given on . It ends on the extremal
MP curve at . Black rings exist in the gray-shaded parameter regions bounded by the
black curves, the segment of the extremal MP dashed line, and the axis with
. 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 , there exist thin
and fat black rings and MP black holes: there is discrete three-fold nonuniqueness. |
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Figure 7:
The phase space covered by doubly-spinning MP black holes and black rings,
obtained by replicating Figure 6 taking . The square 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. |
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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, and (in 5D) potentials , ,
. |
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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 . In the final solution the parameters can
be adjusted so that the metric at on the axis is completely smooth. The (upper) horizon
rod in the final solution has mixed direction , while the other rods are aligned purely
along the or directions. |
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Figure 10:
Curves 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 , .
The cusp of the black-ring curve occurs at , . The cusp of the
black-Saturn curve is at , with area . |
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Figure 11:
Correspondence between phases of black membranes wrapped on a two-torus of side
(left) and quickly-rotating MP black holes with rotation parameter (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]). |
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Figure 12:
Proposal of [80] for the phase curves of thermal equilibrium phases in .
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 seem unlikely. If thermal equilibrium is not
imposed, the whole semi-infinite strip , is covered, and multiple
rings are possible. |
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Figure 13:
against for 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 . |
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Figure 14:
(vertical) against for 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 , so none of the black
holes are BPS. |