VOLUME 88, N
UMBER 10 PHYSICAL REVIEW LETTERS 11M
ARCH 2002
A Rotating Black Ring Solution in Five Dimensions
Roberto Emparan
1,
* and Harvey S. Reall
2
1
Theory Division, CERN, CH-1211 Geneva 23, Switzerland
2
Physics Department, Queen Mary College, Mile End Road, London E1 4NS, United Kingdom
(Received 8 November 2001; published 21 February 2002)
The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary
asymptotically flat spacetime regular on and outside an event horizon of topology
S
1
3 S
2
. It describes
a rotating “black ring.” This is the first example of a stationary asymptotically flat vacuum solution with
an event horizon of nonspherical topology. The existence of this solution implies that the uniqueness
theorems valid in four dimensions do not have simple five-dimensional generalizations. It is suggested
that increasing the spin of a spherical black hole beyond a critical value results in a transition to a black
ring, which can have an arbitrarily large angular momentum for a given mass.
DOI: 10.1103/PhysRevLett.88.101101 PACS numbers: 04.50.+h, 04.20.Jb, 04.70.Bw
Black holes in four spacetime dimensions are highly
constrained objects. A number of classical theorems show
that a stationary, asymptotically flat, vacuum black hole
is completely characterized by its mass and spin [1], and
event horizons of nonspherical topology are forbidden [2].
In this Letter we show explicitly that in five dimen-
sions the situation cannot be so simple by exhibiting an
asymptotically flat, stationary, vacuum solution with a
horizon of topology
S
1
3 S
2
: a black ring. The ring
rotates along the S
1
and this balances its gravitational
self-attraction. The solution is characterized by its mass
M and spin J. The black hole of [3] with rotation in
a single plane (and horizon of topology S
3
) can be ob-
tained as a branch of the same family of solutions. We
show that there exist black holes and black rings with
the same values of M and J. They can be distinguished
by their topology and by their mass dipole measured
at infinity. This shows that there is no obvious five-
dimensional analog of the uniqueness theorems.
S
1
3 S
2
is one of the few possible topologies for the
event horizon in five dimensions that was not ruled out by
the analysis in [4] (although this argument does not apply
directly to our black ring because it assumes time symme-
try). An explicit solution with a regular (but degenerate)
horizon of topology S
1
3 S
2
and spacelike infinity with
S
3
topology has been built recently in [5]. An uncharged
static black ring solution is presented in [6], but it contains
conical singularities. Our solution is the first asymptot-
ically flat vacuum solution that is completely regular on
and outside an event horizon of nonspherical topology.
Our starting point is the following metric, constructed
as a Wick-rotated version of a solution in [7]:
ds
2
苷 2
F共x兲
F共 y兲
µ
dt 1
r
n
j
1
j
2
2 y
A
dc
∂
2
1
1
A
2
共x 2 y兲
2
∑
2F共x兲
µ
G共 y兲dc
2
1
F共 y兲
G共 y兲
dy
2
∂
1 F共 y兲
2
µ
dx
2
G共x兲
1
G共x兲
F共x兲
df
2
∂∏
, (1)
where j
2
is defined below and
F共j兲 苷 1 2j兾j
1
, G共j兲 苷 1 2j
2
1nj
3
. (2)
The solution of [7] was obtained as the electric dual of
the magnetically charged Kaluza-Klein C metric of [8].
Our metric can be related directly to the latter solution by
analytic continuation. When n 苷 0 we recover the static
black ring solution of [6].
We assume that 0 ,n,n
ⴱ
⬅ 2兾共3
p
3兲, which en-
sures that the roots of G共j兲 are all distinct and real. They
will be ordered as j
2
,j
3
,j
4
. It is easy to establish
that 21 ,j
2
, 0 , 1 ,j
3
,j
4
,
1
n
. A double root
j
3
苷 j
4
appears when n 苷 n
ⴱ
. Without loss of generality,
we take A . 0. Taking A , 0 simply reverses the sense
of rotation.
We take x to lie in the range j
2
# x #j
3
and require
that j
1
$j
3
, which ensures that g
xx
, g
ff
$ 0. In order
to avoid a conical singularity at x 苷 j
2
we identify f with
period
Df 苷
4p
p
F共j
2
兲
G
0
共j
2
兲
苷
4p
p
j
1
2j
2
n
p
j
1
共j
3
2j
2
兲共j
4
2j
2
兲
.
(3)
A metric of Lorentzian signature is obtained by taking
y ,j
2
. Examining the behavior of the constant t slices of
(1), one finds that c must be identified with period Dc 苷
Df in order to avoid a conical singularity at y 苷 j
2
fi x.
Regularity of the full metric here can be demonstrated by
converting from the polar coordinates 共 y, c兲 to Cartesian
coordinates —the dtdc term can then be seen to vanish
smoothly at the origin y 苷 j
2
.
There are now two cases of interest depending on the
value of j
1
. One of these will correspond to a black ring
101101-1 0031-9007兾02兾88(10)兾101101(4)$20.00 © 2002 The American Physical Society 101101-1
VOLUME 88, N
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ARCH 2002
and the other to the black hole of [3] with only one non-
vanishing angular momentum.
Case 1 is defined by
j
1
.j
3
. In this case, g
ff
vanishes
at x 苷 j
3
and there will be a conical singularity there
unless f is identified with period
Df
0
苷
4p
p
F共j
3
兲
jG
0
共j
3
兲j
苷
4p
p
j
1
2j
3
n
p
j
1
共j
3
2j
2
兲共j
4
2j
3
兲
.
(4)
We demand Df 苷 Df
0
for consistency with (3). Since
j
2
,j
3
, this is possible only if j
1
is fixed as a function
of the other three roots (i.e., of n)as
j
1
苷
j
2
4
2j
2
j
3
2j
4
2j
2
2j
3
共black ring兲 . (5)
In this case it is easy to show that j
3
,j
1
,j
4
, from
which it follows that the factors of F共x兲 in the metric
are never zero. x and f parametrize a regular surface
of topology S
2
. The sections at constant t, y have the
topology of a ring S
1
3 S
2
. x 苷 j
3
is an axis pointing
“into” the ring (i.e., decreasing S
1
radius) and x 苷 j
2
points out of the ring. This coordinate system is sketched
for the static black ring in [6]; the case considered here is
very similar.
Case 2 is defined by
j
1
苷 j
3
共black hole兲 . (6)
In this case, g
ff
does not vanish at x 苷 j
3
, hence (4) need
not be imposed. When (6) holds, the sections at constant
t, y have the topology of three-spheres S
3
, with c and f
being two independent rotation angles.
The following analysis applies to both cases 1 and 2.
Asymptotic infinity lies at x 苷 y 苷 j
2
.Defining
˜
f 苷
2pf
Df
,
˜
c 苷
2pc
Dc
, and using the coordinate transformation
z 苷
p
j
2
2 y
˜
A共x 2 y兲
, h 苷
p
x 2j
2
˜
A共x 2 y兲
, (7)
with
˜
A 苷 Aj
1
p
n共j
3
2j
2
兲共j
4
2j
2
兲兾关2共j
1
2j
2
兲兴, the
asymptotic metric is brought to the manifestly flat form
ds
2
⬃ 2dt
2
1 dz
2
1z
2
d
˜
c
2
1 dh
2
1h
2
d
˜
f
2
.
(8)
Note that the Killing vector fields k 苷 ≠兾≠t, m 苷 ≠兾≠
˜
c
are canonically normalized near infinity, and
˜
c and
˜
f both
have period 2p.
The Arnowitt-Deser-Misner mass and angular momen-
tum are
M 苷
3p
2GA
2
j
1
2j
2
nj
2
1
共j
3
2j
2
兲共j
4
2j
2
兲
, (9)
J 苷
2p
GA
3
共j
1
2j
2
兲
5兾2
n
3兾2
j
3
1
共j
3
2j
2
兲
2
共j
4
2j
2
兲
2
. (10)
The next limit to consider is y ! 2`. Changing co-
ordinates to Y 苷 21兾y gives a metric regular in a neigh-
borhood of Y 苷 0. Hence there is a new region Y , 0,
in which the coordinate y can be defined as y 苷 21兾Y
and the metric takes the same form as above. The metric in
this region is regular in these coordinates for y .j
4
. k be-
comes spacelike precisely at Y 苷 0, so the region y .j
4
is referred to as the “ergoregion.” The ergosurface at Y 苷
0 has topology S
1
3 S
2
in case 1 and S
3
in case 2. In both
cases, m remains spacelike throughout the ergoregion.
The above coordinates break down at y 苷 j
4
,sodefine
new coordinates x and y by
dx 苷 dc1
p
2F共 y兲
G共 y兲
dy , (11)
dy 苷 dt 1
r
n
j
1
共 y 2j
2
兲
p
2F共 y兲
AG共 y兲
dy , (12)
so x is periodic with period Dx 苷 Dc. In these new
coordinates, the metric takes the form
ds
2
苷 2
F共x兲
F共 y兲
µ
dy2
r
n
j
1
y 2j
2
A
dx
∂
2
1
1
A
2
共x 2 y兲
2
∑
F共x兲共2G共 y兲dx
2
1 2
q
2F共 y兲 dxdy兲 1 F共 y兲
2
µ
dx
2
G共x兲
1
G共x兲
F共x兲
df
2
∂∏
. (13)
This is regular at y 苷 j
4
so the coordinate y can now be
continued into the region y ,j
4
. The surface y 苷 j
4
is
a Killing horizon of the Killing vector field
j 苷
≠
≠y
1
A
p
j
1
p
n 共j
4
2j
2
兲
≠
≠x
(14)
with surface gravity
k 苷
A
p
n
2
j
1
共j
4
2j
3
兲
p
j
4
2j
1
. (15)
Outside this horizon, j 苷 k 1V
H
m, where
V
H
苷
2pA
p
j
1
Df共j
4
2j
2
兲
p
n
苷
A
p
n
2
j
1
共j
3
2j
2
兲
p
j
1
2j
2
. (16)
Note that j is tangent to the null geodesic generators of
the horizon, and j ?≠共
˜
c2V
H
t兲 苷 0 on the horizon. It
follows that the horizon is rotating with angular velocity
V
H
with respect to the inertial frame at infinity.
We have established that the solution possesses a rotat-
ing horizon. The area of a constant time slice through the
horizon is
A 苷
16p
2
A
3
共j
4
2j
1
兲
3兾2
共j
1
2j
2
兲
n
3兾2
j
3
1
共j
4
2j
3
兲共j
3
2j
2
兲共j
4
2j
2
兲
2
.
(17)
In case 1, when the regularity condition (5) is imposed,
the topology of (a constant time slice through) this event
horizon is S
1
3 S
2
: it is a rotating black ring. In case 2,
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VOLUME 88, N
UMBER 10 PHYSICAL REVIEW LETTERS 11M
ARCH 2002
when (6) is imposed, the horizon is a rotating three-sphere.
The latter is actually the five-dimensional rotating black
hole of [3], with one angular momentum parameter set to
zero. To see this, change coordinates to
r
2
苷 m
共j
3
2 y兲共j
4
2 x兲
共j
4
2j
2
兲共x 2 y兲
,
cos
2
u 苷
共j
3
2 y兲共x 2j
2
兲
共j
3
2j
2
兲共x 2 y兲
,
(18)
and define
m 苷
4
A
2
nj
2
3
共j
4
2j
2
兲
, a 苷
2
p
j
3
2j
2
A
p
nj
3
共j
4
2j
2
兲
(19)
(compare [9]). Then one recovers the five-dimensional
Myers-Perry black hole in Boyer-Lindquist coordinates, m
and a being the mass and rotation parameters defined in
[3]. We emphasize that the above expressions for M, J, k,
V
H
, and A are valid for both the black ring and the black
hole.
Physically, (5) is the condition that the rotation bal-
ances the gravitational self-attraction of the ring, and it
fixes a relation between its mass, spin, and radius. There
are two independent parameters for the solutions, n and
A. A has dimensions of inverse length and sets the scale
for the solution. n is dimensionless and determines the
shape of the S
1
3 S
2
horizon. The S
1
can be character-
ized by the inner radius of curvature R
i
at x 苷 j
3
, and
the outer radius R
o
at x 苷 j
2
.Asn ! 0, both radii
tend to the same value R ! 3J兾M 共!
p
2兾A兲. Also,
V
H
R ! 1. Keeping M fixed, the area of the S
2
tends
to zero and R tends to infinity, so small n corresponds
to a large thin ring. In Fig. 1 we plot physical quanti-
ties as a function of n when M is fixed. As n ! n
ⴱ
,
V
H
! 共3p兾8GM兲
1兾2
.Forn near n
ⴱ
the ring is highly
flattened.
If n 苷 n
ⴱ
, then the black ring and the black hole de-
generate to the same solution with j
3
苷 j
1
苷 j
4
. This
is the m 苷 a
2
limit of the five-dimensional rotating black
hole, for which the horizon disappears and is replaced by
a naked singularity.
FIG. 1. Plots, as functions of n at fixed M, of the inner (R
i
)
and outer (R
o
) radii of curvature of the S
1
, total area A of
the ring, surface gravity k, and angular velocity at the horizon
V
H
. All quantities are rendered dimensionless by dividing by
an appropriate power of GM.
The spin of the five-dimensional black holes is bounded
from above [3]:
J
2
M
3
#
32G
27p
, (20)
with equality for the (singular) m 苷 a
2
solution. The cor-
responding ratio for the black ring solutions is
J
2
M
3
苷
32G
27p
共j
4
2j
2
兲
3
共2j
4
2j
2
2j
3
兲
2
共j
3
2j
2
兲
. (21)
These ratios are plotted as a function of n in Fig. 2. Note
that the angular momentum of the ring is bounded from
below,
J
2
M
3
. 0.8437
32G
27p
. (22)
It is known that in six or more dimensions the spin of a
black hole can be arbitrarily large [3]. In five dimensions,
we have shown that the spin can also grow indefinitely, but
only if the spinning object is a ring.
For 0.2164 ,n,n
ⴱ
, there are two black ring solu-
tions with the same values of M and J (but different A).
Moreover, these satisfy the bound (20) so there is also a
black hole with the same values of M and J. This is the
first explicit demonstration that the uniqueness theorems
valid in four dimensions do not have a simple generaliza-
tion to five dimensions.
It is straightforward to check that the classical quantities
M, J, V
H
, k, and A satisfy a Smarr relation
M 苷
3
2
µ
kA
8pG
1V
H
J
∂
. (23)
Another interesting formula relates R
i
to M and k:
k 苷
3pR
i
8GM
. (24)
Since the temperature of the horizon is given by k兾2p,
this formula shows that the temperature of the ring is in-
versely proportional to its mass per unit length (around the
inner S
1
).
FIG. 2. 共27p兾32G兲J
2
兾M
3
as a function of n. Here and in the
following graph, the solid line corresponds to the black ring,
the dashed line to the black hole. The two dotted lines delimit
the values for which a black hole and two black rings with the
same mass and spin can exist.
101101-3 101101-3
VOLUME 88, N
UMBER 10 PHYSICAL REVIEW LETTERS 11M
ARCH 2002
FIG. 3. A兾共GM兲
3兾2
against
p
27p兾32GJ兾M
3兾2
, around the
regime in which a black hole and two black rings with the same
M and J exist. For
p
27p兾32GJ兾M
3兾2
艐 0.942 there exist a
black hole and a black ring with the same mass, spin, and area
A 艐 5.157共GM兲
3兾2
.
If one considers perturbing the black ring in such a way
that it settles down to another black ring solution, then the
first law of black-hole mechanics can be proved in the usual
way [10] since this proof does not depend on the topology
of the event horizon.
The solution with the larger area (entropy) for given val-
ues of
M and J is the one expected to be globally ther-
modynamically stable in the microcanonical ensemble. In
Fig. 3 we have plotted A兾M
3兾2
as a function of J兾M
3兾2
.
As the spin increases (with the mass held fixed), there is
first a small range of spins for which the black hole has lar-
ger area than both black rings. However, at a slightly larger
spin, and before the singular limit is reached, the larger
black ring has a greater area than the black hole and is the
preferred configuration. Hence we conjecture that, as a
five-dimensional black hole is spun up, a phase transition
occurs from the black hole to a black ring. The singular
solution is never reached.
Classically, the second law of black-hole mechanics sug-
gests that a black hole might evolve into a black ring as
angular momentum is added to it. However, this involves
a change in the topology of the horizon, and it is not clear
whether this is possible classically (see Ref. [11] for an ex-
ample in which a classical topology change of the horizon
is forbidden).
If
n ! 0 at fixed M, then the ring becomes large and
thin so one might expect ripples along the c direction
to lead to a classical Gregory-Laflamme [12] instabil-
ity. However, if the above conjecture is correct, then we
would expect a range of values 0 ,n
1
,n,n
2
,n
ⴱ
for which the ring is classically stable. The results of [11]
suggest that the horizon of unstable rings would tend to
become lumpy around the S
1
. The changing quadrupole
moment of such an object would lead to emission of gravi-
tational radiation until a stable end point was reached, pre-
sumably either another ring or a spherical black hole [13].
The extremal limit of the black-hole solution is a naked
singularity, and it appears that the black ring plays a role
in smoothing out the approach to this singularity. The ex-
istence of the ring may be related to cosmic censorship.
In four dimensions, the extremal limit of the black hole is
regular and the third law of black-hole mechanics forbids
violation of the angular momentum bound
jJj # GM
2
[10,14]. In dimension D $ 6, black holes with a single
nonvanishing angular momentum can carry arbitrarily high
J (for a given M). Hence cosmic censorship would not re-
quire black rings to exist in these cases. Five-dimensional
black holes with
two nonzero angular momenta satisfy the
bound (20) with J replaced by jJ
1
j 1 jJ
2
j, but their ex-
tremal limit is nonsingular so the third law suggests that
this bound cannot be violated by throwing matter into the
hole. However, one might wonder what would happen if
one were to throw some matter with J
2
fi 0 into a black
ring (with J
2
苷 0). There may be a generalization of the
black ring solution that carries two angular momenta.
R. E. acknowledges partial support from UPV Grant
No. 063.310-EB187/98 and CICYT AEN99-0315. H. S. R.
was supported by PPARC.
*Also at Departamento de Física Teórica, Universidad del
País Vasco, Bilbao, Spain.
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