Brane World Black Rings.
Anurag Sahay∗, Gautam Sengupta, †,
Department of Physics,
Indian Institute of Technology,
Kanpur 208016, India.
Abstract
Five dimensional neutral rotating black rings are described from a
Randall-Sundrum brane world perspective in the bulk black string frame-
work. To this end we consider a rotating black string extension of a five
dimensional black ring into the bulk of a six dimensional Randall-Sundrum
brane world with a single four brane. The bulk solution intercepts the four
brane in a five dimensional black ring with the usual curvature singular-
ity on the brane. The bulk geodesics restricted to the plane of rotation
of the black ring are constructed and their projections on the four brane
match with the usual black ring geodesics restricted to the same plane.
The asymptotic nature of the bulk geodesics are elucidated with reference
to a bulk singularity at the AdS horizon. We further discuss the descrip-
tion of a brane world black ring as a limit of a boosted bulk black 2 brane
with periodic identification.
April 2007
∗ashaya@iitk.ac.in
†sengupta@iitk.ac.in
http://arxiv.org/abs/0704.0996v2
1 Introduction.
Higher dimensional spacetimes are now an essential aspect of effective field
theories arising from fundamental theories of quantum gravity. The general as-
sumption implicit in such constructions was that the extra spatial dimensions
are compactified to ultrashort length scales. Hence quantum gravity effects were
relegated to very high energy scales. However in recent years the exciting possi-
bility of low scale quantum gravity effects in the brane world models have inspired
considerable interest and interesting phenomenological consequences [1–3]. The
brane world scenario envisaged the gauge sector of the fundamental interactions
to be restricted on a smooth codimension one hypersurface ( refered to as a brane)
embedded in a higher dimensional space-time and the electroweak scale as the
fundamental scale. The usual four dimensional Planck scale was then a derived
scale. In particular the Randall-Sundrum models and their variants based on a
warped non factorable compactification geometry in a bulk Anti deSitter ( AdS)
space time offered a partial resolution to the vexing hierarchy problem [4]. Al-
though the analysis was valid in a linearized framework a full non linear study
from a supergravity perspective confirmed the conclusions and their extension to
any Ricci flat geometry on the brane [5, 6].
For consistency the brane world scenario requires generic four dimensional
gravitational configurations on the brane to arise from a higher dimensional bulk.
The investigation of black hole configurations in this context has been an exciting
aspect of the study of brane world gravity [5]. Such a black hole on the brane
is expected to be a configuration extended in the bulk. Chamblin, Hawking and
Reall [7] attempted the description of a Schwarzschild black hole in a typical single
three brane five dimensional Randall-Sundrum brane world as a bulk black string.
This reproduced the usual Schwarzschild singularity on the brane but additionaly
was also singular at the AdS horizon far away from the three brane. Although a
pathology, this singularity was possibly a linearization artifact and could be shown
to be a mild p-p curvature singularity. The bulk black string was subject to the
usual instabilities against long wavelength perturbations [8,9] and was expected to
pinch off to a cigar geometry before reaching the AdS horizon. However the issue
of stability is contentious and for sphericaly symmetric solutions it was shown
that a more likely scenario is a transition to a non uniform black string [10]
In an earlier article [11]we have generalized the construction of Chamblin et.
al. [7]to consider rotating black holes in a five dimensional single three brane
RS brane world. The bulk configuration proposed was a five dimensional rotat-
ing black string which intercepted the three-brane in a four dimensional rotating
black hole described by a Kerr metric on the three brane. It was found that
the Kerr solution too was singular at the AdS horizon apart from the usual ring
singularity on the brane. The asymptotics of the equatorial geodesics at the AdS
horizon also indicated a p-p curvature singularity although an explicit determina-
tion was computationaly intractable. There have been other approaches to brane
world black holes including numerical studies for off brane metrics and a Hamil-
tonian constraint approach to charged black holes [12–18]. In lower dimensions
exact studies of brane world black holes [19] involving the AdS C-metric have
indicated that the bulk solutions are regular everywhere emphasizing that the
bulk singularity in higher dimension is possibly a linearization artifact. However
absence of exact bulk metrics in higher dimensions requires a linearized approach
and the black string framework is hence physicaly relevant in this context in spite
of such a bulk singularity.
The brane world constructions must be embedded in an appropriate string
theory for consistency, requiring the generalizations of these models to higher
dimensions. The generalization of the Randall-Sundrum construction and its
variants to higher dimensions with a single space like AdS direction and an ap-
propriate codimension one brane is straightforward. Additionaly this may easily
be extended to include the full non linear extensions of a Ricci flat metric [20–22].
In higher dimensions also the consistency of such brane world constructions re-
quire that gravitational configurations arise from appropriate bulk scenarios. In
particular this applies to higher dimensional black holes on the codimension one
brane. In this context in an earlier article [23]we had described the N dimensional
rotating Myers-Perry [24]black hole on a single (N-1) brane in a (N + 1) dimen-
sional RS brane world. The bulk solution in this case was a (N+1) dimensional
rotating black string extended in the AdS direction transverse to the (N-1) brane.
Analysis of equatorial geodesics again indicated a p-p curvature singularity in the
bulk apart from the usual extended singularity on the (N-1) brane.
In the recent past there has been remarkable and surprising progress in under-
standing higher dimensional black holes. In particular it has been realized that
the no hair and the uniqueness theorems are much less restrictive in higher dimen-
sions [27]. In four dimensions the no hair theorem characterizes any stationary
asymptoticaly flat black hole solution of Einstein-Maxwell system only by their
mass, angular momentum and conserved charges whereas the uniqueness theorem
forbids event horizons of non spherical toplogies. However the discovery [25] in
five dimensions of an asymptoticaly flat stationary black hole solution with a non
spherical ring like S2×S1 horizon topology with the possibility of dipole charges,
showed that higher dimensional black holes posess remarkably distinctive prop-
erties. The static black ring solution [28] was first obtained through the Wick
rotation of a neutral solution of an Einstein-Maxwell system [29] although they
involved conical singularities. However the stationary solution rotating in the
S1 direction was regular everywhere except the usual curvature singularity. For
fixed mass the angular momentum of the black ring was bounded below and for a
certain range of parameters two black rings and a usual five dimensional rotating
Myers-Perry black hole all with the same mass and spin coexist. The charged
versions of these black rings were first obtained in the framework of D=5 heterotic
supergravity [30] and fully supersymmeric three charged black ring solutions in
D=5 followed later from compactifications of black supertubes in D=10 [31, 32].
It was seen that these black rings could also support gauge dipoles independent
of the conserved gauge charges entailing an infinite non uniqueness and violating
the no hair theorem [33].
As emphasized earlier, for consistency of the brane world scenario it is im-
perative that gravitational configurations like black holes on the brane should
arise from appropriate bulk solutions. In this context it is but natural to inves-
tigate possible bulk configurations in a higher dimensional brane world scenario
which would describe five dimensional black rings on the brane. This is espe-
cialy relevant for the neutral rotating black rings as they are Ricci flat and hence
satisfy the criteria for embedding in higher dimensional Randall-Sundrum brane
worlds. Naturaly the absence of exact solutions in higher dimensions require the
usual linearized framework to analyse this question. The black string approach
is especialy relevant in this context to highlight the physical aspects of such an
embedding although it suffers from singular pathologies which are possibly lin-
earization artifacts.
In this article we address this issue and show that it is possible to consistently
embed the five dimensional black ring solution on a single four brane in a (5 + 1)
dimensional Randall-Sundrum brane world. Following the black string approach
we consider a six dimensional bulk rotating black string extension of the five
dimensional black ring. This bulk configuration intercepts the four brane in a
five dimensional rotating black ring. In what follows after a brief review of neutral
rotating black rings, we obtain their geodesic equations in the plane of the ring
analogous to the equatorial plane of black holes with spherical topologies. We
further investigate the asymptotic behaviour of both the null and the timelike
geodesics in this plane to elucidate the restricted causal structure of the black ring
space time. In section three we consider a bulk rotating black string extension
of a five dimensional neutral rotating black ring in a six dimensional RS brane
world with a single four brane. The bulk black string intercepts the four brane in
a five dimensional black ring with the usual spacelike curvature singularity on the
brane. Additionaly a curvature singularity also appears at the AdS horizon far
away from the four brane. Following the description of a black ring as a boosted
black string with periodic identification in a certain limit, the bulk solution may
be described as a boosted black two brane with the same periodic identification.
We then construct the six dimensional bulk geodesics in the plane of rotation of
the ring and show that their projections on the four brane reproduces the usual
five dimensional black ring geodesics in the same plane. To study of the nature
of the pathological singularity at the AdS horizon we further investigate the late
time asymptotics of these geodesics. It is shown that the curvature remains
finite along unbound geodesics which reach the AdS horizon. We also discuss the
possibility of the bulk solution to pinch off before reaching the AdS horizon due
to the usual instabilities and comment on the possible stable solution in the light
of the analysis outlined in [9] and [10]. In the last section we provide a summary
of our analysis and results and also discuss certain future open issues in this area.
2 The Rotating Neutral Black Ring .
In this section we first briefly review the neutral rotating black ring and eluci-
date the nature of the adapted coordinate system . We then construct the black
ring geodesics restricted to the plane of rotation of the ring which is analogous
to the equatorial geodesics in solutions with a spherical topology. Furthermore
we analyse the geodesic equations to study the nature of the radial orbits for
this plane and their asymptotics. The static neutral black ring was originally
discovered through a Wick rotation of certain Kaluza Klein C metrics decribing
neutral bubbles [29]. These involved conical singularities and consequent deficit
angles leading to either cosmic string defects joining these singularities or deficit
membranes. However an analytic continuation led to the original neutral rotat-
ing black ring solution which was a five dimensional asymptoticaly flat black hole
with a ringlike S2×S1 horizon topology, regular everywhere except at a spacelik
e curvature singularity. The original solution was further refined through appro-
priate factorizable choice of certain functions appearing in the metric [26,30–32].
The rotating black ring in equlibrium was parametrized by a dimensionless re-
duced angular momentum j = 27π
which was bounded from below for a fixed
mass. It could be shown that in the range 27
≤ j2 < 1 there existed one Myers-
Perry black hole with spherical topology and two black rings with identical mass
and angular momenta, in direct violation of the black hole uniqueness theorem.
2.1 Black Ring Metric
The metric of the neutral rotating five dimensional black ring in a specific adpated
coordinate system which is obtained from the foliation of space-time in terms of
the equipotentials of certain 1-form and 2-form gauge potentials is , [32]
ds2 = −
F (y)
F (x)
dt− C R
1 + y
F (y)
(x− y)2
F (x)
−G(y)
F (y)
dψ2 − dy
F (x)
, (1)
where the functions
F (ξ) = 1 + λξ, G(ξ) = (1− ξ2)(1 + νξ) , (2)
λ(λ− ν)
1 + λ
. (3)
Here R is a length scale which may be interpreted as the radius of the ring in
some limit [32] and the two dimensionless parameters λ and ν which are related
to the shape and the rotation velocity of the ring lie in the range
0 < ν ≤ λ < 1 (4)
. The range of the spatial co-ordinates (x, y) are required to be,
− 1 ≤ x ≤ +1 , −∞ ≤ y ≤ −1 . (5)
respectively.
The constant y hypersurfaces are nested deformed solid toroids with topology
S2 × S1, whereas the coordinate x is like a direction cosine, x = +1 points to
the interior of the ring and x = −1 points to the region outside the ring. The
solution is a stationary axisymmetric solution with rotation in the ψ direction,
and admits t, φ, and ψ Killing isometries.
In order to avoid conical singularities at the fixed points x = −1 and y = −1 of
the Killing isometries ∂φ and ∂ψ the co-ordinates ψ and φ require to be identified
with the equal periods
∆ψ = ∆φ = 4π
F (−1)
|G′(−1)|
. (6)
Furthermore the requirement that the orbits of the isometry ∂φ shows no deficit
angles at x = +1 lead to the condition
1 + ν2
The co-ordinates (x, φ) parametrize a two-sphere S2, the co-ordinate ψ parametrizes
a circle S1 and the solution describes a black ring having a regular horizon of
topology S1 × S2 and rotating in the S1 plane. However the horizon geometry is
not a simple product of S2 and S1 as the two sphere S2 is deformed there and
the deformation grows away from the horizon.
The metric reduces to a conventional five dimensional Myers-Perry black hole
with rotation in a single plane if, instead of (7), we consider the limit, R → 0,
(λ, ν) → 1 and the parameters
, a2 = 2R2
(1− ν)2
, (8)
are held constant. In this case the co-ordinates (x, φ, ψ) characterises a three-
sphere S3 which is a regular horizon of a five dimensional Myers-Perry black hole.
The ergosphere and the event horizon of the black ring are located at y = −1/λ
and y = −1/ν respectively. At y = −∞ there is a spacelike curvature singularity
inside the horizon. Asymptotic infinity is reached as (x, y) → −1.
The ADM mass and angular momentum are given as
λ(λ− ν)(1 + λ)
(1− ν)2
. (10)
The curvature squared for the black ring spacetime is computed to be,
RµνρσR
µνρσ =
6ν2(1 + ν2)2Q(x, y)
R4(1 + ν2 + 2νx)6
(x− y)4, (11)
where Q(x, y) is a poynomial of degree six in x and y. Hence there is a spacelike
curvature singularity at y = −∞ inside the event horizon. In terms of the Myers-
Perry co-ordinates (t, r, θ, ψ, φ) the difference (x−y) goes like 1/r2 at large r, i.e.
towards spatial infinity, so that the curvature squared goes as
RµνρσR
µνρσ ∼
as obtained in the case of five dimensional Myers-Perry black hole.
The rotating black ring in the limit of large radius R may be described after
appropriate coordinate redfinitions as a Schwarzschild black string boosted and
periodically identified along the translation invariant direction with a period 2πR
[30, 32, 33]. The black string metric is given as
ds2 = dw2 − (1−
)dt2 + (1−
)−1dr2 + r2dΩ2
, (13)
where the horizon is at r = r0 and w is the translation invariant direction. The
parameter ν = r0/R is seen to correspond to the thickness of the ring or the
ratio of the radius of the S2 at the horizon and the ring radius R . The ratio
λ/ν then measures the speed of rotation of the ring in the S1 direction and the
coordinate ψ = w/R corresponds to a redefined translation invariant direction of
the black string which is periodically identified as w = w + 2πR. The speed of
rotation is related to the local boost velocity given by
1− (ν/λ) and reduces to
1− (ν2/2) for the black ring space time to exclude any conical singularities. [30]
2.2 Black Ring Geodesics
The first order geodesic equations may be derived using the canonical framework
[34] from the Lagrangian
gµν ẋ
µẋν , (14)
here µ, ν = 0...4 and the covariant components of the metric tensor are as defined
in the previous section and ẋµ = dxµ/dρ with the affine parameter ρ = τ/m
[36]for time like geodesics, τ being the proper time andm the mass of the particle.
Consequently, for both time like and null geodesics the momenta are pµ = ẋµ.
The covariant momenta may be directly obtained from the Lagrangian and are
given as pµ = gµν ẋ
µ. The norm of the conjugate momenta is then given as,
gµνpµpν = −ǫm2 (15)
where gµν are the contravariant components of the black ring metric and ǫ = (0, 1)
for null and time like geodesics respectively.
The black ring spacetime admits three Killing isometries generated by the
vector fields ∂t, ∂ψ, and ∂φ corresponding to time translation and the two rotation
isometries in the coordinates φ, ψ. These isometries provide three conserved
conjugate momenta, pt = −E, pψ = Ψ, pφ = Φ. We consider the geodesics
restricted to the plane of rotation of the black ring, outside the ring, i.e, x = −1.
It is analogous to an equatorial plane in the spherical case in the sense that it
is reflection symmetric and hence geodesics in it with zero initial velocity in the
transverse x direction will continue to remain in the plane. The plane x = −1
being a fixed point of the ∂φ isometry, the gφφ component of the metric tensor
goes to zero smoothly there. The geodesic equations of motion in the equatorial
plane for the t and φ directions are obtained directly from the conserved conjugate
momenta. These turn out to be as follows:
1 + λy
(1− λ)2
(1 + y)4
(1 + λy)G(y)
E − C(1 + y)
R(1− λ)(G(y)
Ψ (16)
CR(1 + y)3
(1− λ)G(y)
(1 + y)2(1 + λy)
R2(1− λ)G(y)
Ψ (17)
The form of the y equation for geodesic motion in the equatorial plane is obtained
directly from eqn. (15) to be,
)2 + gyy
gttE2 − 2gtψEΨ+ gψψΨ2 + ǫm2
= 0, (18)
where gyy = 1/gyy, g
tt = gψψ/D, g
ψψ = gtt/D, g
tψ = −gtψ/D and D = gttgψψ −
Thus, the y equation may be expressed as
ẏ2 = − (1 + y)
(1− λ)2R2
C2(1 + y)3 + (1− λ)2(1 + νy)(1− y)
F (y)
2C(1 + y)2
(1 + λy)(1 + y)
− ǫ(1− λ)(1 + νy)(1− y)m2
where ǫ = (0, 1) for null and timelike geodesics respectively. It should be noted
that the co-efficient of E2 in the r.h.s of the above equation remains finite and
smooth at the ergosphere, y = −1/λ, even though the function F (y) in the
denominator vanishes. The eqn (19) should be compared with that appearing in
[35] for the null geodesics in the plane of the ring, where a a certain normalization
of the metric components have been chosen at asymptotic infinity.
The y co-ordinate ranges over the plane of rotation of the ring from the
curvature singularity to asymptotic infinity and the above equation is analogous
to particle motion in a central potential
ẏ2 + Veff(y;E,Ψ) = 0 (20)
Towards asymptotic infinity, (x, y) → −1, the effective potential for time like
geodesics tends to
Veff(y;E,Ψ) → −
2(1− ν)
R2(1 + λ)
η3(E2 −m2), (21)
where η tends to 0 towards asymptotic infinity and is given by η = −(1 + y).
Unbound time like geodesics can exist only when E2 −m2 > 0 in which case
the effective potential Veff is negative at large distances and approaches zero at
asymptotic infinity (x, y = −1). For the case E2 < m2 only bound geodesics
exist, in the sense that such geodesics do not reach upto asymptotic infinity.
Stable bound orbits are bound orbits which do not end up in the singularity. It is
common knowledge that stable bound orbits do not occur in a higher dimensional
central potential, even in the case of Newtonian gravity. Thus it is expected that
such orbits must be excluded from higher dimensional black hole space times.
This was explicitly shown for the equatorial geodesics of a five dimensional Myers-
Perry black hole in [36]. This conclusion is expected to also hold for the class
of geodesics restricted to the plane of rotation of the ring being considered here.
Their existence is indicated by the presence of stable circular orbits. For circular
orbits, we have the condition
Veff(y = yc) = 0 ,
∂Veff (y)
= 0 (22)
where y = yc is the ‘radius’ of the circular orbit.The condition for stability of the
circular orbit is
∂2Veff
> 0. (23)
−2.2 −2 −1.8 −1.6 −1.4 −1.2 −1
Figure 1: Plot of black ring effective potential for ν = 0.46, L = 4.40145 and
three different values of E as indicated in the box. Motion is allowed only in the
region where Veff < 0. The constants m = R = 1. It is apparent that there are
no stable bound orbits. E = 2.0 is close to having an unstable circular orbit,
whereas for E = 2.02 there are no inaccessible regions. Since E > 1 all the three
curves exhibit unbounded orbits. The case for E < 1 shows an exactly similar
behaviour as regards the bound orbits.
We get two simulataneous biquadratic equations in E and Ψ from Eq(22)
which can be solved in terms of the radius yc for a black ring of specific ν. These
values of Ec and Ψc can be then substituted into (23) to obtain a function of yc
for a specific black ring [36]. It is difficult to interpret the analytic expressions
for Ec,Ψc and that of Eq.(23) in terms of yc. However, numerical plots have been
obtained in Fig. 1 for the effective potential Veff(y) against y which clearly shows
that stable bound orbits are ruled out both for E2 > m2 and E2 < m2 .
3 Brane World Black Ring
In this section we very briefly outline the construction of the Randall-Sundrum
braneworld with a single (N-1)-brane in (N+1) dimensions with a single AdS
direction transverse to the brane. We then consider the specific case of the five
dimensional neutral rotating black ring on a four brane in a (5+1) dimensional
Randall-Sundrum braneworld with a single AdS direction transverse to the brane
hypersurface. We propose that the appropriate bulk description is provided by
a six dimensional rotating black string extension of the five dimensional rotating
black ring. The intercept of the bulk solution on the four brane is a five dimen-
sional black ring with the usual curvature singularity on the brane hypersurface
although an additional bulk singularity also appears at the AdS horizon. We also
compute the six dimensional bulk geodesics restricted to the plane of rotation of
the black ring. The projection of these bulk geodesics on the four brane reduces
to the appropriate class of black ring geodesics on the four brane hypersurface.
The y orbits for the bulk solution which reach the AdS horizon are then analyzed
using the geodesic equation to elucidate the natuer of the bulk singularity at the
AdS horizon. It is seen that the curvature remains finite at the AdS horizon
along the unbounded indicating the presence of a mild p-p curvature singularity.
3.1 Black Ring in a RS Brane World
The bulk metric for single brane RS brane world in (N +1) dimensions, with one
transverse AdS direction to the (N-1) brane is as follows; [19, 21]
ds2 = gmndx
mdxn =
[gµνdx
µdxν + dz2]. (24)
Here µ, ν = 0 . . . (N − 1) and m,n = 0 . . . (N) and l is the AdS length scale. The
transverse coordinate z = 0,∞ are the conformal infinity and the AdS horizon
respectively. The actual RS braneworld geometry is obtained by removing the
small z region at z = z0 and glueing a mirror copy of the large z geometry at the
location of the (N-1) brane which ensures Z2 reflection symmetry. The resulting
topology for the double brane RS scenario is essentialy RN × S1
and in the single
brane variant considered here the S1 direction is essentialy decompactified with
the second regulator brane being at z = ∞. The discontinuity of the extrinsic
curvature at the z = z0 surface corresponds to a thin distributional source of
stress-energy. From the Israel junctions conditions this may be interpreted as
a relativistic (N-1) brane (smooth domain wall) with a corresponding tension
[19,21]. The orginal RS model sliced the AdS space-time both at z = 0 and z = l
and inserted two (N-1) branes with Z2 reflection symmetry at both hypersurfaces.
The Israel junction conditions then required a negative tension for the brane at
z = l. The variant considered here may be obtained from the original RS model
by allowing the negative tension brane to approach the AdS horizon at z = ∞ .
Although we focus here only on the single brane RS model for convenience, our
construction may be generalized to the original RS model with double branes in
a straightforward manner.
The Einstein equations in (N+1) dimensions with a negative cosmological
constant continue to be satisfied for any metric gµν which is Ricci flat. The
curvature of the modified metric now satisfies
RpqrsR
pqrs =
2N(N + 1)
RµνλκR
µνλκ (25)
where (p, q) runs over (N +1) dimensions and (µ, ν) over the N dimensions of the
brane world volume. The perturbations of the (N+1) dimensional metric around
a Ricci flat background are now normalizable modes peaked at the location of
the (N-1) brane.
Having provided this brief introduction to the single brane RS model in (N+1)
dimensions we now specialize to N=5 and consider the bulk description of a five
dimensional neutral rotating black ring on the four brane in a six dimensional RS
braneworld. To this end we consider a bulk six dimensional black string extension
of the five dimensional rotating neutral black ring in the bulk. The black ring
being a Ricci flat space-time the bulk black string extension automaticaly satisfies
the Einstein equation [21] For a reflection symmetric four brane hypersurface fixed
at z = z0 we may introduce the co-ordinate w = z−z0. The bulk metric on either
side of the domain wall may now be expressed as
ds2 =
(z0 + |w|)2
dw2 −
F (y)
F (x)
dt− CR
1 + y
F (y)
(x− y)2
F (x)
−G(y)
F (y)
dψ2 − dy
F (x)
where −∞ < w <∞ and the domain wall is located at w = 0.
The induced metric on the four brane at z = z0 may be recast into the black
ring form by suitably rescaling the coordinates and the parameters. The ADM
mass and angular momentum as measured on the brane, scaled by the conformal
warp factor, are then given as
M , J∗ =
J. (27)
where M,J are the bulk parameters.
The curvature squared for the bulk black string is computed to be;
RjklmR
jklm =
6(1 + ν2)2ν2Q(x, y)
R4(1 + ν2 + 2νx)6
z4(x− y)4
Following Eq(12), towards spatial infinity on the brane the curvature squared
behaves as
RjklmR
jklm ∼
. (29)
The curvature invariant diverges at the spacelike singularity on the brane at
y = −∞. Additionaly, it is also seen to diverge at the AdS horizon z = ∞ for
finite r. As mentioned earlier, such a singularity seems to be a artifact of the
linearized approximation. In order to further investigate this issue we need to
study the geodesics and their behaviour at the AdS horizon.
As mentioned earlier the neutral rotating black ring maybe described in a cer-
tain limit as a Schwarzschild black string boosted in the translationaly invariant
direction and identified periodicaly. In the braneworld construction that we have
developed, this reduces to a six dimensional bulk black two brane boosted along
the extended direction on the four brane and identified periodically. In the 5+1
dimensional brane world Eq. (13) generalizes to;
ds2 =
dz2 + dw2 − (1− r0
)dt2 + (1− r0
)−1dr2 + r2dΩ2
Here u is the translation invariant direction of the black string along the brane
hypersurface and z describes the transverse direction. Apart from the conformal
factor the coordinate z is a spectator dimension and hence we have a six dimen-
sional bulk Schwarzschild black two brane boosted along a translation invariant
direction w and periodicaly identified as w ∼ w+2πR. This bulk black two brane
in the limit of large boost velocity and a large periodicity R intercepts the four
brane in a fast spinning thin five dimensional neutral black ring of large radius R
with the usual curvature singularity on the brane. This is obvious as the boost
does not involve the transverse z direction and the limit of large radius and high
boost velocity are z independent. So in this limit after periodic identification the
event horizon has S2×S1×R topology extended in the bulk and periodic in the
coordinate w on the four brane.
3.2 The Brane World Geodesics.
The geodesic equations for the the bulk spacetime may be obtained as earlier
from the Lagrangian
L = 1
= gjkẋ
j ẋk (31)
where gjk are the covariant components of the 5+1 dimensional metric as in eqn.
(24) and j, k = 0 . . . 5. Also ẋ = dx/dρ and on time like geodesics the affine
parameter ρ = τ/m. Accordingly we have pj = ẋj , pj = gjkẋ
k and
gjkpjpk = −ǫm2 (32)
where ǫ = 0, 1 for null and time like geodesics respectively.
The z equation for geodesic motion is obtained from the Lagrangian as
. (33)
The solution for null geodesics is either z =constant or
z = − z1l
. (34)
For timelike geodesics the solution is
z = −z1cosec(ρm/l). (35)
Herem is the particle mass for timelike geodesics and we should set z1/m=constant
for the null geodesics in this case. The null case z =constant is simply a null
geodesic of the five dimensional rotating black ring. We are interested in the
other solutions which reach the location of the bulk singularity at the AdS hori-
zon z = ∞ for ρ→ 0−.
The bulk spacetime has three killing isometries ∂t, ∂ψ, and ∂φ leading to the
corresponding conserved momenta pt = −E, pψ = Ψ and pφ = Φ for geodesic
motion. Once again we consider only those geodesics in the bulk which, on the
4-brane, are restricted to the plane of rotation of the black ring , i.e in the x = −1
plane. The gφφ component of the 5+1 dimensional metric goes to zero on the
plane of rotation so that E and Ψ are the conserved quantities for such geodesics.
The geodesic equations for the t and ψ co-ordinates in the plane of rotation of
the black ring are given as
z2(1− λ)
l2(1 + λy)
(1− λ)2
(1 + y)4
(1 + λy)G(y)
E − z
2C(1 + y)3
l2R(1− λ)(G(y)
z2CR(1 + y)3
l2(1− λ)G(y)
z2(1 + y)2(1 + λy)
l2R2(1− λ)G(y)
The y equation of motion for time like and null geodesics in the bulk which reach
the AdS horizon is given by
+ gttE2 − gtψEΨ+ gψψΨ2
= 0. (37)
Here the contravariant components of the metric in the equation are essentialy
the black ring metric without the bulk conformal factor.
The bulk timelike or null geodesics when projected onto the brane reduce
to the time like black ring geodesics restricted to the plane of rotation of the
ring. The projection to the four brane hypersurface is effected by scaling out
the z dependence of the geodesics. First, new parameters γ = z2/m2ρ for null
geodesics and γ = (−z2
/lm)cot(mρ/l) for time like geodesics are introduced. We
define the rescaled co-ordinates and parameters x = lx̃/z1, y = lỹ/z1, t = lt̃/z1 ,
R = l2R̃/z2
, λ = z1λ̃/l, ν = z1ν̃/l. The integrals of motion are also rescaled as
E = lẼ/z1,Ψ = l
3Ψ̃/z3
The geodesic equation for the y coordinate in the rescaled quantities may then
be written as,
+ Veff(ỹ; Ẽ, Ψ̃) = 0 (38)
where Veff is the same effective potential as given in eqn. (20). This is pre-
cisely the equation in y for a time like geodesic in the plane of rotation of a five
dimensional rotating black ring with an ADM mass M̃ and angular momentum
M , J̃ =
J (39)
and thus existing on the four brane hypersurface located at z = z0 = l
2/z1. The
parameter γ now serves as the proper time along the time like geodesic.
In order to ascertain the nature of the singularity at the AdS horizon (z = ∞)
we need to study the behaviour of the bulk geodesics near the AdS horizon, i.e
as ρ → 0−. This is equivalent to γ → ∞, so we need to investigate the late
time behaviour of the five dimensional time like geodesics on the four-brane. The
geodesics ending into the black ring singularity will take a finite amount of proper
time to do so. For infinite proper time the geodesics can either reach up to the
asymptotic infinity on the four brane(x̃, ỹ = −z1/l) or remain at a finite distance
from the black ring horizon. The geodesics that reach asymptotic infinity on the
brane have late time behaviour
r̃ ∼ γ
Ẽ2 −m2, (40)
where
r̃2 = − 1
z1/l + ỹ
. (41)
The co-ordinate r̃ is the radial direction on the brane and it is the same as the
radial Myers-Perry coordinate for the black ring in the asymptotic limit modulo
certain constants in the plane of rotation of the black ring.
It is expected that stable bound orbits do not exist in the case of the five
dimensional black rings. So, only unbound geodesics may reach the AdS horizon
at z → ∞. Along such orbits the curvature squared, Eq.(28), remains finite,
thus indicating the presence of a p-p curvature singularity at the AdS horizon.
To explicitly illustrate this, it is necessary to obtain the curvature components
in an orthonormal frame parallely propagated on a timelike geodesic to the AdS
horizon. Although its simple to demonstrate this in the case of the Schwarzschild
black hole in a braneworld for more complicated metrics and higher dimensions
the explicit determination of this frame involves several coupled PDE and renders
this analysis computationaly intractable. Although we have to emphasize that
such frames exist the choice is highly non unique and a specific suitable such
frame is complicated to establish even for four dimensional Kerr black holes in a
braneworld [11].
4 Summary and Discussions.
To summarize we have described a five dimensional neutral rotating black ring on
a four brane in a six dimensional Randall-Sundrum braneworld. As mentioned
earlier this has been motivated by the fact that for consistency the usual grav-
itational configurations on the brane, in particular black holes must arise from
some higher dimensional bulk solutions. The five dimensional black ring being
the first asymptoticaly flat solution with a non spherical horizon topology is an
interesting configuration to study from a bulk brane world perspective. Espe-
cialy as it explicitly violates the no hair and the uniqueness theorem. Due to the
absence of suitable exact bulk metrics in D > 4 a linearized framework around a
fixed solution is necessary for the analysis of the black ring in a brane world. In
this context the bulk black string approach of Chamblin et. al. [7] is especialy
relevant to elucidate the physical issues although the pathology of a singularity at
the AdS horizon persists. However, absence of such a singularity in lower dimen-
sional brane worlds where exact metrics are available shows the bulk singularity
to be a linearization artifact.
To this end we have considered a bulk six dimensional black string extension
of a five dimensional rotating neutral black ring in a 5+1 dimensional Randall-
Sundrum braneworld. This choice is consistent with the usual reflection symmet-
ric junction conditions on the four brane in such warped compactification models.
The bulk black string rotates in the four brane world volume and the induced
five dimensional metric on the four brane describes a neutral rotating black ring.
This reproduces the usual spacelike curvature singularity of the black ring on
the four brane hypersurface. Additionaly a singularity also appears in the bulk
at the AdS horizon. After elucidating the geodesics of the rotating black ring
restricted to the plane of rotation we have obtained both the timelike and the
null geodesics for the black string in the six dimensional bulk. We have further
shown that the restricted bulk geodesics projected on the four brane by scaling
away the AdS direction exactly match the corresponding class of five dimensional
black ring geodesics. The effective potential has been analysed numericaly and
we have shown that stable bound geodesics do n ot exist as is expected in D > 4.
It has been further shown that the curvature invariant remains finite along un-
bounded geodesics which reach the AdS horizon. This clearly indicates that the
bulk curvature singularity at the AdS horizon is possibly a p-p curvature sin-
gularity although an explicit illustration using parallely propogated orthonormal
frames is computationaly intractable.
It is mentioned earlier that a fast spinning thin neutral rotating black ring
may be described as a black string boosted along the translationaly invariant
direction and identified periodically in some limit. We have shown that from the
bulk perspective this description involves naturaly a black two brane in the six
dimensional bulk orthogonal to the four brane hypersurface. To obtain the black
ring on the four brane the black two brane must be boosted along a translationaly
invaraint direction longitudinal to the four brane and identified periodicaly along
this direction. Due to the direct equivalence of the two metrics it is obvious that
the usual matching of the geodesics on the bulk and the brane will continue to
hold in this limit . In the black ring limit the event horzion in the bulk would
constitute a base S2×S1 on the five dimensional brane hypersurface and a trivial
R fibration into the bulk.
The issue of stability of the bulk black string configuration is contentious and
remains unresolved for axialy symmetric stationary solutions. For AdS solutions
one conclusion is that the prefered phase will be an accumulation of a sequence of
lower dimensional black holes with the horzion pinched off at some scale. However
for the usual Schwarzschild black string this conclusion has been contested where
it has been shown that a more likely scenario is an evolution to a translationaly
non invariant stable solution [10]. But this although plausible has not yet been
generalized explicitly to axialy symmetric solutions. It has been argued that the
bulk solution should pinch off due to the instabilities before reaching the singu-
larity at the AdS horzion [7, 9]. However this issue is far from being completely
settled. It is possible that the pathology at the AdS horizon is a linearization
artifact especialy given that lower dimensional exact bulk solutions are regular
everywhere.
There are several open issues for future studies. Charged rotating black ring
solutions have been obtained in the context of string theory through the O(d, d)
transformations. These have been further generalized to rotating black rings with
dipole charges. In the brane world scenario, bulk configurations which reduce to
charged black holes have been investigated. It could be shown in this case that
the black hole on the brane developed a tidal charge due to the extra dimensions
apart from the usual conserved gauge charge [15]. It would be an interesting
exercise to study the brane world formulation of the dipole black rings in this
context. Very recently it has been shown that in higher diemnsions it is possible to
have stable configurations involving combinations of black rings and black holes.
These have been christened black saturn and are remarkably novel solutions of
higher dimensional general relativity [37]. Naturally it would be interesting to
investigate these configurations from a brane world perspective. It is generaly
expected that more such solutions would be possible in the context of higher
dimensions. Some of these issues are being currently studied.
5 Acknowledgements
We would like to thank A.Virmani for collaboration during early stages of this
work. GS would also like to acknowledge J. Maharana for discussions. Both of
us would like to thank D. D. B. Rao and B. N. Tiwari for computational help.
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	Introduction.
	The Rotating Neutral Black Ring .
	Black Ring Metric
	Black Ring Geodesics
	 Brane World Black Ring
	Black Ring in a RS Brane World
	The Brane World Geodesics.
	Summary and Discussions.
	Acknowledgements

  Five dimensional neutral rotating black rings are described from a
Randall-Sundrum brane world perspective in the bulk black string framework. To
this end we consider a rotating black string extension of a five dimensional
black ring into the bulk of a six dimensional Randall-Sundrum brane world with
a single four brane. The bulk solution intercepts the four brane in a five
dimensional black ring with the usual curvature singularity on the brane. The
bulk geodesics restricted to the plane of rotation of the black ring are
constructed and their projections on the four brane match with the usual black
ring geodesics restricted to the same plane. The asymptotic nature of the bulk
geodesics are elucidated with reference to a bulk singularity at the AdS
horizon. We further discuss the description of a brane world black ring as a
limit of a boosted bulk black 2 brane with periodic identification.

Introduction.
Higher dimensional spacetimes are now an essential aspect of effective field
theories arising from fundamental theories of quantum gravity. The general as-
sumption implicit in such constructions was that the extra spatial dimensions
are compactified to ultrashort length scales. Hence quantum gravity effects were
relegated to very high energy scales. However in recent years the exciting possi-
bility of low scale quantum gravity effects in the brane world models have inspired
considerable interest and interesting phenomenological consequences [1–3]. The
brane world scenario envisaged the gauge sector of the fundamental interactions
to be restricted on a smooth codimension one hypersurface ( refered to as a brane)
embedded in a higher dimensional space-time and the electroweak scale as the
fundamental scale. The usual four dimensional Planck scale was then a derived
scale. In particular the Randall-Sundrum models and their variants based on a
warped non factorable compactification geometry in a bulk Anti deSitter ( AdS)
space time offered a partial resolution to the vexing hierarchy problem [4]. Al-
though the analysis was valid in a linearized framework a full non linear study
from a supergravity perspective confirmed the conclusions and their extension to
any Ricci flat geometry on the brane [5, 6].
For consistency the brane world scenario requires generic four dimensional
gravitational configurations on the brane to arise from a higher dimensional bulk.
The investigation of black hole configurations in this context has been an exciting
aspect of the study of brane world gravity [5]. Such a black hole on the brane
is expected to be a configuration extended in the bulk. Chamblin, Hawking and
Reall [7] attempted the description of a Schwarzschild black hole in a typical single
three brane five dimensional Randall-Sundrum brane world as a bulk black string.
This reproduced the usual Schwarzschild singularity on the brane but additionaly
was also singular at the AdS horizon far away from the three brane. Although a
pathology, this singularity was possibly a linearization artifact and could be shown
to be a mild p-p curvature singularity. The bulk black string was subject to the
usual instabilities against long wavelength perturbations [8,9] and was expected to
pinch off to a cigar geometry before reaching the AdS horizon. However the issue
of stability is contentious and for sphericaly symmetric solutions it was shown
that a more likely scenario is a transition to a non uniform black string [10]
In an earlier article [11]we have generalized the construction of Chamblin et.
al. [7]to consider rotating black holes in a five dimensional single three brane
RS brane world. The bulk configuration proposed was a five dimensional rotat-
ing black string which intercepted the three-brane in a four dimensional rotating
black hole described by a Kerr metric on the three brane. It was found that
the Kerr solution too was singular at the AdS horizon apart from the usual ring
singularity on the brane. The asymptotics of the equatorial geodesics at the AdS
horizon also indicated a p-p curvature singularity although an explicit determina-
tion was computationaly intractable. There have been other approaches to brane
world black holes including numerical studies for off brane metrics and a Hamil-
tonian constraint approach to charged black holes [12–18]. In lower dimensions
exact studies of brane world black holes [19] involving the AdS C-metric have
indicated that the bulk solutions are regular everywhere emphasizing that the
bulk singularity in higher dimension is possibly a linearization artifact. However
absence of exact bulk metrics in higher dimensions requires a linearized approach
and the black string framework is hence physicaly relevant in this context in spite
of such a bulk singularity.
The brane world constructions must be embedded in an appropriate string
theory for consistency, requiring the generalizations of these models to higher
dimensions. The generalization of the Randall-Sundrum construction and its
variants to higher dimensions with a single space like AdS direction and an ap-
propriate codimension one brane is straightforward. Additionaly this may easily
be extended to include the full non linear extensions of a Ricci flat metric [20–22].
In higher dimensions also the consistency of such brane world constructions re-
quire that gravitational configurations arise from appropriate bulk scenarios. In
particular this applies to higher dimensional black holes on the codimension one
brane. In this context in an earlier article [23]we had described the N dimensional
rotating Myers-Perry [24]black hole on a single (N-1) brane in a (N + 1) dimen-
sional RS brane world. The bulk solution in this case was a (N+1) dimensional
rotating black string extended in the AdS direction transverse to the (N-1) brane.
Analysis of equatorial geodesics again indicated a p-p curvature singularity in the
bulk apart from the usual extended singularity on the (N-1) brane.
In the recent past there has been remarkable and surprising progress in under-
standing higher dimensional black holes. In particular it has been realized that
the no hair and the uniqueness theorems are much less restrictive in higher dimen-
sions [27]. In four dimensions the no hair theorem characterizes any stationary
asymptoticaly flat black hole solution of Einstein-Maxwell system only by their
mass, angular momentum and conserved charges whereas the uniqueness theorem
forbids event horizons of non spherical toplogies. However the discovery [25] in
five dimensions of an asymptoticaly flat stationary black hole solution with a non
spherical ring like S2×S1 horizon topology with the possibility of dipole charges,
showed that higher dimensional black holes posess remarkably distinctive prop-
erties. The static black ring solution [28] was first obtained through the Wick
rotation of a neutral solution of an Einstein-Maxwell system [29] although they
involved conical singularities. However the stationary solution rotating in the
S1 direction was regular everywhere except the usual curvature singularity. For
fixed mass the angular momentum of the black ring was bounded below and for a
certain range of parameters two black rings and a usual five dimensional rotating
Myers-Perry black hole all with the same mass and spin coexist. The charged
versions of these black rings were first obtained in the framework of D=5 heterotic
supergravity [30] and fully supersymmeric three charged black ring solutions in
D=5 followed later from compactifications of black supertubes in D=10 [31, 32].
It was seen that these black rings could also support gauge dipoles independent
of the conserved gauge charges entailing an infinite non uniqueness and violating
the no hair theorem [33].
As emphasized earlier, for consistency of the brane world scenario it is im-
perative that gravitational configurations like black holes on the brane should
arise from appropriate bulk solutions. In this context it is but natural to inves-
tigate possible bulk configurations in a higher dimensional brane world scenario
which would describe five dimensional black rings on the brane. This is espe-
cialy relevant for the neutral rotating black rings as they are Ricci flat and hence
satisfy the criteria for embedding in higher dimensional Randall-Sundrum brane
worlds. Naturaly the absence of exact solutions in higher dimensions require the
usual linearized framework to analyse this question. The black string approach
is especialy relevant in this context to highlight the physical aspects of such an
embedding although it suffers from singular pathologies which are possibly lin-
earization artifacts.
In this article we address this issue and show that it is possible to consistently
embed the five dimensional black ring solution on a single four brane in a (5 + 1)
dimensional Randall-Sundrum brane world. Following the black string approach
we consider a six dimensional bulk rotating black string extension of the five
dimensional black ring. This bulk configuration intercepts the four brane in a
five dimensional rotating black ring. In what follows after a brief review of neutral
rotating black rings, we obtain their geodesic equations in the plane of the ring
analogous to the equatorial plane of black holes with spherical topologies. We
further investigate the asymptotic behaviour of both the null and the timelike
geodesics in this plane to elucidate the restricted causal structure of the black ring
space time. In section three we consider a bulk rotating black string extension
of a five dimensional neutral rotating black ring in a six dimensional RS brane
world with a single four brane. The bulk black string intercepts the four brane in
a five dimensional black ring with the usual spacelike curvature singularity on the
brane. Additionaly a curvature singularity also appears at the AdS horizon far
away from the four brane. Following the description of a black ring as a boosted
black string with periodic identification in a certain limit, the bulk solution may
be described as a boosted black two brane with the same periodic identification.
We then construct the six dimensional bulk geodesics in the plane of rotation of
the ring and show that their projections on the four brane reproduces the usual
five dimensional black ring geodesics in the same plane. To study of the nature
of the pathological singularity at the AdS horizon we further investigate the late
time asymptotics of these geodesics. It is shown that the curvature remains
finite along unbound geodesics which reach the AdS horizon. We also discuss the
possibility of the bulk solution to pinch off before reaching the AdS horizon due
to the usual instabilities and comment on the possible stable solution in the light
of the analysis outlined in [9] and [10]. In the last section we provide a summary
of our analysis and results and also discuss certain future open issues in this area.
2 The Rotating Neutral Black Ring .
In this section we first briefly review the neutral rotating black ring and eluci-
date the nature of the adapted coordinate system . We then construct the black
ring geodesics restricted to the plane of rotation of the ring which is analogous
to the equatorial geodesics in solutions with a spherical topology. Furthermore
we analyse the geodesic equations to study the nature of the radial orbits for
this plane and their asymptotics. The static neutral black ring was originally
discovered through a Wick rotation of certain Kaluza Klein C metrics decribing
neutral bubbles [29]. These involved conical singularities and consequent deficit
angles leading to either cosmic string defects joining these singularities or deficit
membranes. However an analytic continuation led to the original neutral rotat-
ing black ring solution which was a five dimensional asymptoticaly flat black hole
with a ringlike S2×S1 horizon topology, regular everywhere except at a spacelik
e curvature singularity. The original solution was further refined through appro-
priate factorizable choice of certain functions appearing in the metric [26,30–32].
The rotating black ring in equlibrium was parametrized by a dimensionless re-
duced angular momentum j = 27π
which was bounded from below for a fixed
mass. It could be shown that in the range 27
≤ j2 < 1 there existed one Myers-
Perry black hole with spherical topology and two black rings with identical mass
and angular momenta, in direct violation of the black hole uniqueness theorem.
2.1 Black Ring Metric
The metric of the neutral rotating five dimensional black ring in a specific adpated
coordinate system which is obtained from the foliation of space-time in terms of
the equipotentials of certain 1-form and 2-form gauge potentials is , [32]
ds2 = −
F (y)
F (x)
dt− C R
1 + y
F (y)
(x− y)2
F (x)
−G(y)
F (y)
dψ2 − dy
F (x)
, (1)
where the functions
F (ξ) = 1 + λξ, G(ξ) = (1− ξ2)(1 + νξ) , (2)
λ(λ− ν)
1 + λ
. (3)
Here R is a length scale which may be interpreted as the radius of the ring in
some limit [32] and the two dimensionless parameters λ and ν which are related
to the shape and the rotation velocity of the ring lie in the range
0 < ν ≤ λ < 1 (4)
. The range of the spatial co-ordinates (x, y) are required to be,
− 1 ≤ x ≤ +1 , −∞ ≤ y ≤ −1 . (5)
respectively.
The constant y hypersurfaces are nested deformed solid toroids with topology
S2 × S1, whereas the coordinate x is like a direction cosine, x = +1 points to
the interior of the ring and x = −1 points to the region outside the ring. The
solution is a stationary axisymmetric solution with rotation in the ψ direction,
and admits t, φ, and ψ Killing isometries.
In order to avoid conical singularities at the fixed points x = −1 and y = −1 of
the Killing isometries ∂φ and ∂ψ the co-ordinates ψ and φ require to be identified
with the equal periods
∆ψ = ∆φ = 4π
F (−1)
|G′(−1)|
. (6)
Furthermore the requirement that the orbits of the isometry ∂φ shows no deficit
angles at x = +1 lead to the condition
1 + ν2
The co-ordinates (x, φ) parametrize a two-sphere S2, the co-ordinate ψ parametrizes
a circle S1 and the solution describes a black ring having a regular horizon of
topology S1 × S2 and rotating in the S1 plane. However the horizon geometry is
not a simple product of S2 and S1 as the two sphere S2 is deformed there and
the deformation grows away from the horizon.
The metric reduces to a conventional five dimensional Myers-Perry black hole
with rotation in a single plane if, instead of (7), we consider the limit, R → 0,
(λ, ν) → 1 and the parameters
, a2 = 2R2
(1− ν)2
, (8)
are held constant. In this case the co-ordinates (x, φ, ψ) characterises a three-
sphere S3 which is a regular horizon of a five dimensional Myers-Perry black hole.
The ergosphere and the event horizon of the black ring are located at y = −1/λ
and y = −1/ν respectively. At y = −∞ there is a spacelike curvature singularity
inside the horizon. Asymptotic infinity is reached as (x, y) → −1.
The ADM mass and angular momentum are given as
λ(λ− ν)(1 + λ)
(1− ν)2
. (10)
The curvature squared for the black ring spacetime is computed to be,
RµνρσR
µνρσ =
6ν2(1 + ν2)2Q(x, y)
R4(1 + ν2 + 2νx)6
(x− y)4, (11)
where Q(x, y) is a poynomial of degree six in x and y. Hence there is a spacelike
curvature singularity at y = −∞ inside the event horizon. In terms of the Myers-
Perry co-ordinates (t, r, θ, ψ, φ) the difference (x−y) goes like 1/r2 at large r, i.e.
towards spatial infinity, so that the curvature squared goes as
RµνρσR
µνρσ ∼
as obtained in the case of five dimensional Myers-Perry black hole.
The rotating black ring in the limit of large radius R may be described after
appropriate coordinate redfinitions as a Schwarzschild black string boosted and
periodically identified along the translation invariant direction with a period 2πR
[30, 32, 33]. The black string metric is given as
ds2 = dw2 − (1−
)dt2 + (1−
)−1dr2 + r2dΩ2
, (13)
where the horizon is at r = r0 and w is the translation invariant direction. The
parameter ν = r0/R is seen to correspond to the thickness of the ring or the
ratio of the radius of the S2 at the horizon and the ring radius R . The ratio
λ/ν then measures the speed of rotation of the ring in the S1 direction and the
coordinate ψ = w/R corresponds to a redefined translation invariant direction of
the black string which is periodically identified as w = w + 2πR. The speed of
rotation is related to the local boost velocity given by
1− (ν/λ) and reduces to
1− (ν2/2) for the black ring space time to exclude any conical singularities. [30]
2.2 Black Ring Geodesics
The first order geodesic equations may be derived using the canonical framework
[34] from the Lagrangian
gµν ẋ
µẋν , (14)
here µ, ν = 0...4 and the covariant components of the metric tensor are as defined
in the previous section and ẋµ = dxµ/dρ with the affine parameter ρ = τ/m
[36]for time like geodesics, τ being the proper time andm the mass of the particle.
Consequently, for both time like and null geodesics the momenta are pµ = ẋµ.
The covariant momenta may be directly obtained from the Lagrangian and are
given as pµ = gµν ẋ
µ. The norm of the conjugate momenta is then given as,
gµνpµpν = −ǫm2 (15)
where gµν are the contravariant components of the black ring metric and ǫ = (0, 1)
for null and time like geodesics respectively.
The black ring spacetime admits three Killing isometries generated by the
vector fields ∂t, ∂ψ, and ∂φ corresponding to time translation and the two rotation
isometries in the coordinates φ, ψ. These isometries provide three conserved
conjugate momenta, pt = −E, pψ = Ψ, pφ = Φ. We consider the geodesics
restricted to the plane of rotation of the black ring, outside the ring, i.e, x = −1.
It is analogous to an equatorial plane in the spherical case in the sense that it
is reflection symmetric and hence geodesics in it with zero initial velocity in the
transverse x direction will continue to remain in the plane. The plane x = −1
being a fixed point of the ∂φ isometry, the gφφ component of the metric tensor
goes to zero smoothly there. The geodesic equations of motion in the equatorial
plane for the t and φ directions are obtained directly from the conserved conjugate
momenta. These turn out to be as follows:
1 + λy
(1− λ)2
(1 + y)4
(1 + λy)G(y)
E − C(1 + y)
R(1− λ)(G(y)
Ψ (16)
CR(1 + y)3
(1− λ)G(y)
(1 + y)2(1 + λy)
R2(1− λ)G(y)
Ψ (17)
The form of the y equation for geodesic motion in the equatorial plane is obtained
directly from eqn. (15) to be,
)2 + gyy
gttE2 − 2gtψEΨ+ gψψΨ2 + ǫm2
= 0, (18)
where gyy = 1/gyy, g
tt = gψψ/D, g
ψψ = gtt/D, g
tψ = −gtψ/D and D = gttgψψ −
Thus, the y equation may be expressed as
ẏ2 = − (1 + y)
(1− λ)2R2
C2(1 + y)3 + (1− λ)2(1 + νy)(1− y)
F (y)
2C(1 + y)2
(1 + λy)(1 + y)
− ǫ(1− λ)(1 + νy)(1− y)m2
where ǫ = (0, 1) for null and timelike geodesics respectively. It should be noted
that the co-efficient of E2 in the r.h.s of the above equation remains finite and
smooth at the ergosphere, y = −1/λ, even though the function F (y) in the
denominator vanishes. The eqn (19) should be compared with that appearing in
[35] for the null geodesics in the plane of the ring, where a a certain normalization
of the metric components have been chosen at asymptotic infinity.
The y co-ordinate ranges over the plane of rotation of the ring from the
curvature singularity to asymptotic infinity and the above equation is analogous
to particle motion in a central potential
ẏ2 + Veff(y;E,Ψ) = 0 (20)
Towards asymptotic infinity, (x, y) → −1, the effective potential for time like
geodesics tends to
Veff(y;E,Ψ) → −
2(1− ν)
R2(1 + λ)
η3(E2 −m2), (21)
where η tends to 0 towards asymptotic infinity and is given by η = −(1 + y).
Unbound time like geodesics can exist only when E2 −m2 > 0 in which case
the effective potential Veff is negative at large distances and approaches zero at
asymptotic infinity (x, y = −1). For the case E2 < m2 only bound geodesics
exist, in the sense that such geodesics do not reach upto asymptotic infinity.
Stable bound orbits are bound orbits which do not end up in the singularity. It is
common knowledge that stable bound orbits do not occur in a higher dimensional
central potential, even in the case of Newtonian gravity. Thus it is expected that
such orbits must be excluded from higher dimensional black hole space times.
This was explicitly shown for the equatorial geodesics of a five dimensional Myers-
Perry black hole in [36]. This conclusion is expected to also hold for the class
of geodesics restricted to the plane of rotation of the ring being considered here.
Their existence is indicated by the presence of stable circular orbits. For circular
orbits, we have the condition
Veff(y = yc) = 0 ,
∂Veff (y)
= 0 (22)
where y = yc is the ‘radius’ of the circular orbit.The condition for stability of the
circular orbit is
∂2Veff
> 0. (23)
−2.2 −2 −1.8 −1.6 −1.4 −1.2 −1
Figure 1: Plot of black ring effective potential for ν = 0.46, L = 4.40145 and
three different values of E as indicated in the box. Motion is allowed only in the
region where Veff < 0. The constants m = R = 1. It is apparent that there are
no stable bound orbits. E = 2.0 is close to having an unstable circular orbit,
whereas for E = 2.02 there are no inaccessible regions. Since E > 1 all the three
curves exhibit unbounded orbits. The case for E < 1 shows an exactly similar
behaviour as regards the bound orbits.
We get two simulataneous biquadratic equations in E and Ψ from Eq(22)
which can be solved in terms of the radius yc for a black ring of specific ν. These
values of Ec and Ψc can be then substituted into (23) to obtain a function of yc
for a specific black ring [36]. It is difficult to interpret the analytic expressions
for Ec,Ψc and that of Eq.(23) in terms of yc. However, numerical plots have been
obtained in Fig. 1 for the effective potential Veff(y) against y which clearly shows
that stable bound orbits are ruled out both for E2 > m2 and E2 < m2 .
3 Brane World Black Ring
In this section we very briefly outline the construction of the Randall-Sundrum
braneworld with a single (N-1)-brane in (N+1) dimensions with a single AdS
direction transverse to the brane. We then consider the specific case of the five
dimensional neutral rotating black ring on a four brane in a (5+1) dimensional
Randall-Sundrum braneworld with a single AdS direction transverse to the brane
hypersurface. We propose that the appropriate bulk description is provided by
a six dimensional rotating black string extension of the five dimensional rotating
black ring. The intercept of the bulk solution on the four brane is a five dimen-
sional black ring with the usual curvature singularity on the brane hypersurface
although an additional bulk singularity also appears at the AdS horizon. We also
compute the six dimensional bulk geodesics restricted to the plane of rotation of
the black ring. The projection of these bulk geodesics on the four brane reduces
to the appropriate class of black ring geodesics on the four brane hypersurface.
The y orbits for the bulk solution which reach the AdS horizon are then analyzed
using the geodesic equation to elucidate the natuer of the bulk singularity at the
AdS horizon. It is seen that the curvature remains finite at the AdS horizon
along the unbounded indicating the presence of a mild p-p curvature singularity.
3.1 Black Ring in a RS Brane World
The bulk metric for single brane RS brane world in (N +1) dimensions, with one
transverse AdS direction to the (N-1) brane is as follows; [19, 21]
ds2 = gmndx
mdxn =
[gµνdx
µdxν + dz2]. (24)
Here µ, ν = 0 . . . (N − 1) and m,n = 0 . . . (N) and l is the AdS length scale. The
transverse coordinate z = 0,∞ are the conformal infinity and the AdS horizon
respectively. The actual RS braneworld geometry is obtained by removing the
small z region at z = z0 and glueing a mirror copy of the large z geometry at the
location of the (N-1) brane which ensures Z2 reflection symmetry. The resulting
topology for the double brane RS scenario is essentialy RN × S1
and in the single
brane variant considered here the S1 direction is essentialy decompactified with
the second regulator brane being at z = ∞. The discontinuity of the extrinsic
curvature at the z = z0 surface corresponds to a thin distributional source of
stress-energy. From the Israel junctions conditions this may be interpreted as
a relativistic (N-1) brane (smooth domain wall) with a corresponding tension
[19,21]. The orginal RS model sliced the AdS space-time both at z = 0 and z = l
and inserted two (N-1) branes with Z2 reflection symmetry at both hypersurfaces.
The Israel junction conditions then required a negative tension for the brane at
z = l. The variant considered here may be obtained from the original RS model
by allowing the negative tension brane to approach the AdS horizon at z = ∞ .
Although we focus here only on the single brane RS model for convenience, our
construction may be generalized to the original RS model with double branes in
a straightforward manner.
The Einstein equations in (N+1) dimensions with a negative cosmological
constant continue to be satisfied for any metric gµν which is Ricci flat. The
curvature of the modified metric now satisfies
RpqrsR
pqrs =
2N(N + 1)
RµνλκR
µνλκ (25)
where (p, q) runs over (N +1) dimensions and (µ, ν) over the N dimensions of the
brane world volume. The perturbations of the (N+1) dimensional metric around
a Ricci flat background are now normalizable modes peaked at the location of
the (N-1) brane.
Having provided this brief introduction to the single brane RS model in (N+1)
dimensions we now specialize to N=5 and consider the bulk description of a five
dimensional neutral rotating black ring on the four brane in a six dimensional RS
braneworld. To this end we consider a bulk six dimensional black string extension
of the five dimensional rotating neutral black ring in the bulk. The black ring
being a Ricci flat space-time the bulk black string extension automaticaly satisfies
the Einstein equation [21] For a reflection symmetric four brane hypersurface fixed
at z = z0 we may introduce the co-ordinate w = z−z0. The bulk metric on either
side of the domain wall may now be expressed as
ds2 =
(z0 + |w|)2
dw2 −
F (y)
F (x)
dt− CR
1 + y
F (y)
(x− y)2
F (x)
−G(y)
F (y)
dψ2 − dy
F (x)
where −∞ < w <∞ and the domain wall is located at w = 0.
The induced metric on the four brane at z = z0 may be recast into the black
ring form by suitably rescaling the coordinates and the parameters. The ADM
mass and angular momentum as measured on the brane, scaled by the conformal
warp factor, are then given as
M , J∗ =
J. (27)
where M,J are the bulk parameters.
The curvature squared for the bulk black string is computed to be;
RjklmR
jklm =
6(1 + ν2)2ν2Q(x, y)
R4(1 + ν2 + 2νx)6
z4(x− y)4
Following Eq(12), towards spatial infinity on the brane the curvature squared
behaves as
RjklmR
jklm ∼
. (29)
The curvature invariant diverges at the spacelike singularity on the brane at
y = −∞. Additionaly, it is also seen to diverge at the AdS horizon z = ∞ for
finite r. As mentioned earlier, such a singularity seems to be a artifact of the
linearized approximation. In order to further investigate this issue we need to
study the geodesics and their behaviour at the AdS horizon.
As mentioned earlier the neutral rotating black ring maybe described in a cer-
tain limit as a Schwarzschild black string boosted in the translationaly invariant
direction and identified periodicaly. In the braneworld construction that we have
developed, this reduces to a six dimensional bulk black two brane boosted along
the extended direction on the four brane and identified periodically. In the 5+1
dimensional brane world Eq. (13) generalizes to;
ds2 =
dz2 + dw2 − (1− r0
)dt2 + (1− r0
)−1dr2 + r2dΩ2
Here u is the translation invariant direction of the black string along the brane
hypersurface and z describes the transverse direction. Apart from the conformal
factor the coordinate z is a spectator dimension and hence we have a six dimen-
sional bulk Schwarzschild black two brane boosted along a translation invariant
direction w and periodicaly identified as w ∼ w+2πR. This bulk black two brane
in the limit of large boost velocity and a large periodicity R intercepts the four
brane in a fast spinning thin five dimensional neutral black ring of large radius R
with the usual curvature singularity on the brane. This is obvious as the boost
does not involve the transverse z direction and the limit of large radius and high
boost velocity are z independent. So in this limit after periodic identification the
event horizon has S2×S1×R topology extended in the bulk and periodic in the
coordinate w on the four brane.
3.2 The Brane World Geodesics.
The geodesic equations for the the bulk spacetime may be obtained as earlier
from the Lagrangian
L = 1
= gjkẋ
j ẋk (31)
where gjk are the covariant components of the 5+1 dimensional metric as in eqn.
(24) and j, k = 0 . . . 5. Also ẋ = dx/dρ and on time like geodesics the affine
parameter ρ = τ/m. Accordingly we have pj = ẋj , pj = gjkẋ
k and
gjkpjpk = −ǫm2 (32)
where ǫ = 0, 1 for null and time like geodesics respectively.
The z equation for geodesic motion is obtained from the Lagrangian as
. (33)
The solution for null geodesics is either z =constant or
z = − z1l
. (34)
For timelike geodesics the solution is
z = −z1cosec(ρm/l). (35)
Herem is the particle mass for timelike geodesics and we should set z1/m=constant
for the null geodesics in this case. The null case z =constant is simply a null
geodesic of the five dimensional rotating black ring. We are interested in the
other solutions which reach the location of the bulk singularity at the AdS hori-
zon z = ∞ for ρ→ 0−.
The bulk spacetime has three killing isometries ∂t, ∂ψ, and ∂φ leading to the
corresponding conserved momenta pt = −E, pψ = Ψ and pφ = Φ for geodesic
motion. Once again we consider only those geodesics in the bulk which, on the
4-brane, are restricted to the plane of rotation of the black ring , i.e in the x = −1
plane. The gφφ component of the 5+1 dimensional metric goes to zero on the
plane of rotation so that E and Ψ are the conserved quantities for such geodesics.
The geodesic equations for the t and ψ co-ordinates in the plane of rotation of
the black ring are given as
z2(1− λ)
l2(1 + λy)
(1− λ)2
(1 + y)4
(1 + λy)G(y)
E − z
2C(1 + y)3
l2R(1− λ)(G(y)
z2CR(1 + y)3
l2(1− λ)G(y)
z2(1 + y)2(1 + λy)
l2R2(1− λ)G(y)
The y equation of motion for time like and null geodesics in the bulk which reach
the AdS horizon is given by
+ gttE2 − gtψEΨ+ gψψΨ2
= 0. (37)
Here the contravariant components of the metric in the equation are essentialy
the black ring metric without the bulk conformal factor.
The bulk timelike or null geodesics when projected onto the brane reduce
to the time like black ring geodesics restricted to the plane of rotation of the
ring. The projection to the four brane hypersurface is effected by scaling out
the z dependence of the geodesics. First, new parameters γ = z2/m2ρ for null
geodesics and γ = (−z2
/lm)cot(mρ/l) for time like geodesics are introduced. We
define the rescaled co-ordinates and parameters x = lx̃/z1, y = lỹ/z1, t = lt̃/z1 ,
R = l2R̃/z2
, λ = z1λ̃/l, ν = z1ν̃/l. The integrals of motion are also rescaled as
E = lẼ/z1,Ψ = l
3Ψ̃/z3
The geodesic equation for the y coordinate in the rescaled quantities may then
be written as,
+ Veff(ỹ; Ẽ, Ψ̃) = 0 (38)
where Veff is the same effective potential as given in eqn. (20). This is pre-
cisely the equation in y for a time like geodesic in the plane of rotation of a five
dimensional rotating black ring with an ADM mass M̃ and angular momentum
M , J̃ =
J (39)
and thus existing on the four brane hypersurface located at z = z0 = l
2/z1. The
parameter γ now serves as the proper time along the time like geodesic.
In order to ascertain the nature of the singularity at the AdS horizon (z = ∞)
we need to study the behaviour of the bulk geodesics near the AdS horizon, i.e
as ρ → 0−. This is equivalent to γ → ∞, so we need to investigate the late
time behaviour of the five dimensional time like geodesics on the four-brane. The
geodesics ending into the black ring singularity will take a finite amount of proper
time to do so. For infinite proper time the geodesics can either reach up to the
asymptotic infinity on the four brane(x̃, ỹ = −z1/l) or remain at a finite distance
from the black ring horizon. The geodesics that reach asymptotic infinity on the
brane have late time behaviour
r̃ ∼ γ
Ẽ2 −m2, (40)
where
r̃2 = − 1
z1/l + ỹ
. (41)
The co-ordinate r̃ is the radial direction on the brane and it is the same as the
radial Myers-Perry coordinate for the black ring in the asymptotic limit modulo
certain constants in the plane of rotation of the black ring.
It is expected that stable bound orbits do not exist in the case of the five
dimensional black rings. So, only unbound geodesics may reach the AdS horizon
at z → ∞. Along such orbits the curvature squared, Eq.(28), remains finite,
thus indicating the presence of a p-p curvature singularity at the AdS horizon.
To explicitly illustrate this, it is necessary to obtain the curvature components
in an orthonormal frame parallely propagated on a timelike geodesic to the AdS
horizon. Although its simple to demonstrate this in the case of the Schwarzschild
black hole in a braneworld for more complicated metrics and higher dimensions
the explicit determination of this frame involves several coupled PDE and renders
this analysis computationaly intractable. Although we have to emphasize that
such frames exist the choice is highly non unique and a specific suitable such
frame is complicated to establish even for four dimensional Kerr black holes in a
braneworld [11].
4 Summary and Discussions.
To summarize we have described a five dimensional neutral rotating black ring on
a four brane in a six dimensional Randall-Sundrum braneworld. As mentioned
earlier this has been motivated by the fact that for consistency the usual grav-
itational configurations on the brane, in particular black holes must arise from
some higher dimensional bulk solutions. The five dimensional black ring being
the first asymptoticaly flat solution with a non spherical horizon topology is an
interesting configuration to study from a bulk brane world perspective. Espe-
cialy as it explicitly violates the no hair and the uniqueness theorem. Due to the
absence of suitable exact bulk metrics in D > 4 a linearized framework around a
fixed solution is necessary for the analysis of the black ring in a brane world. In
this context the bulk black string approach of Chamblin et. al. [7] is especialy
relevant to elucidate the physical issues although the pathology of a singularity at
the AdS horizon persists. However, absence of such a singularity in lower dimen-
sional brane worlds where exact metrics are available shows the bulk singularity
to be a linearization artifact.
To this end we have considered a bulk six dimensional black string extension
of a five dimensional rotating neutral black ring in a 5+1 dimensional Randall-
Sundrum braneworld. This choice is consistent with the usual reflection symmet-
ric junction conditions on the four brane in such warped compactification models.
The bulk black string rotates in the four brane world volume and the induced
five dimensional metric on the four brane describes a neutral rotating black ring.
This reproduces the usual spacelike curvature singularity of the black ring on
the four brane hypersurface. Additionaly a singularity also appears in the bulk
at the AdS horizon. After elucidating the geodesics of the rotating black ring
restricted to the plane of rotation we have obtained both the timelike and the
null geodesics for the black string in the six dimensional bulk. We have further
shown that the restricted bulk geodesics projected on the four brane by scaling
away the AdS direction exactly match the corresponding class of five dimensional
black ring geodesics. The effective potential has been analysed numericaly and
we have shown that stable bound geodesics do n ot exist as is expected in D > 4.
It has been further shown that the curvature invariant remains finite along un-
bounded geodesics which reach the AdS horizon. This clearly indicates that the
bulk curvature singularity at the AdS horizon is possibly a p-p curvature sin-
gularity although an explicit illustration using parallely propogated orthonormal
frames is computationaly intractable.
It is mentioned earlier that a fast spinning thin neutral rotating black ring
may be described as a black string boosted along the translationaly invariant
direction and identified periodically in some limit. We have shown that from the
bulk perspective this description involves naturaly a black two brane in the six
dimensional bulk orthogonal to the four brane hypersurface. To obtain the black
ring on the four brane the black two brane must be boosted along a translationaly
invaraint direction longitudinal to the four brane and identified periodicaly along
this direction. Due to the direct equivalence of the two metrics it is obvious that
the usual matching of the geodesics on the bulk and the brane will continue to
hold in this limit . In the black ring limit the event horzion in the bulk would
constitute a base S2×S1 on the five dimensional brane hypersurface and a trivial
R fibration into the bulk.
The issue of stability of the bulk black string configuration is contentious and
remains unresolved for axialy symmetric stationary solutions. For AdS solutions
one conclusion is that the prefered phase will be an accumulation of a sequence of
lower dimensional black holes with the horzion pinched off at some scale. However
for the usual Schwarzschild black string this conclusion has been contested where
it has been shown that a more likely scenario is an evolution to a translationaly
non invariant stable solution [10]. But this although plausible has not yet been
generalized explicitly to axialy symmetric solutions. It has been argued that the
bulk solution should pinch off due to the instabilities before reaching the singu-
larity at the AdS horzion [7, 9]. However this issue is far from being completely
settled. It is possible that the pathology at the AdS horizon is a linearization
artifact especialy given that lower dimensional exact bulk solutions are regular
everywhere.
There are several open issues for future studies. Charged rotating black ring
solutions have been obtained in the context of string theory through the O(d, d)
transformations. These have been further generalized to rotating black rings with
dipole charges. In the brane world scenario, bulk configurations which reduce to
charged black holes have been investigated. It could be shown in this case that
the black hole on the brane developed a tidal charge due to the extra dimensions
apart from the usual conserved gauge charge [15]. It would be an interesting
exercise to study the brane world formulation of the dipole black rings in this
context. Very recently it has been shown that in higher diemnsions it is possible to
have stable configurations involving combinations of black rings and black holes.
These have been christened black saturn and are remarkably novel solutions of
higher dimensional general relativity [37]. Naturally it would be interesting to
investigate these configurations from a brane world perspective. It is generaly
expected that more such solutions would be possible in the context of higher
dimensions. Some of these issues are being currently studied.
5 Acknowledgements
We would like to thank A.Virmani for collaboration during early stages of this
work. GS would also like to acknowledge J. Maharana for discussions. Both of
us would like to thank D. D. B. Rao and B. N. Tiwari for computational help.
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	Introduction.
	The Rotating Neutral Black Ring .
	Black Ring Metric
	Black Ring Geodesics
	 Brane World Black Ring
	Black Ring in a RS Brane World
	The Brane World Geodesics.
	Summary and Discussions.
	Acknowledgements
