We consider nonuniform black strings inside their event horizon. We present
numerical evidence, that the singularity touches the horizon as the horizon
topology changing transition is reached.Comment: 4 pages, 5 figue

Kleihaus, Burkhard

Kunz, Jutta

Physics Letters B

English

arXiv

AbstractWe consider nonuniform black strings inside their event horizon. We present numerical evidence, that the singularity touches the horizon as the horizon topology changing transition is reached

Elsevier - Publisher Connector 

Physics Letters B 664 (2008) 210–213Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Interior of nonuniform black strings
Burkhard Kleihaus ∗, Jutta Kunz
Institut für Physik, Universität Oldenburg, Postfach 2503 D-26111 Oldenburg, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Received 29 November 2007
Received in revised form 9 May 2008
Accepted 12 May 2008
Available online 16 May 2008
Editor: A. Ringwald
We consider nonuniform black strings inside their event horizon. We present numerical evidence, that
the singularity touches the horizon as the horizon topology changing transition is reached.
© 2008 Elsevier B.V. Open access under CC BY license.1. Introduction
Already Kaluza and Klein invoked a ﬁfth dimension to unify
gravity and electromagnetism, hiding the higher dimension subse-
quently by rendering it small and compact. Following these ideas,
today supergravity and string theory also put forward the existence
of higher dimensions in their attempt to unify the fundamental
forces of nature.
In such D-dimensional manifolds with p compact dimensions,
black holes can either be localized in the compact dimensions, or
the black hole horizon can wrap the compact dimensions com-
pletely. Black holes which are localized in the compact dimensions
have the horizon topology of a (D−2)-sphere, SD−2, and are called
‘caged’ black holes [1,2]. In contrast, when the horizon wraps the
compact dimensions, the horizon topology reﬂects the topology of
the compact manifold. In the simplest case, the single compact
dimension is simply a circle, S1. The black holes then have the
horizon topology of a torus, SD−3× S1, and are referred to as black
strings [1,2].
As long as caged black holes are much smaller than the
compact dimension, they resemble more or less D-dimensional
Schwarzschild black holes (when they are static and uncharged).
But as their size and with it their mass grows, they begin to
feel the ﬁnite size of the compact dimension and deform ac-
cordingly, reaching a maximal size beyond which they no longer
ﬁt into the compact dimension [3,4]. Black strings, on the other
hand, exist for all values of the mass. However, as shown by Gre-
gory and Laﬂamme (GL) [5], such ‘uniform’ black strings (UBS),
which do not depend on the compact coordinate, become unsta-
ble below a critical value of the mass. GL therefore suggested, that
unstable UBS would decay to black holes, which possess higher
entropy.
* Corresponding author.
E-mail address: kleihaus@theorie.physik.uni-oldenburg.de (B. Kleihaus).0370-2693 © 2008 Elsevier B.V.
doi:10.1016/j.physletb.2008.05.025
Open access under CC BY license.In contrast, Horowitz and Maeda [6] argued that the horizon
of UBS could not pinch off in ﬁnite aﬃne time. They therefore
conjectured that the solutions would settle down to nontransla-
tionally invariant solutions with the same horizon topology as the
original conﬁgurations, ‘nonuniform’ black strings (NUBS). Shortly
after, such NUBS were found perturbatively [7] and numerically [8].
However, the NUBS cannot serve as endpoints of the instability,
since they are too massive and possess too low an entropy [7], at
least as long as the number of dimensions is smaller than 13 [9],
indicating that black holes are the end-state of the instability, nev-
ertheless [1].
Emerging from the UBS branch at the marginally stable solu-
tion, the NUBS form a branch of solutions along which the de-
formation of the horizon along the compact direction perpetually
increases. This is illustrated in Fig. 1, where we exhibit the spatial
embedding of the horizon of static D = 6 black string solutions for
a sequence of NUBS, yielding a geometrical view of the increas-
ing nonuniformity of the solutions [10]. We here parametrize the
deformation by λ = 12 ( RmaxRmin − 1) [7], where Rmax and Rmin repre-
sent the maximum and minimum radius of a (D − 3)-sphere on
the horizon.
Extrapolating the numerical results to λ → ∞ shows that the
maximal radius Rmax assumes a ﬁnite limiting value, while the ra-
dius Rmin of the ‘waist’ of the black strings shrinks to zero [4,8,10].
In the limit λ → ∞, the horizon will then pinch off, changing the
horizon topology from SD−3 × S1 to SD−2 at a singular ‘merger’
conﬁguration [1,11–13]. Study of the global and horizon properties
of NUBS and black holes in D = 5 and D = 6 dimensions indeed
provides persuasive evidence that the NUBS branch merges with
the black hole branch at such a topology changing transition [4,10].
A deep question associated with the envisaged transition is
whether it is associated with the occurrence of a naked singular-
ity and thus a violation of cosmic censorship. Indeed, it appears
possible, that as the black string pinches, a naked singularity is
formed, because the singularity, which originally winds the com-
pact dimension, gets broken [1]. To address this question, we here
B. Kleihaus, J. Kunz / Physics Letters B 664 (2008) 210–213 211Fig. 1. Spatial embedding of the horizon of D = 6 black strings with increasing de-
formation λ, ranging from λ = 0 to λ = 13.8. (Z denotes the proper length along
the compact direction, R the proper radius of the horizon.)
investigate the inside of nonuniform black strings, to see how the
curvature and in particular the singularity evolve, as λ becomes
large. Our results indicate, that the singularity will touch the hori-
zon at the topology changing transition.
2. Nonuniform black strings
We consider the Einstein action
I = 1
16πG
∫
M
dDx
√−gR − 1
8πG
∫
∂M
dD−1x
√
−hK , (1)
for a D-dimensional spacetime with one compact direction. We
denote the periodic length of the compact direction by L. Here the
last term is the Gibbons–Hawking surface term [14].
A convenient form of the metric is given by [10]
ds2 = −e2A f dt2 + e2B
(
dr˜2
f
+ dz2
)
+ e2C r˜2 dΩ2D−3, (2)
where z is the coordinate of the compact direction. A, B , and C are
functions of r˜ and z only, and f = 1− (r˜0/r˜)D−4. Thus the horizon
resides at r˜ = r˜0. The functions A, B , and C have to be determined
from the Einstein equations Gtt = 0, Gr˜r˜ + Gzz = 0 and Gθθ = 0. The
remaining nontrivial equations Gr˜r˜ − Gzz = 0 and Gr˜z = 0 form the
constraints. (For A = B = C = 0 UBS arise.)
We now focus on black strings in D = 6 dimensions, since
they can be determined with great numerical accuracy up to large
deformations [10]. For solutions outside the horizon, r˜  r˜0, we in-
troduce the coordinate r¯ via r˜ =
√
r˜20 + r¯2, while inside the horizon,
r˜  r˜0, we deﬁne the coordinate r via r˜ =
√
r˜20 − r2. In these coor-
dinates the horizon resides at r¯ = 0 = r. Substitution in the line
element Eq. (2) then yields for r˜  r˜0
ds2 = e2 Aˆr2 dt2 + e2Bˆ(−dr2 + dz2)+ e2Cˆ dΩ23 , (3)
where e2 Aˆ = e2A/(r˜20 − r2), e2Bˆ = e2B , and e2Cˆ = e2C (r˜20 − r2). Inside
the horizon the Einstein equations are hyperbolic, the coordinate r
playing the role of ‘time’,
Aˆ,rr = Aˆ,zz − Aˆ,r
r
−
(
Aˆ,r +1
r
)
( Aˆ,r +3Cˆ,r ) + Aˆ,z ( Aˆ,z +3Cˆ,z ),
Bˆ,rr = Bˆ,zz +3Cˆ,r
(
Aˆ,r +1
r
+ Cˆ,r
)
− 3Cˆ,r ( Aˆ,z +Cˆ,z ) + 3e2(Bˆ−Cˆ),Cˆ,rr = Cˆ,zz −Cˆ,r
(
Aˆ,r +1
r
+ 3Cˆ,r
)
+ Cˆ,z ( Aˆ,z +3Cˆ,z ) − 2e2(Bˆ−Cˆ). (4)
Initial conditions are given at the horizon (setting r˜0 = 1)
Aˆ(0, z) = AH(z), Bˆ(0, z) = BH(z), Cˆ(0, z) = CH(z),
∂r Aˆ|r=0 = ∂r¯ A|r¯=0 = 0, ∂r Bˆ|r=0 = ∂r¯ B|r¯=0 = 0,
∂r Cˆ |r=0 = ∂r¯ C |r¯=0 = 0, (5)
where the functions AH(z), BH(z) and CH(z) are obtained from the
solution outside the horizon. The origin of the z coordinate is cho-
sen such that Rmin = eCH(0) and Rmax = eCH(L/2) .
Setting Gtt = Grr + Gzz = Gθ1θ1 = G
θ2
θ2
= Gϕϕ = 0 in the identities
∇μGμr = 0 and ∇μGμz = 0 reveals that the constraints satisfy the
advection equations
(∂r ± ∂z)
(√−g(Grz ∓ (Grr − Gzz)/2))= 0.
Consequently, the constraints vanish everywhere, since they vanish
at the horizon.
3. Numerics
The hyperbolic differential equations (4) are re-written as a set
of ﬁrst order equations in r. For the evolution in r we use a fourth
order Runge–Kutta method.1 The ‘spatial’ coordinate is scaled via
z → z/L. The difference formulae for z derivatives are obtained
from a polynomial approximation of the functions: If a function y
has values yk at the gridpoints zk , we approximate the qth deriva-
tive of y at zk by
y(q)(zk) =
k+n∑
j=k−n
y j P
(q)
j,k(zk). (6)
Here P (q)j,k(z) is the qth derivative of the Lagrange polynomial of
degree 2n,
P j,k(z) =
k+n∏
l=k−n
l = j
z − zl
z j − zl .
To maintain the same order of approximation at all gridpoints we
include auxiliary gridpoints {−zn,−zn−1, . . . ,−z2} and {1+ z1,1+
z2, . . . ,1+ zn}. The values of the functions at these points are ob-
tained from the symmetry properties y(−z) = y(z) and y(1+ z) =
y(1− z).
However, if a singularity is encountered at some point zs , we
have to restrict to the interval zs < z < 1 − zs . In this case the
approximation of the z derivatives is less ‘symmetric’. Thus, if zks
is the gridpoint next to zs , and ks  k < ks + n, we deﬁne
y(q)(zk) =
ks+2n∑
j=ks
y j P
(q)
j,k(zk), P j,k(z) =
ks+2n∏
l=ks
l = j
z − zl
z j − zl , (7)
and similarly for gridpoints close to 1− zs .
As we argue below, the solution becomes singular if eCˆ tends
to zero. In order to avoid the singularity we have to restrict to
a domain, where eCˆ is larger than some small number, typically
≈ 10−6. Thus the computation is performed in two stages. In the
ﬁrst stage the solution evolves from the horizon (r = 0) until it
1 For the numerical construction of the solutions we have also employed the pa-
rameterization q(r, z) = e−4 Aˆ , w(r, z) = e−2(Bˆ+Cˆ) , γ (r, z) = e3Cˆ+ Aˆ .
212 B. Kleihaus, J. Kunz / Physics Letters B 664 (2008) 210–213Fig. 2. The coordinate zs(r) of the location of the singularity is shown for sev-
eral values of the deformation parameter λ of black strings, ranging from λ = 0
to λ = 13.8.
becomes nearly singular at z = 0 and z = 1 for some r = rs . Dur-
ing this stage we employ the difference formula (6). In the second
stage the solution evolves from r = rs . In each Runge–Kutta step
we check whether the solution is nearly singular at gridpoints zks .
If this is the case, we restrict to zks < zk < 1 − zks in the next
Runge–Kutta step. During this stage the difference formula (7) is
employed. The evolution stops when the z-interval shrinks to zero.
Typical step sizes used are 	r ≈ 1− 4× 10−4 and the number
of gridpoints ranges between N = 100 and N = 2400. The order of
the difference formula is 2n = 6 − 14. We have checked the con-
sistency of the solutions for different choices of 	r, N and n, and
also for different nonequidistant gridpoint distributions.
4. Results
We have constructed NUBS solutions inside their horizon for
several values of the deformation parameter λ. Near the horizon
the solutions extend smoothly into the interior region. The func-
tion eCˆ decreases monotonically for ﬁxed z. When eCˆ tends to zero
at some rs(z), the Kretschmann scalar diverges there, indicating
that the curvature singularity resides at rs(z). For UBS the radial
coordinate of the singularity is constant, rs(z) = 1, while for NUBS
with small deformation rs(z) shows an oscillation about r = 1 with
small amplitude, which then increases with increasing λ. We ex-
hibit the coordinate zs(r) of the singularity in Fig. 2 for several
values of the deformation parameter λ. (Note the similarity with
Fig. 1.)
As seen in the ﬁgure, the singularity approaches the horizon at
z = 0 and z = 1, when λ becomes large. This is further demon-
strated in Fig. 3, where we exhibit the minimal distance Rsing =∫ rs
rH
eBˆ(r,0) dr of the singularity from the horizon at the waist. We
note that for large deformation the distance decreases approx-
imately inversely proportional to the deformation, Rsing ∼ λ−1.
Thus we extrapolate that in the limit λ → ∞ the singularity will
touch the horizon at the waist of the string, i.e., a naked singular-
ity will appear at the topology changing transition [1,12].
We conclude that the deformation of the horizon reﬂects itself
in the location of the singularity. At the symmetric points z = 0
and z = 1, where the function eCH yields the minimal horizon ra-
dius Rmin, the distance Rsing of the singularity from the horizon is
also minimal. In particular, for large λ, Rmin ∼ λ−1 along with Rsing
(see Fig. 3). Extrapolating to the limit of inﬁnite deformation, we
infer that the singularity touches the horizon at z = 0 and z = 1,
but is hidden well inside the horizon everywhere else.
In order to get further insight into the geometry of space in the
NUBS interior, we consider an isometric embedding of surfaces ofFig. 3. The distance Rsing from the horizon to the singularity at the waist (z = 0) is
shown versus the deformation λ for a sequence of NUBS, together with its approxi-
mation at large λ, Rsing = λ−1 (dotted). Also shown are the minimal horizon radius
Rmin and its asymptote (dotted).
Fig. 4. Isometric embedding of surfaces of constant Kretschmann scalar K for NUBS
with λ = 0.03. The straight lines correspond to surfaces of constant K of UBS with
the same temperature.
constant Kretschmann scalar K = Rμνρσ Rμνρσ . Thus, if such a sur-
face has coordinates rK (z), z, θ1, θ2,ϕ (at a ﬁxed time), we deﬁne
coordinates Xi, Z by
dXi dX
i + dZ2 = e2Bˆ(−(drK /dz)2 + 1)dz2 + e2Cˆ dΩ23
= dR2 + dZ2 + R2 dΩ23 , (8)
where the Xi are expressed in terms of spherical coordinates in
the second line. Regarding R and Z as functions of z then yields
R(z) = eCˆ and
Z(z) = L
z∫
1
2
√
e2Bˆ
(
1− (drK /dz)2
)− e2Cˆ (dCˆ/dz)2 dz′,
where Bˆ and Cˆ are taken along the curve (rK (z), z), and the length
scale L is reintroduced.
We exhibit surfaces Z(R) of constant Kretschmann scalar K for
NUBS with deformation λ = 0.03 and λ = 0.47 in Figs. 4 and 5, re-
spectively.2 The limit of inﬁnite K corresponds to the singularity
at R = 0. For UBS, which are also shown, R ∼ K− 18 , independent
of Z . The NUBS surfaces, on the other hand, show an oscillation
about constant R values, and thus a Z dependence of the power
2 Note, that close to the horizon the embedding is pseudo-Euclidean with dZ2 →
−dZ2.
B. Kleihaus, J. Kunz / Physics Letters B 664 (2008) 210–213 213Fig. 5. Same as Fig. 4 for λ = 0.47. The horizon is also shown.
law. Here R tends more slowly to zero for Z = 0 and faster for
Z = ±LK /2, where LK denotes the periodic length of Z . Note, that
LK increases with increasing K . Clearly, the deformation of the
horizon reﬂects itself also in the isometric embeddings of the sur-
faces of constant K .
In the following we argue that the singularity is space-like for
all z. First we note that for the uniform black string the singu-
larity is space-like, since the spacetime is Schwarzschild × S1. For
the nonuniform black strings we consider the worldlines of incom-
ing photons, γ μ = (t(u), r(u), z(u), θ1(u), θ2(u),ϕ(u)), with aﬃne
parameter u. The geodesic equations are obtained as the varia-
tional equations of the Lagrangian K = gμνγ˙ μγ˙ ν , where the dot
denotes the derivative with respect to u. Variation with respect to
t yields t˙e2 Aˆr2 = k/2 = const., and the variational equations w.r.t.
to the angle coordinates admit solutions θ1 = const., θ2 = const.,
ϕ = const., due to the symmetries of the non-uniform black string
solutions. In this case the remaining variational equations can be
cast in the form
d
du
((
r˙2 − z˙2)e2Bˆ − k2
4r2
e−2 Aˆ
)
= 0, (9)
d
du
(
2r˙ z˙e4Bˆ
)+ k2
4
[r˙∂z − z˙∂r]
(
e−2( Aˆ−Bˆ)
r2
)
= 0. (10)
The ﬁrst equation yields r˙2 − z˙2 = k2
4r2
e−2( Aˆ+Bˆ) , consistent with the
constraint K = 0 for photons.
We now argue that the direction of the tangential vector Tμ =
d
duγ
μ(u) is always pointing towards the singularity. We ﬁrst note
that e− Aˆ , eBˆ− Aˆ and e−( Aˆ+Bˆ) tend to zero as the singularity is ap-
proached. For incoming null geodesics r˙  |z˙|, where equality holds
at the singularity. The ratio of the ‘time’ component to the ‘radial’
component,
t˙
r˙
= k
2r
e− Aˆ√
z˙2r2e2 Aˆ + k24 e−2Bˆ
, (11)
vanishes at the singularity: if z˙ vanishes, t˙/r˙ = e(Bˆ− Aˆ)/r tends
to zero, while for ﬁnite z˙, t˙/r˙ ∼ e−2 Aˆ/r2 also tends to zero.Thus at the singularity the direction of the tangential vector
is (0,1,±1,0,0,0), which points always towards the singularity.
Consequently, photons cannot avoid to hit the singularity. The
same conclusion holds for massive particles, since their world-
lines lie inside the lightcone. Hence the singularity is space-like
for all z.
5. Outlook
Concerning the black hole—black string transition we have pro-
vided numerical evidence that at the merger point the NUBS sin-
gularity touches the horizon at the waist of the strings. What is
still missing is the study of the interior of the caged black holes,
as they approach the merger point, to see how their curvature and,
in particular, their singularity evolve. In D = 4 such a study has
been performed recently [15], in higher dimensions, however, this
remains a major numerical challenge.
A GL instability arises also for rotating NUBS [16], and a simi-
lar instability occurs for asymptotically ﬂat rotating black holes in
D  6 dimensions, which have a single angular momentum [17].
This instability then suggests, that as in the case of black strings, a
branch of rotating ‘pinched’ (nonuniform) black holes should arise
at the marginally stable solution [17]. Exploring this analogy fur-
ther leads to an intriguing phase diagram for rotating black holes,
where horizon topology changing transitions from rotating pinched
black holes to black rings, black saturns, and further conﬁgurations
are expected to occur [18].
Acknowledgement
We gratefully acknowledge discussions with V. Frolov, E. Radu
and A. Shoom. B.K. was supported by the German Aerospace Cen-
ter.
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Interior of nonuniform black strings 

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Interior of Nonuniform Black Strings

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