arXiv:gr-qc/0503008v2 13 May 2005
Department of Mathematics and Physics
Osaka City University
Hoop Conjecture in Five-dimensions
-Violation of Cosmic Censorship-
Chul-Moon Yoo∗, Ken-ichi Nakao†
Department of Mathematics and Physics, Graduate School of Science,
Osaka City University, Osaka 558-8585, Japan
Department of Physics, Gakushuin University, Tokyo 171-8588, Japan
We study the condition of black hole formation in five-dimensional space-time. We
analytically solve the constraint equations of five-dimensional Einstein equations for
momentarily static and conformally flat initial data of a spheroidal mass. We numer-
ically search for an apparent horizon in various initial hypersurfaces and find both
necessary and sufficient conditions for the horizon formation in terms of inequalities
relating a geometric quantity and a mass defined in an appropriate manner. In the
case of infinitely thin spheroid, our results suggest a possibility of naked singularity
formation by the spindle gravitational collapse in five-dimensional space-time.
PACS numbers: 04.50.+h, 04.70.Bw
∗E-mail:c m email@example.com
In an attempt to unify fundamental forces including gravity, the possibility that the space-
time dimensions of our universe is higher than four has been much discussed. Such higher-
dimensional theories need mechanism to reduce the space-time dimensions down to four, for
example via Kaluza-Klein type compactifications of extra dimensions, so as to be consistent
with the observed world. The brane world scenario is another attractive idea of dimensional
reduction. In this scenario, the standard model particles are confined to the boundary of
a higher-dimensional space-time and only gravity can propagate in the extra dimensions.
Models of the brane world scenario with large extra dimensions compared to the four-
dimensional Planck scale (≈ 1.6 × 10−33cm) have been considered in some recent works .
According to these models, the fundamental (namely, higher-dimensional) Planck scale may
be set to rather low energy scale, even to 1TeV, of which low energy effects just alter
the short distance behaviour of classical gravitational interactions. The discrepancy in the
gravitational interaction between the four and higher-dimensional theories arises only at the
length scale below 0.1mm so that it is consistent with the gravitational experiments . In
such TeV gravity models, it is suggested that small black holes are produced in accelerators,
such as the CERN Large Hadron Collider  or in high energy cosmic ray events .
In order to understand physical phenomena caused by strong gravitational fields, the
criterion for black hole formation is very crucial. In the case of four-dimensional Einstein
gravity, such a criterion is well known as the hoop conjecture . Hoop conjecture claims that
the necessary and sufficient condition for black hole formation is given by the following; Black
holes with horizons form when and only when a mass M gets compacted into a region whose
circumference in every direction is C ? 4πG4M, where G4is the gravitational constant in
four-dimensional theory of gravity. It is remarkable that no serious counterexample against
hoop conjecture has been presented. However, at first glance, hoop conjecture is not valid
in higher-dimensional Einstein gravity ; there is black string solutions in five or higher-
dimensions, which have infinitely long event horizons, while hoop conjecture claims that
any length scale characterizing black hole should be less than the gravitational length scale
determined by the Schwarzschild radius.
Recently, two of the present authors, DI and KN, proposed a higher-dimensional version
of hoop conjecture . Here we call it the hyperhoop conjecture in the sense that it is a
possible generalization of the original hoop conjecture; Black holes with horizons form when
and only when a mass M gets compacted into a region whose (D−3)-dimensional area VD−3
in every direction is
where GDis the gravitational constant in D-dimensional theory of gravity, and the (D−3)-
dimensional area means the volume of (D−3)-dimensional closed submanifold of a spacelike
hypersurface. Hereafter we call this (D −3)-dimensional closed submanifold the hyperhoop.
The necessity of the condition (1) was confirmed in the case of momentarily static and
conformally flat initial data sets of an axially symmetric line, disk and thin ring source
for the five-dimensional Einstein equations  and for the system of point-particles .
Consistent results with the previous ones were obtained by Barrab´ es et al . They derived
two inequalities for (D−3)-dimensional volume as the necessary and sufficient conditions for
apparent horizon formation in the case of a (D−2)-dimensional convex thin shell collapsing
with the speed of light in a D-dimensional space-time.
The purpose of the present paper is to study both the necessity and in particular suffi-
ciency of the inequality (1) for the horizon formation in different situations from the case
treated in Ref. . We consider the momentarily static and conformally flat four-dimensional
initial hypersurfaces in which a four-dimensional homogeneous spheroid is put as a gravita-
tional source. This procedure has been implemented by Nakamura et al. . We apply their
method to higher-dimensional case. Then, we analytically solve the constraint equations for
five-dimensional Einstein equations. In order to investigate the validity of hyperhoop con-
jecture, we numerically search for an apparent horizon and calculate the ratio V2/G5M for
substantially various hyperhoops.
This paper is organized as follows. In Sec. II, assuming five-dimensional Einstein gravity,
we derive the constraint equations for conformally flat initial hypersurfaces and then give
analytic solutions of these equations for a homogeneous mass of a spheroidal shape. In
Sec. III, we search for an apparent horizon in initial hypersurfaces with various shapes of a
homogeneous spheroid including infinitely thin case by numerically solving a second order
ordinary differential equation. This equation corresponds to the minimum volume condition
for a three-dimensional closed submanifold of an initial hypersurface. The suggestion of the
naked singularity formation is given in this section. In Sec. IV, we define V2/G5M in a
reasonable manner and then give a procedure to select the hyperhoop with minimal value
of V2/G5M. In Sec. V, we show numerical results and their implication to the necessary
and sufficient condition for the horizon formation. Finally, Sec. VI is devoted to summary.
In Appendix A, we derive analytic solutions for the Newtonian gravitational potential of an
ellipsoid in arbitrary space dimension. In Appendix B, the necessary condition of black hole
formation based on Ref. is derived.
In this paper, we adopt the unit of c = 1. We basically follow the notations and sign
conventions in Ref..
II.A MOMENTARILY STATIC SPHEROID IN FIVE-DIMENSIONAL
Let us consider an initial data set (hab,Kab) in a four-dimensional spacelike hypersurface
Σ, where habis the induced metric in Σ and Kabis the extrinsic curvature which represents
how Σ is embedded in the five-dimensional space-time. Denoting the unit normal vector to
Σ by na, haband Kabare, respectively, written as
hab = gab+ nanb,(2)
Kab = −hc
where ∇cis the covariant derivative in the five-dimensional space-time.
The initial data set (hab,Kab) has to satisfy the Hamiltonian and momentum constraints
R − KabKab+ K2= 24π2G5ρ(4)
?Kab− habK?= 12π2G5Ja, (5)
where ρ and Jaare the energy density and energy flux for normal line observers to Σ, Da
and R are the covariant derivative within Σ and the scalar curvature of hab, and G5 is
the gravitational constant in five-dimensional theory of gravity. In this paper, we focus on
momentarily static and conformally flat initial hypersurfaces:
Kab = 0(6)
hab = f2δab,(7)
where (V2)minis the area of the hyperhoop which entirely encircle the source and has the
smallest area. In order to evaluate (V2)min, we focus on the hyperhoop r = ra(ξ) and ϑ = π/2
which satisfy following minimum area condition
δV2= 0, (B6)
where δV2is the small variation of V2for slight deformation of the hyperhoop which keeps
it on ϑ = π/2 and the symmetry holds. Namely, Eq. (B6) leads to the Euler-Lagrange
equation for the Lagrangian L = V2(ra, ˙ ra) as
¨ ra−3˙ ra2
− 2ra+ra2+ ˙ ra2
f(˙ rasinξ + racosξ)∂f
f(˙ racosξ − rasinξ)∂f
We impose following boundary conditions so that every part of the hyperhoop locally satisfy
˙ ra= 0at ξ = 0,
Then r = ra(ξ) is the hyperhoop of the minimum area if and only if ra(ξ) satisfies Eq. (B7)
with the boundary condition (B8). We adopt the area of this hyperhoop as (V2)min.
We numerically search for solutions of Eq. (B7) with (B8). Accordingly, the solutions
always can be found and the hyperhoop always encircle the source in spite of its length. The
value of Γminis depicted in Fig. 7 as a function of b.
The value of Γminmonotonically increases with b but has a finite limit for b → ∞, while
an apparent horizon always covers this line source. Therefore it is necessary for apparent
horizon formation that Γminis smaller than this asymptotic value. The asymptotic value
will be obtained by evaluating the corresponding quantity of the infinitely long source case.
Let us consider the infinitely long singular line source whose density profile is given by
In this case, we can easily solve the Hamiltonian constraint (9) and obtain
f = 1 +πG5M
The area Vcof the cylindrical two-surface R = R0and ϑ = π/2 with coordinate length 2b is
Vc= 2πRf|R=R0× 2bf|R=R0, (B11)
FIG. 7: The value of Γminis plotted as a function of b. The dashed line bound this value above.
This line is the corresponding quantity for the infinitely long spindle source which have line density
where 2πRf|R=R0is the proper length of circle around singular line source and 2bf|R=R0is
the proper length of the cylinder measured along the z-direction. The minimal value of Vc
is realized when R0= πG5M/2b, and in its value is equal to 8π2G5M. This minimal value
might be almost equal to (V2)minof the singular line source (B1) if b is much longer than rs.
As a result, the asymptotic value of Γminfor b → ∞ with the mass M fixed is evaluated as
APPENDIX C: THE DERIVATION OF THE EQUATION FOR A MARGINAL
In this section, we show the derivation of Eq.(26). Here, we generalize Ref. to the
We denote the spacelike unit vector outward from and normal to the marginal surface
by sa, and the spacelike unit vectors spanning the marginal surface are denoted by (eA)a,
where A = 1,..,D − 2. All these vectors are chosen to be orthogonal to each other and to
the unit vector normal to the initial hypersurface na. Then the future directed outward null
vector laorthogonal to the marginal surface is written by
la= na+ sa. (C1)
The expansion χ of this null vector is defined by
χ = δAB(eA)a(eB)b∇bla= (hab− sasb)(Kab− Dbsa), (C2)
where hab and Kab are the induced metric and the extrinsic curvature defined in Sec.II
respectively. The marginal surface is a closed (D−2)-dimensional spacelike submanifold such
that the outward null vector orthogonal to the (D − 2)-dimensional spacelike submanifold
has vanishing expansion. Hence, the equation to define the marginal surface is given by
χ = 0. In the momentarily static case, this equation reduces to
δAB(eA)aDa(eB)bsb= 0. (C3)
In the situation presented in this paper, coordinates of points on the marginal surface
are represented as
xµ= (rm(ξ)cosξ,rm(ξ)sinξ,ϑ,ϕ) (C4)
and following vectors are tangent to the marginal surface as
= (˙ rmcosξ − rsinξ, ˙ rmsinξ + rmcosξ,0,0), (C5)
= (0,0,1,0), (C6)
Hence, we can obtain the components of saand (eA)aas
m+ ˙ r2
(˙ rmsinξ + rmcosξ,−˙ rmcosξ + rmsinξ,0,0),(C8)
m+ ˙ r2
(˙ rmcosξ − rmsinξ, ˙ rmsinξ + rmcosξ,0,0),(C9)
Substituting Eqs.(C8)−(C11) into Eq.(C3), we obtain Eq.(26).
 N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998)
[arXiv:hep-ph/9803315] ; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali,
Phys. Lett. B 436, 257 (1998) [arXiv:hep-ph/9804398] ; L. Randall and R. Sundrum, Phys.
Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221].
 J. C. Long, H. W. Chan and J. C. Price Nucl. Phys. B 539, 23 (1999) [arXiv:hep-ph/9805217];
C. D. Hoyle, D. J. Kapner, B. R. Heckel, E. G. Adelberger, J. H. Gundlach, U. Schmidt and
H. E. Swanson, Phys. Rev. D 70, 042004 (2004) [arXiv:hep-ph/0405262].
 P. C. Argyres, S. Dimopoulos and J. March-Russell, Phys. Lett. B 441, 96 (1998)
[arXiv:hep-th/9808138] ; T. Banks and W. Fischler, arXiv:hep-th/9906038 ; S. Dimopoulos
and G. Landsberg, Phys. Rev. Lett. 87, 161602 (2001) [arXiv:hep-ph/0106295] ; S. B. Gid-
dings and S. Thomas, Phys. Rev. D 65, 056010 (2002) [arXiv:hep-ph/0106219] ; E. J. Ahn,
M. Cavaglia and A. V. Olinto, Phys. Lett. B 551, 1 (2003) [arXiv:hep-th/0201042].
 J. L. Feng and A. D. Shapere, Phys. Rev. Lett. 88, 021303 (2002) [arXiv:hep-ph/0109106]
; R. Emparan,M. Masip and R. Rattazzi, Phys. Rev.D 65, 064023(2002)
[arXiv:hep-ph/0109287] ; L. Anchordoqui and H. Goldberg, Phys. Rev. D 65, 047502 (2002)
 K.S. Thorne, in Magic Without Magic, edited by J. Klauder (Freeman, San Francisco, 1972).
 K. Nakao, K. Nakamura and T. Mishima, Phys. Lett. B 564, 143 (2003) [arXiv:gr-qc/0112067].
 D. Ida and K. Nakao, Phys. Rev. D 66, 064026 (2002) [arXiv:gr-qc/0204082].
 H. Yoshino and Y. Nambu, Phys. Rev. D 67, 024009 (2003) [arXiv:gr-qc/0209003]; H. Yoshino
and Y. Nambu, Phys. Rev. D 70, 084036 (2004) [arXiv:gr-qc/0404109].
 C. Barrabes,V. P. Frolov and E. Lesigne,Phys. Rev. D 69, 101501(R) (2004)
 T. Nakamura, S. L. Shapiro and S. A. Teukolsky, Phys. Rev. D 38, 2972 (1988).
 R.N. Wald, General Relativity, (University of Chicago Press, Chicago, 1984).
 M. Sasaki, K. Maeda, S. Miyama and T. Nakamura, Prog. Theor. Phys. 63, 1051 (1980).
 S. L. Shapiro and S. A. Teukolsky, Phys. Rev. Lett. 66, 994 (1991).
 K. D. Patil, Phys. Rev. D 67, 024017 (2003) ; R. Goswami and P. S. Joshi, Phys. Rev.
D 69, 104002 (2004) [arXiv:gr-qc/0405049]; A. Mahajan, R. Goswami and P. S. Joshi,
 S. Chandrasekhar, “Ellipsoidal Figures of Equilibrium”,(Yale University Press, New Haven,