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arXiv:gr-qc/0503008v2 13 May 2005

Department of Mathematics and Physics

Osaka City University

OCU-PHYS-227

AP-GR-23

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

Daisuke Ida‡

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 yoo@sci.osaka-cu.ac.jp

†E-mail:knakao@sci.osaka-cu.ac.jp

‡E-mail:daisuke.ida@gakushuin.ac.jp

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I.INTRODUCTION

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 [1].

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 [2]. In

such TeV gravity models, it is suggested that small black holes are produced in accelerators,

such as the CERN Large Hadron Collider [3] or in high energy cosmic ray events [4].

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 [5]. 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 [6]; 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 [7]. Here we call it the hyperhoop conjecture in the sense that it is a

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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

VD−3? GDM,(1)

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 [7] and for the system of point-particles [8].

Consistent results with the previous ones were obtained by Barrab´ es et al [9]. 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. [9]. 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. [10]. 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

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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.[7] is derived.

In this paper, we adopt the unit of c = 1. We basically follow the notations and sign

conventions in Ref.[11].

II.A MOMENTARILY STATIC SPHEROID IN FIVE-DIMENSIONAL

SPACE-TIME

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

a∇cnb, (3)

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

given by

R − KabKab+ K2= 24π2G5ρ(4)

and

Db

?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)

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where δabis the metric tensor of four-dimensional Euclidean space. We also require the axial

symmetry in the sense that the metric on Σ have the form

dl2= f2(R,z)?dR2+ R2?dϑ2+ sin2ϑdϕ2?+ dz2?,

where 0 ≤ R < +∞ and −∞ < z < +∞ while ϑ and ϕ are the round coordinates. Then the

momentum constraint leads to zero flux condition Ja= 0, and the Hamiltonian constraint

(8)

becomes

∂2f

∂R2+2

R

∂f

∂R+∂2f

∂z2= −4π2G5f3ρ. (9)

Here we note that the Hamiltonian constraint (9) is equivalent to the Poisson equation

for axi-symmetric Newtonian gravitational potential. Let us consider the density profile

respecting the axial symmetry given by

f3ρ =

2M/π2a3b for R2/a2+ z2/b2≤1,

0for elsewhere,

(10)

where a, b and M are constant parameters.

We consider the gravitational field of an isolated body, so that we assume the asymptotic

condition given by

f → 1 for r → ∞,(11)

where

r =

√R2+ z2. (12)

The regular solution is then obtained as

f = 1 −4G5M [b(2a + b)R2+ 3a2z2− 3a2b(a + b)]

3a3b(a + b)2

for

R2

a2+z2

b2≤ 1, (13)

f = 1 −4G5M

3e4b4

?

2R2− 6z2+ 3e2b2+

√F2− e4b4

×

?

2e2b2R2

(F − e2b2)2−2R2− 3z2+ 3e2b2

F − e2b2

+

3z2

F + e2b2

??

for

R2

a2+z2

b2> 1,(14)