Page 1

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)

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where e is the eccentricity defined by

e =

?

1 −a2

b2

(15)

and F is a function of R and z defined by

F = F(R,z;a,b)

= R2+ z2+

?

4b2R2− 4a2(b2− z2) + (a2+ b2− R2− z2)2.

(16)

The detailed derivation of the above solution is given in Appendix A. Newtonian gravita-

tional potential of an ellipsoid in Euclidean space of arbitrary dimensions is shown there.

Here we only investigate the prolate case a < b.

In the thin limit a → 0 with M and b fixed, two disconnected singularities appear at the

poles (R,z) = (0,±b) of the resultant “singular spheroid”. In order to see this, we evaluate

the Kretschmann invariant

I = RabcdRabcd,(17)

where Rabcdis the four-dimensional Riemann tensor of the spacelike hypersurface. Typical

examples are shown in Figs. 1 and 2. The coordinate values and the Kretschmann invariant

I is normalized by

rs:=

?

G5M.(18)

It will be found in the next section that rsis the coordinate radius of the apparent horizon

in the case of a point source a = b = 0.

Here it should be noted that the Kretschmann invariant I is finite between these two

singularities on the singular spheroid, R = 0 and |z| < b. Further, we can see that the

energy density ρ is also finite there. The conformal factor on the surface of the spheroid is

given by

f = fsf(z)

:= 1 +

4G5M

3ab2(a + b)2

?b2(a + 2b) − 2(b − a)z2?. (19)

Therefore we find that the energy density at the surface becomes in the thin limit a → 0 as

2M

π2a3bf3

sf

ρ = ρsf(z) :=

−→

27b8

5M2(b2− z2)3.

256π2G3

(20)

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00

0.50.5

11

1.51.5

2

R?rs

R?rs

0

1

2

3

z?rs

0

2

4

6

log10?Irs4?

FIG. 1: The logarithm to the base 10 of the Kretschmann invariant I is plotted as function of the

coordinates R and z in the case a = 0 and b = 2rs.

1.5

22.53 3.5

4z?rs

-2

2

4

6

8

10

12

14

log10?Irs4?

FIG. 2: The logarithm to the base 10 of the Kretschmann invariant I on the polar axis (R = 0) is

plotted as function of z in the case a = 0 and b = 2rs. The value of I is diverge at only (0,b).

Here note that the inequality f(R,z) ≥ fsf(z) is satisfied within the spheroid and hence

0 ≤ ρ(R,z) ≤ ρsf(z). Therefore ρ is finite except for the poles z = ±b even in the thin limit

a → 0. This fact means that the scalar polynomials of the five-dimensional Riemann tensor

are also finite if the stress of matter fields is assumed to be reasonable. For example, assuming

the dust as the matter and adopting Gaussian normal coordinate, Einstein equations leads

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to

∂Kab

∂t

= Rab− 4π2G5ρ δab

(21)

on the momentarily static initial hypersurface, where Rab is the four-dimensional Ricci

tensor of this hypersurface. Here note that Rabis finite in the region of finite Kretschmann

invariant I since the metric of the spacelike hypersurface is positive definite. Therefore

the finiteness of the energy density ρ guarantees that the time derivative of the extrinsic

curvature Kabis finite in the region of finite Kretschmann invariant I. This means that the

scalar polynomials of Riemann tensor of five-dimensional space-time are also everywhere

finite except for the poles of the singular spheroid since these are expressed as polynomials

of ∂Kab/∂t, Rab, Kaband DcKabin the Gaussian normal coordinate. Further, we can see

that this singular spheroid except for the poles corresponds to spatial infinity. Consider a

curve z = ζ(R) connecting two points R = R1and R = R2(R1< R2) and assume that both

R1and R2are sufficiently small and 0 < ζ(R) < zmax< b. In this situation, the function F

is written as

F(R,ζ;0,b) = b2+

2b2

b2− ζ2R2+ O(R3). (22)

Hence substituting this for Eq. (14), we find

f (R,ζ) =8(b2− ζ2)3/2G5M

3b4R

+ O(R0).(23)

The proper length between R = R1and R2along the curve z = ζ(R) is bounded below as

?R2

R1

f(R,ζ)

?

1 +

?dζ

dR

?2

dR

≥

≥8G5M

3b4

?R2

R1

f(R,ζ)dR ≃8G5M

3b4

?R2

?R2

dR

R

R1

(b2− ζ2)3/2dR

=8G5M

3b4

R

(b2− z2

max)3/2

R1

(b2− z2

max)3/2lnR2

R1.(24)

We can see from the above equation that the proper length diverges in the limit of R1→ 0

with R2fixed. Therefore each point on the singular spheroid except for the poles, R = 0

and |z| < b, is spacelike infinity.

III.APPARENT HORIZONS

In a momentarily static initial hypersurface in five-dimensional asymptotically flat space-

time, an apparent horizon is a three-dimensional closed marginal surface. Because of the

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axial symmetry of the initial hypersurface, the apparent horizon will also be axially sym-

metric and thus will be expressed by r = rm(ξ) in the present case, where

ξ = arctanR

z.

(25)

Then r = rm(ξ) is a closed marginal surface only if rm(ξ) satisfies

¨ rm−4˙ rm2

rm

?2˙ rm

with boundary conditions ˙ rm= 0 at ξ = 0 and π, where a dot means the derivative with

− 3rm+rm2+ ˙ rm2

cotξ −3

rm

×

rm

f(˙ rmsinξ + rmcosξ)∂f

∂z+3

f(˙ rmcosξ − rmsinξ)∂f

∂R

?

= 0 (26)

respect to ξ. The derivation of Eq. (26) is shown in Appendix C. Since the present system

has a reflection symmetry with respect to z = 0, the apparent horizon should satisfy

˙ rm= 0 at ξ =π

2.(27)

In the case of a point source a = b = 0, we can analytically solve Eq. (26) and find

rm= rs.(28)

Replacing derivatives with respect to ξ in Eq. (26) by finite differences, we numerically

search for solutions of this equation by relaxation method [12]. If apparent horizons exist in

the initial hypersurface, we can find solutions of Eq. (26). Typical examples are shown in

Fig. 3. The coordinate values are normalized by rs.

In the case of singular source a = 0, we also find solutions of Eq. (26) satisfying the

boundary condition (27). In the case of b ≤ 1.48rs, there is an apparent horizon enclosing

whole the singular source. By contrast, for b ≥ 1.49rs, the marginal surface covers only a

central part of the singular source, and the space-time singularities at the poles (R,z) =

(0,±b) are not enclosed by the marginal surface (see Fig. 4). In this case, this marginal

surface is not a closed three-surface and thus is not the apparent horizon since as mentioned

in the previous section, |z| < b on the polar axis R = 0 is the spacelike infinity. This result

implies that a long enough spindle singular source can produce naked singularities, which is

quite different from the singular line source in Ref. [7].

IV.HOW TO CHECK HYPERHOOP CONJECTURE

We have presented the statement of hyperhoop conjecture in Sec. I. Here we should note

that in general, the mass M in hyperhoop conjecture (and also in hoop conjecture) is not

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10

0

0.5

1

1.5

2

2.5

3

00.51 1.522.53

a=0.05rs,b=0.5rs

0

0.5

1

1.5

2

2.5

3

0 0.51 1.52 2.53

a=0.05rs,b=1.5rs

0

0.5

1

1.5

2

2.5

3

0 0.51 1.522.53

a=0.05rs,b=2.5rs

???

?

0

0.5

1

1.5

2

2.5

3

0 0.511.522.53

a=0.1rs,b=0.5rs

0

0.5

1

1.5

2

2.5

3

00.511.52 2.53

a=0.1rs,b=1.5rs

0

0.5

1

1.5

2

2.5

3

00.511.522.53

a=0.1rs,b=2.5rs

0

0.5

1

1.5

2

2.5

3

0 0.511.522.53

a=0.5rs,b=0.5rs

0

0.5

1

1.5

2

2.5

3

00.511.522.53

a=0.5rs,b=1.5rs

0

0.5

1

1.5

2

2.5

3

00.511.52 2.53

a=0.5rs,b=2.5rs

???

?

FIG. 3: Apparent horizons for each shape of the four-dimensional spheroid are depicted in (R,z)-

plane. The solid line shows the surface of the spheroid. The dashed line shows the apparent horizon

?

1.4976

if it is present.

???

0

0.5

1

1.5

2

0 0.511.52

a=0,b=rs

0

0.5

1

1.5

2

00.511.52

a=0,b=1.5rs

1.4978

0 0.0005

0

0.5

1

1.5

2

00.511.52

a=0,b=2rs

1.98

2

00.001

???

?

FIG. 4: In the case of a = 0, apparent horizons for each b are depicted.

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the total mass of the system but the mass encircled by the hyperhoop (the hoop). Therefore

to check the sufficiency of this conjecture, we have to confirm that in an initial hypersurface

without an apparent horizon, there is no hyperhoop satisfying

VD−3? GDMin, (29)

where Minis not the total mass of the system but the one included in the region which is

encircled by this hyperhoop.

Since we cannot try all the possible hyperhoops in Σ, we focus on the axi-symmetric

hyperhoops which have the reflection symmetry with respect to z = 0. Further, these

hyperhoops are expressed in the form r = rh(ξ) and ϑ = π/2 since the spheroid is assumed

to be prolate a < b.(In the oblate case, we should consider two-surface of z = 0 and

R =constant, where the hyperhoop is parameterized by two parameters ξ and ϕ.) The

two-dimensional area V2of the hyperhoop r = rh(ξ) and ϑ = π/2 is then given by

V2=

?2π

0

dϕ

?π

0

dξf2?

˙ rh2+ rh2rhsinξ = 4π

?π/2

0

dξf2?

˙ rh2+ rh2rhsinξ, (30)

where we have taken account of the reflection symmetry in the second equality.

In the framework of general relativity, there is no unique definition of the mass in a

quasilocal manner although the total mass is well defined for the isolated system. This is

one of reasons why it is very difficult to formulate precisely the hoop and also hyperhoop

conjectures. However, despite of this mathematical indefiniteness, the hoop and hyperhoop

conjectures might be useful in understanding black hole formation processes on the physical

ground. In this sense, all the reasonable definitions of quasilocal mass will be meaningful in

the hoop and hyperhoop conjectures since these might give the results not so different from

each other. Here we adopt the following simple definition for the mass Minas

Min= 8π

?π/2

0

dξ

?rh(ξ)

0

drρf3r3sin2ξ.(31)

Then we numerically calculate V2/G5Minfor various hyperhoops in an initial hypersurface

of a spheroid. Hereafter for notational simplicity, we introduce

Γ :=

V2

16πG5Min, (32)

where 16πG5Minis the minimal value of V2in the case of a point source whose mass is Min.

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Our first task is the selection of relevant hyperhoops from all the possible hyperhoops

expressed by r = rh(ξ) and ϑ = π/2 with the reflection symmetry with respect to z = 0. A

hyperhoop is represented as a continuous curve from a point on z-axis to a point on R-axis in

the first quadrant of (R,z)-plane. Thus first, we select a point z = z0on z-axis and consider

various hyperhoops which start from this point. Since the spheroid of the mass is assumed to

be prolate, we expect that the significant hyperhoops are also prolate, and thus we restrict

the curves within the region of R2+ z2≤ z2

Therefore we impose further restrictions. Consider 100 points (R,z) =

0. The number of possible curves is still infinite.

?

R(1)

ji(z0),zi(z0)

?

(i,ji= 1,2,..,10) within this spherical region; zi(z0) is determined in the following manner

zi(z0) =z0

10(10 − i), (33)

and then R(1)

ji(z0) is given as

R(1)

ji(z0) =

?z02− zi2(z0)

10

ji. (34)

We consider the curves composed of ten straight lines connected at (R,z)

?

among ten integers from 1 to 10.

=

R(1)

ji(z0),zi(z0)

?

; i is in order from 1 to 10, and then for each i, jiis appropriately chosen

By investigating several randomly chosen hyperhoops, we found that sharply bended

one might have a value of Γ larger than the ones not so sharply bended. Hence in the

systematic numerical search, a line from the point of (R,z) =

?

jior ji±1 for ji> 1 and is equal to jior ji+1 for ji= 1. All the possible constitutive lines

are depicted in Fig. 5. This means that we consider only the hyperhoops each of which is a

?

R(1)

ji(z0),zi(z0)

?

to (R,z) =

R(1)

ji+1(z0),zi+1(z0)

?

is adopted as a constitutive one of the hyperhoop only if ji+1is equal to

connected set of ten constitutive lines. Then we calculate Γ for each hyperhoop and search

for the minimal one which is specified by a set of ten integers {mi} in the manner of ji= mi.

Further for several values of z0, we carry out the same calculations as the above. We select

the values of z0at even intervals 0.1rsand search for the hyperhoop with the smallest value

of Γ. Finally, we obtain the minimal one which is specified by a set of ten integers and one

real number {mi,q}, where q is the value of z0. The above hyperhoop {mi,q} might not be

exactly minimal since the hyperhoops obtained by the above procedure are too restrictive.

Thus we might find hyperhoops smaller than the one {mi,q} in the following refinement.

We consider a neighbourhood R(1)

mi−1(q0) ≤ R ≤ R(1)

mi+1(q0) of the hyperhoop {mi,q0}, where

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q0is a value which is equal or near to q. In this region, we put further grid points at

R = R(2)

ki(q0) = R(1)

mi−1(q0) +

?q2

0− q2

50

i(q0)

(ki− 1), (ki= 1,2,··· ,11), (35)

where

qi(q0) =q0

10× (10 − i).(36)

Then by the same procedure as in the previous search for the minimal hyperhoop, we will

obtain the hyperhoop with Γ smaller than the previous one (see Fig. 6). Further for several

value of q0in the vicinity of q, we carry out the same calculations as the above. We select

the 11 values

q − 0.1rs+ 0.02rs(l − 1), (l = 1,2,··· ,11) (37)

as q0and search for the hyperhoop with the smallest value of Γ.

z/rs

R/rs

z0

z=z1

z=z2

z=zi

z0/10

(z0

2-zi

2)1/2/10

sphere

z/rs

R/rs

z0

z1

z2

zi

FIG. 5: The connecting points (left figure) and the hyperhoops which we calculate for a value of

z0(right figure) in first search.

V.NUMERICAL RESULTS

For various a and b, numerical results of the minimal value of Γ are listed in TABLE I.

Hereafter we denote the minimal value of Γ by Γminwhich is a function of a and b.

We see from TABLE I that Γminis not smaller than unity in the case of no apparent

horizon. This implies that the inequality (29) is really a sufficient condition in the situations

studied here.

Next, let us study the necessity of hyperhoop conjecture (29). By the investigation of

the singular line source of a “constant” line energy density studied in Ref. [7], we obtain a

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z/rs

R/rs

q0

z=q1

z=q2

z=qi

mi(q2-qi

2)1/2/10

R/rs

q0

z=q1

z=q2

z=qi

FIG. 6: The hyperhoop having the smallest value of Γ in first search (left figure) and the hyper-

hoops which we calculate for a value of q0(right figure) in second search .

TABLE I: The minimal values of Γ (Γmin) is shown for each a and b. The existence of an apparent

horizon is represented by (Yes):exist and (No):not exist.

b\a

1.0 1.05(Yes) 1.05(Yes) 1.05(Yes) 1.05(Yes) 1.04(Yes) 1.03(Yes) 1.01(Yes)

0 0.010.10.30.5 0.70.9

1.5 1.11(No) 1.11(Yes) 1.11(Yes) 1.10(Yes) 1.10(Yes) 1.09(No) 1.08(No)

2.0 1.17(No) 1.17(Yes) 1.17(Yes) 1.17(No) 1.16(No) 1.16(No) 1.17(No)

2.5 1.23(No) 1.23(Yes) 1.23(No) 1.23(No) 1.23(No) 1.24(No) 1.26(No)

3.0 1.29(No) 1.29(Yes) 1.29(No) 1.29(No) 1.30(No) 1.31(No) 1.35(No)

3.5 1.35(No) 1.34(No) 1.34(No) 1.34(No) 1.35(No) 1.38(No) 1.43(No)

4.0 1.40(No) 1.38(No) 1.38(No) 1.38(No) 1.40(No) 1.44(No) 1.51(No)

necessary condition as follows

V2?π

2× 16πG5M.(38)

The derivation of this quantity π/2 is shown in Appendix B. The counterexample for this

condition is not found in TABLE I. Here, it is again noted that a closed marginal surface

is formed only when b ≤ 1.48rs, and thus our result of a = 0 suggests the formation of

naked singularities in the case b ≥ 1.49rs. If it is true, the naked singularities might form

by the gravitational collapse starting from the initial data of nonvanishing but sufficiently

small a and b larger than 1.49rs. If a naked singularity exists, the apparent horizon does not

necessarily mean the existence of a black hole. Therefore, although the apparent horizon

Page 15

15

forms for a ∈ [0.01,0.5] and b = 1.5rs(see TABLE I), these results do not necessarily mean

the black hole formation.

Finally, in the present situation, we find the following inequalities,

Necessary condition : V2?π

216πG5M,(39)

Sufficient condition : V2? 16πG5M.(40)

Our numerical results suggest that the necessary condition is not identical to the sufficient

condition.

VI. SUMMARY

We have investigated the condition of apparent horizon formation in the case of mo-

mentarily static and conformally flat initial data of a spheroidal mass in the framework of

five-dimensional Einstein gravity. All our results are consistent with the hyperhoop con-

jecture. Particularly, we confirmed the sufficiency of the inequality (1). (More precisely,

inequality (29) holds in the present case.)

We also consider the limit of infinitely prolate spheroid. The gravitational field of such

spheroids are singular and have a nontrivial structure. The poles at the both ends of this

singular spheroid are the space-time singularities since the Kretschmann invariant diverges

at the poles of the spheroid. On the other hand, both the Kretschmann invariant and the

energy density are finite elsewhere.

We find that the singular spheroid is spacelike infinity except for the poles. Furthermore,

when the singular spheroid is sufficiently long, in an appropriate sense, no apparent horizon

appears. This property can be regarded as being peculiar to nonuniform distribution of

material energy, because a uniform line energy density is always enclosed by an apparent

horizon in spite of its length [7].

One might wonder if the singular spheroid is a counterexample to the hyperhoop con-

jecture, since it is infinitely thin and hence there seems to be a hyperhoop satisfying the

inequality (1). However as we have shown, the singular spheroid is not a counterexample to

the hyperhoop conjecture. In order to understand its reason, we have to note two important

features of the initial hypersurface studied here. First, the proper area V2of a hyperhoop

Page 16

16

tightly encircling the surface of the spheroid is not necessarily smaller than those encircling

outside of the spheroid, since the conformal factor in the inner region takes the value larger

than that in the outer region. Second, the proper area V2of a hyperhoop tightly encircling

the spheroid does not necessarily become smaller when the coordinate size of the spheroid

characterized by a and b becomes smaller, since the conformal factor in the spheroid of the

smaller size becomes larger if the mass M is fixed.

The difference between the present case and the previous work [7] is that the line energy

density vanishes continuously at the poles in the present case, while it vanishes discontinu-

ously at the poles in the previous case. In general, if an infinitely thin line object forms by

the gravitational collapse, it might have a line energy density which continuously vanishes

at the end of the matter distribution. Therefore, the naked singularity formation seems to

be generic in the axi-symmetric gravitational collapse of highly elongated matter distribu-

tion in five-dimensional space-time, although we would need numerical simulation to have

a definite evidence for the naked singularity formation [13]. This might strongly depend on

the spacetime dimension [14] and this is also a future work.

Acknowledgements

We are grateful to H. Ishihara and colleagues in the astrophysics and gravity group of

Osaka City University for helpful discussion and criticism. This work is supported by the

Grant-in-Aid for Scientific Research (No.16540264) from JSPS.

APPENDIX A: NEWTONIAN POTENTIALS OF A HOMOGENEOUS

ELLIPSOID IN D-DIMENSIONAL SPACE-TIME

In this section, we extend Newtonian potentials of a homogeneous ellipsoid in four-

dimensional space-time to (n+1)-dimensional space-time. The reader may refer to Ref. [15]

about the potentials of four-dimensional space-time.

We want to obtain the potentials of the homogeneous ellipsoid of which bounding ellipsoid

is

n

?

i=1

x2

a2

i

i

= 1.(A1)

At the beginning of derivation, we define the “homoeoid”.

Page 17

17

Definition.A n-dimensional homoeoid is a shell bounded by two similar concentric

n-dimensional ellipsoids in which strata of equal density are also n-dimensional ellipsoids

that are concentric with and similar to the bounding ellipsoids.

Following theorem and corollary is derived in the same way as the case of four-dimensional

space-time [15].

Theorem. The potential at internal point of a n-dimensional homoeoid is constant.

Corollary.The equipotential surfaces external to a thin n-dimensional homoeoid are

n-dimensional ellipsoids confocal to the homoeoid.

We can obtain the potential of a thin n-dimensional homoeoid expressed as

n

?

i=1

x2

a2

i

i

= 1, (a1< a2< ··· < an), (A2)

using n-dimensional ellipsoidal coordinates (y1,y2,··· ,yn) which satisfy

n

?

and

i=1

x2

i

a2

i− yj

= 1, (j = 1,2,··· ,n)(A3)

y1< y2< ··· < yn. (A4)

We can solve Eq. (A3) for x2

iand

x2

i=

?n

j=1(a2

k?=i(a2

i− yj)

i− a2

?

k).(A5)

Therefore

∂xi

∂yj

=

xi

2(yj− a2

i). (A6)

The metric is expressed as

n

?

i=1

dx2

i=1

4

n

?

i=1

?

k?=i(yi− yk)

?n

?n?n

j=1(a2

j− yi)dy2

i,(A7)

thus

detg =

?1

4

j=1

??

?

k?=j(yj− yk)

i,j(a2

?

i− yj)

. (A8)

Page 18

18

In order to obtain the potential ΦN of a thin n-dimensional homoeoid, we only have to

solve following equation because of the theorem and the corollary.

△ΦN(y1) = 0(A9)

with

ΦN→ −

M

(n − 2)rn−2

forr → ∞, (A10)

where M is constant which correspond to the mass of the thin n-dimensional homoeoid and

r is the length from the center of the homoeoid. It can be seen that −y1∼ r2at infinity.

Using ellipsoidal coordinates to Eq. (A9), we obtain

∂

∂y1

?

n?

i=1

?

a2

i− y1∂ΦN(y1)

∂y1

?

= 0. (A11)

The solution of Eq. (A11) with (A10) is

ΦN(u) = −n − 2

2

M

?∞

u

1

?n

i=1

?a2

i+ udu,

(A12)

where u = −y1.

Integrating this potentials of thin homoeoid which is foliated in all region of ellipsoid in

the same manner as four-dimensional space-time [15], we can finally obtain the Newtonian

potential of a D-dimensional homogeneous ellipsoid. The integration can be done as follows.

We can express the thin homoeoids which is similar and concentric to the ellipsoid (A1) as

n

?

i=1

x2

a2

i

i

= m2,(A13)

where m is constant and we assume 0 ≤ m ≤ 1. Let us consider the homogeneous homoeoid

bounded by two ellipsoids?n

deviation. The mass of this homoeoid is

i=1

x2

a2

i

i

= m2and?n

i=1

x2

a2

i

i

= (m + dm)2, where dm is small

ωna1a2···anρmn−1dm,(A14)

where ωnand ρ is n-dimensional solid angle and the density of the homoeoid respectively.

First, we derive the potential in outer region of homogeneous ellipsoid. Substituting

(A14) into M of (A12), we can obtain the potential of the homoeoidal element (A14) at

(x′

1,x′

2,··· ,x′

n)

−n − 2

2

ωna1a2···anρmn−1dm

?∞

λ(m2)

du′

?n

i=1

?a2

im2+ u′,(A15)

Page 19

19

where λ is the largest root of?n

we have

i=1

x′

i

2

i+λ= 1. Integrating this equation about 0 ≤ m ≤ 1,

m2a2

−n − 2

2

ωna1a2···anρ

?1

0

dmmn−1

?∞

?∞

λ(m2)

du′

?a2

du

?a2

?n

?n

i=1

i+ u′

= −n − 2

4

ωna1a2···anρ

?1

0

dm2

µ(m2)

i=1

i+ u,

(A16)

where u and µ defined by u′= m2u and λ(m2) = m2µ(m2) respectively. Now, we can invert

the order of integrations because µ(m2) is the monotone decreasing function of m2. Since

µ → ∞ when m → 0 and µ = λ when m = 1, we can write

ΦN= −n − 2

4

ωna1···anρ

?∞

λ(1)

(1 − m2(u))

?n

i=1

?a2

i+ udu

(A17)

in outer region, where

m2(u) =

n

?

i=1

x2

i+ u.

i

a2

(A18)

Next, we derive the potential in inner region of homogeneous ellipsoid at the point

(x′

1,x′

2,··· ,x′

n) which satisfy

?n

i=1

x′2

i

a2

i

= m′2, where m′is constant and 0 ≤ m′< 1.

On the one hand, the potential of homogeneous ellipsoid bounded by?n

(x′

−n − 2

4

i=1

x2

im′2 = 1 at

i

a2

1,x′

2,··· ,x′

n) is

ωna1···anρ

?∞

0

(m′2− m2(u))

?n

i=1

?a2

i+ udu,

(A19)

where we use (A17). On the other hand, the potential of homogeneous homoeoid bounded

by?n

from m = m′to m = 1, and we have

i=1

x2

a2

i

i= m′2and?n

i=1

x2

a2

i

i= 1 at (x′

1,x′

2,··· ,x′

n) is obtained by integration of (A15)

−n − 2

4

ωna1a2···anρ(1 − m′2)

?∞

0

du

?a2

?n

i=1

i+ u,

(A20)

Adding together (A20) and (A19), we can obtain the required potential

ΦN= −n − 2

4

ωna1···anρ

?∞

0

(1 − m2(u))

?n

i=1

?a2

i+ udu

(A21)

in inner region.

If we have same radiuses to the directions corresponding to coordinates xi,xi+1,··· in

(A1), we can introduce multipolar coordinates to these and calculate as same.

Page 20

20

APPENDIX B: THE NECESSARY CONDITION OF BLACK HOLE

FORMATION IN FIVE-DIMENSIONAL SPACE-TIME

We consider the singular line source studied in Ref. [7]. The energy density is given by

f3ρ =

1

4πR2

G5M

2b

δ(R)θ(b − |z|) (B1)

where θ is the Heaviside’s step function and the “length” of this line source is given by 2b.

In this case, the solution of the Hamiltonian constraint (9) is given by

f = 1 +G5M

2bR

?

arctanz + b

R

− arctanz − b

R

?

.(B2)

As shown in Ref. [7], this line source is always covered by an apparent horizon.

In order to obtain the necessary condition of an apparent horizon formation, we have

to calculate the values of Γmindefined in Sec. V. Here, we consider the hyperhoops which

are expressed in the form r = rh(ξ) and ϑ = π/2 with the same symmetry as discussed in

Sec. IV (axi-symmetry and reflection symmetry with respect to z = 0).

In this case, the hyperhoop which intersects the line source has infinite two-dimensional

area V2. In order to see this, consider a hyperhoop expressed as ϑ = π/2 and z = η(R),

where |η(0)| < b so that the hyperhoop intersects the line source. We focus on a segment

R1< R < R2and z > 0 of this hyperhoop. We assume that R1and R2are sufficiently small

and 0 < η(R) < b on this segment. The conformal factor is approximately given by

f ≃π (2η − b)G5M

4bRη

(B3)

on this segment of the hyperhoop. Hence, the area of this segment is bounded below as

?R2

R1

?2π

?R2

5M2

8b2

0

f(R,η)2R

?

1 +

?dη

dR

?2

5M2

8b2

dϕdR

≥ 2π

≥π3G2

R1

f(R,η)2RdR ≃π3G2

?R2

?R2

R1

?

2 −b

η

?2dR

R

R1

dR

R

=π3G2

5M2

8b2

lnR2

R1.(B4)

We can see from the above equation that the area of the hyperhoop diverges in the limit

of R1→ 0 with R2fixed. Therefore we have to consider only the hyperhoop which entirely

encircle the line source, and hence we find

Γmin=

?

V2

16πG5Min

?

min

=

(V2)min

16πG5M,

(B5)

Page 21

21

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

?˙ ra

ra

− 2ra+ra2+ ˙ ra2

racotξ −2

ra

×

f(˙ rasinξ + racosξ)∂f

∂z+2

f(˙ racosξ − rasinξ)∂f

∂R

?

= 0.(B7)

We impose following boundary conditions so that every part of the hyperhoop locally satisfy

Eq. (B7)

˙ ra= 0at ξ = 0,

π

2. (B8)

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

f3ρ =

1

4πR2

G5M

2b

δ(R). (B9)

In this case, we can easily solve the Hamiltonian constraint (9) and obtain

f = 1 +πG5M

2bR

. (B10)

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)

Page 22

22

0

0.5

1

1.5

2

2.5

3

02468 10

Γmin

b/rs

length=2b

infinitely long

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

M/2b.

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

π/2.

APPENDIX C: THE DERIVATION OF THE EQUATION FOR A MARGINAL

SURFACE

In this section, we show the derivation of Eq.(26). Here, we generalize Ref.[12] to the

D-dimensional case.

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)

Page 23

23

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

∂xµ

∂ξ

∂xµ

∂ϑ

∂xµ

∂ϕ

= (˙ rmcosξ − rsinξ, ˙ rmsinξ + rmcosξ,0,0), (C5)

= (0,0,1,0), (C6)

= (0,0,0,1).(C7)

Hence, we can obtain the components of saand (eA)aas

sµ=

1

f?r2

f?r2

fR(0,0,1,0),

1

fRsinϑ(0,0,0,1).

m+ ˙ r2

1

m

(˙ rmsinξ + rmcosξ,−˙ rmcosξ + rmsinξ,0,0),(C8)

(e1)µ=

m+ ˙ r2

m

(˙ rmcosξ − rmsinξ, ˙ rmsinξ + rmcosξ,0,0),(C9)

(e2)µ=

1

(C10)

(e3)µ=

(C11)

Substituting Eqs.(C8)−(C11) into Eq.(C3), we obtain Eq.(26).

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