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arXiv:gr-qc/0602109v2 3 May 2006

Final fate of spherically symmetric gravitational collapse of a dust cloud in

Einstein-Gauss-Bonnet gravity

Hideki Maeda

∗

Advanced Research Institute for Science and Engineering,

Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan

(Dated: February 7, 2008)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust

cloud including the perturbative effects of quantum gravity. The n(≥ 5)-dimensional action with

the Gauss-Bonnet term for gravity is considered and a simple formulation of the basic equations is

given for the spacetime M ≈ M2× Kn−2with a perfect fluid and a cosmological constant. This

is a generalization of the Misner-Sharp formalism of the four-dimensional spherically symmetric

spacetime with a perfect fluid in general relativity. The whole picture and the final fate of the

gravitational collapse of a dust cloud differ greatly between the cases with n = 5 and n ≥ 6. There

are two families of solutions, which we call plus-branch and the minus-branch solutions. A plus-

branch solution can be attached to the outside vacuum region which is asymptotically anti-de Sitter

in spite of the absence of a cosmological constant. Bounce inevitably occurs in the plus-branch

solution for n ≥ 6, and consequently singularities cannot be formed. Since there is no trapped

surface in the plus-branch solution, the singularity formed in the case of n = 5 must be naked. On

the other hand, a minus-branch solution can be attached to the outside asymptotically flat vacuum

region. We show that naked singularities are massless for n ≥ 6, while massive naked singularities

are possible for n = 5. In the homogeneous collapse represented by the flat Friedmann-Robertson-

Walker solution, the singularity formed is spacelike for n ≥ 6, while it is ingoing-null for n = 5. In

the inhomogeneous collapse with smooth initial data, the strong cosmic censorship hypothesis holds

for n ≥ 10 and for n = 9 depending on the parameters in the initial data, while a naked singularity

is always formed for 5 ≤ n ≤ 8. These naked singularities can be globally naked when the initial

surface radius of the dust cloud is fine-tuned, and then the weak cosmic censorship hypothesis is

violated.

PACS numbers: 04.20.Dw, 04.40.Nr, 04.50.+h

I. INTRODUCTION

Einstein’s general theory of relativity has successfully passed many observational tests and is now a central paradigm

in gravitation physics. General relativity explains such gravitational phenomena as the perihelion shift of Mercury’s

orbit, gravitational lensing, redshift in the light spectrum from extragalactic objects, and so on. One of the most

intriguing predictions of the theory is the existence of a spacetime region from which nothing can escape, i.e., a black

hole.

It has been considered that black holes are formed from the gravitational collapse in the last stage of heavy stars’

life or in high-density regions of the density perturbations in the early universe. The first analytic model of black-hole

formation in general relativity was obtained by Oppenheimer and Snyder in 1939, which represents the spherically

symmetric gravitational collapse of a homogeneous dust cloud in asymptotically flat vacuum spacetime [1]. In this

spacetime, the singularity formed is spacelike and hidden inside the black-hole event horizon, so that it is not visible

to any observer. However, it was shown later that this is not a typical model and the singularities formed in generic

collapse are naked, i.e., observable [2, 3, 4, 5].

In general relativity, it was proven that spacetime singularities inevitably appear in general situations and under

physical energy conditions [6]. Gravitational collapse is one of the presumable scenarios in which singularities are

formed. Where a naked singularity exists, the spacetime is not globally hyperbolic, so that the future predictability

of the spacetime breaks down. In this context, Penrose proposed the cosmic censorship hypothesis (CCH), which

prohibits the formation of naked singularities in gravitational collapse of physically reasonable matters with generic

regular initial data [7, 8]. The weak version of CCH prohibits only the formation of globally naked singularities,

i.e., those which can be seen by an observer at infinity. If the weak CCH is correct, singularities formed in generic

gravitational collapse are hidden inside black holes, and the future predictability of the spacetime outside the black-

hole event horizon is guaranteed. On the other hand, the strong version of CCH prohibits the formation of locally

∗Electronic address: hideki@gravity.phys.waseda.ac.jp

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naked singularities also, which can be seen by some local observer. The strong CCH asserts the future predictability

of the whole spacetime, i.e., global hyperbolicity of the spacetime.

The CCH is one of the most attractive and important unsolved problems in gravitation physics. Validity of the CCH

is assumed in the many strong theorems such as the black-hole uniqueness theorem or the positive energy theorem

in asymptotically flat spacetime. At present, however, the general proof of the CCH is far from complete. On the

contrary, there are many counterexample “candidates” in general relativity. (See [9] for a review.)

The formation of a singularity means that a spacetime region with infinitely high curvature can be realized in the

very final stage of gravitational collapse. It is naturally considered that quantum effects of gravity cannot be neglected

in such regions, so that the classical theory of gravity cannot be applied there. Therefore, naked singularities give us

a chance to observe the quantum effects of gravity. From this point of view, Harada and Nakao proposed a concept

named the spacetime border, which is the spacetime region where classical theories of gravity cannot be applied [10].

The spacetime border is an effective “singularity” in classical theory, and then the CCH can be naturally modified

to more a practical version, which prohibits the formation of naked spacetime borders. If the modified CCH is true,

spacetime regions where quantum effects of gravity dominate are never observed. On the other hand, if it is violated,

there is a possibility in principle for us to observe such regions and obtain information useful to the construction of the

quantum theory of gravity, which is still incomplete. From this point of view, studies of the final fate of gravitational

collapse are quite important.

Up to now, many quantum theories of gravity have been proposed. Among them, superstring/M-theory is the most

promising candidate and has been intensively investigated, which predicts higher-dimensional spacetime (more than

four dimensions). In this theory, when the curvature radius of the central high-density region in gravitational collapse

is comparable with the compactification radius of extra dimensions, the effects of extra dimensions will be important.

Such regions can be modeled effectively by higher-dimensional gravitational collapse.

A recent attractive proposal for a new picture of our universe, which is called the braneworld universe [11, 12, 13], is

based on superstring/M-theory [14]. In the braneworld universe, we live on a four-dimensional timelike hypersurface

embedded in the higher-dimensional bulk spacetime. Because the fundamental scale could be around the TeV scale

in this scenario, the braneworld suggests that the creation of tiny black holes in the upcoming high-energy collider

is possible [15]. From this point of view, the effects of superstring/M-theory on black holes or gravitational collapse

should be investigated.

However, the non-perturbative aspects of superstring/M-theory are not understood completely so far, although the

progress in recent years has been remarkable. Given the present circumstances, taking their effects perturbatively

into classical gravity is one possible approach to studying the quantum effects of gravity. The Gauss-Bonnet term in

the Lagrangian is the higher curvature correction to general relativity and naturally arises as the next leading order

of the α′-expansion of heterotic superstring theory, where α′is the inverse string tension [16]. Such a theory is called

the Einstein-Gauss-Bonnet gravity.

In a previous paper, the author presented a model of the n(≥ 5)-dimensional spherically symmetric gravitational

collapse of a null dust fluid in Einstein-Gauss-Bonnet gravity [17]. It was shown that the spacetime structure of the

gravitational collapse differs greatly between n = 5 and n ≥ 6. In five dimensions, massive timelike naked singularities

can be formed, which never appear in the general relativistic case, while massless ingoing-null naked singularities are

formed in the n(≥ 6)-dimensional case.

In this paper, we consider the n(≥ 5)-dimensional spherically symmetric gravitational collapse of a dust fluid with

smooth initial data in Einstein-Gauss-Bonnet gravity. In general relativity, the same system has been analyzed by

many researchers both for n = 4 [2, 3, 4, 5] and for n ≥ 5 [18]. They showed that the singularity formed is censored

for n(≥ 6), while it is naked for n = 4. For n = 5, the singularity can be censored depending on the parameters in

the initial data.

This paper is organized as follows. In Sec. II, for the n(≥ 5)-dimensional spacetime M ≈ M2× Kn−2with

a perfect fluid and a cosmological constant, where Kn−2is the (n − 2)-dimensional Einstein space, we define a

scalar on M2, of which dimension is mass, and give a simple formulation of the basic equations in Einstein-Gauss-

Bonnet gravity. In Sec. III, using this formalism, we investigate the final fate of the n(≥ 5)-dimensional spherically

symmetric gravitational collapse of a dust cloud without a cosmological constant. Section V is devoted to discussion

and conclusions. In Appendix A, we review the study of the general relativistic case for comparison and give some

complements. Throughout this paper we use units such that c = 1. As for notation we follow [19]. The Greek indices

run µ = 0,1,··· ,n − 1.

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II.MODEL AND BASIC EQUATIONS

We begin with the following n-dimensional (n ≥ 5) action:

S =

?

dnx√−g

?

1

2κ2

n

(R − 2Λ + αLGB)

?

+ Smatter,(2.1)

where R and Λ are the n-dimensional Ricci scalar and the cosmological constant, respectively. κn≡√8πGn, where

Gnis the n-dimensional gravitational constant. The Gauss-Bonnet term LGBis the combination of the Ricci scalar,

Ricci tensor Rµν, and Riemann tensor Rµνρσas

LGB= R2− 4RµνRµν+ RµνρσRµνρσ. (2.2)

α is the coupling constant of the Gauss-Bonnet term. This type of action is derived in the low-energy limit of heterotic

superstring theory [16]. In that case, α is regarded as the inverse string tension and positive definite, and thus we

assume α > 0 in this paper. We consider a perfect fluid as a matter field, whose action is represented by Smatterin

Eq. (2.1). We do not consider the case with n ≤ 4, in which the Gauss-Bonnet term does not contribute to the field

equations.

The gravitational equation of the action (2.1) is

Gµ

ν+ αHµ

ν+ Λδµ

ν= κ2

nTµ

ν, (2.3)

where

Gµν ≡ Rµν−1

2gµνR,(2.4)

Hµν ≡ 2

?

RRµν− 2RµαRα

ν− 2RαβRµανβ+ Rαβγ

µ

Rναβγ

?

−1

2gµνLGB. (2.5)

The energy-momentum tensor of a perfect fluid is

Tµν= (p + ρ)uµuν+ pgµν, (2.6)

where uµ, ρ and p are the n-velocity of the fluid element, energy density, and pressure, respectively.

Lemma 1 If p = −ρ, then ρ is constant.

Proof. The energy-momentum conservation equation Tν

µ ;ν= 0 becomes

ρ;νuν+ (ρ + p)uµ

(ρ + p)uµ;νuν= −p,νhν

;µ= 0, (2.7)

(2.8)

µ,

where hµν≡ gµν+ uµuνis the projection tensor. From Eqs. (2.7) and (2.8), ρ is constant if p = −ρ. 2

By Lemma 1, a perfect fluid obeying p = −ρ is equivalent to a cosmological constant. We assume p ?= −ρ in this

paper.

We consider the n-dimensional spacetime as a product manifold M ≈ M2× Kn−2, where Kn−2is the (n − 2)-

dimensional Einstein space, with the line element

ds2= −e2Φ(t,r)dt2+ e2Ψ(t,r)dr2+ S(t,r)2γijdxidxj, (2.9)

where γijis the unit curvature metric on Kn−2. Hereafter, a dot and a prime denote the differentiation with respect

to t and r, respectively. k denotes the curvature of Kn−2and takes 1 (positive curvature), 0 (zero curvature), and

−1 (negative curvature). We adopt the comoving coordinates such that the n-velocity of the fluid element is

uµ∂

∂xµ= e−Φ∂

∂t.

(2.10)

The following Lemma is necessary to give our formalism of the basic equations.

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Lemma 2 If p ?= −ρ, then S2+ 2(n − 3)(n − 4)α(k + e−2Φ˙S2− e−2ΨS′2) cannot be zero.

Proof. If the relation

S2+ 2(n − 3)(n − 4)α(k + e−2Φ˙S2− e−2ΨS′2) = 0 (2.11)

is satisfied at a moment, then the (t,t) and (r,r) components of the field equation (2.3) give

(n − 1)(n − 2)

8α(n − 3)(n − 4)+ Λ = −κ2

nρ (2.12)

and

(n − 1)(n − 2)

8α(n − 3)(n − 4)+ Λ = κ2

np, (2.13)

respectively. Eqs. (2.12) and (2.13) give a contradiction p = −ρ. 2

Here we give a definition of a scalar on M2with the dimension of mass such that

m ≡

(n − 2)Vk

2κ2

n−2

n

[−˜ΛSn−1+ Sn−3(k − S,µS,µ) + ˜ αSn−5(k − S,µS,µ)2],(2.14)

where ˜ α ≡ (n − 3)(n − 4)α,˜Λ ≡ 2Λ/[(n − 1)(n − 2)], and a comma denotes the partial differentiation. A constant

Vk

2π(n−1)/2/Γ((n−1)/2) is the surface area of the (n−2)-dimensional unit sphere, where Γ(x) is the gamma function.

In the four-dimensional spherically symmetric case without a cosmological constant, m is reduced to the Misner-Sharp

mass [20].

Then, the field equations are written in the following simple form:

n−2is the surface area of the (n − 2)-dimensional unit Einstein space if it is compact. For example, V1

n−2≡

p′= −(ρ + p)Φ′, (2.15)

˙ ρ = −(ρ + p)

?

˙Ψ + (n − 2)

˙S

S

?

,(2.16)

m′= Vk

˙ m = −Vk

0 = −˙S′+ Φ′˙S +˙ΨS′,

(n − 2)Vk

2κ2

n−2ρS′Sn−2,

n−2p˙SSn−2,

(2.17)

(2.18)

(2.19)

m =

n−2

n

[−˜ΛSn−1+ Sn−3(k + e−2Φ˙S2− e−2ΨS′2) + ˜ αSn−5(k + e−2Φ˙S2− e−2ΨS′2)2].(2.20)

The first two equations are the energy-momentum conservation equations.

component of Eq. (2.3) with Lemma 2. Eq. (2.20) is obtained from Eq. (2.14). Eqs. (2.17) and (2.18) are obtained

from the (t,t) and (r,r) components of Eq. (2.3) by using Eqs. (2.19) and (2.20). Five of the above six equations are

independent.

Eq. (2.19) is derived from the (t,r)

III. SPHERICALLY SYMMETRIC DUST CLOUD

After this, we only consider the spherically symmetric collapse of a dust fluid without a cosmological constant, i.e.,

p = 0 and Λ = 0, for simplicity. We assume the positive energy density, i.e., ρ > 0. Then Eq. (2.15) implies that

Φ = Φ(t), so that we can set Φ = 0 by redefinition of our time coordinate without a loss of generality. Throughout

this paper, we call the direction of increasing (decreasing) t future (past). Eq. (2.18) implies that m = m(r), which is

an arbitrary function. m is naturally interpreted as the mass inside the comoving radius r because Eq. (2.17) implies

m =

?r

V1

n−2ρSn−2∂S

∂rdr

(3.1)

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on the hypersurface with a constant t. From Eq. (2.19), we obtain

e2Ψ=

S′2

1 + f(r), (3.2)

where f(r) is an arbitrary function satisfying f > −1. Eq. (2.17) gives

ρ =

m′

V1

n−2S′Sn−2. (3.3)

From this equation, we find that there may exist both shell-crossing singularities, where S′= 0, and shell-focusing

singularities, where S = 0. Hereafter we assume S′> 0, or equivalently m′> 0, in order that shell-crossingsingularities

may be removed from our consideration.

From Eq. (2.20), we obtain a master equation of the system:

˙S2= f −S2

2˜ α

?

1 ∓

?

1 +

8˜ ακ2

nm

n−2Sn−1

(n − 2)V1

?

. (3.4)

For later convenience, we call the solution of Eq. (3.4) the Gauss-Bonnet-Lemaˆitre-Tolman-Bondi (GB-LTB) solution

although the explicit form of the solution is not obtained in this paper. There are two families of solutions which

correspond to the sign in front of the square root in Eq. (3.4). We call the family with the minus (plus) sign the

minus-branch (plus-branch) solution. In the general relativistic limit ˜ α → 0, the minus-branch solution is reduced

to the n-dimensional Lemaˆitre-Tolman-Bondi solution. The case with n = 4 has been intensively studied by many

authors [4, 5, 21], while the case with n ≥ 5 has also been studied by several authors [18]. On the contrary, there is

no general relativistic limit for the plus-branch solution.

We can consider that the GB-LTB solution is attached at a finite constant comoving radius r = r0> 0, where we

represent this timelike hypersurface as Σ, to the outside vacuum region. The outside vacuum region is represented by

the solution independently discovered by Boulware and Deser [22] and Wheeler [23], whose metric is

ds2= −F(S)dT2+

dS2

F(S)+ S2dΩ2

n−2, (3.5)

where dΩ2

n−2is the line element of the (n − 2)-dimensional unit sphere. F(S) is defined by

F(S) ≡ 1 +S2

2˜ α

?

1 ∓

?

1 +

8˜ ακ2

nM

n−2Sn−1

(n − 2)V1

?

, (3.6)

where M is a constant. We call this solution the GB-Schwarzschild solution. There are also two branches of the

solution which correspond to the sign in front of the square root in Eq. (3.6). The global structures of the GB-

Schwarzschild solution are presented in [24]. The minus-branch GB-Schwarzschild solution is asymptotically flat,

while the plus-branch solution is asymptotically anti-de Sitter in spite of the absence of a cosmological constant. The

constant M in the minus-branch GB-Schwarzschild solution is the higher-dimensional ADM mass [25].

The position of Σ and the relation between T and t on Σ are represented by S = SΣ(t) and T = TΣ(t), respectively.

Now we derive the governing equations for TΣand SΣ[26].

As seen from inside of Σ, the metric on Σ is obtained by

ds2

Σ= −dt2+ S2

ΣdΩ2

n−2. (3.7)

As seen from outside of Σ, on the other hand, it is

ds2

Σ= −

?

F(SΣ)˙T2

Σ−

˙S2

Σ

F(SΣ)

?

dt2+ S2

ΣdΩ2

n−2. (3.8)

Because the induced metric must be the same on both sides of the hypersurface Σ, we have

1 = F(SΣ)˙T2

Σ−

˙S2

Σ

F(SΣ).(3.9)