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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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where the function χ2(v) is now obtained by
χ2(v) ≡ −
?1
v
?b0+ M0v3−n?−3/2?b2+ M2v3−n?dv,(A8)
where the integrand is finite for 0 ≤ v ≤ 1. The time when the central shell reaches the singularity is given from
Eq. (A4) by
ts0=
?1
0
dv
(b0+ M0v3−n)1/2. (A9)
From Eqs. (A7) and (A9), the time when other shells close to r = 0 reach the singularity is given by
ts(r) = ts0+r2
2χ2(0) + ··· .(A10)
By Lemma 8, χ2(0) is non-negative and we assume χ2(0) > 0 here.
From Eqs. (A4), (A7) and (4.46), we obtain
v′=?b0+ M0v3−n?1/2(rχ2(v) + ···) (A11)
near r = 0.
Theorem 7 Let us consider the gravitational collapse of a spherical dust cloud with positive mass and smooth initial
profiles satisfying χ2(0) > 0 in general relativity without a cosmological constant. Then, the strong cosmic censorship
hypothesis holds for n ≥ 6 and n = 5 if 0 < χ2(0) < M1/2
Proof. From Eq. (A1), only the singularity at r = 0 has the possibility of being naked. A decreasing apparent horizon
in the (r,t)-plane is a sufficient condition for the formed singularity to be censored because it shows the entrapment
of the neighborhood of the center before the singularity. Expanding the integrand in Eq. (A6) in a power series in v
and keeping only the leading order term, we obtain
0 .
tah(r) = ts0+r2
2χ2(0) + ···
n − 1M1/(n−3)
−
2
0
r(n−1)/(n−3). (A12)
For n ≥ 6, the last term dominates the second term, and consequently the apparent horizon is decreasing near r = 0
in the (r,t)-plane. For n = 5 with χ2(0) < M1/2
2
0 , the apparent horizon is also decreasing near r = 0 in the (r,t)-plane.
Next we consider the nakedness of singularities for n = 4 and 5. We show the nakedness of singularities in the
similar manner to the case in Einstein-Gauss-Bonnet gravity.
Theorem 8 Let us consider the n(≥ 4)-dimensional gravitational collapse of a spherical dust cloud with positive mass
and smooth initial profiles satisfying χ2(0) > 0 in general relativity without a cosmological constant. Then, the strong
cosmic censorship hypothesis is violated for n = 4.
Proof. Using Eqs. (A3) and (A11), we obtain from Eq. (4.62) the desired root equation
x0= lim
r→0
1
qrq−1
?
v + M1/2
0 v(3−n)/2r2χ2(0)
??
1 − rM1/2
0
v(3−n)/2?????
v=x0rq−1. (A13)
For n = 4, the positive finite root of Eq. (A13) is obtained with q = (n + 3)/(n − 1) = 7/3 as
x0=
?
3M1/2
0χ2(0)
4
?2/3
. (A14)
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For n = 5, it is obtained with q = (n + 3)/(n − 1) = 2 for
χ2(0) ≥11 + 5√5
2
M1/2
0
(A15)
by solving the algebraic equation
x3
0+ M1/2
0 x2
0− χ2(0)M1/2
0 x0+ χ2(0)M0= 0. (A16)
Thus, we have found the possible behaviors of a future-directed outgoing radial null geodesic near r = 0 as
S ≃ x0r(n+3)/(n−1), (A17)
where x0is given by Eqs. (A14) and (A16) for n = 4 and n = 5, respectively.
By Lemma 10, we prove the existence of such a null geodesic. We consider the null geodesic equation (4.50) with
q = (n + 3)/(n − 1). By Eqs. (A3) and (A11), we expand Ξ around r = 0 as
1
r
+o(r2(3−n)/(n−1)).
Ξ(x,r) =
?
x + M1/2
0 x(3−n)/2χ2(0)
?
−M1/2
0 x(3−n)/2r2(3−n)/(n−1)
?
x + M1/2
0 x(3−n)/2χ2(0)
?
(A18)
We note that 2(3−n)/(n−1) ≥ −1 with equality holding for n = 5. Then, by Lemma 10, we can show the existence
of a null geodesic which behaves as Eq. (A17) around r = 0 for n = 4, where x0 is given by Eq. (A14). However,
as the case of n = 9 in Einstein-Gauss-Bonnet gravity, we cannot apply Lemma 10 to the case of n = 5 in general
relativity, so another method is required to show the existence in that case.
Finally we show that the curvature scalars actually diverge along the null geodesic. If ρ diverges, the curvature
scalars also diverge because Eq. (2.3) gives
ρ =(n − 2)R
2κ2
n
(A19)
in general relativity. The shell-focusing singularity is characterized by v = 0, r = 0, M = M0, and M′= 0 for n ≥ 4.
From Eq. (A11), we obtain
v′≃ M1/2
0 x(3−n)/2
0
r(5−n)/(n−1)χ2(0) (A20)
along the null geodesic (A17), so that rv′→ 0 for r → 0. Therefore, ρ and the curvature scalars diverge along the
null geodesic (A17). 2
Unfortunately, the fixed-point method cannot be applied to show the existence of the singular null geodesics for
n = 5. If they exist, they behave as Eq. (A17) around r = 0, where x0is given by Eq. (A16).
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