Violation of cosmic censorship in the gravitational collapse of a dust cloud in five dimension

We analyze the null geodesic equations in five dimensional spherically symmetric spacetime with collapsing inhomogeneous dust cloud. By using a new method, we prove the existence and non-existence of solutions to null geodesic equation emanating from central singularity for smooth initial distribution of dust. Moreover, we also show that the null geodesics can extend to null infinity in a certain case, which imply the violation of cosmic censorship conjecture.


Introduction
Black hole spacetime is one of the most fascinating objects in gravitational theory. In particular, it is quite interesting that black holes can contain singularities inside their event horizons. At singularities the spacetime curvature often diverges, and the physics breaks down because gravitational theory is described in terms of curvature.
Observers outside black holes cannot see such breakdown because no information can come out of black holes, at least at the classical level. However, black hole singularities may cause serious effects on an observer inside a black hole and/or the final fate of black hole evapolation due to Hawking radiation. Here we define a singularity that is visible for an observer inside a black hole as a "locally naked singularity." If a singularity is not wrapped by the horizon, it may cause serious effects on the physics. We define this as a "globally naked singularity." In order to certify the predictability of physics, it is very important to ask whether they can be naked or not. It is usually supposed that no naked singularity will appear in a physical situation. In this context, Penrose proposed the so-called cosmic censorship conjecture (CCC) [1]. More precisely, there are two types of CCC, strong CCC and weak CCC. Strong CCC states that no locally naked singularities form during gravitational collapse; weak CCC states that no globally naked singularities form during collapse.
The cosmic censorship conjecture is assumed in proving several key theorems on black hole spacetime, such as the event horizon topology theorem, uniqueness theorem, and so on (see [2] for the details). The conjecture has been investigated by many authors in a variety of setups, but it is still controversial. In [3], Oppenheimer and Snyder considered a spherical collapse of PTEP 2016, 103E01 R. Mizuno et al. homogeneous pressureless fluid, which is called dust; they found that no naked singularities form. But many subsequent works, for example [4][5][6][7][8][9][10][11], reported the violation of strong CCC in several situations. In [12], Christodoulou examined the global nakedness of singularities in fourdimensional spacetime. He considered the spherical collapse of inhomogeneous dust and proved that a singularity can be globally naked in some situations, i.e. violation of weak CCC occurs in general.
Motivated by fundamental theories such as string theory, those works have been extended to higher-dimensional spacetime [13][14][15][16]. It was shown that strong CCC always holds for the spherical collapse of a dust cloud with smooth initial data in spacetimes with dimensions higher than five, i.e. any observer cannot see singularities formed. One simple explanation for this fact is that, in higher dimensions, gravity near the singularity is stronger than the lower-dimensional case and the event horizon appears earlier. On the other hand, in general, it was also shown that the strong CCC does not hold in five-dimensional spacetimes.
As far as we know, there has been no work on a global analysis of collapsing spacetime in five dimensions. It is natural to ask whether the weak CCC is actually violated in five dimensions. And, if violated, it is important to clarify in what conditions naked singularities form. In this paper we focus on analysis of five-dimensional inhomogeneous spherically symmetric dust collapse and give a new method to examine the nakedness of a singularity. To do so we have to know whether a causal geodesic emanating from the singularity exists or not. So, we give a method to investigate the existence of a solution of the null geodesic equation. Furthermore, we examine the spacetime structure in detail, and give a necessary and sufficient condition for naked singularity formation. We also examine the condition that a singularity is not only locally naked but also globally naked. Finally, we will see the dependence of the global nakedness of a singularity on the initial density distribution.
The organization of this paper is as follows. In Sect. 2, we present the setting and the fundamental nature of inhomogeneous spherically symmetric dust collapse in five-dimensional spacetime. Then, we derive the differential equation for the null geodesic in terms of dimensionless quantities. In Sect. 3, we examine the existence condition for a solution to the differential equation for the null geodesic. By virtue of the Schauder fixed-point theorem [17], we show that a solution of the differential equation for the null geodesic exists near the singularity. This means that the singularity can be at least locally naked. In Sect. 4, we analyze the spacetime structure around the singularity. We identify the earliest null geodesic emanating from the central singularity, and give a necessary and sufficient condition for the singularity to be naked. In Sect. 5, we consider the global nakedness of the singularity. We will show that a class of initial density distributions leads to a globally naked singularity.

Five-dimensional Lemaître-Tolman-Bondi spacetime and the equation of the null line
We consider spherically symmetric dust collapse in five dimensions. This is known as the Lemaître-Tolman-Bondi (LTB) solution in higher dimensions. In the comoving coordinate of the dust, the metric of this spacetime is written as ds 2 = −dt 2 + e 2ω(t,r) dr 2 + R(t, r) 2 d 2 , where R is the area radius of the r = const. three-sphere and we set r by R(0, r) = r. Then, in this coordinate, the Einstein equations become where ρ(t, r) is the energy density of dust, which is proportional to the Ricci scalar, M (r) is an arbitrary C 1 function of r, and E(r) is an arbitrary function of r. Dot "˙" and prime " " mean partial derivatives with respect to t and r, respectively. M (r) corresponds to the total mass in the region surrounded by the r = const. surface. Actually, M (r) is proportional to the Misner-Sharp quasi-local mass [18], and E(r) is the initial energy of the dust shell. As mentioned in the introduction, the point where spacetime curvature diverges is a singularity. From Eq. (2), we can find the two types of singularities, i.e. R = 0 and R = 0. A singularity at R = 0 (R = 0) is called a shell focusing singularity (a shell crossing singularity). If we introduce pressure to the fluid, the shell crossing singularity may disappear, so they are regarded as unphysical singularities. Throughout this paper, we only consider the shell focusing singularity, that is, we assume R > 0. In addition, we also assume that the initial velocity of the shells at t = 0 is zero, that is,Ṙ Now we can solve Eq. (3) as From this equation and (2), we see that the singularity appears at Using Eqs. (1), (3), and (4), we can compute the expansion of outgoing null geodesics on the r = const. surface (r) as In the second equality we used the equation −dt 2 + e 2ω dr 2 = 0, which holds for null geodesics along the outer radial direction. This equation implies that the apparent horizon ( = 0) is located at R = √ M (r). Since we are interested in naked singularity formation during the dust collapse, we 3 assume that there is no apparent horizon initially. Then, Eq. (8) with the setting of R(0, r) = r tells us that is required for all r in this coordinate patch. From Eqs. (1), (4), and (5), this is equivalent to the condition that r is a spacelike coordinate. In addition, from Eq. (6), we also see that the apparent horizon appears at From Eqs. (7) and (10), if r = 0, it is easy to see that holds. This means that the apparent horizon appears before the singularity does, and the singularity is surrounded by the apparent horizon. So only the singularity at r = 0 could be naked, and another singularities are covered by the event horizon [2]. In order to examine the possibility of the occurrence of a naked singularity, we assume that the singularity at r = 0 appears within non-zero finite time t S (0), that is, Therefore, using a C 0 function A(r) with A(0) = 0, we can write M (r) as In the above, A(r) corresponds to the mean density for the region surrounded by the r = const. shell. From Eqs. (2) and (13), we have Moreover, we assume that the initial density ρ(0, r) is a C ∞ function of compact support which monotonically decreases with respect to r, and ρ (0, r) is continuous at r = 0 on the initial time slice [ρ (0, r) ≤ 0, ρ (0, 0) = 0]. Then, we see that A(r) satisfies 1 near r = 0, and where α, β ∈ R are parameters satisfying α > 0, β ≥ 0. Using A(r), t S (r) and t AH (r) are rewritten as If β = 0 in Eq. (15), we can immediately show that the central singularity must be covered by the apparent horizon. THEOREM 1 If β = 0, the strong cosmic censorship holds. Then, Thus, there exists r 0 ∈ R such that holds for arbitrary r ∈ [0, r 0 ]. Therefore, there exists an apparent horizon around the past of the central singularity, so null geodesics cannot emanate from the singularity and the strong cosmic censorship holds.
This fact is already known in Refs. [13][14][15]. Accordingly, in order to figure out the condition for naked singularity formation, we suppose β > 0 in the following discussion. Let us consider the future-directed null geodesics along the outer radial direction. Because of the spherical symmetry, the differential equation for future-directed null geodesics along the outer radial direction is given by We used Eq. (1) in the first line, Eqs. (4), (5), and (13) in the second line, and the explicit expression of R derived from Eq. (6) in the third. Now, we introduce the dimensionless functions and parameters to write Eq. (22) In the above, m and l are dimensionless parameters that are related to the total mass of the dust and the dust cloud radius, respectively. These conditions imply that a(x) is written as Using a(x), we write A(r) as We can easily show that this A(r) satisfies the conditions (15) and (16). In the above, α and β are non-zero positive parameters satisfying This condition comes from Eq. (9). In this way, A(r) is parameterized by the two parameters α, β. From Eq. (14), the initial density ρ(0, r) is also parameterized as In the following, the function a(x) is given so that it satisfies Eqs. (23)-(27), and the initial density distribution is parameterized by α and β. Using α, β, and a(x), Eq. (22) is rewritten as Moreover, using dimensionless coordinates we obtain the dimensionless equation for the null line where Equation (30) is also rewritten as Note here that the right-hand side of Eq. (35) is not a Lipschitz continuous function in the region that contains x = ζ = 0. If this equation has a solution that starts from x = ζ = 0, then, at least, the singularity is locally naked. Moreover, if the solution could extend to x → ∞, the singularity would be visible at null infinity, that is, it would be globally naked. From now on we will ask if this differential equation has a solution. For convenience, we define the dimensionless coordinate θ as Then Eq. (35) becomes (39) In the current expression, from Eqs. (17), (18), and the relation θ = √ αt−1 x 2 , we see that the singularity and the apparent horizon are located at and respectively, and Note that, from the coordinate transformations (34) and (38), we see that the region x = 0 is singular for arbitrary θ. But any future-directed curve does not emanate from the central singularity located in θ < 0 because any point in the region satisfying x = 0 and θ < 0 is not in the future time slice of the central singularity. Then, for our purposes, we focus on the central singularity located in θ ≥ 0.
If the differential equation (39) has a C 1 solution θ(x) for x ∈ [0, l], the multiplication of x with the right-hand side of Eq. (35) for its solution behaves around x = 0 as Thus, if holds, then the first-order pole of Eq. (39) is cancelled out and does not appear. As with Christodoulou [12], let us introduce a real number λ satisfying Since γ is a positive real number [see below Eqs. (29) and (36)], we have In addition, since we see where that is, γ has the minimum at λ = λ M . If γ > γ min holds, we have the two solution to Eq. (46) for given γ , λ − (γ ), and λ + (γ ), satisfying 0 < λ − (γ ) < λ M < λ + (γ ) < 1 4 . If γ = γ min , Eq. (46) has the single solution λ ± (γ ) = λ M . If γ < γ min , Eq. (46) has no solution. From Eq. (46), it is easy to see that λ − (γ ) and λ + (γ ) satisfy lim γ →∞ λ − (γ ) = 0 and lim γ →∞ λ + (γ ) = 1 4 . If λ satisfying Eq. (46) exists, we rewrite Eq. (39) as The formal solution to this equation is given by where T λ is a nonlinear map on a functional space. Equations (51) and (52) imply that the fixed point of T λ can be a solution to Eq. (51). Thus, if T λ has a fixed point on a proper subset of C 0 on a proper domain, we can prove the existence of the solution satisfying (51), which emanates from the central singularity, and this means that the singularity is naked. In the next section, using the fixed-point theorem for a compact operator introduced soon, we examine the condition that the central singularity is naked.

Preparation
In the four-dimensional case [12], the existence of null geodesics is shown by using the fixed-point theorem for contraction mapping [19]. In the five-dimensional case, however, we cannot use the same method (see Appendix B). Thus, we have to innovate. First, we introduce a fixed-point theorem that is suitable for the current issue [20].
THEOREM (Schauder fixed-point theorem [17]) Let D be a nonempty, closed, bounded, convex subset of a Banach space X , and suppose T : D → D is a compact operator. Then T has a fixed point.
In the above, a compact operator is defined as follows.

DEFINITION (Compact operator) An operator T is compact if and only if:
(1) T is continuous.
(2) T maps a bounded set into a relatively compact set.
Here, "relatively compact" means that the closure is compact. Note that T does not have to be a linear operator.
We also use the Arzelà-Ascoli theorem to show the relative compactness of the image of T . (i) Uniform boundedness: (ii) Equicontinuity: For arbitrary > 0, there exists δ > 0, which depends only on , such that for each x, y ∈Ḡ satisfying |x − y| < δ.
We apply the Schauder fixed-point theorem to the nonlinear operator T λ defined in Eq. (52) and ask if the equation for the null line has a solution. First of all, we introduce a domain such that T λ maps its domain into itself. So let us define The last inequality can always be satisfied for sufficiently small d because θ S is a continuous function and λ < 1 4 = θ S (0) always holds. We can control the maximum norm of the elements of D λ,b,c,d by parameter b and the speed of their convergence as x → 0 by parameter c, respectively. Here we introduce the uniform norm θ ≡ sup x∈ [0,d] Then we can show the following lemma. Proof. For θ ∈ D λ,b,c,d , we estimate the right-hand side of Eq. (51) as where we used (46) and (55) (the details are shown in Appendix A) and the terms O(1) and O(x 2c−1 ) do not depend on θ. Then we see that hold and d is sufficiently small, the second and third terms in the right-hand side of Eq. (57) can be much smaller than the first term and ignorable, and This means that there exists a positive numberd such that There exists c such that Eqs. (58) and (59) and then we can take the parameter c in this range so as to satisfy (58) and (59). For example, c(λ) in Lemma 1 is a number slightly larger than Remark: the restriction for c, c < 1, in Lemma 1 comes from the circumstance that one wants to control the matter initial distribution by Eq. (15) or (28).
From (50) and (60), for all λ − (γ ), we can take some c satisfying Eqs. (58) and (59) because of 16 . Thus, we can choose c and d such that . Next, we evaluate T λ (θ) and show that it is uniformly continuous on D λ,b,c,d .
Proof. First, we evaluate the absolute value of the difference of the integrand in T λ (θ) for different θ 1 and θ 2 in D λ,b,c,d . For convenience, let us define where the index i takes 1 or 2. Since we have θ i (x) < θ S (x) from the definition (55), and Eq. (40) is a positive function that does not depend on d, θ 1 , or θ 2 , and we used Eqs. (26) and (46) and the fact that (see Appendix A for the details). Using (63), we obtain The integral of the right-hand side of the inequality is finite because Moreover, for specific initial conditions, T λ becomes a contraction mapping. In this case, as below, we can immediately show the existence of a solution to Eq. (52). THEOREM 2 For all λ < 1 6 , there exist d ∈ (0, l] (l corresponds to the surface of the dust cloud) and a unique solution θ ∈ C ∞ (0, d] to the integral equation (52), which is continuous at x = 0 and satisfies θ(0) = λ.
Proof. The last term of (65) is estimated as Here, if we can choose sufficiently small d such that Thus, by the fixed-point theorem for contraction mapping [19], T λ has a unique fixed point θ ∈ D λ,b,c,d . Note that the condition (67) is equivalent to λ < 1 6 .
Since the integrand of the right-hand side of Eq. (52) is a C ∞ function in the region except for the singularity, the solution θ must be a C ∞ function in (0, d]. On the other hand, the solution (35). This means that the differential equation of the null line (35) has a solution ζ(x) ∈ C 1 [0, d] which is a future-directed outgoing null geodesic emanating from the central singularity for all λ < 1 6 . For γ > 24, Eq. (46) tells us λ − (γ ) < 1 6 . Then, in this case, there exists a null line emanating from the central singularity, that is, it is a locally naked singularity at least.

A proof of the existence of the null geodesics
In the case of γ min ≤ γ ≤ 24, that is, 1 6 ≤ λ − (γ ) holds, we cannot use Theorem 2 to show the existence of a solution to Eq. (52). Then we have to develop another method. As we mentioned already, we can show the existence of a solution to Eq. (52) by using the Schauder fixed-point theorem. Proof. From Schauder fixed-point theorem, if D λ,b,c,d is a nonempty, closed, bounded, convex subset of a Banach space, and T λ maps D λ,b,c,d into itself and is a compact operator, then T λ has a fixed point. We already showed in Lemma 1 that we can take certain numbers d ∈ (0, l] and c ∈ (0, 1) such that T λ maps D λ,b,c,d into itself for all λ < 9− √ 33 16 . Moreover, Lemma 2 tells us that T λ is continuous. By the definition (55), D λ,b,c,d is obviously a nonempty, closed, bounded subset of This means that D λ,b,c,d is convex. Then all we have to show is that T λ (D λ,b,c,d ) is a relatively compact set. By virtue of the Arzelà-Ascoli theorem, the remaining task is to show uniform boundedness and equicontinuity of T λ (D λ,b,c,d ). Uniform boundedness results from the boundedness of D λ,b,c,d as follows: Next, to show equicontinuity, we evaluate Here, note that we can assume x < y without loss of generality. Then, using Eq. (56), where h(x) is a C 0 positive function in the range [0, d] that does not depend on θ , and n is an arbitrary natural number (the details of the calculation are in Appendix A). So we take n such that Then, since h(x), y c , and 2 n+1 h(x)y , there exist real numbers 1 , 2 , and 3 such that Since all continuous functions on a compact set are uniformly continuous, for all > 0 there exist This means that T λ (D λ,b,c,d ) is equicontinuous. Therefore, T λ (D λ,b,c,d ) is a relatively compact set. Since any closed set included in a compact set is also compact, T maps any bounded set into a relatively compact set. Thus T is a compact operator.
In the same way as the discussion below Theorem 2, Theorem 3 means that the differential equation for null line (35) has a solution ζ ∈ C 1 [0, d] which is a future-directed outgoing null geodesic emanating from the central singularity for all λ < 9− √ 33 16 . Thus, in the following, if the function that is a solution to Eq. (51) converges to a finite value as x → 0, we consider the function as a solution to Eq. (51) that is also defined at x = 0. Note that the solution found in Theorem 2 is unique, but it is not necessary that the solution found in Theorem 3 is unique.

Spacetime structure around the singularity
In this section, we show the existence of the earliest null geodesic θ n 0 emanating from the central singularity for all γ ≥ γ min = 11 + 5 √ 5. Since such a null geodesic determines the causal structure around the naked singularity and the global nakedness of the singularity, θ n 0 plays an important role in our analysis. On the other hand, for γ < γ min , we also show that there is no causal geodesic emanating from the central singularity.
In the case of four dimensions, we can show that g rr of the LTB spacetime is a strictly monotonically decreasing function with respect to t near the singularity. Using this nature, we can immediately 14/32 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/10/103E03/2468905 by guest on 29 July 2018 specify θ n 0 (see Ref. [12]). By contrast, in the case of five dimensions, g rr becomes a monotonically increasing function with respect to t near the singularity. Therefore, we need to develop another method to specify θ n 0 that is general to some extent.
First, we show that any future-directed null geodesic along the outer radial direction cannot emanate from the central singularity located at θ < λ − (γ ).
where κ 1 is a nonzero real number. Thus, in the same way as the case that λ satisfying Eq. (46) exists, we can show that any solution θ(x) cannot be bounded. Therefore, any future-directed null geodesic along the outer radial direction must diverge to −∞ as x → 0. Since the future-directed null geodesic along the outer radial direction is obviously the earliest line that emerged from x = 0 at arbitrary time, any future-directed outgoing causal line must diverge to −∞ at x = 0.
Thus, for the case that λ satisfying Eq. (46) does not exist, there is no causal line which emanates from the central singularity, that is, strong cosmic censorship holds. In contrast, for the case that λ satisfying Eq. (46) exists, we have just discussed converged null geodesics and have not yet shown anything about other causal lines that emanate from the central singularity. To address this point, we will first present Lemmas 4 and 5.
Thus, from Lemmas 3 and 4, we conclude that any future-directed outgoing null geodesic along the radial direction, θ(x), satisfies (i) lim does not immediately mean that any future-directed outgoing causal line comes to satisfy θ ≥ λ − (γ ) or θ → −∞ as x → 0, because a null line that converges to the central singularity exists in this case. Hence, we have to carefully examine the geodesics in this case.
For d > 0, let us define G λ − (γ ),d ⊂ C 0 [0, d] as the set of the solutions to Eq. (51) for γ that converge to λ − (γ ) as x → 0 and do not enter the singularity at a point in (0, d]. Then, we can show the following lemma for G λ − (γ ),d . Proof. We suppose that λ satisfying Eq. (46) exists. Let us define where . Since the right-hand side of Eq. (51) satisfies the Lifshitz condition on an arbitrary closed set that does not contain the singularity, all solutions to Eq. (51) do not intersect each other and can extend arbitrarily in any open set that does not contain the singularity. This fact means that the ordering of the solution orbits with respect to the coordinate θ is conserved.
From Theorem 4, if θ n 0 (x; γ ) can extend to x = l and θ n 0 (l; γ ) < θ AH (l; γ ) holds, the central singularity must be globally naked. Using this fact, in the next section we consider the global structure of this spacetime.

Global spacetime structure and the globally naked singularity
In this section, we consider global properties of a singularity. We will see the dependence of the nakedness of the central singularity on the initial density distribution characterized by γ and a(x) (see Eq. (31) for the definitions). The discussion in this section is similar to the four-dimensional case [12]. LEMMA 6 For any initial density distribution parameterized as (31), there exists γ 0 such that the solution θ n 0 (x; γ ) can extend to x = l (corresponding to the surface of the dust cloud) and θ n 0 (l; γ ) < θ AH (l; γ ) holds for all γ ∈ [γ 0 , ∞]. θ n 0 is defined in Lemma 5.
Proof. Since the outer region of the x = l surface is the Schwarzschild spacetime and the event horizon is identical to the apparent horizon in the Schwarzschild spacetime, θ n 0 (l; γ ) < θ AH (l; γ ) means that the null line corresponding to θ n 0 (x; γ ) arrives at the outer region of the event horizon of the Schwarzschild spacetime, that is, the null line θ n 0 (x; γ ) will attain the future null infinity and then the central singularity is globally naked. 19 To prove this lemma, by virtue of Theorem 3, it is enough to show that, for sufficiently large γ , there exist b and c such that (i) λ − (γ )+bx c < θ S (x) holds for all x in [0, l]; (ii) |T λ − (γ ) (θ)(x)−λ − (γ )| ≤ bx c holds for all x in [0, l] and all θ in C 0 [0, l] that satisfy |θ(x) − λ − (γ )| ≤ bx c ; and (iii) θ n 0 (l; γ ) < θ AH (l; γ ) holds (see the proof of Theorem 3 for the details).
Therefore, for all γ ∈ [γ 0 , ∞), θ n 0 (x; γ ) arrives at the surface of the dust cloud before the event horizon appears there; that is, the central singularity is globally naked in this case.
On the other hand, for γ sufficiently close to η defined by (30), we show that the central singularity is surrounded by the event horizon, that is, the central singularity is only locally naked. Proof. We suppose that λ satisfying Eq. (46) exists. In this case, from Theorem 3, the central singularity is locally naked at least. Let us define x η as Note that x η = 0 because a(0) is finite and a(x)x 2 > 0 except for x = 0. At x = x η , the apparent horizon appears at If η ≥ γ min holds, there exists γ 1 such that the right-hand side of this equation becomes negative for arbitrary γ in (η, γ 1 ]. Additionally, since 1 − η γ < 1 always holds, θ AH (x η ; γ ) would be negative if a(x η ) were equal to 1. This fact and a(l) ≤ a(x η ) ≤ 1 tell us that γ 1 → ∞ for a(l) → 1. Since θ AH (x η ; γ ) < 0 means that the apparent horizon can exist at an earlier timeslice than the central singularity appears, null geodesics emanating from the central singularity cannot arrive at future null infinity for arbitrary γ in (η, γ 1 ]; that is, the central singularity is only locally naked. On the other hand, for η < γ min , γ cannot approach η because of the condition (37). But if there exists x 0 in [0, l] that satisfies γ min γ min +x 2 0 < a(x 0 ), then θ AH (x 0 ; γ min ) < 0 holds from Eq. (111). Since θ AH (x; γ ) is continuous with respect to γ , there exists γ 2 such that θ AH (x 0 ; γ ) < 0 holds for arbitrary γ ∈ [γ min , γ 2 ], that is, the central singularity is only locally naked in these cases. In addition, since γ min γ min +x 2 0 < 1 and a(l) ≤ a(x 0 ) ≤ 1 always hold, we have γ 2 → ∞ for a(l) → 1.
Furthermore, we can show the monotonicity of θ n 0 (x; γ ) with respect to γ at each x. Let us define θ(x; γ ) as a solution to Eq. (51) for γ , which converges to λ − (γ ) as x → 0. LEMMA 8 For any initial density distribution parameterized as (31), θ(x; γ s ) > θ(x; γ l ) holds for γ s and γ l such that γ s < γ l , and all x such that θ(x; γ s ) exists. In particular, θ n 0 (x; γ ) defined in Lemma 5 is a monotonically decreasing function of γ at each x.
Proof. We suppose that γ s < γ l and θ(x; γ s ) exists in the range [0, d s ). Now let us define where F(x) is the positive function defined as For the first inequality in the above, we used the fact that As x 2 → d 0 , this inequality becomes Proof. It is obvious from Eq. (41).
THEOREM 5 (i) For any initial density distribution which is parameterized as (31) and satisfies η ≥ γ min , there exists γ C which satisfies η < γ C and γ C → ∞ for a(l) → 1 such that (a) for arbitrary γ ∈ (γ C , ∞), θ n 0 (x; γ ) defined in Lemma 5 goes to future null infinity, that is, the central singularity is globally naked and weak CCC does not hold, and (b) for all γ ∈ (η, γ C ), the central singularity is only locally naked, that is, weak CCC holds and the outer region of the event horizon is regular.
(ii) Let us assume that the initial density distribution is parameterized as (31) and satisfies η < γ min = 11 + 5 √ 5. If there exists x 0 in [0, l] such that γ min γ min +x 2 0 < a(x 0 ) holds, in the same way as in the proof of (i), we can show that (ii) holds. On the other hand, if such x 0 does not exist in [0, l], we cannot use Lemma 7. Then all we could show in this regard is Lemma 6 only.

Conclusion and discussion
In this paper, we analyzed five-dimensional inhomogeneous spherically symmetric dust collapse. By virtue of the Schauder fixed-point theorem, we proved an existence theorem for null geodesics in singular spacetime. Moreover, by using it, we showed a necessary and sufficient condition for the singularity to be naked and saw the dependence of the global nakedness of the central singularity on the initial density distribution. In Sect. 2, we fixed the initial energy distribution of the dust so that the initial velocity of the shells is zero. This assumption is not critical for our method. Therefore, we can also discuss the nakedness of the singularity without this assumption. To prove the existence of a null geodesic emanating from the central singularity in this general case, we have to find an appropriate domain such that the 26/32 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/10/103E03/2468905 by guest on 29 July 2018 operator T λ maps its domain into itself. We expect that, for some class of energy distribution, D λ,b,c,d defined by (55) can be such domain for certain b, c, and d, and the argument will follow in a similar manner to this paper.
In specific dimensional spherically symmetric dust collapse in Lovelock gravity, or particularly in nine-dimensional spherically symmetric dust collapse in Einstein-Gauss-Bonnet gravity [21], 2 we cannot use Christodoulou's method and discussion to show the existence of null geodesics emanating from the central singularity because the singular term in the differential equation for the null geodesic does not take the form of a simple function. In contrast, our method may be used to examine the nakedness of a singularity for the above cases because our existence theorem improved Christodoulou's method [12].
where B 1 (x) is introduced as in the text.

Appendix B. Four-dimensional case
In this appendix, we give an overview of Christodolou's paper [12] which examined the global nakedness of a singularity in four-dimensional LTB spacetime, and see the difference between Christodoulou's and our discussions on the existence of null geodesics near the singuality. In the four-dimensional case, after change of variables, the dimensionless differential equation for futuredirected null geodesics along the outer radial direction is given as whereθ andx are dimensionless coordinates, which correspond to θ and x defined by (38) and (33) respectively,λ is a certain constant, and f 4 is a C ∞ function.λ and f 4 are also the variables that correspond to λ and f defined in (46) and (51) in the five-dimensional case, respectively. In order not to contain a noncentral singularity,θ is restricted in the range 0 ≤θ < σ(x), where σ is a certain function which satisfies σ (x) ≥ 4 x for a positive constant 4 . The formal solution to this differential equation is given bŷ θ(x) = λ 1 +x where μ is a positive real number satisfying μ < σ (x) for allx ∈ [0,d]. Dd ,μ becomes a subset of a Banach space by the uniform norm. After some discussion on the nature of T 4,λ , as with the five-dimensional case, we can conclude that T 4,λ maps Dd ,μ into itself for sufficiently smalld. Furthermore, we obtain then T 4,λ becomes a contraction mapping from Dd 0 ,μ into itself. Therefore, by the fixed-point theorem for contraction mappings [19], we can conclude that T 4,λ has a unique fixed point, that is, a null geodesic emanating from the central singularity exists and the singularity is naked.
By contrast, in the five-dimensional case, what we can do is only to deform the differential equation for the null geodesic near the central singularity like where g(θ; γ ) is defined by (77) and f 5 is a function such that xf 5 (x, θ; γ ) converges to 0 as x → 0 in the region θ < θ S (x). In the four-dimensional case, the right-hand side of the differential equation for the null geodesic has a constant coefficient pole at first order only. However, in the five-dimensional case, the coefficient of the pole of the right-hand side of (B.7) is a function with respect to θ. Thus, the variable that corresponds to in (B.4) is not finite in five dimensions and we cannot directly use the method employed for the four-dimensional case [12]. As above, we can apply the method in [12] to the case that the geodesic equation has a constant coefficient pole at first order only. On the other hand, our method can be applied to the more general case that the geodesic equation can be deformed to an expression having a general pole at first order.