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Naked Singularities in Spherically Symmetric Inhomogeneous Tolman-Bondi Dust Cloud Collapse

We investigate the occurrence and nature of naked singularity for the inhomogeneous gravitational collapse of Tolman-Bondi dust clouds.It is shown that the naked singularities form at the center of the collapsing cloud in a wide class of collapse models which includes the earlier cases considered by Eardley and Smarr and Christodoulou. This class also contains self-similar as well as non-self-similar models. The structure and strength of this singularity is examined and the question is investigated as to when a non-zero measure set of non-spacelike trajectories could be emitted from the singularity as opposed to isolated trajectories coming out. It is seen that the weak energy condition and positivity of energy density ensures that the families of non-spacelike trajectories come out of the singularity. The curvature strength of the naked singularity is examined which provides an important test for its physical significance and powerful curvature growth near the naked singularity is pointed out for several subclasses considered. The conditions are discussed for the naked singularity to be globally naked. Implications for the basic issue of the final fate of gravitational collapse are considered once the inhomogeneities in the matter distribution are taken into account. It is argued that a physical formulation for the cosmic censorship may be evolved which avoids the features above. Possibilities in this direction are discussed while indicating that the analysis presented here should be useful for any possible rigorous formulation of the cosmic censorship hypothesis.

models. The structure and strength of this singularity is examined and the question is investigated as to when a non-zero measure set of non-spacelike trajectories could be emitted from the singularity as opposed to isolated trajectories coming out. It is seen that the weak energy condition and positivity of energy density ensures that the families of non-spacelike trajectories come out of the singularity. The curvature strength of the naked singularity is examined which provides an important test for its physical significance. This is done in terms of the strong curvature condition which ensures that all the volume forms must be crushed to zero size in the limit of approach to the singularity; and also the divergence of the Kretschmann scalar K = R abcd R abcd is pointed out. We show that the class considered here contains sub-classes of solutions which admit strong curvature naked singularities in either of the senses stated above. The conditions are discussed for the naked singularity to be globally naked. An implication for the fundamental issue of the final fate of gravitational collapse is that naked singularities need not be considered as artifacts of geometric symmetries of space-time such as self-similarity, but arise in a wide range of gravitational collapse scenarios once the inhomogeneities in the matter distribution are taken into account. It is argued that a physical formulation for the cosmic censorship may be evolved which avoids the features above. Possibilities in this direction are suggested while indicating that the analysis presented here should be useful for any possible rigorous formulation of the cosmic censorship hypothesis.

INTRODUCTION
It is generally believed that a generic gravitational collapse would commence from a highly inhomogeneous initial state. This will be described in terms of an inhomogeneous energy density distribution in terms of a regular initial data on a spacelike hypersurface. The general class of solutions of Einstein's field equations describing spherically symmetric dust clouds, independent of the homogeneity assumption, was given by Tolman [1], which was further developed and studied by Bondi [2].
This class could be used to model the gravitational collapse of matter from general inhomogeneous initial conditions and one can study the fundamentally important issue of the final fate of gravitational collapse of a massive star which has exhausted its nuclear fuel within this framework. The assumptions involved here are the vanishing pressure and the spherical symmetry of the matter distribution which is in the form of dust. One could argue that in the final stages of collapse, the matter distibution would become almost spherically symmetric, and that the pressures should play a minor role to justify the dust approximation. From our view point however, the main advantage is that subject to these conditions these models allow us to describe the evolution of inhomogeneous distributions of matter, which offers a very general class for the study of the gravitational collapse phenomena.
A special case of the these Tolman-Bondi class of solutions is the Oppenheimer-Snyder [3] study of a completely homogeneous dust cloud collapse with zero pressure. This example has been studied in great detail and has provided much insight towards understanding the final fate of a continually collapsing massive body such as a star, which could achieve no equilibrium state because of the dominance of gravitational forces. This case provides the basic motivation for the idea of formation of black holes as the final state of collapse, and the related cosmic censorship hypothesis [4] which broadly states that the singularities forming in gravitational collapse must necessarily be hidden behind the event horizons of gravity and hence permanently invisible to the outside observers. This cosmic censorship hypothesis plays a fundamental role in both the theory and applications of the black hole physics and has been recognized as one of the most important open problems in the general theory of relativity and gravitation physics to-day.
As it turns out, despite several attempts no proof or any precise mathematical formulation of the cosmic censorship has been available so far. Further, the completely homogeneous dust collapse mentioned above could also be viewed as a special case which forms a set of zero measure in the general inhomogeneous class represented by the Tolman-Bondi solutions. It thus becomes imperative to study the general class of Tolman-Bondi models in greater detail in order to understand the final fate of a gravitationally collapsing massive body when the effects of inhomogeneities are taken into account. In fact, it was pointed out by the numerical simulations of Eardley and Smarr [5] that naked singularities not covered by event horizons arise in the marginally bound Tolman-Bondi collapse, and subsequently a class of such models was studied in detail analytically by Christodoulou [6] to draw the same conclusion. However, these singularities were shown to be gravitationally weak by Newman [7], who studied the curvature strengths of such naked singularities and conjectured that nature avoids strong curvature naked singularities.
Our purpose here is to study the Tolman-Bondi inhomogeneous collapse for a rather general class of models, which includes the above classes, and to study the formation and structure of the naked singularity occurring at the center of the collapsing cloud. We show that the formation of naked singularity is a generic feature for a very wide range of solutions considered here. We have recently shown [8] such a result for the general class of self-similar models describing the gravitational collapse of a perfect fluid, where it is shown that a powerfully strong curvature naked singularity forms from which families of non-spacelike geodesics escape in the space-time. Further to this, the class considered here is shown to include all the self-similar Tolman-Bondi models as well as a wide range of non-self-similar models.
This indicates that the naked singularity may not be regarded as the consequence of the geometric property of self-similarity only [9]. A naked singularity may not be treated as a serious enough situation if only a single non-spacelike trajectory escaped from it. Thus, we examine the sufficient conditions when families of nonspacelike geodesics could escape from the naked singularity. Interestingly, it turns out that the validity of weak energy condition in the space-time ensures the existence of such families. This is analogous to the results of Ref.8 for the self-similar class. We also discuss the issue as to when the naked singularity will be globally naked, i.e. visible to far away observers.
The organization of the paper is as below. In section 2, the basic parameters of the Tolman-Bondi models describing the inhomogeneous dust collapse are specified.
The existence and structure of the naked singularity is analyzed in section 3. We also characterize here the conditions that ensure that families of non-spacelike geodesics, rather than a single isolated trajectory, are emitted from the naked singularity. In particular, it is shown that the weak energy condition, together with the positivity of energy density, implies that a non-zero measure set of non-spacelike geodesics comes out from the naked singularity. The global versus local nakedness of the singularity is also discussed here. The curvature strength of the naked singularity provides an important test of the physical significance for the same. This issue is examined in section 4, where it is shown that the models considered here include both selfsimilar as well as non-self-similar classes admitting a strong curvature singularity in a powerful sense. The final section 5 summarizes some of the implications and conclusions.

TOLMAN-BONDI SPACE-TIMES
The Tolman-Bondi metric representing collapse of a spherically symmetric inhomogeneous dust cloud in the comoving coordinates (i.e. u i = δ i t ) is given by where T ij is the stress-energy tensor, ǫ is the energy density and R is a function of both t and r given byṘ Here the dot and the prime denote partial derivatives with respect to the parameters t and r respectively and as we are only concerned with the gravitational collapse of dust, we requireṘ(t, r) < 0. The quantities F and f are arbitrary functions of r. The quantity 4πR 2 (t, r) gives the proper area of the mass shells and the area of such a shell at r = const. goes to zero when R(t, r) = 0. Integration of equation (3) gives where G(y) is a strictly real positive and bound function which has the range 1 ≥ y ≥ −∞ and is given by and t 0 (r) is a constant of integration. We thus have in all three arbitrary functions of r namely f (r), F (r) and t 0 (r). One could however, use the remaining coordinate freedom left in the choice of scaling of r in order to reduce the number of such arbitrary functions to two. We therefore rescale R using this coordinate freedom such that Then t 0 (r) is evaluated by using the equation above and (4) to give The time t = t 0 (r) corresponds to the value R = 0 where the area of the shell of matter at a constant value of the coordinate r vanishes. It follows that the singularity curve t = t 0 (r) corresponds to the time when the matter shells meet the physical singularity. Thus, the range of the coordinates is given by It follows that unlike the collapsing Friedmann case, where the physical singularity occurs at a constant epoch of time (say, at t = 0), the singular epoch now is a function of r as a result of inhomogeneity in the matter distribution. One could recover the Friedmann case from the above equations if we set t 0 (r) = t ′ 0 (r) = 0.
The function f (r) classifies the space-time as bound, marginally bound, or unbound depending on the range of its values which are respectively. The function F (r) can be interpreted as the weighted mass (weighted by the factor √ 1 + f ) within the dust ball B of coordinate radius r which is conserved in the following sense.
where ǫ(0, r) = ρ(r). For physical reasonableness the weak energy condition would be assumed throughout the space-time, i.e. T ij V i V j ≥ 0 for all non-spacelike vectors V i . This implies that the energy density ǫ is everywhere positive, (ǫ ≥ 0) including the region near r = 0. Partial derivatives of R like R ′ andṘ ′ are of importance in our analysis. We get from the equations (3) to (7) where we have put The function β(r) is defined to be zero when f is constant and zero. The factor r α has been introduced here for the sake of convenience in examining the structure of the naked singularity. The exact value of the positive constant α ≥ 1 is to be determined and will depend on the different models of the space-time which allow naked singularities. Functions H(X, r) and N (X, r) are defined by equations (10) and (11). Using the scaling given by (6), the energy density ǫ on the hypersurface t = 0 is written as ǫ = F ′ /r 2 . Since the weak energy conditions are satisfied and F is a function of r only, it follows that F ′ ≥ 0 through out the space-time. One can write the energy density as Since F ′ = ηΛr α−1 , it follows from the above that everywhere H(X, r) ≥ 0 and ηΛ ≥ 0 everywhere as a consequence of the weak energy condition.
Singularities are the boundary points of the space-time where the normal differentiability and manifold structures break down. In other words, these are the points where the energy density given by equation (2), or the curvature quantities such as the scalar polynomials constructed out of the metric tensor and the Riemann tensor diverge. One example of such a quantity is the Kretschmann scalar K = R abcd R abcd , which is given in the Tolman-Bondi case by Such singularities are indicated by the existence of incomplete future or past directed non-spacelike geodesics in the space-time which terminate at the singularity. Then one requires that the curvature quantities stated above assume unboundedly large values in the limit of approach to the singularity along the non-spacelike geodesics terminating there. If such a condition is satisfied, then one would like to consider the singularity to be a physically significant curvature singularity.
In Tolman-Bondi space-times singularities occur, as one can see from equations (2) and (15), at points where R = 0, which are called shell focusing singularities, and also at points where R ′ = 0. At the points where R ′ = 0 the Tolman-Bondi metric is degenerate and these are called shell crossings. In the context of Tolman-Bondi space-times the points R > 0, F ′ > 0, where R ′ = 0, are called the shell crossing singularities [7]. Such shell crossing singularities in Tolman-Bondi spacetimes have been analyzed in detail in the literature [10,11], and their nature appears to be fairly well understood. Even though such shell-crossing singularities could be locally naked, the important point is they have been shown to be gravitationally weak [7]. Thus it is generally believed that such shell crossing singularities need not be taken seriously as far as the cosmic censorship conjecture is concerned. The absence of shell-crossing singularities in a space-time turns out to be related to the condition that the function t 0 (r) giving the proper time for the shells to fall into the physical singularity should be a monotonically increasing function. The dust density and certain components of the curvature blow up near such a singularity.
However, the causal structure of the space-time can be extended through such a singularity and the space-time metric also can be defined in the neighborhood of such a point in a distributional sense [12]. In the context of such a situation, we do not consider here such shell crossings, and assume that there are no shell crossing singularities in the space-time (except probably right at the center r = 0 [10]). This does not involve any loss of generality as our basic purpose here is to examine the formation and local structure of the shell focusing naked singularity at the center of the collapsing cloud. Whereas the existence of shell crossings will not affect the qualitative nature of these general conclusions, the above assumption allows the calculations to be presented in a more transparent manner.
Unlike the shell crossings, the space-time metric however, admits no extension through a shell focusing singularity occurring at R = 0 which is more difficult to ignore. A shell-focusing singularity can be avoided only by rejecting the forms of matter such as dust as the fundamental forms of matter ( see e.g. [5]). Hence, we investigate here the occurrence of such shell focusing singularities at he center of the collapsing dust cloud and examine their nature and structure for the Tolman-Bondi space-times. It has been shown earlier [6] that a shell focusing singularity occurring at r > 0, R = 0 is totally spacelike and therefore our discussion would be confined to the singularity at r = 0.
The points (t 0 , r 0 ) where a shell focusing singularity R(t 0 , r 0 ) = 0 occurs, are related by equation (4). The singularity here occurs at r = r 0 at the coordinate time t = t 0 and we would call the singularity to be a central singularity if it occurs at r = 0. Earlier work [3,4] has shown that this central shell focusing singularity is naked, though gravitationally weak, for a class of Tolman-Bondi space-time for which the energy density (which is assumed to be positive every where and is taken to be non-zero at r = 0) and the metric are even smooth functions of t and r.
Translated in terms of parameters defined above, this corresponds to the class for which η(0) = 3, β(0) = 2, and p(r) is an even smooth function of r. In terms of functions F (r) and f (r) it amounts to the conditions, It was however, pointed out by Waugh and Lake [13] and Ori and Piran [14] that this class of gravitationally weak naked singularities excludes the self-similar Tolman-Bondi models, where they showed the singularity to be gravitationally strong along the Cauchy horizon, which is a null geodesic coming out of the singularity. Further, Grillo [15] pointed out an example in the case of unbound Tolman-Bondi models which are non-self-similar and the naked singularity is gravitionally strong. In fact, we have pointed out recently [16] that the naked singularity exists and is gravitationally strong for a wide class of Tolman-Bondi models which are non-selfsimilar in general and include all the self-similar models as a special subclass. In the notation used here, these models are characterized by the conditions η(0) = 1 with F (r) and f (r) being analytic at r = 0.
Through out the present consideration we would require rather general differentiability conditions on the functions F (r) and f (r) in that they will be assumed to be atleast C 1 at the center r = 0, ∞ > η(0) > 0, and β(0) is finite [17]. We note that the function f , and also its first derivatives (through R ′ ) enter the metric potentials. One might actually argue that the above is a more general condition than should be required, because it is often a customary practice to assume that the metric is C 2 -differentiable (which ensures again the above requirement), so that the metric transformations and other functions connected with the metric are welldefined to do regular physics. Hence, such a condition may be considered to be physically reasonable and a rather general differentiability requirement which includes practically all the inhomogeneous collapse Tolman-Bondi models of interest.
In fact, one could argue that if the metric is not C 2 -differentiable, but say only C 1 on initial surface, it may be considered as being already naked singular and not defining a regular initial data on an initial spacelike hypersurface.
In order to represent the gravitational collapse scenario, we assume the energy density ǫ to have a compact support on an initial spacelike hypersurface and the Tolman-Bondi space-times given by (1) can be matched at some r = const = r c to the exterior Schwarzschild field where dΩ 2 = dθ 2 + sin 2 θdφ 2 . The value of the Schwarzschild radial coordinate is r s = R(t, r c ) at the boundary r = r c . We have m(r c ) = M where M is the total Schwarzschild mass enclosed within the dust ball of coordinate radius of r = r c .
Without going into further details of the matching conditions we would like to say a few words regarding the apparent horizon. The apparent horizon in the interior dust ball lies at R = F (r). From (4) and (7) one can see that the corresponding time t = t ah (r) is given by It has been shown earlier [6,7] that emissions from the shell focusing singularity R(t 0 , r 0 ) = 0 for all r 0 > 0 would lie in the region above t = t ah i.e. t 0 > t ah for all r 0 > 0, t 0 being the time when singularity at r = r 0 occurs. Hence all radiations would be future trapped from shell focusing singularities at r > 0. At r = 0 however, t(0) = t ah (0) and the singularity could be atleast locally naked.
Any light ray terminating at this singularity in the past goes to the future infinity if it reaches the surface of the cloud r = r c earlier than the apparent horizon at r = r c . In such a case the singularity would be globally naked. We now examine this central singularity in the section below.

THE EXISTENCE AND STRUCTURE OF NAKED SINGULARITY
In this section we investigate the existence of the naked central singularity for the general class of Tolman-Bondi space-times under consideration. The singularity is naked if there are future directed non-spacelike curves in the space-time with their past end-point at the singularity. The existence of such curves implies that either photons or timelike particles can be emitted from the singularity. In particular we will examine the future directed non-spacelike geodesics for their past end-point at the singularity. Other related issues examined here are when a non-zero measure set of non-spacelike trajectories will meet the singularity in the past, rather than a single isolated geodesic; and when such a singularity will be globally naked.
A. The Existence The tangents K a = dx a /dk for the outgoing non-spacelike geodesics in the Tolman-Bondi space-time given by (1) can be written as below, Here ℓ is an impact parameter that labels different geodesics and vanishes(ℓ = 0) for radial trajectories, B characterizes the type of geodesics i.e. B = 0 for null and B = −1 for timelike curves, and the function P = P(t, r) satisfies the differential The parameter k is an affine parameter along the geodesics. For future directed non-radial trajectories that meet the central singularity at R = 0 in past, it follows from equation (20) that P ≥ ℓ near the singularity.
If the outgoing non-spacelike geodesics are to terminate in the past at the central singularity r = 0, which occurs at some time t = t 0 at which R(t 0 , 0) = 0, then along such geodesics we have R → 0 as r → 0. The following is satisfied along non-spacelike geodesics where we have put u = r α . The function H(X, r) in the above equation is strictly positive and non-zero for all r > 0 as a consequence of (10) and (14). For an outgoing geodesic (dR/du) must be positive while the negative value for this quantity means the geodesic is ingoing. The point u = r α = 0, R = 0 is a singularity of the above differential equation.
It is now essential to understand the exact nature of this singularity. If the functions appearing in the numerator and denominator of (23) are expandable and contain linear terms, then one can apply the standard analysis on the classification of singular points of first order differential equations [18] to understand the nature of this singularity. However, in the case otherwise, the same could be understood only by means of studying the detailed behavior of the characteristic curves in the vicinity of the singularity. If these characteristics terminate at the singularity in past with a definite tangent, this is determined by the limiting value of X = R/r α = R/u at R = 0, u = 0. If the non-spacelike geodesics meet the singularity with a definite value of tangent, then using equation (23) and l'Hospital rule we get for the value of X 0 , If a real and positive value of X 0 satisfies the above equation then the singularity could be naked. Real and positive roots of the above equation gives the possible values of tangents the outgoing geodesics can have at the singularity. Thus, if a real and positive value of X = X 0 satisfying above equation exists, then integral curves of the differential equation (23) i.e. outgoing non-spacelike geodesics, can terminate in the past at the singularity with a definite value of the tangent given by X = X 0 .
Clearly if no real positive root of the above exists then the singularity is not naked.
In order to make the discussion transparent, at this point we would limit ourselves to radial null geodesics only. Similar consideration can be given for the non-radial non-spacelike geodesics as well in terms of (23) and (24), which will be more complicated in view of the assumed generality of the functions involved. The equations (23) and (24) could then be written as where we have introduced the notation Along an outgoing null geodesic from the singularity, r increases and so does the area coordinate R. A point to note is that dR/du is positive for X > Λ, which implies R > F , and the geodesic is outgoing. If the geodesics cross and get inside the curve R = F , which represents the apparent horizon, dR/du becomes negative and hence the geodesics are ingoing (in the sense that area coordinate R starts decreasing). Since the apparent horizon R = F is the boundary of all trapped surfaces, if a null geodesic terminating at the singularity is to be outgoing it must have R > F at the singularity along the geodesic. The null geodesic would also reach the future infinity if it does not get inside the apparent horizon(i.e. R < F ) within the boundary of the dust cloud and reaches this boundary at r = r c with R > F along the same geodesic.
In the description given here the constant α actually represents the behavior of singular geodesics near the singularity i.e. R ∝ r α near the singularity. In fact we can write R = X 0 r α in the neighborhood of singularity, X 0 being the real and positive root of equation (26). Thus the determination of α really means determining the behavior of possible singular geodesics terminating at the singularity.
The alogrithem for evaluation of the value of α is as follows: Given the functions F (r) and f (r) (which specify the Tolman-Bondi model), the unique value of α is determined by the condition that Θ(r) P + 1 X does not vanish or goes to infinity identically as r → 0 in the limit of approach to the central singularity along any X = const direction. This condition ensures that the quantity H(X, 0) will not be identically zero or infinite in (26) and (27). Note that Θ(r) vanishes identically only for the case of Friedmann models (η(r) = 3, β(r) = 2) where the singularity is spacelike. Once such a value of α is determined the values of positive roots of the equation (22) are then determined if there are any. There remains a possibility when such a value of α can not be found. Such a case can arise only in the some of the situations where β(0) = 2, η(0) > 3. In this case, actually one has Θ(r) ∝ (r η 0 −2−α ln r) near the singularity at r = 0. However, in this situation one can use a suitable change of the variable R, namely,R = R +ar η 0 −2 (ln r +b) andX = X +ar η 0 −2−α (ln r +b), (a and b are some constants). This reduces equation (25) and (26) in the desired form and the value of α can then again be determined. Once the value of α is known in this manner, one can easily establish whether the singularity could possibly be naked. That is, if for this value of α the quantity Λ 0 diverges then clearly the space-time does not permit a naked singularity as X 0 = −∞. In fact, this puts an upper bound on the possible values of α if the singularity is to be naked, which is given by α ≤ η 0 . It follows from equation (26) that V (0) = 0 hence X = 0 can not be the root of V (X) = 0 and this implies that H 0 = H(X 0 , 0) = 0. We specify the values of α for some specific classes. If for example, η 0 = 1 then α = 1. It should be noted that for the case when η 0 = 3, β 0 = 2 (the cases that have been discussed in [5] and [6,7]), and F and f are even functions of r, the value of α turns out to be In these cases a shell crossing singularity also occurs at the central singularity (i.e. R ′ = 0) along with the shell focusing singularity. Again, α determines the occurrence of a shell crossing singularity at the central singularity. It follows from equations (10) and (26) that near the central singularity at r = 0, R ′ = r α−1 H(X 0 , 0).
Hence, for α > 1 the shell crossing singularity would also occur along with a shell focusing one. This actually happens in the cases already discussed by the references [5] and [6]. On the other hand, if α = 1 no shell crossing singularity occurs at the central singularity as the cases discussed in Ref. 15.
This determination of the value of α allows one to determine the existence of real and positive roots of equation (26). If the equation V (X) = 0 has a real and positive root, the singularity could be naked and the geodesics could terminate at the singularity in past with the tangent X = X 0 in the (u, R) plane. Therefore, existence of atleast one real positive root of (26) is the necessary condition for the space-time to admit naked singularity. The positive root X = X 0 actually represents the value of the tangent to null geodesics at the singularity, and it follows from equation (26) that X 0 > Λ 0 . Since Λ 0 is the value of the tangent of the apparent horizon R = F [19] at the singularity, it is clear that the geodesics in such cases could be at least locally naked. Clearly if no real positive root of the above is found, the singularity R = 0, r = 0 is not naked. It should be noted that many real positive roots of the equation (26) may exist which give the possible values of tangents to the null geodesics at the singularity. It is possible however, that the integral curves may or may not realize a given value X 0 at the singularity.
To determine whether a value X 0 is realized at the naked singularity along any outgoing singular geodesic, which establishes the nakedness of the singularity, consider the equation of radial null geodesics in the form u = r α = u(X). From equation (25) we have The solution of the above gives trajectories of radial null geodesics in the form u = u(X). The necessary condition for a null geodesic to terminate at the singularity at R = 0, u = 0 is that V (X) = 0 must have a real positive root X = X 0 . In such a case, non-spacelike curves could terminate at the singularity with the tangent X 0 . Therefore, if the null geodesics do terminate at the singularity then u → 0 as X → X 0 along the same. Let X = X 0 be a simple root of the equation (26). We could then write where h 0 is a constant the value of which is determined in terms of the quantities defined earlier as, The function h(X) is so chosen that i.e. h(X) contains higher order terms in (X − X 0 ) and H 0 = H(X 0 , 0), N 0 = N (X 0 , 0). Equation (30) could then be written as where S = S(X, u) = U (X, u) − U (X, 0) + h(X) is such that S(X 0 , 0) = 0, i.e. in the limit as u = 0, X = X 0 we have S → 0. Integration of (33) gives the equation of geodesics as u = u(X). Multiplying (33) by u −h 0 +1 and integrating gives where D is a constant of integration that labels different geodesics. If the singularity is the end point of these geodesics with tangent X = X 0 , we must have X → X 0 as u → 0 in (34). Note that as X → X 0 , u → 0, the last term in equation (34) always vanishes near the singularity regardless of the value of the constant h 0 (this is due to the reason that as u → 0, X → X 0 , we have S → 0). The first term In the above F 0 is to be treated as a constant. In this case, the relevant functions and equation (26) are as below.
In the case n > 1, where Λ(0) = 0, the above equation has only one positive root X = 1 which satisfies the equation V (X) = 0 for all n > 1, thus establishing the existence of naked singularity for all these space-times. These results agree with the earlier numerical calculations of [5] for the cases n = 9 5 and , n = 9 7 . In case n = 1 the space-time is self-similar with Λ(0) = F 0 and the equation (36) becomes where we have put x 2 = X. The above has real and positive roots if For example, for √ F 0 = 7/17 there are two positive roots x = 0.5 and x = 0.658.
Hence for all such values given by equation (38) the singularity is naked.

Next, consider the Tolman-Bondi space-times defined by the values of F and
f given by, Here f 0 , F 0 and f 1 are to be treated as some constants. For this second example the relevant quantities are written as Where we have again put X = x 2 and we see that for a wide range of constants f 0 , F 0 , f 1 the positive root of the above would exist and the singularity would be naked. In fact, for Θ 0 (F 0 ) 3/2 > 13 + of such a condition is seen very clearly in the case of self-similar models [13,8], where this derivative of V is determined directly by the Einstein field equations in terms of the energy density and the components of the metric tensor. It turns out in that case that this derivative will be positive with h 0 > 1 provided the weak energy condition is satisfied and the energy density is always greater than a certain lower bound in the neighborhood of the singularity, which gives a sufficient condition for families to meet the naked singularity in the past.
Suppose now that equation (26) has two simple positive roots X 1 and X 2 . In such a case at least one singular geodesic would always terminate along each of the tangents X = X 1 and X = X 2 at the singularity. In fact, Lemos [21] has pointed out recently several parallels between the self-similar where we have put and note that in the limit Thus we see that infinite many integral curves (each characterized by a different value of the constant D) would terminate at the singularity provided h 0 > 0. Hence we deduce that future directed null geodesics would terminate at the singularity in the past, as long as If positivity of energy in the near regions of singularity is respected as stated in equation (42) i.e. Λ 0 > 0, then using equations (32) and (26) and the fact that if f (0) = 0 then β(0) = 0 we get for the value of h 0 when Λ 0 = 0, Hence, we conclude that h 0 > 0 as long as the positivity of energy holds in the near regions of the singularity. Therefore families of geodesics would always terminate at the singularity when it is naked and provided the positivity of energy holds.
It is illustrative at this point to note the examples given in the earlier section in the context of families meeting the singularity. Note that for the first example given by equation (35), in the case n = 2 for example, Λ 0 = 0 and V (X) = 0 has only one root given by X = 1 and h 0 = 1/2. Therefore no families of integral curves terminate at the singularity with the tangent X = 1 . On the other hand, for n = 1, Λ 0 = 0 the space-time is self-similar and the families or infinitely many non-spacelike curves terminate at the singularity. The same is the case with the second example in which Λ 0 = 0 where families would terminate at the singularity when it is naked.

C. Global Visibility
It is seen that the existence of a real positive root V (X) = 0 establishes that the singularity would be atleast locally naked. Such a locally naked singularity could be globally naked as well. To examine this issue note that the apparent horizon lies at R(t, r) = F (r), and therefore if a geodesic gets inside the apparent horizon it becomes ingoing (i.e. R < F along geodesics and dR/dr is negative) Eventually this trajectory falls back to the singularity. Therefore, if a light ray is to reach future infinity in order for the singularity to be globally naked, it must cross r = r c , which is the boundary of the dust cloud before the apparent horizon. Hence all escaping non-spacelike geodesics that reach the boundary r = r c with R(r c ) > F (r c ) would reach the future infinity. Since geodesics emerge from the singularity with the tangent value X 0 and the apparent horizon has the tangent at the singularity Λ 0 , it follows from equation (26)  with the results already obtained. Earlier, this example has been analyzed using a special null trajectory which is the Cauchy horizon [13] which is given by X = const.
We show below however, that actually one can integrate the geodesic equations completely for this self-similar case to obtain radial null families. As it was pointed out earlier, in this case if the condition (38) is satisfied than V (X) = 0 has two real positive and two complex roots. Let x 1 , x 2 (x 1 > x 2 ) be two such positive roots of this equation. Equation of geodesics, in the form r = r(x), X = x 2 is given by where Here n 1 , n 2 , A, B, D 1 , D 2 are constants given by 3x 3 and D is the constant which labels the different geodesics. The constants n 1 , n 2 are positive. In fact for the case Λ 0 = 7 17 they are given by and hence the equation of such a geodesic can be written as The event horizon is represented by the geodesic for which X c = Λ(r c ). Since it is outgoing dR/d(r α ) is positive at r = 0 and ejected into the region R > F where dR/dr is positive. Therefore all the geodesics that reach the line r = r c (the line at which the the metric (1) is matched with the Schwarzschild exterior) with X c > Λ(r c ) would escape to infinity, while others would become ingoing.
It follows that the geodesics that reach future infinity with their past end point at the singularity are given by the equation (54) with X c > Λ c . Hence, in case when a family of geodesics terminates at the singularity with tangent X = X 0 and η(r)Λ(r) < αX 0 , for r c ≥ r > 0, the singularity would be globally naked as there would always be some geodesics that would escape to infinity.
Consider the case now when the equation V (X) = 0 has two positive roots X 1 and X 2 (X 1 > X 2 ). In such a case, as shown earlier, families of curves would emerge from the singularity with the tangent either X 1 or X 2 . Let η(r)Λ(r) < αX 2 for r c ≥ r > 0, then it ensures that some geodesics would cross the boundary of the cloud with X c > Λ(r c ) making the singularity globally naked. The same holds even in case when more than two positive roots exist. Thus if the family of geodesics do terminate at the singularity with tangent X 0 , then the condition η(r)Λ(r) < αX 0 for r c ≥ r > 0 implies the global nakedness of the singularity. curvature singularity [7,22].
The gravitational strength and physical seriousness of a space-time singularity have been discussed in detail and characterized precisely in the literature. In particular, Clarke and Krolak [23] have provided a sufficient condition for a singularity to be strong in the sense of Tipler [24], which is that atleast along one null geodesic with the affine parameter k, with k = 0 at the singularity, the following should be satisfied in the limit of approach to the singularity, This provides a sufficient condition for all the 2-forms µ(k) defined along the singular null geodesic to vanish as singularity is approached and implies a very powerful curvature growth establishing a strong curvature singularity. For the timelike geodesics this will imply that all the volume forms defined by the Jacobi fields along these trajectories must vanish in the limit of approach to the singularity or they must vanish infinitely many times in this limit.
The criteria on the strength of a singularity are of course subject to further refinement. However, the important physical consequences of the existence of a singularity are related to its strength. The point is if the singularity is gravitationally weak, it may be possible to extend the space-time through the same classically. On the other hand, when there is a strong curvature singularity forming in the above sense, the gravitational tidal forces associated with this singularity are so strong that any object trying to cross it gets destroyed. Thus, as argued by Ori [25], the extension of space-time becomes meaningless for such a strong singularity which destroys to zero size all the objects terminating at the singularity. From this point of view, the strength of singularity may be considered crucial to the issue of classically extending the space-time and thus avoiding the singularity; because for a strong curvature singularity defined in the above sense, no continuous extension of the space-time may be possible.
For the general class of Tolman-Bondi models under consideration, using (2) we get where K a is tangent to null geodesics. For radial null geodesics,using L'hospital rule and equations (4) to (14) and (19) to (22) and the fact that at the singularity Hence it is seen from the definition of Λ in (12) that lim k→0 k 2 Ψ = 0 for α < η 0 (58) lim k→0 k 2 Ψ = 0 for α ≥ η 0 However, from our earlier conclusions naked singularity occurs only when α ≤ η 0 , therefore the strong curvature condition is satisfied along singular geodesics only for the classes where α = η 0 . As noted earlier, for the special class considered by Newman and Christodoulou, α = 7/3 and η = 3 and hence the naked singularity turns out to be gravitationally weak as concluded earlier. On the other hand, it is clear form the above that for a wide variety of Tolman-Bondi solutions satisfying the condition α = η 0 , the singularity will be a strong curvature singularity in the above sense. In general it is also possible that non-radial null or timelike curves could terminate at the naked singularity. Then, a similar calculation along non-spacelike geodesics in general gives Hence as discussed above one concludes that condition for strong curvature is satisfied along non-spacelike geodesics as well if α = η 0 and if such families meet the naked singularity in the past.
The Kretschmann scalar R abcd R abcd along the geodesics goes in the Tolman-Bondi space-times as Hence the singularity is a scalar polynomial singularity as long as α > η 0 3 . The self-similar Tolman-Bondi models are defined by the conditions f (r) = const. and η(r) = 1 = η(0) = α. It follows from the above that the naked singularity forming in this class will be a strong curvature singularity along all the families of radial null geodesics. As shown in Ref. 8, other families of non-spacelike geodesics also do terminate at the naked singularity along which as well the strong curvature condition is satisfied.

CONCLUDING REMARKS
We have analyzed here the Tolman-Bondi models for the existence and structure of the naked singularities. As stated earlier, these are dust models assuming the pressure p = 0, and also the exact spherical symmetry of the space-time. Would the introduction of pressure change the qualitative nature of the conclusions obtained here ? This does not seem to be the case atleast for the self-similar gravitational collapse of a perfect fluid incorporating pressure as indicatd by the analysis of [14] and [8]. It is possible, on the other hand, that in the final stages of collapse, the dust equation of state could be relevant (see e.g. Penrose [26], Hagerdorn [27]) and at higher and higher densities the matter may behave more and more like dust.
Again, there is some case for the argument that eventually in the final stages of collapse, the matter distribution should become almost spherically symmetric (see e.g. Nakamura and Sato [28] ). Hence, it is clearly useful to examine the inhomogeneous dust collapse as modelled by the Tolman-Bondi space-times. Further, a situation analogous to the singularity theorems might develop here where the conclusions derived under the assumption of spherical symmetry are preserved when small perturbations are taken into account. Thus, spherical symmetry may be a good model to represent a certain class of gravitational collapse. Also, we have not addressed the issue of the stability of naked singularity. If these are not stable (in a sense to be defined suitably) such naked singularities need not be considered as counter-examples to the cosmic censorship hypothesis. As far as the issue of stability is concerned, one needs to develop a precise criterion for stability in general relativity. In this connection it may be noted however, that for self-similar Tolman-Bondi models the Cauchy horizon is stable atleast against the blue shift mode of instability [21].
Subject to these reservations, it is seen here that Tolman-Bondi space-times admit naked singularities under fairly general conditions, from which a non-zero measure set of non-spacelike trajectories emanate in the future direction. Certain examples of particular classes where non-radial non-spacelike geodesics terminate at the naked singularity in the past are also explicitly worked out. An interesting point is that in the case of the strong curvature condition being satisfied along radial null trajectories, the same conclusion also holds along all other non-spacelike geodesics. For various other classes of naked singularity space-times, even though the strong curvature condition may not be satisfied along radial curves, they could still be regarded as strong curvature singularity in the sense that the Kretschmann scalar diverges.
Another feature one would like to note here is that while strong curvature naked singularities have been found to occur in self-similar gravitational collapse as indicated earlier, the present consideration gives a wide class of inhomogeneous collapse models which need not be self-similar in general. A wide class of spacetimes has been pointed out, namely the ones for which α = η 0 , which gives a set of solutions of the field equations which admit a strong curvature naked singularity.
The suggestion that seems to be coming is that the phenomena of naked singularity is probably not related to the space-times with any particular geometric properties such as the self-similarity of the models. It may be that the existence of naked singularity is not just a geometric phenomena and the answer to cosmic censorship conjecture could lie in the dynamics of the Einstein equations. Of course, if one rules out the matter fields such as the dust and perfect fluid etc. from consideration because they may create singularity even without gravity, then such naked singularities are ruled out (see however, [29] where the occurrence of naked singularity is pointed out for a wide range of matter satisfying the weak energy condition in self-similar gravitational collapse).
To summarize, the conclusions on the final fate of gravitational collapse are rather different in the generally inhomogeneous Tolman-Bondi models as compared to the Oppenheimer-Snyder case of a completely homogeneous dust collapse, which forms a set of zero measure in the general Tolman-Bondi class considered here. In fact, the similarity in conclusions concerning the nature and structure of the naked singularity for the radiation collapse [20], the general self-similar collapse [8] of perfect fluid, and the results here appear suggestive of a certain general property of Einstein equations. It would be worth while to isolate and study this feature as that might help towards a definite mathematical formulation of the cosmic censorship by pointing out the precise feature one wants to rule out. Such a study would be of independent interest any way because not much is understood on the global properties of the Einstein equations except the results on the existence of space-time singularities as predicted by the singularity theorems.
While the analysis we have presented here should be useful towards arriving at any rigorous formulation of cosmic censorship in a provable form as pointed out above, we would like to argue here that a physical formulation of the cosmic censorship may be evolved which avoids features such as above. For example, an interesting feature that emerges from the presently available examples is the role of energy conditions in determining the escape of families of non-spacelike trajectories from the naked singularity, which is an important criteria for the physical significance of the same. In all the presently available collapse scenarios, it is the weak energy condition together with the positivity of of energy, which leads to the existence of families of non-spacelike geodesics terminating at the naked singularity in the past. Could one then argue that some how the energy conditions must be violated in the very final stages of gravitational collapse so as to avoid the formation of naked singularity ? In fact, in the case of self-similar collapse [8], it can be shown that the violation of the energy condition near the singularity no longer allow the families of non-spacelike geodesics to come out but only an isolated trajectory can emerge. Hence, for all practical purposes, the singularity is no longer naked preserving the effective censorship. Again, as emphasized by Israel [30], many of the naked singularities arising in the spherically symmetric collapse are massless (with the mass being defined in a suitable manner, see also Lake [18]); and as a consequence these may not violate the basic physical spirit of the cosmic censorship.
Such possibilities need a serious investigation.