We present here an overview of our basic understanding and recent developments on spacetime singularities in the Einstein theory of gravity. Several issues related to physical significance and implications of singularities are discussed. The nature and existence of singularities are considered which indicate the formation of super ultra-dense regions in the universe as predicted by the general theory of relativity. Such singularities develop during the gravitational collapse of massive stars and in cosmology at the origin of the universe. Possible astrophysical implications of the occurrence of singularities in the spacetime universe are indicated. We discuss in some detail the profound and key fundamental issues that the singularities give rise to, such as the cosmic censorship and predictability in the universe, naked singularities in gravitational collapse and their relevance in black hole physics today, and their astrophysical implications in modern relativistic astrophysics and cosmology.
After Einstein proposed the general theory of relativity describing the gravitational force in terms of the spacetime curvatures, the proposed field equations related the spacetime geometry to the matter content of the universe. The early solutions found for these equations were the Schwarzschild metric and the Friedmann models. While the first represented the gravitational field around an isolated body such as a spherically symmetric star, the later solutions described the geometry of the universe. Both these models contained a spacetime singularity where the curvatures as well as the matter and energy densities were diverging getting arbitrarily high, and the physical description would then break down. In the Schwarzschild solution such a singularity was present at the center of symmetry whereas for the Friedmann models it was found at the epoch which is beginning of the universe and origin of time where the scale factor for the universe vanishes and all objects are crushed to a zero volume due to infinite gravitational tidal forces.
Even though the physical problem posed by the existence of such a strong curvature singularity was realized immediately in these solutions, which turned out to have several important implications towards the experimental verification of the general relativity theory, initially this phenomenon was not taken seriously. It was generally thought that the existence of such a singularity must be a consequence of the very high degree of symmetry imposed on the spacetime while deriving and obtaining these solutions. Subsequently, the distinction between a genuine spacetime singularity and a mere coordinate singularity became clear and it was realized that the singularity at in the Schwarzschild spacetime was only a coordinate singularity which could be removed by a suitable coordinate transformation. It was clear, however, that the genuine curvature singularity at cannot be removed by any coordinate transformations. The hope was then that when more general solutions to the field equations are considered with a lesser degree of symmetry, such singularities will be avoided.
This issue was sorted out when a detailed study of the structure of a general spacetime and the associated problem of singularities was taken up by Hawking, Penrose, and Geroch (see for example, Hawking and Ellis, 1973, and references therein). It was shown by this work that a spacetime will admit singularities within a rather general framework provided it satisfies certain reasonable physical assumptions such as the positivity of energy, a suitable causality assumption and a condition implying strong gravitational fields, such as the existence of trapped surfaces. It thus followed that the spacetime singularities form a general feature of the relativity theory. In fact, these considerations ensure the existence of singularities in other theories of gravity also which are based on a spacetime manifold framework and that satisfy the general conditions such as those stated above.
Therefore the scenario that emerges is the following: Essentially for all classical spacetime theories of gravitation, the occurrence of singularities form an inevitable and integral part of the description of the physical reality. In the vicinity of such a singularity, typically the energy densities, spacetime curvatures, and all other physical quantities would blow up, thus indicating the occurrence of super ultra-dense regions in the universe. The behaviour of such regions may not be governed by the classical theory itself, which may breakdown having predicted the existence of the singularities, and a quantum gravitation theory would be the most likely description of the phenomena created by such singularities.
Further to the general relativity theory in 1915, the gravitation physics was a relatively quiet field with few developments till about 1950s. However, the 1960s saw the emergence of new observations in high energy astrophysics, such as quasars and high energy phenomena at the center of galaxies such as extremely energetic jets. These observations, together with important theoretical developments such as studying the global structure of spacetimes and singularities, led to important results in black hole physics and relativistic astrophysics and cosmology.
Our purpose here is to review some of these rather interesting as well as intriguing developments, in a somewhat pedagogic and elementary fashion providing a broad perspective. Specifically, we would like to highlight here several recent issues and challenges that have emerged related to spacetime singularities, which appear to have a considerable physical significance and may have interesting astrophysical implications. We take up and consider here important topics such as, what is meant by a singular spacetime and specify the notion of a singularity. It turns out that it is the notion of geodesic incompleteness that characterizes a singularity in an effective manner for a spacetime and enables their existence to be proved by means of certain general enough theorems. We highlight here several recent developments which deal with certain exciting current issues related to spacetime singularities on which research in gravitation and cosmology is happening today. These include the work on final endstates of gravitational collapse, cosmic censorship, and black holes and naked singularities. Related major cosmic conundrums such as the issue of predictability in the universe are discussed, and observational implications of naked singularities are indicated.
Ii What is a singularity?
In the Einstein theory of gravitation, the universe is modeled as a spacetime with a mathematical structure of a four dimensional differentiable manifold. In that case, locally the spacetime is always flat in a sufficiently small region around any point, but on a larger scale this need not be the case and it can have a rich and varied structure. An example of a differentiable manifold is a sphere, which is flat enough in the vicinity of any single point on its surface, but has a global curvature. For a more detailed discussion on spacetime manifolds and their key role in general relativity, we refer to (Wald 1984).
When should we say that a spacetime universe , which is a differentiable manifold with a Lorentzian metric, has become singular? What we need is a specification and a specific criterion for the existence of a singularity for any given universe model in general relativity.
As stated above, several examples of singular behaviour in spacetime models of general relativity are known. Important exact solutions such as the FriedmannRobertsonWalker (FRW) cosmological models and the Schwarzschild spacetime contain a singularity where the energy density or curvatures diverge strongly and the usual description of the spacetime breaks down. In the Schwarzschild spacetime there is an essential curvature singularity at in that along any nonspacelike trajectory falling into the singularity, the Kretschmann scalar . Also, all future directed nonspacelike geodesics which enter the event horizon at must fall into this curvature singularity within a finite value of the proper time, or finite value of the affine parameter, so all such curves are future geodesically incomplete. Similarly, for FRW models, if at all times, where is the total energy density and the pressure, there is a singularity at which could be identified as the origin of the universe. Then along all the past directed trajectories meeting this singularity, and the curvature scalar . Again, all the past directed nonspacelike geodesics are incomplete. This essential singularity at cannot be transformed away by any coordinate transformations. Similar behaviour was generalized to the class of spatially homogeneous cosmological models by Ellis and King (1974) which satisfy the positivity of energy conditions.
Such singularities, where the curvature scalars and densities diverge imply a genuine spacetime pathology where the usual laws of physics break down. The existence of the geodesic incompleteness in these spacetimes imply, for example, that a timelike observer suddenly disappears from the spacetime after a finite amount of proper time. Of course, singular behaviour can also occur without bad behaviour of curvature. For example, consider the Minkowski spacetime with a point deleted. Then there will be timelike geodesics running into the hole which will be future incomplete. Clearly this is an artificial situation one would like to rule out by requiring that the spacetime is inextendible, that is, it cannot be isometrically embedded into another larger spacetime as a proper subset. But one could give a non-trivial example of the singular behaviour where a conical singularity exists (see e.g. Ellis and Schmidt, 1977). Here spacetime is inextendible but curvature components do not diverge near the singularity, as in a Weyl type of solution. The metric is given by,
with coordinates given by with , with and identified and . There is a conical singularity at through which the spacetime cannot be extended and the singular boundary is related to the timelike two-plane of the Minkowski spacetime.
An important question then is, whether such singularities develop even when a general model is considered, and if so under what conditions. To answer this, it is first necessary to characterize precisely what one means by a spacetime singularityfor a general spacetime. Then it is seen that singularities must exist for a very wide class of spacetimes under reasonable general set of conditions. Such singularities may develop as the end state of gravitational collapse of a massive star, or in the cosmological situations such as the origin of the universe.
The first point to note here is by very definition, the metric tensor has to be well defined at all the regular points of the spacetime. This is no longer true at a spacetime singularity such as those discussed above and a singularity cannot be regarded as a regular point of spacetime but is a boundary point attached to . This causes difficulty when one tries to characterize a singularity by the criterion that the curvatures must blow up near the singularity. The trouble is, since the singularity is not part of the spacetime, it is not possible to define its neighborhood in the usual sense to discuss the behaviour of curvature quantities in that region. One may try to characterize a singularity in terms of the divergence of components of the Riemann curvature tensor along non-spacelike trajectories. Then the trouble is, the behaviour of such components will in general change with the frames used so this is not of much help. One can try the curvature scalars or the scalar polynomials in the metric and the Riemann tensor and require them to achieve unboundedly large values. This is encountered in Schwarzschild and the Friedmann models. But it is possible that such a divergence occurs only at infinity for a given nonspacelike curve. In general it looks reasonable to demand that some sort of curvature divergence must take place along the nonspacelike curves which encounter a singularity. However, a general characterization of singularity in terms of the curvature divergence runs into various difficulties.
Considering these and similar situations, the occurrence of nonspacelike geodesic incompleteness is generally agreed upon as the criterion for the existence of a singularity for a spacetime. This may not cover all types of singular behaviours possible but it is clear that if a spacetime manifold contains incomplete nonspacelike geodesics, there is a definite singular behaviour present, as a timelike observer or a photon suddenly disappears from the spacetime after a finite amount of proper time or after a finite value of the affine parameter. The singularity theorems which result from an analysis of gravitational focusing and global properties of a spacetime prove this incompleteness property for a wide class of spacetimes under a set of rather general conditions.
Physically, a singularity in any physics theory typically means that the theory breaks down either in the vicinity or at the singularity. This means that a broader and more comprehensive theory is needed, demanding a revision of the given theory. A similar reasoning apply to spacetime singularities, which may be taken to imply that a quantum gravity description is warranted in those regions of the universe, rather than using merely a classical framework.
Iii Gravitational focusing
The simple model solutions such as the Schwarzschild and the FRW universes give very useful indications on what is possible in general relativity, as opposed to the Newtonian gravity. In particular, these solutions told us on important indicators on the existence and nature of the spacetime singularities.
The key to occurrence of singularities in these solutions is really the gravitational focusing that the matter fields and spacetime curvature causes in the congruences of null and timelike curves, which represent the light paths and the material particle trajectories in any given spacetime universe. It would then be important to know how general and generic such a feature and property is for a general spacetime.
The matter fields with positive energy density which create the curvature in spacetime affect the causality relations in a spacetime and create focusing in the families of nonspacelike trajectories. The phenomenon that occurs here is matter focuses the nonspacelike geodesics of the spacetime into pairs of focal points or conjugate points. The property of conjugate points is that if are two conjugate points along a nonspacelike geodesic, then and must be timelike related. Now, there are three-dimensional null hypersurfaces such as the boundary of the future of an event such that no two points of such a hypersurface can be joined by a timelike curve. Thus, the null geodesic generators of such surfaces cannot contain conjugate points and they must leave the hypersurface before encountering any conjugate point. This puts strong limits on such null surfaces and the singularity theorems result from an analysis of such limits.
If we consider a congruence of timelike geodesics in the spacetime, this is a family of curves and through each event there passes precisely one timelike geodesic trajectory. Choosing the curves to be smooth, this defines a smooth timelike vector field on the spacetime. The rate of change of volume expansion for a given congruence of timelike geodesics can be written as
where, for a given congruence of timelike (or null) geodesics, the quantities , and are expansion, shear, and rotation tensors respectively. The above equation is called the Raychaudhuri equation (Raychaudhuri, 1955), which describes the rate of change of the volume expansion as one moves along the timelike geodesic curves in the congruence.
We note that the second and third term on the right-hand side involving and are positive always. For the term , by Einstein equations this can be written as
The term above represents the energy density measured by a timelike observer with unit tangent , which is the four-velocity of the observer. For all reasonable classical physical fields this energy density is generally taken as non-negative and we may assume for all timelike vectors ,
Such an assumption is called the weak energy condition. It is also considered reasonable that the matter stresses will not be so large as to make the right-hand side of the equation above negative. This will be satisfied when the following is satisfied, . Such an assumption is called the strong energy condition and it implies that for all timelike vectors , . By continuity it can be argued that the same will then hold for all null vectors as well. Both the strong and weak energy conditions will be valid for well-known forms of matter such as the perfect fluid provided the energy density is non-negative and there are no large negative pressures which are bigger or comparable to .
With the strong energy condition, the Raychaudhuri equation implies that the effect of matter on spacetime curvature causes a focusing effect in the congruence of timelike geodesics due to gravitational attraction. This causes the neighbouring geodesics in the congruence to cross each other to give rise to caustics or conjugate points. Such a separation between nearby timelike geodesics is governed by the geodesic deviation equation,
where is the separation vector between nearby geodesics of the congruence. Solutions of the above equation are called the Jacobi fields along a given timelike geodesic.
Suppose now is a timelike geodesic, then two points and along are called conjugate points if there exists a Jacobi field along which is not identically zero but vanishes at and . From the Raychaudhuri equation given above it is clear that the occurrence of conjugate points along a timelike geodesic is closely related to the behaviour of the expansion parameter of the congruence. In fact, it can be shown that the necessary and sufficient condition for a point to be conjugate to is that for the congruence of timelike geodesics emerging from , we must have at (see for example, Hawking and Ellis, 1973). The conjugate points along null geodesics are also similarly defined.
Iv Geodesic incompleteness
It was widely believed that for more general solutions of the Einstein equations which incorporate several other physical features and not necessarily symmetric, the existence of singularities would be avoided. Further investigations, however, showed that singularities in the form of geodesic incompleteness do exist for general spacetimes. These results used the gravitational focusing considerations mentioned above and global properties of a general spacetime.
The behaviour of the expansion parameter is governed by the Raychaudhuri equation as pointed out above. Consider for example the situation when the spacetime satisfies the strong energy condition and the congruence of timelike geodesics is hypersurface orthogonal. Then and the corresponding term vanishes. Then, the expression for the covariant derivative of implies that it must vanish for all future times as well. It follows that we must have then which means that the volume expansion parameter must be necessarily decreasing along the timelike geodesics. If denotes the initial expansion then the above can be integrated as . It is clear from this that if the congruence is initially converging and is negative then within a proper time distance .
It then follows that if is a spacetime satisfying the strong energy condition and is a spacelike hypersurface with at , then if is a timelike geodesic of the congruence orthogonal to passing through then there exists a point conjugate to along within a proper time distance . This is provided can be extended to that value of the proper time.
The basic implication of the above results is that once a convergence occurs in a congruence of timelike geodesics, the conjugate points or the caustics must develop in the spacetime. These can be interpreted as the singularities of the congruence. Such singularities could occur even in Minkowski spacetime and similar other perfectly regular spacetimes. However, when combined with certain causal structure properties of spacetime the results above imply the existence of singularities in the form of geodesic incompleteness. One could similarly discuss the gravitational focusing effect for the congruence of null geodesics or for null geodesics orthogonal to a spacelike two-surface.
There are several singularity theorems available which establish the nonspacelike geodesic incompleteness for a spacetime under different sets of conditions and applicable to different physical situations. However, the most general of these is the HawkingPenrose theorem (Hawking and Penrose, 1970), which is applicable in both the collapse situation and cosmological scenario. The main idea of the proof of such a theorem is the following. Using the causal structure analysis it is shown that there must be maximal length timelike curves between certain pairs of events in the spacetime. Now, a causal geodesic which is both future and past complete must contain pairs of conjugate points if satisfies an energy condition. This is then used to draw the necessary contradiction to show that must be non-spacelike geodesically incomplete.
Theorem A spacetime cannot be timelike and null geodesically complete if the following are satisfied: (1) for all non-spacelike vectors ;
(2) the generic condition is satisfied, that is, every non-spacelike geodesic contains a point at which , where is the tangent to the nonspacelike geodesic;
(3) the chronology condition holds on ; that is, there are no closed timelike curves in the spacetime, and,
(4) there exists in either a compact achronal set (i.e. a set no two points of which are timelike related) without edge or a closed trapped surface, or a point such that for all past directed null geodesics from , eventually must be negative.
The main idea of the proof is the following. One shows that the following three cannot hold simultaneously: (a) every inextendible non-spacelike geodesic contains pairs of conjugate points;
(b) the chronology condition holds on ;
(c) there exists an achronal set in such that or is compact.
In the above, and indicate the future and past horismos for the set (for further definitions and detail we refer to Hawking and Ellis (1973), or Joshi (1993).
We note that while the geodesic incompleteness, as a definition of spacetime singularities, allows various theorems to be proved on the existence of singularities, it does not capture all possible singular behaviors for a spacetime. It also does not imply that the singularity predicted is necessarily a physically relevant powerful curvature singularity. It does of course include many cases where that will be the case. We discuss below such a scenario and the criterion for the singularity to be physically relevant and important.
V Strong Curvature Singularities
As we see above, the existence of an incomplete nonspacelike geodesic or the existence of an inextendible nonspacelike curve which has a finite length as measured by a generalized affine parameter, implies the existence of a spacetime singularity. The generalized affine length for such a curve is defined as (Hawking and Ellis, 1973),
which is a finite quantity. The s are the components of the tangent to the curve in a parallel propagated tetrad frame along the curve. Each such incomplete curve defines a boundary point of the spacetime which is a singularity.
The important point now is, in order to call this a genuine physical singularity, one would typically like to associate such a singularity with unboundedly growing spacetime curvatures. If all the curvature components and the scalar polynomials formed out of the metric and the Riemann curvature tensor remained finite and well-behaved in the limit of approach to the singularity along an incomplete non-spacelike curve, it may be possible to remove such a singularity by extending the spacetime when the differentiability requirements are lowered (Clarke, 1986).
There are several ways in which such a requirement can be formalized. For example, a parallely propagated curvature singularity is the one which is the end point of at least one nonspacelike curve on which the components of the Riemann curvature tensor are unbounded in a parallely propagated frame. On the other hand, a scalar polynomial singularity is the one for which a scalar polynomial in the metric and the Riemann tensor takes an unboundedly large value along at least one nonspacelike curve which has the singular end point. This includes the cases such as the Schwarzschild singularity where the Kretschmann scalar blows up in the limit as .
What is the guarantee that such curvature singularities will at all occur in general relativity? The answer to this question for the case of parallely propagated curvature singularities is provided by a theorem of Clarke (1975) which establishes that for a globally hyperbolic spacetime which is inextendible, when the Riemann tensor is not very specialized in the sense of not being type-D and electrovac at the singular end point, then the singularity must be a parallely propagated curvature singularity.
Curvature singularities to be characterized below, also arise for a wide range of spacetimes involving gravitational collapse. This physically relevant class of singularities, called the strong curvature singularities was defined and analyzed by Tipler (1977); Tipler, Clarke and Ellis (1980), and Clarke and Królak (1986). The idea here is to define a physically all embracing strong curvature singularity in such a way so that all the objects falling within the singularity are destroyed and crushed to zero volume by the infinite gravitational tidal forces. The extension of spacetime 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 spacetime, thus avoiding the singularity. This is because for a strong curvature singularity defined in the above sense, no continuous extension of the spacetime may be possible.
Vi Can we avoid spacetime singularities?
Given the scenario above, it is now clear that spacetime singularities are an inevitable feature for most of the physically reasonable models of universe and gravitational systems within the framework of the Einstein theory of gravity. It is also seen that near such a spacetime singularity, the classical description that predicted it must itself breakdown. The existence of singularities in most of the classical theories of gravity, under reasonable physical conditions, imply that in a sense the Einstein gravity itself predicts its own limitations, namely that it predicts regions of universe where it must breakdown and a new and revised physical description must take over.
As the curvatures and all other physical quantities must diverge near such a singularity, the quantum effects associated with gravity are very likely to dominate such a regime. It is possible that these may resolve the classical singularity itself. But we have no viable and consistent quantum theory of gravity available as of today despite serious attempts. Therefore the issue of resolution of singularities as produced by classical gravity remains open.
The other possibility is of course that some of the assumptions of the singularity theorems may be violated so as to avoid the singularity occurrence. Even when these are fairly general, one could inquire whether some of these could actually breakdown and do not hold in physically realistic models. This could save us from the occurrence of singularities at the classical level itself. Such possibilities mean a possible violation of causality in the spacetime, or no trapped surfaces occurring in the dynamical evolution of universe, or possible violation of energy conditions.
The singularity theorem stated above and also other singularity theorems contain the assumption of causality or strong causality, or some other suitable causality condition. Then the alternative is that causality may be violated rather than a singularity occurring in the spacetime. So the implication of the singularity theorem stated above is when there is enough matter present in the universe, either the causality is violated or a boundary point must exist for the spacetime. In the cosmological case, such stress-energy density will be provided by the microwave background radiation, or in the case of stellar collapse trapped surfaces may form (Schoen and Yau, 1983), providing a condition leading to the formation of a singularity.
The Einstein equations by themselves do not rule out causality violating configurations which really depend on the global topology of the spacetime. Hence the question of causality violations versus spacetime singularity needs a careful examination as to whether causality violation could offer an alternative to singularity formation. Similarly, it also has to be inquired if the violation of energy conditions or non-occurrence of trapped surfaces may be realized so as to achieve the singularity avoidance in a spacetime. We discuss briefly some of these points below.
Vii Causality violations
The causal structure in a spacetime specifies what events can be related to each other by means of timelike or light signals. A typical causality violation would mean that an event could be in its own past, which is contrary to our normal understanding of time, and that of past and future. This has been examined in considerable detail in general relativity, and that there is no causality violation taking place in the spacetime is one of the important assumptions used by singularity theorems. However, general relativity allows for situations where causality violation is permitted in a spacetime. The Gödel solution (Gödel, 1949) allows the existence of a closed timelike curve through every point of the spacetime.
One would of course like to rule out if possible the causality violations on physical grounds, treating them as a very pathological behaviour in that in such a case one would be able to enter one’s own past. However, as they are allowed in principle in general relativity, so one must rule them out only by an additional assumption. The question then is, can one avoid the spacetime singularities if one allows for the violation of causality? This has been considered by researchers and it is seen that the causality violation in its own right creates spacetime singularities again under certain reasonable conditions. Thus, this path of avoiding spacetime singularities does not appear to be very promising.
Specifically, the question of finite causality violations in asymptotically flat spacetime was examined by Tipler (1976, 1977). This showed that the causality violation in the form of closed timelike lines is necessarily accompanied by incomplete null geodesics, provided the strong energy condition is satisfied for all null vectors and if the generic condition is satisfied. It was assumed that the energy density has a positive minimum along past directed null geodesics.
There is in fact a heirarchy of causality conditions available for a spacetime. It may be causal in the sense of having no closed nonspacelike curves. But given an event, future directed nonspacelike curves from the same could return to its arbitrarily close neighborhood in the spacetime. This is as bad as a causality violation itself. The higher order causality conditions such as strong causality and stable causality rule out such behaviour. Of the higher order causality conditions, much physical importance is attached to the stable causality which ensures that if is causal, its causality should not be disturbed with small perturbations in the metric tensor. Presumably, the general relativity is a classical approximation to some, as yet unknown, quantum theory of gravity in which the value of the metric at a point will not be exactly known and small fluctuations in the value must be taken into account.
Results on causality violations and higher-order causality violations with reference to occurrence of singularities were obtained by Joshi (1981), and Joshi and Saraykar (1986), who show that the causality violations must be accompanied by singularities even when the spacetime is causal but the higher order causality conditions are violated. Thus we know that for a causal spacetime the violations of higher order causality conditions give rise to spacetime singularities. Another question examined was that of measure of causality violating sets when such a violation occurs. It turns out that in many cases, the causality violating sets in a spacetime will have a zero measure, and thus such a causality violation may not be taken very seriously. Also, Clarke and Joshi (1988) studied global causality violation for a reflecting spacetime and the theorems of Kriele (1990) improved some of the conditions under which the results on chronology violations implying the singularities have been obtained. Also, global causality violating spacetimes were studied by Clarke and de Felice (1982). What we discussed above implies that if the causality of breaks down with the slightest perturbation of the metric then this must be accompanied by the occurrence of spacetime singularities.
As a whole, the above results imply that violating either causality or any of the higher-order causality conditions may not be considered a good alternative to the occurrence of spacetime singularities. There are also philosophical problems connected with the issue of causality violation such as entering one’s own past. But even if one allowed for the causality violations, the above results show that these are necessarily accompanied by spacetime singularities again.
Viii Energy conditions and trapped surfaces
Another possibility to avoid singularities is a possible violation of energy conditions. This is another of the assumptions in the proofs for singularity theorems. In fact, this possibility has also been explored in some detail and it turns out that as long as there is no gross or very powerful violation of energy conditions over global regions in the universe, this would not help avoid singularities either.
For example, the energy condition could be violated locally at certain spacetime points, or in certain regions of spacetimes due to peculiar physics there. But as long as it holds on an average, in the sense that the stress-energy density is positive in an integrated sense, then still spacetime singularities do occur (for a discussion and references, see e.g. Joshi, 1993).
On a global scale, there is evidence now that the universe may be dominated by a dark energy field. There is no clarity as to what exactly such a field would be and what would be its origin. It could be due to scalar fields or ghost fields floating in the universe, or due to a non-zero positive cosmological constant present in the Einstein equations. In such a case, the weak or strong energy conditions may be violated depending on the nature of these exotic fields. However, in the earlier universe of a matter dominated phase, the positive matter fields would again dominate, thus respecting the energy conditions even if they are violated at the present epoch.
Again, the above discussion is in the context of a cosmological scenario. When it comes to the gravitational collapse of massive stars, clearly their density and overall energy content are dominated by the ordinary matter fields we are much more familiar with. Such matter certainly respects the energy conditions modulo some minor violations if at all any. Thus for gravitational collapse of massive stars, one would expect the energy conditions to hold and the conclusions on singularity occurrence stated above would apply.
Yet another possibility to avoid singularity is to avoid trapped surfaces occurring in the spacetime. Indeed, such a route can give rise to geodesically complete spacetimes, as was shown by Senovilla (1990). As for the cosmological scenario, basically this means and amounts to the condition that the matter energy densities must fall off sufficiently rapidly on any given spacelike surface, and in an averaged sense, in order to avoid the cosmological trapped surfaces. Whether such a condition is realizable in the universe would have to be checked through observational tests. A sufficiently uniform energy density, such as say the microwave background radiation could in turn cause the cosmic trapping. As for the massive stars, the densities are of course very high indeed and would only grow for example in a gravitational collapse. Therefore in collapse scenarios, the trapped surfaces are unlikely to be avoided.
Further to the considerations such as above, if we accept on the whole that spacetime singularities do occur under fairly general conditions in the framework of the Einstein theory of gravity, or for classical gravity in general, then one must consider physical implications and consequences of such a scenario for physics and cosmology. As we noted earlier, two main arenas of physical relevance where spacetime singularities will be of interest are the cosmological situation and the gravitational collapse scenarios.
Ix Fundamental implications and challenges
The existence of spacetime singularities in Einstein gravity and other classical theories of gravitation poses intriguing challenges and fundamental questions in physics as well as cosmology. These would have farreaching consequences on our current understanding of the universe and how we try to further model it, as we shall try to bring out in rest of this article.
The inevitable existence of singularities for wide classes of rather general models of spacetimes means that the classical gravity evolutions necessarily give rise to regions in the spacetime universe where the densities and spacetime curvatures would really grow arbitrarily high without any bounds, and where all other relevant physical parameters also would diverge.
To take the first physical scenario, such a phenomenon in cosmology would correspond to a singularity that will represent the origin of the universe. Secondly, whenever locally a large quantity of matter and energy collapses under the force of its own gravity, a singularity will occur. This later situation will be effectively realized in the gravitational collapse of massive stars in the universe, which collapse and shrink catastrophically under their self-gravity, when the star has exhausted its nuclear fuel within that earlier supplied the internal pressures to halt the infall due to gravity.
Over past decades, once the existence of spacetime singularities was accepted, there have been major efforts to understand the physics in the vicinity of the same. In the cosmological case, this has given rise to an entire physics of the early universe, which depicts the few initial moments immediately after the big bang singularity from which the universe is supposed to have emerged. The complexities in this case have been enormous, both physicswise, and conceptually. The physics complexities arise because when trying to understand physics close to the hot big bang singularity, we are dealing with the highest energy scales, never seen earlier in any laboratory physics experiments. Our particle physics theories are then to be stretched to the extreme where there is no definite or unique framework available to deal with these phenomena. Understanding early universe physics has of course very big consequences in that it governs the most important physical phenomena such as the later galaxy formation in the universe, and other issues related to the large scale structure of the universe.
As for the conceptual issues, the simultaneous big bang singularity gives rise to a host of problems and puzzles. One of these is the ‘horizon problem’ which arises due to the causal structure of this spacetime. Distinct regions of the universe simply cannot interact with each other due to the cosmic horizons and it becomes extremely difficult to explain the average overall current homogeneity of the universe on a large enough scale. There are also other issues such as why the current universe is looking so flat, which is called the ‘flatness’ problem. As a possible resolve to these dilemmas, the inflationary models for early universe have been proposed, various facets of which are still very much under an active debate.
The key issue, as far as the big bang singularity is concerned, is that it happened only once in the past and there is no way to probe it any further other then current observations on universe and their extrapolation in the past. One must observe deeper and deeper into the space and back into time to understand the nature and physics of this early universe singularity.
As we mentioned above, the other class of such spacetime singularities will occur in the gravitational collapse of massive stars in the universe. Unlike the big bang, such a singularity will occur whenever a massive star in the universe collapses. This is therefore more amenable to observational tests.
There are rather fundamental cosmic conundrums associated with singularities of gravitational collapse. One of the most intriguing ones of these is the question whether such a singularity will be visible to external faraway observers in the universe. The big bang singularity is visible to us in principle, as we get to see the light from the same. But as we discuss below, the singularity of collapse can sometimes be hidden below the event horizons of gravity, and therefore not visible. The possibility that all singularities of collapse will be necessarily hidden inside horizons is called the cosmic censorship conjecture. As we discuss below this is not yet proved, and in fact singularities of collapse can be visible under many physical circumstances.
When visible or naked singularities develop in gravitational collapse, they give rise again to extremely intriguing physical possibilities and problems. The opportunity offered in that case is, we may have the possibility to observe the possible ultra-high energy physical processes occurring in such a region of universe, including the quantum gravity effects. Such observations of ultra-high energy events in the universe could provide observational tests and guide our efforts for a possible quantum theory of gravity. But a conundrum that is presented is, whether this would break the so called ‘classical predictability’ of the spacetime universe. We shall discuss this further below. On the other hand, even when the singularity is necessarily hidden within a black hole, that still gives rise to profound puzzles such as the ‘information paradox’, issues with unitarity and such other problems. So the point is, even if the cosmic censorship was correct and all singularities were hidden inside black holes only, still we shall be faced with many deep paradoxes, which are not unique to naked singularities only.
It would be only reasonable to say that all these deep physical as well as conceptual issues are closely connected with the existence and formation of spacetime singularities in the dynamical gravitational processes taking place in the universe. While the big bang singularity happened only once in the past, the singularities of collapse have in fact a repeated occurrence, and hence they possess an interesting observational perspective and potential. We shall therefore discuss the same in some detail below, while also providing the key ingredients of black hole physics in the process.
X Gravitational collapse
When a massive star more than about ten solar masses exhausted its internal nuclear fuel, it is believed to enter the stage of a continual gravitational collapse without any final equilibrium state. The star then goes on shrinking in its radius, reaching higher and higher densities. What would be the final fate of such an object according to the general theory of relativity? This is one of the central questions in relativistic astrophysics and gravitation theory today. It is suggested that the ultra-dense object that forms as a result of the collapse could be a black hole in space and time from which not even light rays escape. Alternatively, if an event horizon of gravity fails to cover the final super ultra-dense crunch, it could be a visible singularity in the spacetime which could causally interact with the outside universe and from which region the emissions of light and matter may be possible.
The issue is of importance from both the theoretical as well as observational point of view. At the theoretical level, working out the final fate of collapse in general relativity is crucial to the problem of asymptotic predictability, namely, whether the singularities forming as the collapse endstate will be necessarily covered by the event horizons of gravity. Such a censorship hypothesis remains fundamental to the theoretical foundations of black hole physics and its many recent astrophysical applications. These include the area theorem for black holes, laws of black hole thermodynamics, Hawking radiation, predictability in a spacetime, and on observational side the accretion of matter by black holes, massive black holes at the center of galaxies etc. On the other hand, existence of visible or naked singularities offer a new approach on these issues requiring modification and reformulation of our usual theoretical conception on black holes.
We mention and discuss below some of these recent developments in these directions, examining the possible final fate of gravitational collapse. To investigate this issue, dynamical collapse scenarios have been examined in the past decade or so for many cases such as clouds composed of dust, radiation, perfect fluids, or matter with more general equations of state (see e.g. Joshi 2008 for references and details). We discuss these developments and the implications for a possible formulation of cosmic censorship are indicated, mentioning the open problems in the field.
Xi Spherical collapse and the black hole
To understand the final fate of a massive gravitationally collapsing object we first outline here the spherically symmetric collapse situation. Though this is an idealization, the advantage is one can solve the case analytically to get exact results when matter is homogeneous dust. In fact, the basic motivations for the idea and theory of black holes come from this collapse model, first worked out by Oppenheimer and Snyder (1939) and Datt (1938).
Consider a gravitationally collapsing spherical massive star. The interior solution for the object depends on the properties of matter, equation of state, and the physical processes taking place within the stellar inside. But assuming the matter to be pressureless dust allows the problem to be solved analytically, providing important insights. The energy-momentum tensor is given by , and the Einstein equations are to be solved for the spherically symmetric metric. The metric potentials can be solved and the interior geometry of the collapsing dust ball is given by,
where is the metric on two-sphere. The interior geometry of cloud matches at the boundary with the exterior Schwarzschild spacetime.
Fig 1: The gravitational collapse of a spherically symmetric homogeneous dust cloud. The event horizon forms prior to the singularity and the collapse endstate is a black hole.
The basic features of such a configuration are given in Fig 1. The collapse is initiated when the star surface is outside the Schwarzschild radius and a light ray from the surface of the star can escape to infinity. But once the star has collapsed below , a black hole region of no escape develops in the spacetime which is bound by the event horizon at . Any point in this empty region represents a trapped surface, a two-sphere for which both the outgoing and ingoing families of null geodesics emitted from this point converge and so no light comes out of this region. Then, the collapse to an infinite density and curvature singularity at becomes inevitable in a finite proper time as measured by an observer on the surface of the star. The black hole region in the resulting vacuum Schwarzschild geometry is given by with the event horizon as the outer boundary at which the radial outwards photons stay where they are but all the rest are dragged inwards. No information from the black hole can propagate outside to observers far away. We thus see that the collapse gives rise to a black hole in the spacetime which covers the resulting spacetime singularity. The ultimate fate of the star undergoing such a collapse is an infinite curvature singularity at , completely hidden within the black hole. No emissions or light rays from the singularity go out to observer at infinity and the singularity is causally disconnected from the outside spacetime.
The question now is whether one could generalize these conclusions on the occurrence of spacetime singularity in collapse for more general forms of matter or for non-spherical situations, or possibly for small perturbations away from spherical symmetry. It is known using the stability of Cauchy development in general relativity that the formation of trapped surfaces is indeed a stable property when departures from spherical symmetry are taken into account. The argument essentially is the following: Considering a spherical collapse evolution from given initial data on a partial Cauchy surface , we find the formation of trapped surfaces in the form of all the spheres with in the exterior Schwarzschild geometry. The stability of Cauchy development then implies that for all initial data sufficiently near to the original data in the compact region , where and denote the causal futures or pasts of respectively, the trapped surfaces still must occur. Then, the curvature singularity of spherical collapse also turns out to be a stable feature, as implied by the singularity theorems which show that the closed trapped surfaces always imply the existence of a spacetime singularity under reasonable general enough conditions.
Xii Cosmic censorship hypothesis
The real stars in the universe are not made of pressureless homogeneous dust. They are inhomogeneous, typically with density higher at center, may have non-trivial matter with an equation of state as yet unknown, and there is spin. Will a physically realistic collapse of such a star necessarily end up in the black hole final state only, just as in the idealized case of the Oppenheimer- Snyder-Datt model above? In other words, while the more general gravitational collapse will also end up in a spacetime singularity, the question is whether the singularity will be again necessarily covered inside an event horizon of gravity.
In fact, there is no proof available that such a singularity will continue to be hidden within a black hole and remain causally disconnected from outside observers, even when the collapse is not exactly spherical or when the matter does not have the form of exact homogeneous dust. Therefore, in order to generalize the notion of black holes to more general gravitational collapse situations, it becomes necessary to rule out such naked or visible singularities by means of an explicit assumption. This is stated as the cosmic censorship hypothesis, which essentially says that if is a partial Cauchy surface from which the collapse commences, then there are no naked singularities to the future of which could be seen from the future null infinity. This is true for the spherical homogeneous dust collapse, where the resulting spacetime is future asymptotically predictable and the censorship holds. In such a case, the breakdown of physical theory at the spacetime singularity does not disturb prediction in future for the outside asymptotically flat region.
The corresponding scenario for other more general collapse situations, when inhomogeneities or non-sphericity and such other physically realistic features are allowed for has to be investigated. The answer in general is not known either as a proof of the future asymptotic predictability for general spacetimes or in the form of any general theorem on cosmic censorship. It is clear that the assumption of censorship in a suitable form is crucial to the basic results in black hole physics. Actually when one considers the gravitational collapse in a generic situation, the very existence of black holes requires this hypothesis.
To establish the censorship by means of a rigorous proof certainly requires a much more precise formulation of the hypothesis. The statement that result of complete gravitational collapse must be a black hole only and not a naked singularity, or all singularities of collapse must be hidden inside black holes, is not rigorous enough. Because under general circumstances, the censorship or asymptotic predictability is false as one could always choose a spacetime manifold with a naked singularity which would be a solution to Einstein’s equations if we define . So certain conditions on the stress-energy tensor are required at the minimum, say for example, an energy condition. However, to obtain an exact set of conditions on matter fields to prove the censorship hypothesis turns out to be an extremely difficult task as yet not accomplished.
The requirements in the black hole physics and general predictability have led to several different possible formulations of cosmic censorship hypothesis, none of which proved as yet. The weak cosmic censorship, or asymptotic predictability, postulates that the singularities of gravitational collapse cannot influence events near the future null infinity. The other version called the strong cosmic censorship, is a general predictability requirement on any spacetime, stating that all physically reasonable spacetimes must be globally hyperbolic (see e.g. Penrose, 1979). The global hyperbolicity here means that we must be able to predict the entire future and past evolutions in the universe by means of the Einstein equations, given the initial data on a three-dimensional spacelike hypersurface in the spacetime.
On a further analysis, it however becomes clear that such formulations need much more sharpening if at all any concrete proof is to be obtained. In fact, as for the cosmic censorship, it is a major problem in itself to find a satisfactory and mathematically rigorous formulation of what is physically desired to be achieved. Presently, there is no general proof available for any suitably formulated version of the weak censorship. The main difficulty seems to be that the event horizon is a feature depending on the whole future behavior of the solution over an infinite time period, but the present theory of quasi-linear hyperbolic equations guarantee the existence and regularity of solutions over only a finite time internal. It is clear that even if true, any proof for a suitable version of weak censorship seems to require much more knowledge on general global properties of Einstein equations than is known currently.
To summarize the situation, cosmic censorship is clearly a crucial assumption underlying all of the black hole physics and gravitational collapse theory and related important areas. The first major task here would be to formulate rigorously a satisfactory statement for cosmic censorship, which if not true would throw the black hole dynamics into serious doubt. That is why censorship is one of the most important open problems for gravitation theory today. No proof, however, seems possible unless some major theoretical advances by way of mathematical techniques and understanding global structure of Einstein equations are made, and direction for such theoretical advances needed is far from clear at present.
We therefore conclude that the first and foremost task at the moment is to carry out a detailed and careful examination of various gravitational collapse scenarios to examine for their end states. It is only such an investigation of more general collapse situations which could indicate what theoretical advances to expect for any proof, and what features to avoid while formulating the cosmic censorship. Basically, we still do not have sufficient data and information available on the various possibilities for gravitationally collapsing configurations so as to decide one way or other on the issue of censorship.
In recent years, many investigations have been carried out from such a perspective on gravitational collapse, either for inhomogeneous dust collapse or with more general matter fields. It turns out that the collapse outcome is not a black hole always and the naked singularity final state can arise in a variety of situations. In the next sections we discuss some of these developments.
Xiii Inhomogeneous Dust Collapse
Since we are interested in collapse, we require that the spacetime contains a regular initial spacelike hypersurface on which the matter fields as represented by the stress-energy tensor , have a compact support and all physical quantities are well-behaved on this surface. Also, the matter should satisfy a suitable energy condition and the Einstein equations are satisfied. We say that the spacetime contains a naked singularity if there is a future directed nonspacelike curve which reaches a far away observer at infinity in future, which in the past terminates at the singularity.
As an immediate generalization of the Oppenheimer-Snyder-Datt homogeneous dust collapse, one could consider the collapse of inhomogeneous dust and examine the nature and structure of resulting singularity with special reference to censorship and the occurrence of black holes and naked singularities. The main motivation to discuss this situation is this provides a clear picture in an explicit manner of what is possible in gravitational collapse. One could ask how the conclusions given above for homogeneous collapse are modified when the inhomogeneities of matter distribution are taken into account. Clearly, it is important to include effects of inhomogeneities because typically a physically realistic collapse would start from an inhomogeneous initial data with a centrally peaked density profile.
This question of inhomogeneous dust collapse has attracted attention of many researchers and it is seen that the introduction of inhomogeneities leads to a rather different picture of gravitational collapse. It turns out that while homogeneous collapse leads to a black hole formation, introduction of any physically realistic inhomogeneity, e.g. the density peaked at the center of the cloud and slowly decreasing away, leads to a naked singularity final state for the collapse. This is certainly an intriguing result implying that the black hole formation in gravitational collapse need not be such a stable phenomenon as it was thought to be the case.
The problem was investigated in detail using the Tolman-Bondi-Lemaitre models, which describe gravitational collapse of an inhomogeneous spherically symmetric dust cloud (Joshi and Dwivedi 1993). This is an infinite dimensional family of asymptotically flat solutions of Einstein equations, which is matched to the Schwarzschild spacetime outside the boundary of the collapsing star. The Oppenheimer-Snyder-Datt model is a special case of this class of solutions.
It is seen that the introduction of inhomogeneities leads to a rather different picture of gravitational collapse. The metric for spherically symmetric collapse of inhomogeneous dust, in comoving coordinates , is given by,
where is the stress-energy tensor, is the energy density, and is a function of both and given by
Here the dot and prime denote partial derivatives with respect to the parameters and respectively. As we are considering collapse, we require The quantities and are arbitrary functions of and is the proper area of the mass shells. The physical area of such a shell at goes to zero when . For gravitational collapse situation, we take to have compact support on an initial spacelike hypersurface and the spacetime is matched at some to the exterior Schwarzschild field with total Schwarzschild mass enclosed within the dust ball of coordinate radius of . The apparent horizon in the interior dust ball lies at .
Using this framework, the nature of the singularity can be examined. In particular, the problem of nakedness or otherwise of the singularity can be reduced to the existence of real, positive roots of an algebraic equation , constructed out of the free functions and and their derivatives, which constitute the initial data of this problem. If the equation has a real positive root, the singularity could be naked. In order to be the end point of null geodesics at least one real positive value of should satisfy the above. If no real positive root of the above is found, the singularity is not naked. It should be noted that many real positive roots of the above equation may exist which give the possible values of tangents to the singular null geodesics terminating at the singularity in the past. Suppose now is a simple root to . To determine whether is realized as a tangent along any outgoing singular geodesics to give a naked singularity, one can integrate the equation of the radial null geodesics in the form and it is seen that there is always at least one null geodesic terminating at the singularity , with . In addition there would be infinitely many integral curves as well, depending on the values of the parameters involved, that terminate at the singularity. It is thus seen that the existence of a positive real root of the equation is a necessary and sufficient condition for the singularity to be naked. Finally, to determine the curvature strength of the naked singularity at , , one may analyze the quantity near the singularity. Standard analysis shows that the strong curvature condition is satisfied, in that the above quantity remains finite in the limit of approach to the singularity. The spacetime picture for a collapse terminating in a naked singularity is given in Fig 2.
Fig 2: The gravitational collapse of a spherical but inhomogeneous dust cloud with a density profile peaked at the center. The event horizon no longer forms prior to the singularity and the collapse endstate is a naked singularity.
The assumption of vanishing pressures, which could be important in the final stages of the collapse, may be considered as the limitation of dust models. On the other hand, it is also argued sometimes that in the final stages of collapse the dust equation of state could be relevant and at higher and higher densities the matter may behave much more like dust. Further, if there are no large negative pressures (as implied by the validity of the energy conditions), then the pressure also might contribute gravitationally in a positive manner to the overall effect of dust and may not alter the final conclusions.
Xiv Collapse with general matter fields
It is clearly important to consider collapse situations which consider matter with non-zero pressures and with reasonable equations of state. It is possible that pressures may play an important role for the later stages of collapse and one must investigate the possibility if pressure gradients could prevent the occurrence of naked singularity.
Many collapse scenarios have been considered by now with non-zero pressures and physically reasonable equations of state. What one needs to examine here again is the existence, the termination of future directed nonspacelike geodesic families at the singularity in the past, and the strength of such a singularity for collapse with non-zero pressure.
A useful insight into this issue is provided by self-similar collapse for a perfect fluid with a linear equation of state . A numerical treatment of self-similar perfect fluid collapse was given by Ori and Piran (1987) and the analytic consideration for the same was given by Joshi and Dwivedi (1992). It is seen that the collapse evolutions as allowed by the Einstein equations permit for both the black hole and naked singularity final states. If in a self-similar collapse a single null radial geodesic escapes the singularity, then in fact an entire family of nonspacelike geodesics would also escape provided the positivity of energy density is satisfied. It also follows that no families of nonspacelike geodesics would escape the singularity, even though a single null trajectory might, if the weak energy condition is violated. The singularity will be globally visible to faraway observers in the spacetime for a wide set of conditions. What these results show is, naked singularity is not avoided by introduction of non-zero pressures or a reasonable equation of state.
Actually, consideration of matter forms such as directed radiation, dust, perfect fluids etc imply a similar general pattern emerging as far as the final outcome of collapse is concerned. Basically, the result that emerges is, depending on the nature of the regular initial data in terms of the density and pressure profiles, the Einstein equations permit both the classes of dynamical evolutions, those leading to either of the black hole or naked singularity final states.
Hence one could ask the question whether the final fate of collapse would be independent of the form of the matter under consideration. An answer to this is useful because it was often thought that once a suitable form of matter with an appropriate equation of state, also satisfying energy conditions is considered then there may be no naked singularities. Of course, there is always a possibility that during the final stages of collapse the matter may not have any of the forms such as dust or perfect fluids considered above, because such relativistic fluids are phenomenological and perhaps one must treat matter in terms of some fundamental field, such as for example, a massless scalar field. In that context, a naked singularity is also seen to form for the scalar field collapse (Choptuik, 1993), though for a fine-tuned initial data.
In the above context, efforts in the direction of understanding collapse final states for general matter fields are worth mentioning, which generalize the above results on perfect fluid to matter forms without any restriction on the form of , with the matter satisfying the weak energy condition. A consideration to a general form of matter was given by Lake (1992); and by Szekeres and Iyer (1993), who do not start by assuming an equation of state but a class of metric coefficients is considered with a certain power law behavior. Also, Joshi and Dwivedi (1999) and Goswami and Joshi (2007), gave results in this direction. The main argument is along the following lines. It was pointed out above that naked singularities could form in gravitational collapse from a regular initial data, from which non-zero measure families of nonspacelike trajectories come out. The criterion for the existence of such singularities was characterized in terms of the existence of real positive roots of an algebraic equation constructed out of the field variables. A similar procedure is developed now for general form of matter. In comoving coordinates, the general matter can be described by three free functions, namely the energy density and the radial and tangential pressures. The existence of naked singularity is again characterized in terms of the real positive roots of an algebraic equation, constructed from the equations of nonspacelike geodesics which involve the three metric functions. The field equations then relate these metric functions to the matter variables and it is seen that for a subspace of this free initial data in terms of matter variables, the above algebraic equation will have real positive roots, producing a naked singularity in the spacetime. When there are no such roots existing, the endstate is a black hole.
It follows that the occurrence or otherwise of the naked singularity is basically related to the choice of initial data to the Einstein field equations as determined by the allowed evolutions. Therefore these occur from regular initial data within the general context considered, subject to the matter satisfying weak energy condition. The condition on initial data which leads to the formation of black hole is also characterized.
It would then appear that the occurrence of naked singularity or a black hole is more a problem of choice of the initial data for field equations rather than that of the form of matter or the equation of state (see Fig 3).
Fig 3: For a generic collapse of a general matter field, the collapse final state can be either a black hole or a naked singularity depending on the dynamical evolution chosen as allowed by the Einstein equations.
Such a conclusion has an important implication for cosmic censorship in that in order to preserve the same one has to avoid all such regular initial data causing naked singularity, and hence a much deeper understanding of the initial data space is required in order to determine such initial data and the kind of physical parameters they would specify. In other words, this classifies the range of physical parameters to be avoided for a particular form of matter. Such an understanding would also pave the way for black hole physics to use only those ranges of allowed parameter values which produce black holes only, thus putting black hole physics on a more firm footing.
Xv Non-spherical Collapse and numerical simulations
Basically, the results and detailed studies such as above on gravitational collapse show that the cosmic censorship cannot hold in an unqualified general form. It must be properly fine-tuned and only under certain suitably restrictive conditions on collapse the black holes will form.
An important question at the same time is: What will be the final fate of gravitational collapse which is not spherically symmetric? The main phases of spherical collapse of a massive star would be typically instability, implosion of matter, and subsequent formation of an event horizon and a spacetime singularity of infinite density and curvature with infinite gravitational tidal forces. This singularity may or may not be fully covered by the horizon as we discussed above.
As noted, small perturbations over the sphericity would leave the situation unchanged in the sense that an event horizon will continue to form in the advanced stages of the collapse. The next question then is, do horizons still form when the fluctuations from the spherical symmetry are high and the collapse is highly non-spherical? It was shown by Thorne (1972), for example, that when there is no spherical symmetry, the collapse of infinite cylinders do give rise to naked singularities in general relativity, which are not covered by horizons. This situation motivated Thorne to propose the hoop conjecture for finite systems in an asymptotically flat spacetime for the final fate of non-spherical collapse: The horizons of gravity form when and only when a mass gets compacted in a region whose circumference in every direction obeys . Thus, unlike the cosmic censorship, the hoop conjecture does not rule out all naked singularities but only makes a definite assertion on the occurrence of event horizons in gravitational collapse. The hoop conjecture is concerned with the formation of event horizons, and not with naked singularities. Thus, even when event horizons form, say for example in the spherically symmetric case, it does not rule out the existence of naked singularities, or it does not imply that such horizons must always cover the singularities.
When the collapse is sufficiently aspherical, with one or two dimensions being sufficiently larger than the others, the final state of collapse could be a naked singularity, according to the hoop conjecture. Such a situation is inspired by the Lin, Mestel and Shu (1965) instability consideration in Newtonian gravity, where a non-rotating homogeneous spheroid collapses maintaining its homogeneity and spheroidicity but with growing deformations. If the initial condition is that of a slightly oblate spheroid, the collapse results into a pancake singularity through which the evolution could proceed. But for a slightly prolate spheroidal configuration, the matter collapses to a thin thread which results into a spindle singularity. The gravitational potential and the tidal forces blow up as opposed to only density blowing up so it is a serious singularity. Even in the case of an oblate collapse, the passing of matter through the pancake causes prolateness and subsequently a spindle singularity again results without the formation of any horizon.
It is clear though, that the non-spherical collapse scenario is rather complex to understand, and a recourse to the numerical simulations of evolving collapse models may greatly enhance our understanding on possible collapse final states in this case. In such a context, the numerical calculations of Shapiro and Teukolsky (1991) indicated conformity with the hoop conjecture. They evolved collissionless gas spheroids in full general relativity which collapse in all cases to singularities. When the spheroid is sufficiently compact a black hole may form, but when the semimajor axis of the spheroid is sufficiently large, a spindle singularity forms without an apparent horizon forming. This gives rise to the possibility of occurrence of naked singularities in collapse of finite systems in asymptotically flat spacetimes which violate weak cosmic censorship but are in accordance with the hoop conjecture.
We note that the Kerr black hole is believed to be the unique stationary solution in Einstein gravity when the mass and rotation parameters are included. But it is to be noted that while the Schwarzschild black hole is the final endstate of the homogeneous dust collapse, we have no interior solution for a rotating collapsing cloud. In other words, an exterior Kerr geometry has no internal solution in general relativity. We therefore do not really know the final fate of a gravitational collapse with rotation. To understand the same, numerical simulations in full general relativity will be of great value. There are many such numerical programs in the making currently to deal with this problem of modeling a rotating collapsing massive star. The idea here is to include rotation in collapse and then to let the Einstein equations evolve the collapse to see if the Kerr black hole necessarily emerges as final state (see e.g. Giacomazzo et al, 2011 and references therein). and references therein. It is worth noting that numerical simulations in higher dimensions have also recently produced some very intriguing naked singularity formation scenarios Lehner et al (2010).
We finally note that apart from such numerical simulations, certain analytic treatments of aspherical collapse are also available. For example, the non-spherical Szekeres models for irrotational dust without any Killing vectors, generalizing the spherical Tolman-Bondi-Lemaitre collapse, were studied by Joshi and Krolak (1986) to deduce the existence of strong curvature naked singularities. While this indicates that naked singularities are not necessarily confined to spherical symmetry only, it is to be noted that dynamical evolution of a non-spherical collapse still remains a largely uncharted territory.
Xvi Are naked singularities stable and generic?
Naked singularities may develop in gravitational collapse, either spherical or otherwise. However, if they are not either generic or stable in some suitable sense, then they may not be necessarily physically relevant. An important question then is the genericity and stability of naked singularities arising from regular initial data. Will the initial data subspace, which gives rise to naked singularity as end state of collapse, have a vanishing measure in a suitable sense? In that case, one would be able to reformulate more suitably the censorship hypothesis, based on a criterion that naked singularities could form in collapse but may not be generic.
We note here that the genericity and stability of the collapse outcomes, in terms of black holes and naked singularities need to be understood carefully and in further detail. It is by and large well-accepted now, that the general theory of relativity does allow and gives rise to both black holes and naked singularities as final fate of a continual gravitational collapse, evolving from a regular initial data, and under reasonable physical conditions. What is not fully clear as yet is the distribution of these outcomes in the space of all allowed outcomes of collapse. The collapse models discussed above and considerations we gave here would be of some help in this direction, and may throw some light on the distribution of black holes and naked singularity solutions in the initial data space. For some considerations on this issue, especially in the context of scalar field collapse, we refer to Christodoulou (1999), and Joshi, Malafarina and Saraykar (2011), and references therein, for further discussion. For the case of inhomogeneous dust collapse, the black hole and naked singularity spaces are shown in Fig 4.
Fig 4: In the spaces of mass functions and energy functions, the initial data leading to black holes and naked singularities are shown.
The important point, however, is in general relativity there is no well defined concept or formulation as to what to call generic and stable outcomes, unlike the Newtonian case. In other words, there are no well-defined criteria or definition available as to what is meant by stability in general relativity. The ambiguity mainly arises because of non-unique topologies on the space of all Lorentzian metrics on a given spacetime manifold, and a similar non-uniqueness of measures. Under the situation, there is no easy way to answer this question in any unique and definite manner, and people generally resort to the physical meaningfulness of the collapse scenario which gives rise to either the black hole or the naked singularity outcome.
From such a perspective, it is natural and meaningful to ask here, what is really the physics that causes a naked singularity to develop in collapse, rather than a black hole? We need to know how at all particles and energy are allowed to escape from extremely strong gravity fields. We have examined this issue in some detail to bring out the role of inhomogeneities and spacetime shear towards distorting the geometry of horizons that form in collapse (see Fig 5).
Fig 5: The apparent horizon formation is delayed depending on the amount of inhomogeneity present as the collapse proceeds (from Joshi, Dadhich and Maartens, 2002).
In Newtonian gravity, it is only the matter density that determines the gravitational field. In Einstein theory, however, density is only one attribute of the overall gravitational field, and the various curvature components and scalar quantities play an equally important role to dictate what the overall nature of the field is. What we showed is, once the density is inhomogeneous or higher at the center of the collapsing star, this rather naturally delays the trapping of light and matter during collapse, which can in principle escape away. This is a general relativistic effect to imply that even if the densities are very high, there are paths available for light or matter to escape due to inhomogeneously collapsing matter fields. These physical features then naturally lead to a naked singularity formation (Joshi, Dadhich and Maartens, 2002).
As it turns out, it is the amount of inhomogeneity that counts towards distorting the apparent horizon formation. If it is very small, below a critical limit, a black hole will form, but with sufficient inhomogeneity the trapping is delayed to cause a naked singularity. This criticality also comes out in the Vaidya class of radiation collapse models, where it is the rate of collapse, that is how fast or slow the cloud is collapsing, that determines the black hole or naked singularity formation.
Xvii Astrophysical and observational aspects
It is clear that the black hole and naked singularity outcomes of a complete gravitational collapse for a massive star are very different from each other physically, and would have quite different observational signatures. In the naked singularity case, if it occurs in nature, we have the possibility to observe the physical effects happening in the vicinity of the ultra dense regions that form in the very final stages of collapse. However, in a black hole scenario, such regions are necessarily hidden within the event horizon of gravity.
There have been attempts where researchers explored physical applications and implications of the naked singularities (see e.g. Joshi and Malafarina 2011 and references in there). If we could find astrophysical applications of the models that predict naked singularities as collapse final fate, and possibly try to test the same through observational methods and the signatures predicted, that could offer a very interesting avenue to get further insight into the problem as a whole. An attractive recent possibility in that connection is to explore the naked singularities as possible particle accelerators (Patil and Joshi 2011), where the possibility also emerges that the Cauchy Horizons may not be innocuous, and high energy collisions could occur in the vicinity of the same if they are generated by naked singularity (see Fig 6).
Fig 6: Very high energy particle collisions can occur in the vicinity of the Cauchy horizon emerging from the naked singularity.
Also, the accretion discs around a naked singularity, wherein the matter particles are attracted towards or repulsed away from the singularities with great velocities could provide an excellent venue to test such effects and may lead to predictions of important observational signatures to distinguish the black holes and naked singularities in astrophysical phenomena. The question of what observational signatures would then emerge and distinguish the black holes from naked singularities is then necessary to be investigated, and we must explore what special astrophysical consequences the latter may have.
One may ask several intriguing questions such as:Where could the observational signatures of naked singularities lie? If we look for the sign of singularities such as the ones that appear at the end of collapse, we have to consider explosive and high energy events. In fact such models expose the ultra-high density region at the time of formation of the singularity while the outer shells are still falling towards the center. In such a case, shock-waves emanating from the superdense region at scales smaller than the Schwarzschild radius (that could be due to quantum effects or repulsive classical effects) and collisions of particles near the Cauchy horizon could have effects on the outer layers. These would be considerably different from those appearing during the formation of a black hole. If, on the other hand, we consider singularities such as the super-spinning Kerr solution we can look for different kinds of observational signatures. Among these the most prominent features deal with the way the singularity could affect incoming particles, either in the form of light bending, such as in gravitational lensing, particle collisions close to the singularity, or properties of accretion disks.
Essentially what we ask is: Whether we could test censorship using astronomical observations. With so many high technology power missions to observe the cosmos, can we not just observe the skies carefully to determine the validity or otherwise of the cosmic censorship? In this connection, several proposals to measure the mass and spin ratio for compact objects and for the galactic center have been made by different researchers. In particular, using pulsar observations it is suggested that gravitational waves and the spectra of X-rays binaries could test the rotation parameter for the center of our galaxy. Also, the shadow cast by the compact object can be used to test the same in stellar mass objects, or X-ray energy spectrum emitted by the accretion disk can be used. Using certain observable properties of gravitational lensing that depend upon rotation is also suggested (for references, see Joshi and Malafarina, 2011).
The basic issue here is that of sensitivity, namely how accurately and precisely can we measure and determine these parameters. A number of present and future astronomical missions could be of help. One of these is the Square-Kilometer Array (SKA) radio telescope, which will offer a possibility here, with a collecting area exceeding a factor of hundred compared to existing ones. The SKA astronomers point out they will have the sensitivity desired to measure the required quantities very precisely to determine the vital fundamental issues in gravitation physics such as the cosmic censorship, and to decide on its validity or otherwise. Other missions that could in principle provide a huge amount of observational data are those that are currently hunting for the gravitational waves. Gravitational wave astronomy has yet to claim its first detection of waves, nevertheless in the coming years it is very likely that the first observations will be made by the experiments such as LIGO and VIRGO that are currently still below the threshold for observation. Then gravitational wave astronomy will become an active field with possibly large amounts of data to be checked against theoretical predictions and it appears almost certain that this will have a strong impact on open theoretical issues such as the Cosmic Censorship problem.
There are three different kinds of observations that one could devise in order to distinguish a naked singularity from a black hole. The first one relies on the study of accretion disks. The accretion properties of particles falling onto a naked singularity would be very different from those of black hole of the same mass (see for example (Pugliese et al, 2011), and the resulting accretion disks would also be observationally different. The properties of accretion disks have been studied in terms of the radiant energy, flux and luminosity, in a Kerr-like geometry with a naked singularity, and the differences from a black hole accretion disk have been investigated. Also, the presence of a naked singularity gives rise to powerful repulsive forces that create an outflow of particles from the accretion disk on the equatorial plane. This outflow that is otherwise not present in the black hole case, could be in principle distinguished from the jets of particles that are thought to be ejected from black hole’s polar region and which are due to strong electromagnetic fields. Also, when charged test particles are considered the accretion disk’s properties for the naked singularity present in the Reissner-Nordstrom spacetime are seen to be observationally different from those of black holes.
The second way of distinguishing black holes from naked singularities relies on gravitational lensing. It is argued that when the spacetime does not possess a photon sphere, then the lensing features of light passing close to the singularity will be observationally different from those of a black hole. This method, however, does not appear to be very effective when a photon sphere is present in the spacetime (see e.g. Virbhadra et al 1998, and for recent update and further references, Sahu et al, 2012, 2013, and Joshi and Malafarina, 2011). Assuming that a Kerr-like solution of Einstein equations with massless scalar field exists at the center of galaxies, its lensing properties are studied and it was found that there are effects due to the presence of both the rotation and scalar field that would affect the behavior of the bending angle of the light ray, thus making those objects observationally different from black holes (see e.g. Schee and Stuchlik, 2009).
Finally, a third way of distinguishing black holes from naked singularities comes from particle collisions and particle acceleration in the vicinity of the singularity. In fact, it is possible that the repulsive effects due to the singularity can deviate a class of infalling particles, making these outgoing eventually. These could then collide with some ingoing particle, and the energy of collision could be arbitrarily high, depending on the impact parameter of the outgoing particle with respect to the singularity. The net effect is thus creating a very high energy collision that resembles that of an immense particle accelerator and that would be impossible in the vicinity of a Kerr black hole.
It was pointed our recently by Joshi, Malafarina and Ramesh Narayan (2011) that one could obtain equilibrium configurations as final outcome of a gravitational collapse. If such an object arises without trapped surfaces but with a singularity at the center (see Fig 7) then again the accretion disk properties are very different from a black hole of the same mass.
Fig 7: An equilibrium configuration can be obtained from gravitational collapse which halts asymptotically, and which would contain a central naked singularity.
Xviii Predictability and other cosmic puzzles
What then is the status of naked singularities versus censorship today? Can cosmic censorship survive in some limited and specialized form, and firstly, can we properly formulate it after all these studies in recent years on gravitational collapse? While this continues to be a major cosmic puzzle, recent studies on formation of naked singularities as collapse end states for many realistic models have brought to forefront some of the most intriguing basic questions, both at classical and quantum level, which may have significant physical relevance. Some of these are: Can the super ultra-dense regions forming in a physically realistic collapse of a massive star be visible to far away observers in space-time? Are there any observable astrophysical consequences? What is the causal structure of spacetime in the vicinity of singularity as decided by the internal dynamics of collapse which evolves from a regular initial data at an initial time? How early or late the horizons will actually develop in a physically realistic gravitational collapse, as determined by the astrophysical conditions within the star? When a naked singularity forms, is it possible to observe the quantum gravity effects taking place in the ultra-strong gravity regions? Can one possibly envisage a connection to observed ultra-high energy phenomena such as cosmic gamma ray bursts?
A continuing study of collapse phenomena within a general and physically realistic framework may be the only way to answers on some of these issues. This could lead us to novel physical insights and possibilities emerging out of the intricacies of gravitational force and nature of gravity, as emerging from examining the dynamical evolutions as allowed by Einstein equations.
Apart from its physical relevance, the collapse phenomena also have profound philosophical implications such as on the issue of predictability in the universe. We summarize below a few arguments, for and against it in the classical general relativity.
It is sometimes argued that breakdown of censorship means violation of predictability in spacetime, because we have no direct handle to know what a naked singularity may radiate and emit unless we study the physics in such ultra-dense regions. One would not be able then to predict the universe in the future of a given epoch of time as would be the case, for example, in the case of the Schwarzschild black hole that develops in Oppenheimer-Snyder collapse. A concern usually expressed is if naked singularities occurred as the final fate of gravitational collapse, predictability is violated in the spacetime, because the naked singularity is characterized by the existence of light rays and particles that emerge from the same. Typically, in all the collapse models discussed above, there is a family of future directed non-spacelike curves that reach external observers, and when extended in the past these meet the singularity. The first light ray that comes out from the singularity marks the boundary of the region that can be predicted from a regular initial Cauchy surface in the spacetime, and that is called the Cauchy horizon for the spacetime. The causal structure of spacetime would differ significantly in the two cases, when there is a Cauchy horizon and when there is none.
In general relativity, a given ‘epoch’ of time is sometimes represented by a spacelike surface, which is a three-dimensional space section. For example, in the standard Friedmann models of cosmology, there is such an epoch of simultaneity, from which the universe evolves in future, given the physical variables and initial data on this surface. The Einstein equations govern this evolution of universe, and there is thus a predictability which one would expect to hold in a classical theory. The concern then is one would not be able to predict in the future of naked singularity, and that unpredictable inputs may emerge from the same.
Given a regular initial data on a spacelike hypersurface, one would like to predict the future and past evolutions in the spacetime for all times (see for example, Hawking and Ellis 1973). Such a requirement is termed as the global hyperbolicity of the spacetime. A globally hyperbolic spacetime is a fully predictable universe, it admits a Cauchy surface, which is a three dimensional spacelike surface the data on which can be evolved for all times in the past as well as in future. Simple enough spacetimes such as the Minkowski or Schwarzschild are globally hyperbolic, but the Reissner-Nordstrom or Kerr geometries are not globally hyperbolic. For further details on these issues, we refer to (Joshi, 2008).
The key role that the event horizon of a black hole plays is that it hides the super-ultra-dense region formed in collapse from us. So the fact that we do not understand such regions has no effect on our ability to predict what happens in the universe at large. But if no such horizon exists, then the ultra-dense region might, in fact, play an important and even decisive role in the rest of the universe, and our ignorance of such regions would become of more than merely academic interest.
Yet such an unpredictability is common in general relativity, and not always directly related to censorship violation. Even black holes themselves need not fully respect predictability when they rotate or have some charge. For example, if we drop an electric charge into an uncharged black hole, the spacetime geometry radically changes and is no longer predictable from a regular initial epoch of time. A charged black hole admits a naked singularity which is visible to an observer within the horizon, and similar situation holds when the black hole is rotating. There is an important debate in recent years, if one could over-charge or over-rotate a black hole so that the singularity visible to observers within the horizon becomes visible to external far away observers too (see e.g. Joshi 2009).
Also, if such a black hole was big enough on a cosmological scale, the observer within the horizon could survive in principle for millions of years happily without actually falling into the singularity, and would thus be able to observe the naked singularity for a long time. Thus, only purest of pure black holes with no charge or rotation at all respect the full predictability, and all other physically realistic ones with charge or rotation actually do not. As such, there are many models of the universe in cosmology and relativity that are not totally predictable from a given spacelike hypersurface in the past. In these universes, the spacetime cannot be neatly separated into space and time foliation so as to allow initial data at a given moment of time to fully determine the future.
Actually the real breakdown of predictability is the occurrence of spacetime singularity itself we could say, which indicates the true limitation of the classical gravity theory. It does not matter really whether it is hidden within an event horizon or not. The real solution of the problem would then be the resolution of singularity itself, through either a quantum theory of gravity or in some way at the classical level.
In fact the cosmic censorship way to predictability, that of ‘hiding the singularity within a black hole’, and then thinking that we restored the spacetime predictability may not be the real solution, or at best it may be only a partial solution to the key issue of predictability in spacetime universes. It may be just shifting the problem elsewhere, and some of the current major paradoxes faced by the black hole physics such as the information paradox, the various puzzles regarding the nature of the Hawking radiation, and other issues could as well be a manifestation of the same.
No doubt, the biggest argument in support of censorship would be that it would justify and validate the extensive formalism and laws of black hole physics and its astrophysical applications made so far. Censorship has been the foundation for the laws of black holes such as the area theorem and others, and their astrophysical applications. But these are not free of major paradoxes. Even if we accept that all massive stars would necessarily turn into black holes, this still creates some major physical paradoxes. Firstly, all the matter entering a black hole must of necessity collapse into a space-time singularity of infinite density and curvatures, where all known laws of physics break down, which is some kind of instability at the classical level itself. This was a reason why many gravitation theorists of 1940s and 1950s objected to black hole formation, and Einstein also repeatedly argued against such a final fate of a collapsing star, writing a paper in 1939 to this effect. Also, as is well-known and has been widely discussed in the past few years, a black hole, by potentially destroying information, appears to contradict the basic principles of quantum theory. In that sense, the very formation of a black hole itself with a singularity within it appears to come laden with inherent problems. It is far from clear how one would resolve these basic troubles even if censorship were correct.
In view of such problems with the black hole paradigm, a possibility worth considering is the delay or avoidance of horizon formation as the star collapses under gravity. This happens when collapse to a naked singularity takes place, namely, where the horizon does not form early enough or is avoided. In such a case, if the star could radiate away most of its mass in the late stages of collapse, this may offer a way out of the black hole conundrum, while also resolving the singularity issue, because now there is no mass left to form the singularity. While this may be difficult to achieve purely classically, such a phenomenon could happen when quantum gravity effects are taken into account (Fig 8, see also the next section for a further discussion).
Fig 8: If the star could radiate away very considerable mass, especially through negative quantum pressures close to the classical singularity, this may effectively resolve the singularity.
What this means is, such an ‘unpredictability’ is somewhat common in general relativity. For example, if we drop a slight charge in a Schwarzschild black hole, the spacetime geometry completely changes into that of a charged black hole that is no longer predictable in the above sense. Similar situation holds when the black hole is rotating. In fact, there are very many models of universe in use in relativity which are not ‘globally hyperbolic’, that is, not totally predictable in the above sense where space and time are neatly separated so as to allow initial data to fully determine future for all times.
In any case, a positive and useful feature that has emerged from work on collapse models so far is, we already have now several important constraints for any possible formulation of censorship. It is seen that several versions of censorship proposed earlier would not hold, because explicit counter-examples are available now. Clearly, analyzing gravitational collapse plays a crucial role here. Only if we understand clearly why naked singularities do develop as collapse endstates in many realistic models, there could emerge any pointer or lead to any practical and provable version of censorship.
Finally, it may be worth noting that even if the problem of singularity was resolved somehow, possibly by invoking quantum gravity which may smear the singularity, we still have to mathematically formulate and prove the black hole formation assuming an appropriate censorship principle, which is turning out to be most difficult task with no sign of resolve. As discussed, the detailed collapse calculations of recent years show that the final fate of a collapsing star could be a naked singularity in violation to censorship. Finally, as is well-known and widely discussed by now, a black hole creates the information loss paradox, violating unitarity and making contradiction with basic principles of quantum theory. It is far from clear how one would resolve these basic troubles even if censorship were correct.
Xix A Lab for quantum gravity–Quantum stars?
It is believed that when we have a reasonable and complete quantum theory of gravity available, all spacetime singularities, whether naked or those hidden inside black holes, will be resolved away. As of now, it remains an open question if the quantum gravity will remove naked singularities. After all, the occurrence of spacetime singularities could be a purely classical phenomenon, and whether they are naked or covered should not be relevant, because quantum gravity will possibly remove them all any way. It is possible that in a suitable quantum gravity theory the singularities will be smeared out, though this has been not realized so far.
In any case, the important and real issue is, whether the extreme strong gravity regions formed due to gravitational collapse are visible to faraway observers or not. It is quite clear that the gravitational collapse would certainly proceed classically, at least till the quantum gravity starts governing and dominating the dynamical evolution at the scales of the order of the Planck length, i.e. till the extreme gravity configurations have been already developed due to collapse. The point is, it is the visibility or otherwise of such ultra-dense regions that is under discussion, whether they are classical or quantum (see Fig 9).
Fig 9: The naked singularity may be resolved by quantum gravity effects but the ultra-strong gravity region that developed in gravitational collapse will still be visible to external observers in the universe.
What is important is, classical gravity implies necessarily the existence of ultra-strong gravity regions, where both classical and quantum gravity come into their own. In fact, if naked singularities do develop in gravitational collapse, then in a literal sense we come face-to-face with the laws of quantum gravity, whenever such an event occurs in the universe.
In this way, the gravitational collapse phenomenon has the potential to provide us with a possibility of actually testing the laws of quantum gravity. In the case of a black hole developing in the collapse of a finite sized object such as a massive star, such strong gravity regions are necessarily hidden behind an event horizon of gravity, and this would be well before the physical conditions became extreme near the spacetime singularity. In that case, the quantum effects, even if they caused qualitative changes closer to singularity, will be of no physical consequences as no causal communications are then allowed from such regions. On the other hand, if the causal structure were that of a naked singularity, then the communications from such a quantum gravity dominated extreme curvature ball would be visible in principle. This will be so either through direct physical processes near a strong curvature naked singularity, or via the secondary effects, such as the shocks produced in the surrounding medium. It is possible that a spacetime singularity basically represents the incompleteness of the classical theory and when quantum effects are combined with the gravitational force, the classical singularity may be resolved.
Therefore, more than the existence of a naked singularity, the important physical issue is whether the extreme gravity regions formed in the gravitational collapse of a massive star are visible to external observers in the universe. An affirmative answer here would mean that such a collapse provides a good laboratory to study quantum gravity effects in the cosmos, which may possibly generate clues for an as yet unknown theory of quantum gravity. Quantum gravity theories in the making, such as the string theory or loop quantum gravity in fact are badly in need of some kind of an observational input, without which it is nearly impossible to constrain the plethora of possibilities.
We could say quite realistically that a laboratory similar to that provided by the early universe is created in the collapse of a massive star. However, the big bang, which is also a naked singularity in that it is in principle visible to all observers, happened only once in the life of the universe and is therefore a unique event. But a naked singularity of gravitational collapse could offer an opportunity to explore and observe the quantum gravity effects every time a massive star in the universe ends its life.
The important questions one could ask are: If in realistic astrophysical situations the star terminates as a naked singularity, would there be any observable consequences which reflect the quantum gravity signatures in the ultra-strong gravity region? Do naked singularities have physical properties different from those of a black hole? Such questions underlie our study of gravitational collapse.
In view of recent results on gravitational collapse, and various problems with the black hole paradigm, a possibility worth considering is the delay or avoidance of horizon formation as the star evolves collapsing under gravity. This happens when collapse to a naked singularity takes place, where the horizon does not form early enough or is avoided. In such a case, in the late stages of collapse if the star could radiate away most of its mass, then this may offer a way out of the black hole conundrum, while also resolving the singularity issue, because now there is no mass left to form the curvature singularity. The purpose is to resolve the black hole paradoxes and avoid the singularity, either visible or within a black hole, which actually indicates the breakdown of physical theory. The current work on gravitational collapse suggests possibilities in this direction.
In this context, we considered a cloud that collapsed to a naked singularity final state, and introduced loop quantum gravity effects (Goswami, Joshi and Singh, 2006). It turned out that the quantum effects generated an extremely powerful repulsive force within the cloud. Classically the cloud would have terminated into a naked singularity, but quantum effects caused a burstlike emission of matter in the very last phases of collapse, thus dispersing the star and dissolving the naked singularity. The density remained finite and the spacetime singularity was eventually avoided. One could expect this to be a fundamental feature of other quantum gravity theories as well, but more work would be required to confirm such a conjecture.
For a realistic star, its final catastrophic collapse takes place in matter of seconds. A star that lived millions of years thus collapses in only tens of seconds. In the very last fraction of a microsecond, almost a quarter of its total mass must be emitted due to quantum effects, and therefore this would appear like a massive, abrupt burst to an external observer far away. Typically, such a burst will also carry with it specific signatures of quantum effects taking place in such ultra-dense regions. In our case, these included a sudden dip in the intensity of emission just before the final burstlike evaporation due to quantum gravity. The question is, whether such unique astrophysical signatures can be detected by modern experiments, and if so, what they tell on quantum gravity, and if there are any new insights into other aspects of cosmology and fundamental theories such as string theory. The key point is, because the very final ultra-dense regions of the star are no longer hidden within a horizon as in the black hole case, the exciting possibility of observing these quantum effects arises now, independently of the quantum gravity theory used. An astrophysical connection to extreme high energy phenomena in the universe, such as the gamma-rays bursts that defy any explanations so far, may not be ruled out.
Such a resolution of naked singularity through quantum gravity could be a solution to some of the paradoxes mentioned above. Then, whenever a massive star undergoes a gravitational collapse, this might create a laboratory for quantum gravity in the form of a Quantum Star (see e.g. Joshi, 2009), that we may be able to possibly access. This would also suggest intriguing connections to high energy astrophysical phenomena. The present situation poses one of the most interesting challenges which have emerged through the recent work on gravitational collapse.
We hope the considerations here have shown that gravitational collapse, which essentially is the investigation of dynamical evolutions of matter fields under the force of gravity in the spacetime, provides one of the most exciting research frontiers in gravitation physics and high energy astrophysics. In our view, there is a scope therefore for both theoretical as well as numerical investigations in these areas, which may have much to tell for our quest on basic issues in quantum gravity, fundamental physics and gravity theories, and towards the expanding frontiers of modern high energy astrophysical observations.
Xx Concluding remarks
We considered here several aspects of spacetime singularities and the physical scenarios where these may be relevant, playing an interesting and intriguing role. We hope this creates a fairly good view of the exciting new physics that the spacetime singularities are leading us to, presenting a whole spectrum of new possibilities towards our search of the universe.
After discussing their existence and certain key basic properties, we discussed in some detail the gravitational collapse scenarios and the useful conclusions that have emerged so far in this context. In the first place, singularities not covered fully by the event horizon do occur in several collapsing configurations from regular initial data, with reasonable equations of state such as describing radiation, dust or a perfect fluid with a non-zero pressure, or also for general forms of matter. These naked singularities are physically significant in that densities and curvatures diverge powerfully near the same. Such results on the final fate of collapse, generated from study of different physically reasonable collapse scenarios, may provide useful insights into black hole physics and may be of help for any possible formulation of the cosmic censorship hypothesis.
An insight that seems to emerge is, the final state of a collapsing star, in terms of either a black hole or a naked singularity, may not really depend on the form or equation of state of collapsing matter, but is actually determined by the physical initial data in terms of the initial density profiles and pressures.
As an example, for inhomogeneous dust collapse, the final fate could be a black hole or a naked singularity depending on the values of initial parameters. The collapse ends in a naked singularity if the leading nonvanishing derivative of density at the center is either the first one or the second one. There is a transition from the naked singularity phase to the black hole phase as the initial density profile is made more and more homogeneous near the center. As one progresses towards more homogeneity, and hence towards a stronger gravitational field, there first occurs a weak naked singularity, then a strong naked singularity, and finally a black hole.
The important question then is the genericity and stability of such naked singularities arising from regular initial data. Will the initial data subspace giving rise to naked singularity have zero measure in a suitable sense? In that case, one would be able to reformulate more suitably the censorship hypothesis, based on a criterion that naked singularities could form in collapse but may not be generic. As we pointed out, the answer is far from clear due to ambiguities in the definitions of measures and the stability criteria.
One may try to evolve some kind of a physical formulation for cosmic censorship, where the available studies on various gravitational collapse scenarios such as above may provide a useful guide. The various properties of naked singularities collectively may be studied as they emerge from the studies so far and one would then argue that objects with such properties are not physical. But the way forward is again far from clear.
One could also invoke quantum effects and quantum gravity. While naked singularities may form in classical general relativity, quantum gravity presumably removes them. The point is that even though the final singularity may be removed in this way, still there would be very high density and curvature regions in the classical regime which would be causally communicating with outside observers, as opposed to the black hole case. If quantum effects could remove the naked singularity, this would then be some kind of quantum cosmic censorship.
We hope the considerations here have shown that gravitational collapse, which essentially is the investigation of dynamical evolutions of matter fields under the force of gravity in the spacetime, provides one of the most exciting research frontiers in gravitation physics and high energy astrophysics. There are issues here which have deep relevance both for theory as well as observational aspects in astrophysics and cosmology. Also these problems are of relevance for basics of gravitation theory and quantum gravity, and these inspire a philosophical interest and inquiry into the nature and structure of spacetime, causality, and profound issues such as predictability in the universe, as we indicated here.
Research is already happening in many of these areas as the discussion here pointed out. Some of the most interesting questions from my personal perspective are: Genericity and stability of collapse outcomes, examining the quantum gravity effects near singularities, observational and astrophysical signatures of the collapse outcomes, and other related issues. In particular, one of the most interesting questions would be, if naked singularities which are hypothetical astrophysical objects, did actually form in nature, what distinct observational signatures they would present. That is, how one distinguishes the black holes from naked singularities would be an important issue. There have been some efforts on this issue in recent years as we indicated above. The point is, there are already very high energy astrophysical phenomena being observed today, with several observational missions working both from ground and space. The black holes and naked singularities which are logical consequences of star collapse in general relativity, would appear to be the leading candidates to explain these phenomena. The observational signatures that each of these would present, and their astrophysical consequences would be of much interest for the future theoretical and computational research, and for their astrophysical applications.
In our view, there is therefore a scope for both theoretical as well as numerical investigations in these frontier areas, which may have much to tell for our quest on basic issues in quantum gravity, fundamental physics and gravity theories, and towards the expanding frontiers of modern high energy astrophysical observations.
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