Springer Handbook of Spacetime || Spacetime Singularities

Spacetime Sin409

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20. Spacetime Singularities

Pankaj S. Joshi

We give here an overview of our basic under-standing of and recent developments in spacetimesingularities in the Einstein theory of gravity. Sev-eral issues related to physical significance andimplications of singularities are discussed. The na-ture and existence of singularities are consideredwhich indicate the formation of super ultra-denseregions in the universe as predicted by the generaltheory of relativity. Such singularities develop dur-ing the gravitational collapse of massive stars andin cosmology at the origin of the universe. Possi-ble astrophysical implications of the occurrence ofsingularities in the spacetime universe are indi-cated. We discuss in some detail the profound andkey fundamental issues that the singularities giverise to, such as the cosmic censorship and pre-dictability in the universe, naked singularities ingravitational collapse and their relevance in blackhole physics today, and their astrophysical im-plications in modern relativistic astrophysics andcosmology.

20.1 Space, Time and Matter . ....................... 409

20.2 What Is a Singularity? ........................... 411

20.3 Gravitational Focusing .......................... 412

20.4 Geodesic Incompleteness ...................... 413

20.5 Strong Curvature Singularities ............... 414

20.6 Can We Avoid Spacetime Singularities? .. 415

20.7 Causality Violations .............................. 416

20.8 Energy Conditionsand Trapped Surfaces ........................... 417

20.9 Fundamental Implicationsand Challenges .................................... 417

20.10 Gravitational Collapse ........................... 419

20.11 Spherical Collapse and the Black Hole ... 419

20.12 Cosmic Censorship Hypothesis ............... 420

20.13 Inhomogeneous Dust Collapse .............. 422

20.14 Collapse with General Matter Fields ...... 423

20.15 Nonspherical Collapseand Numerical Simulations ................... 425

20.16 Are Naked Singularities Stableand Generic?. ....................................... 426

20.17 Astrophysical and Observational Aspects 427

20.18 Predictability and Other Cosmic Puzzles. 429

20.19 A Lab for Quantum Gravity–QuantumStars?. .................................................. 432

20.20 Concluding Remarks ............................. 434

References ..................................................... 435

20.1 Space, Time and Matter

After Einstein proposed the general theory of relativitydescribing the gravitational force in terms of space-time curvatures, the proposed field equations related thespacetime geometry to the matter content of the uni-verse. The early solutions found for these equationswere the Schwarzschild metric and the Friedmann mod-els. While the first represented the gravitational fieldaround an isolated body such as a spherically symmet-

ric star, the later solutions described the geometry of theuniverse. Both these models contained a spacetime sin-gularity where the curvatures as well as the matter andenergy densities diverged and became arbitrarily high,and the physical description would then break down.In the Schwarzschild solution such a singularity waspresent at the center of symmetry rD 0, whereas forthe Friedmann models it was found at the epoch tD 0,

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410 Part C Spacetime Structure and Mathematics

which is the beginning of the universe and origin oftime where the scale factor for the universe vanishesand all objects are crushed to a zero volume due to infi-nite gravitational tidal forces.

Even though the physical problem posed by theexistence of such a strong curvature singularity wasrealized immediately in these solutions, which turnedout to have several important implications for the ex-perimental verification of the general relativity theory,initially this phenomenon was not taken seriously. Itwas generally thought that the existence of such a sin-gularity must be a consequence of the very high degreeof symmetry imposed on the spacetime while these so-lutions were being derived and obtained. Subsequently,the distinction between a genuine spacetime singular-ity and a mere coordinate singularity became clear andit was realized that the singularity at r D 2m in theSchwarzschild spacetime was only a coordinate singu-larity which could be removed by a suitable coordinatetransformation. It was clear, however, that the genuinecurvature singularity at r D 0 cannot be removed byany coordinate transformations. The hope was then thatwhen more general solutions to the field equations areconsidered with a lesser degree of symmetry, such sin-gularities will be avoided.

This issue was sorted out when a detailed studyof the structure of a general spacetime and the associ-ated problem of singularities was taken up by Hawking,Penrose, and Geroch (see, for example, [20.1] andreferences therein). It was shown by this work thata spacetime will admit singularities within a rathergeneral framework provided that it satisfies certain rea-sonable physical assumptions such as the positivity ofenergy, a suitable causality assumption and a condi-tion implying strong gravitational fields, such as theexistence of trapped surfaces. It thus followed that thespacetime singularities form a general feature of the rel-ativity theory. In fact, these considerations ensure theexistence of singularities in other theories of gravity,which are also based on a spacetime manifold frame-work and satisfy the general conditions such as thosestated above.

Therefore, the scenario that emerges is the follow-ing: essentially for all classical spacetime theories ofgravitation, the occurrence of singularities forms aninevitable and integral part of the description of the

physical reality. In the vicinity of such a singularity, typ-ically the energy densities, spacetime curvatures, andall other physical quantities would blow up, thus indi-cating the occurrence of super ultra-dense regions in theuniverse. The behavior of such regions may not be gov-erned by the classical theory itself, which may breakdown having predicted the existence of the singular-ities, and a quantum gravitation theory would be themost likely description of the phenomena created bysuch singularities.

Further to the general relativity theory in 1915,gravitation physics was a relatively quiet field withfew developments until about the 1950s. However, the1960s saw the emergence of new observations in highenergy astrophysics, such as quasars and high energyphenomena at the center of galaxies such as extremelyenergetic jets. These observations, together with im-portant theoretical developments such as studying theglobal structure of spacetimes and singularities, led toimportant results in black hole physics and relativisticastrophysics and cosmology.

Our purpose here is to review some of theserather interesting as well as intriguing developments,in a somewhat pedagogic and elementary fashion toprovide a broad perspective. Specifically, we wouldlike to highlight here several recent issues and chal-lenges that have emerged related to spacetime singu-larities, which appear to have a considerable physicalsignificance and may have interesting astrophysical im-plications. We take up and consider here importanttopics such as what is meant by a singular spacetimeand specify the notion of a singularity. It turns outthat it is the notion of geodesic incompleteness thatcharacterizes a singularity in an effective manner fora spacetime and enables their existence to be provedby means of certain relatively general theorems. Wehighlight here several recent developments which dealwith certain exciting current issues related to space-time singularities on which research in gravitation andcosmology is being carried out today. These includethe work on final end states of gravitational collapse,cosmic censorship, and black holes and naked singu-larities. Related major cosmic conundrums such as theissue of predictability in the universe are discussed andobservational implications of naked singularities areindicated.

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20.2 What Is a Singularity?

In the Einstein theory of gravitation, the universe ismodeled as a spacetime with a mathematical structureof a four-dimensional differentiable manifold. In thatcase, locally the spacetime is always flat in a sufficientlysmall region around any point, but on a larger scale thisneed not be the case and it can have a rich and var-ied structure. An example of a differentiable manifold isa sphere, which is flat enough in the vicinity of any sin-gle point on its surface, but has a global curvature. Fora more detailed discussion on spacetime manifolds andtheir key role in general relativity, we refer to [20.2].

When should we say that a spacetime uni-verse .M; g/, which is a differentiable manifold witha Lorentzian metric, has become singular? What weneed is a specification and a specific criterion for theexistence of a singularity for any given universe modelin general relativity.

As stated above, several examples of singular be-havior in spacetime models of general relativity areknown. Important exact solutions such as the Fried-mann–Robertson–Walker (FRW) cosmological mod-els and Schwarzschild spacetime contain a singularitywhere the energy density or curvatures diverge stronglyand the usual description of the spacetime breaks down.In the Schwarzschild spacetime there is an essen-tial curvature singularity at rD 0 in that along anynonspace-like trajectory falling into the singularity, theKretschmann scalar ˛ D RijklRijkl!1. Also, all futuredirected nonspace-like geodesics which enter the eventhorizon at rD 2m must fall into this curvature singular-ity within a finite value of the proper time, or finite valueof the affine parameter, so all such curves are futuregeodesically incomplete. Similarly, for FRW models, if�C3p > 0 at all times, where � is the total energy den-sity and p the pressure, there is a singularity at tD 0which could be identified as the origin of the universe.Then along all the past directed trajectories meetingthis singularity, �!1 and the curvature scalar RDRijRij!1. Again, all the past directed nonspace-likegeodesics are incomplete. This essential singularity attD 0 cannot be transformed away by any coordinatetransformations. Similar behavior was generalized tothe class of spatially homogeneous cosmological mod-els by Ellis and King [20.3] which satisfy the positivityof energy conditions.

Such singularities, where the curvature scalars anddensities diverge imply a genuine spacetime pathologywhere the usual laws of physics break down. The ex-istence of the geodesic incompleteness in these space-

times implies, for example, that a time-like observersuddenly disappears from the spacetime after a finiteamount of proper time. Of course, singular behaviorcan also occur without bad behavior of curvature. Forexample, consider Minkowski spacetime with a pointdeleted. Then there will be time-like geodesics runninginto the hole which will be future incomplete. Clearly,this is an artificial situation one would like to rule outby requiring that the spacetime is inextendible, that is,it cannot be isometrically embedded into another largerspacetime as a proper subset. However, one could givea nontrivial example of singular behavior where a coni-cal singularity exists (see, e.g., [20.4]). Here spacetimeis inextendible but curvature components do not divergenear the singularity, as in a Weyl-type solution. Themetric is given by

ds2 D�dt2C dr2C r2.d 2C sin2 d�2/ ;

with coordinates given by �1 < t <1; 0< r <1; 0< < � with 0< � < a, with � D 0 and � D aidentified and a¤ 2� . There is a conical singularity atrD 0 through which the spacetime cannot be extendedand the singular boundary is related to the time-liketwo-plane rD 0 of Minkowski spacetime.

An important question, then, is whether such singu-larities develop even when a general model is consid-ered, and if so under what conditions. To answer this,it is first necessary to characterize precisely what onemeans by a spacetime singularity for a general space-time. Then it is seen that singularities must exist fora very wide class of spacetimes under a reasonable gen-eral set of conditions. Such singularities may develop asthe end state of the gravitational collapse of a massivestar, or in cosmological situations such as the origin ofthe universe.

The first point to note here is by very definition, themetric tensor must be well-defined at all regular pointsof the spacetime. This is no longer true at a spacetimesingularity such as those discussed above and a singu-larity cannot be regarded as a regular point of spacetimebut is a boundary point attached to M. This causesdifficulties when one tries to characterize a singular-ity by the criterion that the curvatures must blow upnear the singularity. The trouble is, since the singu-larity is not part of the spacetime, it is not possibleto define its neighborhood in the usual sense to dis-cuss the behavior of curvature quantities in that region.One may try to characterize a singularity in terms of

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the divergence of components of the Riemann curva-ture tensor along nonspace-like trajectories. Then thetrouble is that the behavior of such components will,in general, change with the frames used, so this is notof much help. One can try the curvature scalars or thescalar polynomials in the metric and the Riemann tensorand require them to achieve unboundedly large values.This is encountered in Schwarzschild and Friedmannmodels. However, it is possible that such a divergenceoccurs only at infinity for a given nonspace-like curve.In general, it looks reasonable to demand that somesort of curvature divergence must take place alongthe nonspace-like curves which encounter a singular-ity. However, a general characterization of singularityin terms of the curvature divergence runs into variousdifficulties.

Considering these and similar situations, the oc-currence of nonspace-like geodesic incompleteness isgenerally agreed upon as the criterion for the existenceof a singularity for a spacetime. This may not cover all

types of singular behaviors possible, but it is clear thatif a spacetime manifold contains incomplete nonspace-like geodesics, there is a definite singular behaviorpresent, as a time-like observer or a photon suddenlydisappears from the spacetime after a finite amountof proper time or after a finite value of the affine pa-rameter. The singularity theorems which result from ananalysis of gravitational focusing and global propertiesof a spacetime prove this incompleteness property fora wide class of spacetimes under a set of rather generalconditions.

Physically, a singularity in any physics theory typ-ically means that the theory breaks down either in thevicinity or at the singularity. This means that a broaderand more comprehensive theory is needed, demandinga revision of the given theory. Similar reasoning applyto spacetime singularities, which may be taken to im-ply that a quantum gravity description is warranted inthose regions of the universe, rather than using merelya classical framework.

20.3 Gravitational Focusing

Simple model solutions such as Schwarzschild andFRW universes give very useful indications as to whatis possible in general relativity, as opposed to Newto-nian gravity. In particular, these solutions are importantindicators on the existence and nature of the spacetimesingularities.

The key to occurrence of singularities in these solu-tions is really the gravitational focusing that the matterfields and spacetime curvature cause in the congruencesof null and time-like curves, which represent the lightpaths and the material particle trajectories in any givenspacetime universe. It would then be important to knowhow general and generic such a feature and property isfor a general spacetime.

The matter fields with positive energy density whichcreate the curvature in spacetime affect the causality re-lations in a spacetime and create focusing in families ofnonspace-like trajectories. The phenomenon that occurshere is that matter focuses the nonspace-like geodesicsof the spacetime into pairs of focal points or conju-gate points. The property of conjugate points is thatif p; q are two conjugate points along a nonspace-likegeodesic, then p and q must be time-like related. Now,there are three-dimensional null hypersurfaces such asthe boundary of the future of an event such that no twopoints of such a hypersurface can be joined by a time-like curve. Thus, the null geodesic generators of such

surfaces cannot contain conjugate points and they mustleave the hypersurface before encountering any conju-gate point. This puts strong limits on such null surfacesand the singularity theorems result from an analysis ofsuch limits.

If we consider a congruence of time-like geodesicsin the spacetime, this is a family of curves and througheach event there passes precisely one time-like geodesictrajectory. Choosing the curves to be smooth, this de-fines a smooth time-like vector field on the spacetime.The rate of change of volume expansion for a given con-gruence of time-like geodesics can be written as

d

dD�RlkVlVk � 1

3 2 � 2�2C 2!2 ;

where, for a given congruence of time-like (or null)geodesics, the quantities , � , and ! are expansion,shear, and rotation tensors, respectively. The aboveequation is called the Raychaudhuri equation [20.5],which describes the rate of change of the volume expan-sion as one moves along the time-like geodesic curvesin the congruence.

We note that the second and third term on the right-hand side involving and � are always positive. For theterm RijViVj, by Einstein equations this can be writtenas

RijViVj D 8�

TijV

iVjC 12 T�:

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The term TijViVj above represents the energy densitymeasured by a time-like observer with unit tangent Vi,which is the four-velocity of the observer. For all rea-sonable classical physical fields this energy density isgenerally taken as nonnegative and we may assume forall time-like vectors Vi

TijViVj � 0 :

Such an assumption is called the weak energy con-dition. It is also considered reasonable that the matterstresses will not be so large as to make the right-handside of the equation above negative. This will be sat-isfied when the following is satisfied: TijViVj � � 1

2 T .Such an assumption is called the strong energy con-dition and it implies that for all time-like vectors Vi,RijViVj � 0. By continuity it can be argued that thesame will then hold for all null vectors as well. Boththe strong and weak energy conditions will be valid forwell-known forms of matter such as the perfect fluidprovided that the energy density � is nonnegative andthere are no large negative pressures which are biggeror comparable to �.

With the strong energy condition, the Raychaudhuriequation implies that the effect of matter on spacetimecurvature causes a focusing effect in the congruence

of time-like geodesics due to gravitational attraction.This causes neighboring geodesics in the congruenceto cross each other to give rise to caustics or conju-gate points. Such a separation between nearby time-like geodesics is governed by the geodesic deviationequation

D2Zj D�RjkilV

kZiVl ;

where Zi is the separation vector between nearbygeodesics of the congruence. Solutions of the aboveequation are called the Jacobi fields along a given time-like geodesic.

Suppose now that � is a time-like geodesic, then twopoints p and q along � are called conjugate points ifthere exists a Jacobi field along � which is not identi-cally zero but vanishes at p and q. From the Raychaud-huri equation given above it is clear that the occurrenceof conjugate points along a time-like geodesic is closelyrelated to the behavior of the expansion parameter of the congruence. In fact, it can be shown that thenecessary and sufficient condition for a point q to beconjugate to p is that for the congruence of time-likegeodesics emerging from p, we must have !�1 at q(see, for example, [20.1]). The conjugate points alongnull geodesics are also similarly defined.

20.4 Geodesic Incompleteness

It was widely believed that for more general solutions ofthe Einstein equations which incorporate several otherphysical features and are not necessarily symmetric, theexistence of singularities would be avoided (see, e.g.,[20.6]; and for recent development and reviews we referto [20.7] and [20.8]). Further investigations, however,showed that singularities in the form of geodesic in-completeness do exist for general spacetimes. Theseresults used the gravitational focusing considerationsmentioned above and global properties of a generalspacetime.

The behavior of the expansion parameter is gov-erned by the Raychaudhuri equation as pointed outabove. Consider, for example, the situation when thespacetime satisfies the strong energy condition and thecongruence of time-like geodesics is hypersurface or-thogonal. Then !ij D 0 and the corresponding term!2 vanishes. Further, the expression for the covariantderivative of !ij implies that it must vanish for all fu-ture times as well. It follows that we must then haved =d �. 2=3/, which means that the volume ex-pansion parameter must be necessarily decreasing along

the time-like geodesics. If 0 denotes the initial ex-pansion then the above can be integrated as �1 �

�10 C =3. It is clear from this that if the congruence

is initially converging and 0 is negative, then !�1within a proper time distance 3= j 0 j.

It then follows that if M is a spacetime satisfy-ing the strong energy condition and S is a space-likehypersurface with < 0 at p 2 S, then if � is a time-like geodesic of the congruence orthogonal to S passingthrough p there exists a point q conjugate to S along �within a proper time distance 3= j j. This is pro-vided that � can be extended to that value of the propertime.

The basic implication of the above results is thatonce a convergence occurs in a congruence of time-like geodesics, the conjugate points or the causticsmust develop in the spacetime. These can be inter-preted as the singularities of the congruence. Suchsingularities could occur even in Minkowski space-time and similar other perfectly regular spacetimes.However, when combined with certain causal structureproperties of spacetime, the results above imply the

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existence of singularities in the form of geodesic incom-pleteness. One could similarly discuss the gravitationalfocusing effect for the congruence of null geodesicsor for null geodesics orthogonal to a space-like two-surface.

There are several singularity theorems availablethat establish nonspace-like geodesic incompletenessfor a spacetime under different sets of conditions andare applicable to different physical situations. How-ever, the most general of these is the Hawking–Penrosetheorem [20.9], which is applicable in both the col-lapse situation and cosmological scenario. The mainidea of the proof of such a theorem is the following.Using the causal structure analysis it is shown that theremust be maximal length time-like curves between cer-tain pairs of events in the spacetime. Now, a causalgeodesic which is both future and past complete mustcontain pairs of conjugate points if M satisfies an en-ergy condition. This is then used to draw the necessarycontradiction to show that M must be nonspace-likegeodesically incomplete.

Theorem 20.1A spacetime .M; g/ cannot be time-like and nullgeodesically complete if the following are satisfied:

1. RijKiKj � 0 for all nonspace-like vectors Ki.2. The generic condition is satisfied, that is, every

nonspace-like geodesic contains a point at whichKŒiRjelŒmKnKeKl ¤ 0, where K is the tangent to thenonspace-like geodesic.

3. The chronology condition holds on M; that is, thereare no closed time-like curves in the spacetime.

4. There exists in M either a compact achronal set(i. e., a set no two points of which are time-likerelated) without edge or a closed trapped surface,or a point p such that for all past directed nullgeodesics from p, eventually must be negative.

The main idea of the proof is the following. Oneshows that the following cannot hold simultaneously:

a) Every inextendible nonspace-like geodesic containspairs of conjugate points.

b) The chronology condition holds on M.c) There exists an achronal set S in M such that EC.S/

or E�.S/ is compact.

In the above, EC and E� indicate the future andpast horismos for the set S (for further definitionsand details we refer to Hawking and Ellis [20.1], orJoshi [20.10]).

We note that while geodesic incompleteness, asa definition of spacetime singularities, allows varioustheorems to be proved on the existence of singularities,it does not capture all possible singular behaviors fora spacetime. It also does not imply that the singularitypredicted is necessarily a physically relevant powerfulcurvature singularity. It does, of course, include manycases where that will be the case. Below, we discusssuch a scenario and the criterion for the singularity tobe physically relevant and important.

20.5 Strong Curvature Singularities

As we saw above, the existence of an incomplete non-space-like geodesic or the existence of an inextendiblenonspace-like curve which has a finite length as mea-sured by a generalized affine parameter, implies theexistence of a spacetime singularity. The generalizedaffine length for such a curve is defined as [20.1]

L./D

aZ0

"3X

iD0

.Xi/2

#1=2

ds ;

which is a finite quantity. The Xis are the componentsof the tangent to the curve in a parallel propagatedtetrad frame along the curve. Each such incompletecurve defines a boundary point of the spacetime whichis a singularity.

The important point now is, in order to call thisa genuine physical singularity, one would typicallylike to associate such a singularity with unboundedlygrowing spacetime curvatures. If all the curvature com-ponents and the scalar polynomials formed out of themetric and the Riemann curvature tensor remained fi-nite and well-behaved in the limit of approach to thesingularity along an incomplete nonspace-like curve,it may be possible to remove such a singularity byextending the spacetime when the differentiability re-quirements are lowered [20.11].

There are several ways in which such a requirementcan be formalized. For example, a parallely propagatedcurvature singularity is the one which is the end pointof at least one nonspace-like curve on which the com-

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ponents of the Riemann curvature tensor are unboundedin a parallely propagated frame. On the other hand,a scalar polynomial singularity is the one for whicha scalar polynomial in the metric and the Riemann ten-sor takes an unboundedly large value along at least onenonspace-like curve which has a singular end point.This includes cases such as Schwarzschild singularity,where the Kretschmann scalar RijklRijkl blows up in thelimit as r! 0.

What is the guarantee that such curvature singulari-ties will at all occur in general relativity? The answerto this question for the case of parallely propagatedcurvature singularities is provided by a theorem ofClarke [20.12], which establishes that for a globally hy-perbolic spacetime M which is C0� inextendible, whenthe Riemann tensor is not very specialized in the senseof not being type-D and electrovac at the singular endpoint, then the singularity must be a parallely propa-gated curvature singularity.

Curvature singularities to be characterized below,also arise for a wide range of spacetimes involvinggravitational collapse. This physically relevant class ofsingularities, called the strong curvature singularitieswas defined and analyzed by Tipler [20.13], Tipleret al. [20.14], and Clarke and Królak [20.15]. The ideahere is to define a physically all embracing strong cur-vature singularity in such a way so that all the objectsfalling within the singularity are destroyed and crushedto zero volume by the infinite gravitational tidal forces.The extension of spacetime becomes meaningless forsuch a strong singularity which destroys to zero sizeall the objects terminating at the singularity. Fromthis point of view, the strength of singularity may beconsidered crucial to the issue of classically extendingthe spacetime, thus avoiding the singularity. This isbecause for a strong curvature singularity defined in theabove sense, no continuous extension of the spacetimemay be possible.

20.6 Can We Avoid Spacetime Singularities?

Given the scenario above, it is now clear that space-time singularities are an inevitable feature for most ofthe physically reasonable models of universe and grav-itational systems within the framework of the Einsteintheory of gravity. It is also seen that near such a space-time singularity, the classical description that predictedit must itself break down. The existence of singulari-ties in most of the classical theories of gravity, underreasonable physical conditions, imply that in a sensethe Einstein gravity itself predicts its own limitations,namely that it predicts regions of the universe whereit must breakdown and a new and revised physical de-scription must take over.

As the curvatures and all other physical quantitiesmust diverge near such a singularity, the quantum ef-fects associated with gravity are very likely to dominatesuch a regime. It is possible that these may resolve theclassical singularity itself. However, we currently donot have any viable and consistent quantum theory ofgravity despite serious attempts. Therefore, the issueof resolution of singularities as produced by classicalgravity remains open.

The other possibility is, of course, that some of theassumptions of the singularity theorems may be vio-lated so as to avoid the singularity occurrence. Evenwhen these are fairly general, one could inquire whether

some of them could actually breakdown and do nothold in physically realistic models. This could save usfrom the occurrence of singularities at the classical levelitself. Such possibilities mean a possible violation ofcausality in the spacetime, or no trapped surfaces oc-curring in the dynamical evolution of the universe, orpossible violation of energy conditions.

The singularity theorem stated above and also othersingularity theorems contain the assumption of causal-ity or strong causality, or some other suitable causalitycondition. Then the alternative is that causality maybe violated rather than a singularity occurring in thespacetime. So the implication of the singularity theo-rem stated above is that when there is enough matterpresent in the universe, either the causality is violatedor a boundary point must exist for the spacetime. Inthe cosmological case, such stress-energy density willbe provided by microwave background radiation, orin the case of stellar collapse trapped surfaces mayform [20.16], providing a condition leading to the for-mation of a singularity.

The Einstein equations by themselves do not ruleout causality violating configurations which really de-pend on the global topology of the spacetime. Hencethe question of causality violations versus spacetimesingularity needs a careful examination as to whether

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causality violation could offer an alternative to sin-gularity formation. Similarly, it must also be inquiredwhether the violation of energy conditions or nonoc-

currence of trapped surfaces may be realized so as toachieve singularity avoidance in a spacetime. We brieflydiscuss some of these points below.

20.7 Causality Violations

The causal structure in a spacetime specifies whatevents can be related to each other by means of time-like or light signals. A typical causality violation wouldmean that an event could be in its own past, which iscontrary to our normal understanding of time, and thatof past and future. This has been examined in consid-erable detail in general relativity, and that no causalityviolation takes place in the spacetime is one of theimportant assumptions used by singularity theorems.However, general relativity allows for situations wherecausality violation is permitted in a spacetime. TheGödel solution [20.17] allows the existence of a closedtime-like curve through every point of the spacetime.

One would, of course, like to rule out if possiblecausality violations on physical grounds, treating themas very pathological behavior in that in such a case onewould be able to enter one’s own past. However, as theyare allowed in principle in general relativity, so onemust rule them out only by an additional assumption.The question then is, can one avoid spacetime singular-ities if one allows for the violation of causality? Thishas been considered by researchers and it was seen thatthe causality violation in its own right creates spacetimesingularities again under certain reasonable conditions.Thus, this path of avoiding spacetime singularities doesnot appear to be very promising.

Specifically, the question of finite causality viola-tions in asymptotically flat spacetime was examined byTipler [20.13, 18, 19]. This showed that the causal-ity violation in the form of closed time-like lines isnecessarily accompanied by incomplete null geodesics,provided the strong energy condition is satisfied for allnull vectors and the generic condition is satisfied. It wasassumed that the energy density � has a positive mini-mum along past directed null geodesics.

There is, in fact, a hierarchy of causality condi-tions available for a spacetime. It may be causal inthe sense of having no closed nonspace-like curves.However, given an event, future directed nonspace-likecurves from the same event could return to its arbi-trarily close neighborhood in the spacetime. This isas bad as a causality violation itself. The higher-ordercausality conditions such as strong causality and sta-

ble causality rule out such behavior. Of the higher-ordercausality conditions, much physical importance is at-tached to stable causality, which ensures that if M iscausal, its causality should not be disturbed with smallperturbations in the metric tensor. Presumably, gen-eral relativity is a classical approximation to some, asyet unknown, quantum theory of gravity in which thevalue of the metric at a point will not be exactly knownand small fluctuations in the value must be taken intoaccount.

Results on causality violations and higher-ordercausality violations with reference to occurrence of sin-gularities were obtained by Joshi [20.20], and Joshi andSaraykar [20.21], who showed that the causality viola-tions must be accompanied by singularities even whenthe spacetime is causal but the higher-order causalityconditions are violated. Thus we know that for a causalspacetime, the violations of higher-order causality con-ditions give rise to spacetime singularities. Anotherquestion examined was that of the measurement ofcausality violating sets when such a violation occurs. Itturns out that in many cases, the causality violating setsin a spacetime will have a zero measure, and thus sucha causality violation may not be taken very seriously.Also, Clarke and Joshi [20.22] studied global causalityviolation for a reflecting spacetime and the theorems ofKriele [20.23] improved some of the conditions underwhich the results on chronology violations implying thesingularities have been obtained. Also, global causalityviolating spacetimes were studied by Clarke and de Fe-lice [20.24]. What we discussed above implies that ifthe causality of M breaks down with the slightest per-turbation of the metric, then this must be accompaniedby the occurrence of spacetime singularities.

As a whole, the above results imply that violat-ing either causality or any of the higher-order causalityconditions may not be considered a good alternativeto the occurrence of spacetime singularities. There arealso philosophical problems connected with the issueof causality violation, such as entering one’s own past.However, even if one allowed for the causality viola-tions, the above results show that these are necessarilyaccompanied by spacetime singularities again.

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20.8 Energy Conditions and Trapped SurfacesAnother possibility to avoid singularities is a possibleviolation of energy conditions. This is another of theassumptions in the proofs for singularity theorems. Infact, this possibility has also been explored in some de-tail and it turns out that as long as there is no grossor very powerful violation of energy conditions overglobal regions in the universe, this would not help avoidsingularities either.

For example, the energy condition could be violatedlocally at certain spacetime points, or in certain regionsof spacetimes due to peculiar physics there. However,as long as it holds on an average, in the sense that thestress-energy density is positive in an integrated sense,then spacetime singularities still occur (for a discussionand references, see, e.g., [20.10]).

On a global scale, there is evidence now that the uni-verse may be dominated by a dark energy field. Thereis no clarity as to what exactly such a field would beand what would be its origin. It could be due to scalarfields or ghost fields floating in the universe, or due toa nonzero positive cosmological constant present in theEinstein equations. In such a case, the weak or strongenergy conditions may be violated depending on thenature of these exotic fields. However, in the earlier uni-verse of a matter dominated phase, the positive matterfields would again dominate, thus respecting the en-ergy conditions even if they are violated at the presentepoch.

Again, the above discussion is in the context ofa cosmological scenario. When it comes to the gravi-tational collapse of massive stars, clearly their densityand overall energy content are dominated by the or-dinary matter fields with which we are much more

familiar. Such matter certainly respects the energy con-ditions modulo with some minor violations if at all any.Thus for gravitational collapse of massive stars, onewould expect the energy conditions to hold and the con-clusions on singularity occurrence stated above wouldapply.

Yet another possibility to avoid singularity is toavoid trapped surfaces occurring in the spacetime. In-deed, such a route can give rise to geodesically com-plete spacetimes, as was shown by Senovilla [20.25].As for the cosmological scenario, basically this meansand amounts to the condition that the matter energydensities must fall off sufficiently rapidly on any givenspace-like surface, and in an averaged sense, in order toavoid the cosmological trapped surfaces. Whether sucha condition is realizable in the universe would have tobe checked through observational tests. A sufficientlyuniform energy density, such as, say, the microwavebackground radiation could in turn cause cosmic trap-ping. As for massive stars, the densities are, of course,very high indeed and would only grow, for example, ina gravitational collapse. Therefore, in collapse scenar-ios, the trapped surfaces are unlikely to be avoided.

Further to the above considerations, if we accepton the whole that spacetime singularities do occur un-der fairly general conditions in the framework of theEinstein theory of gravity, or for classical gravity ingeneral, then we must consider physical implicationsand consequences of such a scenario for physics andcosmology. As we noted earlier, two main arenas ofphysical relevance where spacetime singularities will beof interest are the cosmological situation and the gravi-tational collapse scenarios.

20.9 Fundamental Implications and Challenges

The existence of spacetime singularities in Einsteingravity and other classical theories of gravitation posesintriguing challenges and fundamental questions inphysics as well as cosmology. These would have far-reaching consequences for our current understanding ofthe universe and how we try to model it further, as weshall try to bring out in rest of this article.

The inevitable existence of singularities for wideclasses of rather general models of spacetimes meansthat the classical gravity evolutions necessarily give riseto regions in the spacetime universe where the densities

and spacetime curvatures would really grow arbitrarilyhigh without any bounds, and where all other relevantphysical parameters would also diverge.

To take the first physical scenario, such a phe-nomenon in cosmology would correspond to a singu-larity that will represent the origin of the universe.Secondly, whenever locally a large quantity of matterand energy collapses under the force of its own grav-ity, a singularity will occur. This later situation will beeffectively realized in the gravitational collapse of mas-sive stars in the universe, which collapse and shrink

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catastrophically under their self-gravity, when the starhas exhausted its nuclear fuel within that earlier sup-plied the internal pressures to halt the infall due togravity.

Over the past decades, once the existence of space-time singularities was accepted, there have been majorefforts to understand the physics in the vicinity of thesame. In the cosmological case, this has given rise toan entire physics of the early universe, which depictsthe few initial moments immediately after the big bangsingularity from which the universe is supposed to haveemerged. The complexities in this case have been enor-mous, both physics-wise, and conceptually. The physicscomplexities arise because when trying to understandphysics close to the hot big bang singularity, we aredealing with the highest energy scales, never seen ear-lier in any laboratory physics experiments. Our particlephysics theories are then to be stretched to the extremewhere there is no definite or unique framework avail-able to deal with these phenomena. Understanding earlyuniverse physics has, of course, very big consequencesin that it governs the most important physical phenom-ena such as the later galaxy formation in the universeand other issues related to the large-scale structure ofthe universe.

As for the conceptual issues, simultaneous big bangsingularity gives rise to a host of problems and puzzles.One of these is the horizon problem which arises due tothe causal structure of this spacetime. Distinct regionsof the universe simply cannot interact with each otherdue to the cosmic horizons and it becomes extremelydifficult to explain the average overall current homo-geneity of the universe on a large enough scale. Thereare also other issues such as why the current universelooks so flat, which is called the flatness problem. Asa possible means to resolve these dilemmas inflationarymodels for the early universe have been proposed, vari-ous facets of which are still very much under an activedebate.

The key issue, as far as big bang singularity is con-cerned, is that it happened only once in the past andthere is no way to probe it any further other than currentobservations on the universe and their extrapolation inthe past. One must look deeper and deeper into spaceand back into time to understand the nature and physicsof this early universe singularity.

As we mentioned above, the other class of suchspacetime singularities will occur in the gravitationalcollapse of massive stars in the universe. Unlike the big

bang, such a singularity will occur whenever a massivestar in the universe collapses. This is, therefore, moreamenable to observational tests.

There are rather fundamental cosmic conundrumsassociated with singularities of gravitational collapse.One of the most intriguing of these is the question ofwhether such a singularity will be visible to externalfaraway observers in the universe. The big bang singu-larity is visible to us in principle, as we get to see thelight from the same. However, as we discuss below, thesingularity of collapse can sometimes be hidden belowthe event horizons of gravity, and are therefore not visi-ble. The possibility that all singularities of collapse willbe necessarily hidden inside horizons is called the cos-mic censorship conjecture. As we discuss below this isnot yet proved, and, in fact, singularities of collapse canbe visible under many physical circumstances.

When visible or naked singularities develop in grav-itational collapse, they give rise again to extremelyintriguing physical possibilities and problems. The op-portunity offered in that case is that we may have thepossibility to observe the possible ultra-high energyphysical processes occurring in such a region of theuniverse, including quantum gravity effects. Such ob-servations of ultra-high energy events in the universecould provide observational tests and guide our effortsfor a possible quantum theory of gravity. However, a co-nundrum that is presented is whether this would breakthe so-called classical predictability of the spacetimeuniverse. We shall discuss this further below. On theother hand, even when the singularity is necessarily hid-den within a black hole, this still gives rise to profoundpuzzles such as the information paradox, issues withunitarity and other such problems. So the point is that,even if the cosmic censorship was correct and all sin-gularities were hidden inside black holes only, we shallstill be faced with many deep paradoxes, which are notunique to naked singularities only.

It would be only reasonable to say that all these deepphysical as well as conceptual issues are closely con-nected with the existence and formation of spacetimesingularities in the dynamical gravitational processestaking place in the universe. While the big bang singu-larity happened only once in the past, the singularitiesof collapse have, in fact, a repeated occurrence, andhence they possess an interesting observational per-spective and potential. We shall, therefore, discuss thesame in some detail below, while also providing the keyingredients of black hole physics in the process.

Spacetime Singularities 20.11 Spherical Collapse and the Black Hole 419Part

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20.10 Gravitational CollapseWhen a massive star of more than about ten solarmasses exhausts its internal nuclear fuel, it is believed toenter the stage of continual gravitational collapse with-out any final equilibrium state. The star then goes onshrinking in its radius, reaching higher and higher den-sities. What would be the final fate of such an objectaccording to the general theory of relativity? This is oneof the central questions in relativistic astrophysics andgravitation theory today. It is suggested that the ultra-dense object that forms as a result of the collapse couldbe a black hole in space and time from which not evenlight rays escape. Alternatively, if an event horizon ofgravity fails to cover the final super ultra-dense crunch,it could be a visible singularity in the spacetime whichcould causally interact with the outside universe andfrom which region the emissions of light and mattermay be possible.

The issue is of importance from both the theo-retical as well as the observational point of view. Atthe theoretical level, working out the final fate of col-lapse in general relativity is crucial to the problemof asymptotic predictability, namely, whether the sin-gularities forming as the collapse end state will be

necessarily covered by the event horizons of gravity.Such a censorship hypothesis remains fundamental tothe theoretical foundations of black hole physics and itsmany recent astrophysical applications. These includethe area theorem for black holes, laws of black holethermodynamics, Hawking radiation, predictability ina spacetime, and on the observational side the accre-tion of matter by black holes, massive black holes at thecenter of galaxies, etc. On the other hand, the existenceof visible or naked singularities offers a new approachto these issues requiring modification and reformulationof our usual theoretical conception of black holes.

We mention and discuss below some of the recentdevelopments in these directions, examining the pos-sible final fate of gravitational collapse. To investigatethis issue, dynamical collapse scenarios have been ex-amined in the past decade or so for many cases suchas clouds composed of dust, radiation, perfect fluids,or matter with more general equations of state (see,e.g., [20.26] for references and details). We discussthese developments and the implications for a possibleformulation of cosmic censorship are indicated, men-tioning the open problems in the field.

20.11 Spherical Collapse and the Black Hole

To understand the final fate of a massive gravitationallycollapsing object we first outline here the sphericallysymmetric collapse situation. Although this is an ide-alization, the advantage is that one can solve the caseanalytically to obtain exact results when matter is ho-mogeneous dust. In fact, the basic motivations for theidea and theory of black holes come from this col-lapse model, first worked out by Oppenheimer andSnyder [20.27] and Datt [20.28].

Consider a gravitationally collapsing spherical mas-sive star. The interior solution for the object dependson the properties of matter, equations of state, and thephysical processes taking place within the stellar inte-rior. However, assuming the matter to be pressurelessdust allows us to solve the problem analytically, whichprovides important insights. The energy–momentumtensor is given by Tij D �uiuj and the Einstein equa-tions are to be solved for the spherically symmetricmetric. The metric potentials can be solved and theinterior geometry of the collapsing dust ball is given

by

ds2 D�dt2CR2.t/

�dr2

1� r2C r2 d˝2

�;

where d˝2 D d 2C sin2 d�2 is the metric on thetwo-sphere. The interior geometry of the cloud matchesat the boundary r D rb with the exterior Schwarzschildspacetime.

The basic features of such a configuration are givenin Fig. 20.1. The collapse is initiated when the star sur-face is outside the Schwarzschild radius rD 2m anda light ray from the surface of the star can escape toinfinity. However, once the star has collapsed belowrD 2m, a black hole region of no escape develops inthe spacetime which is bound by the event horizonat rD 2m. Any point in this empty region representsa trapped surface, a two-sphere for which both the out-going and ingoing families of null geodesics emittedfrom this point converge and thus no light comes out

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Spacetimesingularity

Apparenthorizon

Eventhorizon

Boundaryof the star

Initial surface

Light rayr

= 2

M

r =

0t

r

Collapsingmatter

Fig. 20.1 The gravitational collapse of a spherically sym-metric homogeneous dust cloud. The event horizon formsprior to the singularity and the collapse end state is a blackhole

of this region. Then, the collapse to an infinite densityand curvature singularity at rD 0 becomes inevitablein a finite proper time as measured by an observer onthe surface of the star. The black hole region in theresulting vacuum Schwarzschild geometry is given by0< r < 2m with the event horizon as the outer bound-ary at which the radial outwards photons stay wherethey are but all the others are dragged inwards. No in-formation from the black hole can propagate outsiderD 2m to observers far away. We thus see that the col-lapse gives rise to a black hole in the spacetime which

covers the resulting spacetime singularity. The ultimatefate of the star undergoing such a collapse is an infi-nite curvature singularity at rD 0, completely hiddenwithin the black hole. No emissions or light rays fromthe singularity go out to an observer at infinity and thesingularity is causally disconnected from the outsidespacetime.

The question now is whether one could generalizethese conclusions on the occurrence of spacetime sin-gularity in collapse for more general forms of matter orfor nonspherical situations, or possibly for small per-turbations away from spherical symmetry. It is knownusing the stability of Cauchy development in generalrelativity that the formation of trapped surfaces is in-deed a stable property when departures from sphericalsymmetry are taken into account. The argument essen-tially is the following. Considering a spherical collapseevolution from given initial data on a partial Cauchysurface S, we find the formation of trapped surfaces Tin the form of all the spheres with r < 2m in the exte-rior Schwarzschild geometry. The stability of Cauchydevelopment then implies that for all initial data suf-ficiently near the original data in the compact regionJC.S/\J�.T/, where JC and J� denote the causal fu-tures or pasts of S, respectively, the trapped surfaces stillmust occur. Then, the curvature singularity of sphericalcollapse also turns out to be a stable feature, as im-plied by the singularity theorems which show that theclosed trapped surfaces always imply the existence ofa spacetime singularity under reasonable relatively gen-eral conditions.

20.12 Cosmic Censorship Hypothesis

Real stars in the universe are not made of pressurelesshomogeneous dust. They are inhomogeneous, typicallywith density higher at the center, may have nontrivialmatter with an equation of state that is as yet un-known, and there is spin. Will a physically realisticcollapse of such a star necessarily end up in the blackhole final state only, just as in the idealized case ofthe Oppenheimer–Snyder–Datt model above? In otherwords, while more general gravitational collapse willalso end up in a spacetime singularity, the question iswhether the singularity will be again necessarily cov-ered inside an event horizon of gravity.

In fact, there is no proof available that such a sin-gularity will continue to be hidden within a black holeand remain causally disconnected from outside ob-

servers, even when the collapse is not exactly sphericalor when the matter does not have the form of exact ho-mogeneous dust. Therefore, in order to generalize thenotion of black holes to more general gravitational col-lapse situations, it becomes necessary to rule out suchnaked or visible singularities by means of an explicitassumption. This is stated as the cosmic censorshiphypothesis [20.29], which essentially says that if S isa partial Cauchy surface from which the collapse com-mences, then there are no naked singularities to thefuture of S which could be seen from the future nullinfinity. This is true for spherical homogeneous dustcollapse, where the resulting spacetime is future asymp-totically predictable and the censorship holds. In sucha case, the breakdown of physical theory at the space-

Spacetime Singularities 20.12 Cosmic Censorship Hypothesis 421Part

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time singularity does not disturb prediction in future forthe outside asymptotically flat region.

The corresponding scenario for other more gen-eral collapse situations, when inhomogeneities or non-sphericity and such other physically realistic featuresare allowed for must be investigated. The answer ingeneral is not known either as a proof of the futureasymptotic predictability for general spacetimes or inthe form of any general theorem on cosmic censorship.It is clear that the assumption of censorship in a suitableform is crucial to the basic results in black hole physics.Actually when one considers the gravitational collapsein a generic situation, the very existence of black holesrequires this hypothesis.

To establish censorship by means of a rigorousproof certainly requires a much more precise formu-lation of the hypothesis. The statement that the resultof complete gravitational collapse must be a black holeonly and not a naked singularity, or all singularities ofcollapse must be hidden inside black holes, is not rigor-ous enough. Because under general circumstances, thecensorship or asymptotic predictability is false as onecould always choose a spacetime manifold with a nakedsingularity which would be a solution to Einstein’sequations if we define Tij � .1=8�/Gij. So certain con-ditions on the stress–energy tensor are required at theminimum, say, for example, an energy condition. How-ever, to obtain an exact set of conditions on matter fieldsto prove the censorship hypothesis turns out to be anextremely difficult task that has as yet not been accom-plished.

The requirements in black hole physics and gen-eral predictability have led to several different possibleformulations of cosmic censorship hypothesis, none ofwhich has been proved as yet. Weak cosmic censor-ship, or asymptotic predictability, postulates that thesingularities of gravitational collapse cannot influenceevents near the future null infinity. The other versioncalled the strong cosmic censorship is a general pre-dictability requirement on any spacetime, stating that allphysically reasonable spacetimes must be globally hy-perbolic (see, e.g., [20.30]). Global hyperbolicity heremeans that we must be able to predict the entire fu-ture and past evolutions in the universe by means ofthe Einstein equations, given the initial data on a three-dimensional space-like hypersurface in spacetime.

However, on further analysis it becomes clear thatsuch formulations need much more sharpening if any

concrete proof is to be obtained at all. In fact, as forthe cosmic censorship, it is a major problem in itselfto find a satisfactory and mathematically rigorous for-mulation of what it is physically desired to achieve.Presently, there is no general proof available for anysuitably formulated version of weak censorship. Themain difficulty seems to be that the event horizon isa feature depending on the whole future behavior of thesolution over an infinite time period, but the present the-ory of quasi-linear hyperbolic equations guarantees theexistence and regularity of solutions over only a finitetime internal. It is clear that even if true, any proof fora suitable version of weak censorship seems to requiremuch more knowledge on general global properties ofEinstein equations than is currently known.

To summarize the situation, cosmic censorship isclearly a crucial assumption underlying all of black holephysics and gravitational collapse theory, and relatedimportant areas. The first major task here would be toformulate rigorously a satisfactory statement for cos-mic censorship, which if not true would throw blackhole dynamics into serious doubt. This is why censor-ship is one of the most important open problems forgravitation theory today. No proof, however, seems pos-sible unless some major theoretical advances by way ofmathematical techniques and understanding the globalstructure of Einstein equations are made, and the direc-tion needed for such theoretical advances is far fromclear at present.

We, therefore, conclude that the first and foremosttask at the moment is to carry out a detailed and carefulexamination of various gravitational collapse scenariosto examine for their end states. It is only such an in-vestigation of more general collapse situations whichcould indicate what theoretical advances to expect forany proof, and what features to avoid while formulat-ing the cosmic censorship. Basically, we still do nothave sufficient data and information available on thevarious possibilities for gravitationally collapsing con-figurations so as to decide one way or other on the issueof censorship.

In recent years, many investigations have been car-ried out from such a perspective on gravitational col-lapse, either for inhomogeneous dust collapse or withmore general matter fields. It turns out that the collapseoutcome is not always a black hole and the naked singu-larity final state can arise in a variety of situations. In thenext sections we discuss some of these developments.

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20.13 Inhomogeneous Dust CollapseSince we are interested in collapse, we require that thespacetime contains a regular initial space-like hyper-surface on which the matter fields as represented bythe stress-energy tensor Tij have compact support andall physical quantities are well-behaved on this sur-face. Also, the matter should satisfy a suitable energycondition and the Einstein equations. We say that thespacetime contains a naked singularity if there is a fu-ture directed nonspace-like curve which reaches a faraway observer at infinity in future, which in the pastterminates at the singularity.

As an immediate generalization of the Oppen-heimer–Snyder–Datt homogeneous dust collapse, onecould consider the collapse of inhomogeneous dust andexamine the nature and structure of resulting singularitywith special reference to censorship and the occurrenceof black holes and naked singularities. The main moti-vation to discuss this situation is that it provides a clearpicture in an explicit manner of what is possible in grav-itational collapse. One could ask how the conclusionsgiven above for homogeneous collapse are modifiedwhen the inhomogeneities of matter distribution aretaken into account. Clearly, it is important to include ef-fects of inhomogeneities because typically a physicallyrealistic collapse would start from an inhomogeneousinitial data with a centrally peaked density profile.

This question of inhomogeneous dust collapse hasattracted the attention of many researchers and it isseen that the introduction of inhomogeneities leads toa rather different picture of gravitational collapse. Itturns out that while homogeneous collapse leads toblack hole formation, the introduction of any physicallyrealistic inhomogeneity, e.g., the density peaked at thecenter of the cloud and slowly decreasing away, leadsto a naked singularity final state for the collapse. Thisis certainly an intriguing result implying that the blackhole formation in gravitational collapse may not be sucha stable phenomenon as was thought to be the case.

The problem was investigated in detail using theTolman–Bondi–Lemaître models, which describe grav-itational collapse of an inhomogeneous sphericallysymmetric dust cloud [20.31]. This is an infinite dimen-sional family of asymptotically flat solutions of Ein-stein equations, which is matched to the Schwarzschildspacetime outside the boundary of the collapsing star.The Oppenheimer–Snyder–Datt model is a special caseof this class of solutions.

It is seen that the introduction of inhomogeneitiesleads to a rather different picture of gravitational

collapse. The metric for spherically symmetric col-lapse of inhomogeneous dust, in comoving coordinates.t; r; ; �/ is given by

ds2 D�dt2CR02

1C fdr2CR2.d 2C sin2 d�2/

Tij D �ıitı

jt; � D �.t; r/D

F0

R2R0;

where Tij is the stress–energy tensor, � is the energydensity, and R is a function of both t and r given by

PR2 DF

RC f :

Here the dot and prime denote partial derivatives withrespect to the parameters t and r, respectively. As we areconsidering collapse, we require PR.t; r/ < 0: The quan-tities F and f are arbitrary functions of r and 4�R2.t; r/is the proper area of the mass shells. The physicalarea of such a shell at rD const. goes to zero whenR.t; r/D 0. For the gravitational collapse situation, wetake � to have compact support on an initial space-likehypersurface and the spacetime is matched at some rDconst.D rc to the exterior Schwarzschild field with thetotal Schwarzschild mass m.rc/DM enclosed withinthe dust ball of coordinate radius of rD rc. The appar-ent horizon in the interior dust ball lies at RD F.r/.

Using this framework, the nature of the singular-ity RD 0 can be examined. In particular, the problem

Spacetimesingularity

Eventhorizon

Apparenthorizon

Initial surface

Outgoingnull geodesics

r =

2M

r =

0

t

r

Collapsingmatter

Fig. 20.2 The gravitational collapse of a spherical but in-homogeneous dust cloud with a density profile peaked atthe center. The event horizon no longer forms prior to thesingularity and the collapse end state is a naked singularity

Spacetime Singularities 20.14 Collapse with General Matter Fields 423Part

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of nakedness or otherwise of the singularity can bereduced to the existence of real, positive roots of an al-gebraic equation V.X/D 0, constructed out of the freefunctions F and f and their derivatives, which consti-tute the initial data of this problem. If the equationV.X/D 0 has a real positive root, the singularity couldbe naked. In order to be the end point of null geodesicsat least one real positive value of X0 should satisfy theabove. If no real positive root of the above is found,the singularity is not naked. It should be noted thatmany real positive roots of the above equation may ex-ist which give the possible values of tangents to thesingular null geodesics terminating at the singularityin the past. Suppose now X D X0 is a simple root toV.X/D 0. To determine whether X0 is realized as a tan-gent along any outgoing singular geodesics to givea naked singularity, one can integrate the equation ofthe radial null geodesics in the form rD r.X/ and itis seen that there is always at least one null geodesicterminating at the singularity tD 0; rD 0, with X DX0. In addition, there would be infinitely many inte-gral curves as well, depending on the values of theparameters involved, which terminate at the singular-

ity. It is thus seen that the existence of a positivereal root of the equation V.X/D 0 is a necessary andsufficient condition for the singularity to be naked. Fi-nally, to determine the curvature strength of the nakedsingularity at tD 0, rD 0, one may analyze the quan-tity k2RabKaKb near the singularity. Standard analysisshows that the strong curvature condition is satisfied,in that the above quantity remains finite in the limit ofapproach to the singularity. The spacetime picture fora collapse terminating in a naked singularity is givenin Fig. 20.2.

The assumption of vanishing pressures, which couldbe important in the final stages of the collapse, maybe considered as the limitation of dust models. On theother hand, it is also argued sometimes that in the fi-nal stages of collapse the dust equation of state couldbe relevant and at higher and higher densities the mat-ter may behave much more like dust. Further, if thereare no large negative pressures (as implied by the va-lidity of the energy conditions), then the pressure mightalso contribute gravitationally in a positive manner tothe overall effect of dust and may not alter the finalconclusions.

20.14 Collapse with General Matter Fields

It is clearly important to consider collapse situationswhich consider matter with nonzero pressures and withreasonable equations of state. It is possible that pres-sures may play an important role for the later stagesof collapse and one must investigate the possibilitywhether pressure gradients can prevent the occurrenceof naked singularity.

Many collapse scenarios have been considered bynow with nonzero pressures and physically reasonableequations of state. What one needs to examine hereagain is the existence, the termination of future directednonspace-like geodesic families at the singularity in thepast, and the strength of such a singularity for collapsewith nonzero pressure.

A useful insight into this issue is provided byself-similar collapse for a perfect fluid with a linearequation of state pD k�. A numerical treatment of self-similar perfect fluid collapse was given by Ori andPiran [20.32], and the analytic consideration for thesame was given by Joshi and Dwivedi [20.33]. It canbe seen that the collapse evolutions as allowed by theEinstein equations permit both the black hole and nakedsingularity final states. If in a self-similar collapse a sin-

gle null radial geodesic escapes the singularity, then,in fact, an entire family of nonspace-like geodesicswould also escape provided the positivity of energydensity is satisfied. It also follows that no families ofnonspace-like geodesics would escape the singularity,even though a single null trajectory might, if the weakenergy condition is violated. The singularity will beglobally visible to faraway observers in the spacetimefor a wide set of conditions. These results show thatnaked singularity is not avoided by the introduction ofnonzero pressures or a reasonable equation of state.

Actually, consideration of matter forms such as di-rected radiation, dust, perfect fluids, etc., imply thata similar general pattern emerges, as far as the final out-come of collapse is concerned. Basically, the result thatemerges is that, depending on the nature of the regularinitial data in terms of the density and pressure profiles,the Einstein equations permit both classes of dynamicalevolutions, those leading to either of the black hole ornaked singularity final states.

Hence one could ask the question whether the finalfate of collapse would be independent of the form of thematter under consideration. An answer to this is useful

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0(r), pθ0(r), pr0

(r),

E'(r) E(r)

Blackhole

Initialsurface

Nakedsingularity

Fig. 20.3 For a generic collapse of a general matter field, the col-lapse final state can be either a black hole or a naked singularitydepending on the dynamical evolution chosen as allowed by theEinstein equations

because it was often thought that once a suitable form ofmatter with an appropriate equation of state, also satis-fying energy conditions, is considered then there maybe no naked singularities. Of course, there is alwaysa possibility that during the final stages of collapse thematter may not have any of the forms such as dust orperfect fluids considered above, because such relativis-tic fluids are phenomenological and perhaps one musttreat matter in terms of some fundamental field, suchas, for example, a massless scalar field. In that con-text, a naked singularity is also seen to form for thescalar field collapse [20.34], although for fine-tuned ini-tial data.

In the above context, it is worth mentioning effortsin the direction of understanding collapse final statesfor general matter fields, which generalize the aboveresults on perfect fluid to matter forms without any re-striction on the form of Tij, with the matter satisfyingthe weak energy condition. A consideration to a gen-eral form of matter was given by Lake [20.35] and bySzekeres and Iyer [20.36], who do not start by assumingan equation of state, but consider a class of metric coef-ficients with a certain power law behavior. Also, Joshiand Dwivedi [20.37] and Goswami and Joshi [20.38],and Giambo et al. [20.39] had results in this direction.The main argument is along the following lines. It was

pointed out above that naked singularities could formin gravitational collapse from regular initial data fromwhich nonzero measure families of nonspace-like tra-jectories come out. The criterion for the existence ofsuch singularities was characterized in terms of the ex-istence of real positive roots of an algebraic equationconstructed out of the field variables. A similar proce-dure is developed now for the general form of matter.In comoving coordinates, the general matter can bedescribed by three free functions, namely the energydensity and radial and tangential pressures. The ex-istence of naked singularity is again characterized interms of the real positive roots of an algebraic equa-tion, constructed from the equations of nonspace-likegeodesics which involve the three metric functions. Thefield equations then relate these metric functions to thematter variables and it is seen that for a subspace of thisfree initial data in terms of matter variables, the abovealgebraic equation will have real positive roots, produc-ing a naked singularity in the spacetime. When no suchroots exist, the end state is a black hole.

It follows that the occurrence or otherwise of nakedsingularity is basically related to the choice of initialdata to the Einstein field equations as determined by theevolutions allowed. Therefore, these occur from regularinitial data within the general context considered, sub-ject to the matter satisfying the weak energy condition.The condition on initial data which leads to the forma-tion of black hole is also characterized.

It would then appear that the occurrence of nakedsingularity or a black hole is more a problem of thechoice of the initial data for field equations rather thanthat of the form of matter or the equation of state(Fig. 20.3).

Such a conclusion has an important implication forcosmic censorship in that in order to preserve the same,one must avoid all such regular initial data causingnaked singularity, and hence a much deeper under-standing of the initial data space is required in orderto determine such initial data and the kind of physi-cal parameters they would specify. In other words, thisclassifies the range of physical parameters to be avoidedfor a particular form of matter. Such an understand-ing would also pave the way for black hole physicsto use only those ranges of allowed parameter valueswhich produce black holes only, thus putting black holephysics on a firmer footing.

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20.15 Nonspherical Collapse and Numerical Simulations

Basically, the results and detailed studies such as theabove on gravitational collapse show that cosmic cen-sorship cannot hold in an unqualified general form. Itmust be properly fine-tuned and the black holes willform only under certain suitably restrictive conditionson collapse.

An important question at the same time is: what willbe the final fate of gravitational collapse which is notspherically symmetric? The main phases of sphericalcollapse of a massive star would be typically instabil-ity, implosion of matter, and subsequent formation ofan event horizon and a spacetime singularity of infinitedensity and curvature with infinite gravitational tidalforces. This singularity may or may not be fully cov-ered by the horizon, as we discussed above.

As noted, small perturbations over the sphericitywould leave the situation unchanged in the sense thatan event horizon will continue to form in the ad-vanced stages of the collapse. The next question thenis, do horizons still form when the fluctuations from thespherical symmetry are high and the collapse is highlynonspherical? It was shown by Thorne [20.40], for ex-ample, that when there is no spherical symmetry, thecollapse of infinite cylinders gives rise to naked singu-larities in general relativity, which are not covered byhorizons. This situation motivated Thorne to proposethe hoop conjecture for finite systems in an asymptot-ically flat spacetime for the final fate of nonsphericalcollapse. The horizons of gravity form when and onlywhen a mass M gets compacted in a region whose cir-cumference in every direction obeys C 2�.2GM=c2/.Thus, unlike cosmic censorship, the hoop conjecturedoes not rule out all naked singularities but only makesa definite assertion on the occurrence of event hori-zons in gravitational collapse. The hoop conjecture isconcerned with the formation of event horizons andnot with naked singularities. Thus, even when eventhorizons form, say, for example, in the spherically sym-metric case, it does not rule out the existence of nakedsingularities, or it does not imply that such horizonsmust always cover the singularities.

When the collapse is sufficiently aspherical, withone or two dimensions being sufficiently larger than theothers, the final state of collapse could be a naked singu-larity, according to the hoop conjecture. Such a situationis inspired by the Lin et al. [20.41] instability considera-tion in Newtonian gravity, where a nonrotating homoge-neous spheroid collapses maintaining its homogeneityand spheroidicity but with growing deformations. If the

initial condition is that of a slightly oblate spheroid, thecollapse results in a pancake singularity through whichthe evolution could proceed. However, for a slightlyprolate spheroidal configuration, the matter collapses toa thin thread which results into a spindle singularity.The gravitational potential and the tidal forces blow upas opposed to only density blowing up so it is a seri-ous singularity. Even in the case of an oblate collapse,the passing of matter through the pancake causes pro-lateness and, subsequently, a spindle singularity againresults without the formation of any horizon.

It is clear though that the nonspherical collapse sce-nario is rather complex to understand, and a recourse tothe numerical simulations of evolving collapse modelsmay greatly enhance our understanding of possible finalcollapse states in this case. In such a context, the nu-merical calculations of Shapiro and Teukolsky [20.42]indicated conformity with the hoop conjecture. Theyevolved collissionless gas spheroids in full general rela-tivity, which in all cases collapse to singularities. Whenthe spheroid is sufficiently compact a black hole mayform, but when the semimajor axis of the spheroid issufficiently large, a spindle singularity forms withoutthe formation of an apparent horizon. This gives rise tothe possibility of the occurrence of naked singularitiesin the collapse of finite systems in asymptotically flatspacetimes which violate weak cosmic censorship butare in accordance with the hoop conjecture.

We note that the Kerr black hole is believed to bethe unique stationary solution in Einstein gravity whenmass and rotation parameters are included. However, itis to be noted that while the Schwarzschild black holeis the final end state of homogeneous dust collapse, wehave no interior solution for a rotating collapsing cloud.In other words, an exterior Kerr geometry has no in-ternal solution in general relativity. We, therefore, donot really know the final fate of gravitational collapsewith rotation. To understand the same, numerical sim-ulations in full general relativity will be of great value.There are many such numerical programs in the makingcurrently to deal with this problem of modeling a rotat-ing collapsing massive star. The idea here is to includerotation in collapse and then to let the Einstein equa-tions evolve the collapse to see if the Kerr black holenecessarily emerges as a final state (see, e.g., [20.43]and references therein). It is worth noting that numeri-cal simulations in higher dimensions have also recentlyproduced some very intriguing naked singularity forma-tion scenarios (Lehner and Pretorius [20.44]).

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We finally note that apart from such numericalsimulations, certain analytic treatments of asphericalcollapse are also available. For example, the non-spherical Szekeres models for irrotational dust withoutany Killing vectors, generalizing spherical Tolman–Bondi–Lemaître collapse, were studied by Joshi and

Królak [20.45] to deduce the existence of strong curva-ture, naked singularities. While this indicates that nakedsingularities are not necessarily confined to sphericalsymmetry only, it is to be noted that dynamical evo-lution of a nonspherical collapse still remains a largelyuncharted territory.

20.16 Are Naked Singularities Stable and Generic?

Naked singularities may develop in gravitational col-lapse, either spherical or otherwise. However, if theyare 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 sta-bility of naked singularities arising from regular initialdata. Will the initial data subspace, which gives rise tonaked singularity as end state of collapse, have a vanish-ing measure in a suitable sense? In that case, one wouldbe able to reformulate more suitably the censorship hy-pothesis, based on the criterion that naked singularitiescould form in collapse but may not be generic.

We note here that the genericity and stability of thecollapse outcomes, in terms of black holes and nakedsingularities need to be understood carefully and in fur-ther detail. It is by and large well accepted now thatthe general theory of relativity allows and gives riseto both black holes and naked singularities as the finalfate of a continual gravitational collapse, evolving fromregular initial data and under reasonable physical condi-tions. What is not fully clear as yet is the distribution ofthese outcomes in the space of all allowed outcomes ofcollapse. The collapse models discussed above and theconsiderations we have given here would be of somehelp in this direction and may throw some light on thedistribution of black holes and naked singularity solu-tions in the initial data space. For some considerationson this issue, especially in the context of scalar fieldcollapse, we refer to Christodoulou [20.46] and Joshiet al. [20.47], and references therein, for further discus-sion. For the case of inhomogeneous dust collapse, theblack hole and naked singularity spaces are shown inFig. 20.4.

The important point, however, is that in general rel-ativity there is no well-defined concept or formulationas to what to call generic and stable outcomes, unlikethe Newtonian case. In other words, there are no well-defined criteria or definitions available as to what ismeant by stability in general relativity. The ambiguitymainly arises because of nonunique topologies on the

space of all Lorentzian metrics on a given spacetimemanifold and a similar nonuniqueness of measures. Inthis situation there is no easy way to answer the ques-tion in any unique and definite manner, and peoplegenerally resort to the physical meaningfulness of thecollapse scenario, which gives rise to either the blackhole or the naked singularity outcome.

From such a perspective, it is natural and meaning-ful to ask here, what is really the physics that causesa naked singularity to develop in collapse, rather thana black hole? We need to know how at all parti-cles and energy are allowed to escape from extremelystrong gravity fields. We have examined this issue insome detail to bring out the role of inhomogeneitiesand spacetime shear towards distorting the geometry ofhorizons that form in collapse (Fig. 20.5).

In Newtonian gravity, it is only the matter densitythat determines the gravitational field. In Einstein the-

Naked singularityBlack hole

33 34 35 36 37

M3

E5

0.96

0.94

0.92

0.9

Fig. 20.4 Initial data leading to black holes and nakedsingularities shown in the spaces of mass and energyfunctions

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F3 = –30F3 = –26F3 = –20F3 = –2F3 = –0.5

0 0.001 0.002 0.003 0.004 0.005

t

r

1.2×10–6

1×10–6

0.8×10–6

0.6×10–6

0.4×10–6

0.2×10–6

0

–0.2×10–6

Fig. 20.5 The apparent horizon formation is delayed de-pending on the amount of inhomogeneity present as thecollapse proceeds (after [20.48])

ory, however, density is only one attribute of the overallgravitational field, and the various curvature compo-nents and scalar quantities play an equally important

role to dictate what the overall nature of the field is.What we have shown is that once the density is inho-mogeneous or higher at the center of the collapsingstar, this rather naturally delays the trapping of lightand matter during collapse, which can in principle es-cape. This is a general relativistic effect to imply thateven if the densities are very high, there are pathsavailable for light or matter to escape due to inho-mogeneously collapsing matter fields. These physicalfeatures then naturally lead to a naked singularity for-mation [20.48].

As it turns out, it is the amount of inhomogene-ity that counts towards distorting the apparent horizonformation. If it is very small, below a critical limit,a black hole will form, but with sufficient inhomogene-ity the trapping is delayed to cause a naked singularity.This criticality also comes out in the Vaidya class ofradiation collapse models, where it is the rate of col-lapse, that is how fast or slow the cloud is collapsing,which determines the black hole or naked singularityformation.

20.17 Astrophysical and Observational Aspects

It is clear that the black hole and naked singularity out-comes of a complete gravitational collapse for a mas-sive star are very different from each other physicallyand would have quite different observational signatures.In the naked singularity case, if it occurs in nature, wehave the possibility to observe the physical effects oc-curring in the vicinity of the ultra-dense regions thatform in the very final stages of collapse. However, ina black hole scenario, such regions are necessarily hid-den within the event horizon of gravity.

There have been attempts where researchers ex-plored physical applications and implications of thenaked singularities (see, e.g., [20.49] and referencestherein). If we could find astrophysical applications ofthe models that predict naked singularities as collapsefinal fate and possibly try to test these through observa-tional methods and the signatures predicted, this couldoffer a very interesting avenue to obtain further insightinto the problem as a whole. An attractive recent possi-bility in this connection is to explore naked singularitiesas possible particle accelerators [20.50], where the pos-sibility also emerges that the Cauchy horizons may notbe innocuous, and high energy collisions could occur inthe vicinity of the same if they are generated by nakedsingularity (Fig. 20.6).

Also, the accretion discs around a naked singular-ity, wherein the matter particles are attracted towards orrepulsed away from the singularities with great veloci-ties could provide an excellent venue to test such effectsand may lead to predictions of important observationalsignatures to distinguish the black holes and nakedsingularities in astrophysical phenomena. It is then nec-essary to investigate the question of what observational

A B

Naked singularity

Collision

Cauchyhorizon

Fig. 20.6 Very high energy particle collisions can occurin the vicinity of the Cauchy horizon emerging from thenaked singularity

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signatures would then emerge and to distinguish theblack holes from naked singularities, and we must ex-plore what special astrophysical consequences the lattermay have.

One may ask several intriguing questions such as:Where could the observational signatures of naked sin-gularities lie? If we look for the sign of singularitiessuch as the ones that appear at the end of collapse, wemust consider explosive and high energy events. In fact,such models expose the ultra-high density region at thetime of the formation of the singularity while the outershells are still falling towards the center. In such a case,shock waves emanating from the superdense region atscales smaller than the Schwarzschild radius (whichcould be due to quantum effects or repulsive classicaleffects) and collisions of particles near the Cauchy hori-zon could have effects on the outer layers. These wouldbe considerably different from those appearing duringthe formation of a black hole. If, on the other hand, weconsider singularities such as the super-spinning Kerrsolution we can look for different kinds of observationalsignatures. Among these the most prominent featuresdeal with the way the singularity may affect incomingparticles, either in the form of light bending, such asin gravitational lensing, particle collisions close to thesingularity, or properties of accretion disks.

Essentially what we ask is: Could we test censor-ship using astronomical observations? With so manyhigh technology power missions to observe the cosmos,can we not just observe the skies carefully to deter-mine the validity or otherwise of cosmic censorship? Inthis connection, several proposals to measure the massand spin ratio for compact objects and for the galacticcenter have been made by different researchers. In par-ticular, using pulsar observations it has been suggestedthat gravitational waves and the spectra of x-ray bina-ries could test the rotation parameter for the center ofour galaxy. Also, the shadow cast by the compact ob-ject can be used to test the same in stellar mass objects,or the x-ray energy spectrum emitted by the accretiondisk can be used. Using certain observable propertiesof gravitational lensing that depend upon rotation hasalso been suggested (for references, see [20.49]).

The basic issue here is that of sensitivity, namelyhow accurately and precisely we can measure and deter-mine these parameters. A number of present and futureastronomical missions may be of help. One of theseis the Square-Kilometer Array (SKA) radio telescope,which will offer a possibility here, with a collecting areaexceeding a factor of 100 compared to existing ones.SKA astronomers point out they will have the sensitiv-

ity desired to measure the required quantities to veryprecisely determine vital fundamental issues in gravita-tion physics, such as cosmic censorship, and to decideon its validity or otherwise. Other missions that could inprinciple provide a huge amount of observational dataare those that are currently hunting for gravitationalwaves. Gravitational wave astronomy has yet to claimits first detection of waves, nevertheless in the comingyears it is very likely that the first observations willbe made by experiments such as LIGO and VIRGO,which are currently still below the threshold for obser-vation. Then gravitational wave astronomy will becomean active field with possibly large amounts of data to bechecked against theoretical predictions, and it appearsalmost certain that this will have a strong impact onopen theoretical issues such as the cosmic censorshipproblem.

There are three different kinds of observations thatone could devise in order to distinguish a naked sin-gularity from a black hole. The first one relies onthe study of accretion disks. The accretion propertiesof particles falling onto a naked singularity would bevery different from those of black hole of the samemass (see, for example, [20.51]), and the resulting ac-cretion disks would also be observationally different.The properties of accretion disks have been studiedin terms of the radiant energy, flux, and luminosity,in a Kerr-like geometry with a naked singularity, andthe differences from a black hole accretion disk havebeen investigated. Also, the presence of a naked sin-gularity gives rise to powerful repulsive forces thatcreate an outflow of particles from the accretion diskon the equatorial plane. This outflow that is otherwisenot present in the black hole case, could, in princi-ple, be distinguished from the jets of particles thatare thought to be ejected from a black hole’s polarregion and which are due to strong electromagneticfields. Also, when charged test particles are consideredthe accretion disk’s properties for the naked singular-ity present in the Reissner–Nordstrom spacetime areseen to be observationally different from those of blackholes.

The second way of distinguishing black holes fromnaked singularities relies on gravitational lensing. It isargued that when the spacetime does not possess a pho-ton sphere, the lensing features of light passing closeto the singularity will be observationally different fromthose of a black hole. This method, however, does notappear to be very effective when a photon sphere ispresent in the spacetime. Assuming that a Kerr-like so-lution of Einstein equations with massless scalar field

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0 3MTOT

Interior of the cloud

High density region

Schwarzschild exterior

t

t–

R

R(rb, t)

Light rays

Rb

Fig. 20.7 An equilibrium configuration can be obtainedfrom gravitational collapse which halts asymptotically andwhich would contain a central naked singularity

exists at the center of galaxies, its lensing properties arestudied and it was found that there are effects due tothe presence of both the rotation and scalar field thatwould affect the behavior of the bending angle of the

light ray, thus making those objects observationally dif-ferent from black holes.

Finally, a third way of distinguishing black holesfrom naked singularities comes from particle collisionsand particle acceleration in the vicinity of the singu-larity. In fact, it is possible that the repulsive effectsdue to singularity can deviate a class of infalling par-ticles, making these outgoing eventually. These couldthen collide with some ingoing particle, and the energyof collision could be arbitrarily high, depending on theimpact parameter of the outgoing particle with respectto the singularity. The net effect is thus the creation ofa very high energy collision that resembles that of animmense particle accelerator and that would be impos-sible in the vicinity of a Kerr black hole.

It was pointed our recently by Joshi et al. [20.52]that one could obtain equilibrium configurations as thefinal outcome of a gravitational collapse. If such anobject arises without trapped surfaces but with a sin-gularity at the center (see Fig. 20.7) then again theaccretion disk properties are very different from a blackhole of the same mass.

20.18 Predictability and Other Cosmic Puzzles

What then is the status of naked singularities versuscensorship today? Can cosmic censorship survive insome limited and specialized form, and firstly, can weproperly formulate it after all these studies on gravi-tational collapse in recent years? While this continuesto be a major cosmic puzzle, recent studies on the for-mation of naked singularities as collapse end states formany realistic models have brought to forefront someof the most intriguing basic questions, both at clas-sical and quantum levels, which may have significantphysical relevance. Some of these are: can the superultra-dense regions forming in a physically realistic col-lapse of a massive star be visible to far away observersin spacetime? Are there any observable astrophysicalconsequences? What is the causal structure of space-time in the vicinity of singularity as decided by theinternal dynamics of collapse which evolves from regu-lar initial data at an initial time? How early or late willthe horizons actually develop in a physically realisticgravitational collapse, as determined by the astrophysi-cal conditions within the star? When a naked singularityforms, is it possible to observe the quantum gravity ef-fects 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 withina general and physically realistic framework may bethe only way to find answers to some of these issues.This could lead us to novel physical insights and possi-bilities emerging out of the intricacies of gravitationalforce and the nature of gravity, as emerging from exam-ining the dynamical evolutions as allowed by Einsteinequations.

Apart from their physical relevance, the collapsephenomena also have profound philosophical impli-cations such as on the issue of predictability in theuniverse. Below, we summarize a few arguments forand against this in classical general relativity.

It is sometimes argued that the breakdown of cen-sorship means violation of predictability in spacetime,because we have no direct handle to know what a nakedsingularity may radiate and emit unless we study thephysics in such ultra-dense regions. One would then notbe able to predict the universe in the future of a givenepoch of time as would be the case, for example, inthe case of the Schwarzschild black hole that devel-ops in Oppenheimer–Snyder collapse. A concern that

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is usually expressed is if naked singularities occurredas the final fate of gravitational collapse, predictabil-ity is violated in spacetime because naked singularityis characterized by the existence of light rays and par-ticles that emerge from the same. Typically, in all thecollapse models discussed above, there is a family offuture directed nonspace-like curves that reach externalobservers, and when extended in the past they met thesingularity. The first light ray that comes out from thesingularity marks the boundary of the region that canbe predicted from a regular initial Cauchy surface in thespacetime, and this is called the Cauchy horizon for thespacetime. The causal structure of spacetime would dif-fer significantly in the two cases when there is a Cauchyhorizon and when there is none.

In general relativity, a given epoch of time is some-times represented by a space-like surface, which isa three-dimensional space section. For example, in thestandard Friedmann models of cosmology, there is suchan epoch of simultaneity, from which the universeevolves in future, given the physical variables and initialdata on this surface. The Einstein equations govern thisevolution of universe, and there is thus a predictabilitywhich one would expect to hold in a classical theory.The concern then is that one would not be able to predictthe future of naked singularity, and that unpredictableinputs may emerge from the same.

Given regular initial data on a space-like hyper-surface, one would like to predict the future and pastevolutions in spacetime for all times (see, for exam-ple, [20.1]). Such a requirement is termed the globalhyperbolicity of the spacetime. A globally hyperbolicspacetime is a fully predictable universe; it admitsa Cauchy surface, which is a three-dimensional space-like surface, the data on which can be evolved for alltimes in the past as well as in the future. Simple enoughspacetimes such as Minkowski or Schwarzschild areglobally hyperbolic, but Reissner–Nordstrom or Kerrgeometries are not globally hyperbolic. For further de-tails on these issues, we refer to [20.26].

The key role that the event horizon of a black holeplays is that it hides the super ultra-dense region formedin collapse from us. So the fact that we do not under-stand such regions has no effect on our ability to predictwhat happens in the universe at large. However, if nosuch horizon exists, then the ultra-dense region might,in fact, play an important and even decisive role in therest of the universe, and our ignorance of such regionswould become of more than merely academic interest.

Yet such an unpredictability is common in generalrelativity and not always directly related to censorship

violation. Even black holes themselves need not fullyrespect predictability when they rotate or have somecharge. For example, if we drop an electric chargeinto an uncharged black hole, the spacetime geome-try changes radically and is no longer predictable froma regular initial epoch of time. A charged black hole ad-mits a naked singularity that is visible to an observerwithin the horizon, and a similar situation holds whenthe black hole is rotating. There has been an importantdebate in recent years whether one could over-charge orover-rotate a black hole so that the singularity visible toobservers within the horizon would also become visibleto external far away observers (see, e.g., [20.53]).

Also, if such a black hole were big enough on a cos-mological scale, the observer within the horizon could,in principle, survive happily for millions of years with-out actually falling into the singularity, and would thusbe able to observe the naked singularity for a long time.Thus, only the purest of pure black holes with no chargeor rotation at all respects the full predictability, and allother physically realistic ones with charge or rotationactually do not. As such, there are many models of theuniverse in cosmology and relativity that are not totallypredictable from a given space-like hypersurface in thepast. In these universes, the spacetime cannot be neatlyseparated into space and time foliation so as to allowinitial data at a given moment of time to fully determinethe future.

Actually the real breakdown of predictability is theoccurrence of spacetime singularity itself we could say,which indicates the true limitation of the classical grav-ity theory. It does not matter really whether or not it ishidden within an event horizon. The real solution of theproblem would then be the resolution of singularity it-self, through either a quantum theory of gravity or insome way at the classical level.

In fact, the cosmic censorship way to predictabil-ity, that of hiding the singularity within a black hole,and then thinking that we restored the spacetime pre-dictability may not be the real solution, or at best itmay be only a partial solution to the key issue ofpredictability in spacetime universes. It may be justshifting the problem elsewhere, and some of the cur-rent major paradoxes faced by black hole physics suchas the information paradox, the various puzzles regard-ing the nature of Hawking radiation, and other issuescould also be a manifestation of the same.

No doubt, the biggest argument in support of cen-sorship is that it would justify and validate the extensiveformalism and laws of black hole physics and the as-trophysical applications made so far. Censorship has

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r = 0 Generalised Vaidya metric

Interiorgeometry

Outgoingradiation

Fig. 20.8 If the star could radiate away very considerablemass, especially through negative quantum pressures closeto the classical singularity, this may effectively resolve thesingularity

been the foundation for the laws of black holes suchas the area theorem and others, and their astrophysi-cal applications. However, these are not free of majorparadoxes. Even if we accept that all massive starswould necessarily turn into black holes, this still createssome major physical paradoxes. Firstly, all the matterentering a black hole must of necessity collapse intoa spacetime singularity of infinite density and curva-tures, where all known laws of physics break down,which is some kind of instability at the classical levelitself. This was a reason why many gravitation theo-rists of the 1940s and the 1950s objected to black holeformation, and Einstein also repeatedly argued againstsuch a final fate of a collapsing star, writing a paper in1939 to this effect. Also, as is well known and has beenwidely discussed in the past few years, a black hole,by potentially destroying information, appears to con-tradict the basic principles of quantum theory. In thatsense, the very formation of a black hole itself witha singularity within it appears to come laden with inher-ent problems. It is far from clear how one would resolvethese basic troubles even if censorship were correct.

In view of such problems with the black holeparadigm, a possibility worth considering is the delayor avoidance of horizon formation as the star collapsesunder gravity. This happens when collapse to a nakedsingularity takes place, namely, where the horizon doesnot 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 theblack hole conundrum, while also resolving the singu-larity issue, because now there is no mass left to formthe singularity. While this may be difficult to achievepurely classically, such a phenomenon could happenwhen quantum gravity effects are taken into account(Fig. 20.8, see also the next section for a further dis-cussion).

What this means is that such an unpredictability issomewhat common in general relativity. For example, ifwe drop a slight charge in a Schwarzschild black hole,the spacetime geometry completely changes into that ofa charged black hole that is no longer predictable in theabove sense. A similar situation holds when the blackhole is rotating. In fact, there are very many models ofuniverse in use in relativity that are not globally hyper-bolic, that is, not totally predictable in the above sensewhere space and time are neatly separated so as to allowinitial data to fully determine the future for all times.

In any case, a positive and useful feature thathas emerged from work on collapse models so faris, we already now have several important constraintsfor any possible formulation of censorship. It can beseen that several versions of censorship proposed ear-lier would not hold, because explicit counter-examplesare now available. Clearly, analyzing gravitational col-lapse plays a crucial role here. Only if we understandclearly why naked singularities develop as collapse endstates in many realistic models, could there emerge anypointer or lead to any practical and provable version ofcensorship.

Finally, it may be worth noting that even if theproblem of singularity were resolved somehow, possi-bly by invoking quantum gravity which may smear thesingularity, we still have to mathematically formulateand prove the black hole formation assuming an ap-propriate censorship principle, which is turning out tobe a most difficult task with no sign of resolution. Ashas been discussed, the detailed collapse calculationsof recent years show that the final fate of a collapsingstar could be a naked singularity in violation of cen-sorship. Finally, as is well known and has been widelydiscussed by now, a black hole creates an informationloss paradox, violating unitarity and contradicting basicprinciples of quantum theory. It is far from clear howone would resolve these basic troubles even if censor-ship were correct.

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20.19 A Lab for Quantum Gravity–Quantum Stars?

It is believed that when we have a reasonable andcomplete quantum theory of gravity, all spacetime sin-gularities, whether naked or those hidden inside blackholes, will be resolved. As of now, it remains an openquestion whether quantum gravity will remove nakedsingularities. After all, the occurrence of spacetime sin-gularities could be a purely classical phenomenon andwhether they are naked or covered should not be rel-evant, because quantum gravity will possibly removethem all anyway. It is possible that in a suitable quan-tum gravity theory the singularities will be smeared out,although this has not been realized so far.

In any case, the important and real issue is whetheror not the extreme strong gravity regions formed due togravitational collapse are visible to faraway observers.It is quite clear that gravitational collapse would cer-tainly proceed classically, at least until quantum gravitystarts governing and dominating the dynamical evolu-tion at scales of the order of the Planck length, i. e.,until the extreme gravity configurations have alreadybeen developed due to collapse. The point is that it isthe visibility or otherwise of such ultra-dense regionsthat is under discussion, whether they are classical orquantum (see Fig. 20.9).

What is important is that classical gravity necessar-ily implies the existence of ultra-strong gravity regions,where both classical and quantum gravity come intotheir own. In fact, if naked singularities develop ingravitational collapse, then in a literal sense we comeface-to-face with the laws of quantum gravity wheneversuch an event occurs in the universe.

In this way, the gravitational collapse phenomenonhas the potential to provide us with a possibility of ac-tually testing the laws of quantum gravity. In the caseof a black hole developing in the collapse of a finitesized object such as a massive star, such strong gravityregions are necessarily hidden behind an event horizonof gravity, and this would be well before the physicalconditions become extreme near the spacetime singu-larity. In that case, the quantum effects, even if theycause qualitative changes closer to singularity, will beof no physical consequence as no causal communica-tions are then allowed from such regions. On the otherhand, if the causal structure were that of a naked singu-larity, then the communications from such a quantumgravity dominated extreme curvature ball would be vis-ible in principle. This would be so either through directphysical processes near a strong curvature naked singu-larity, or via the secondary effects, such as the shocks

produced in the surrounding medium. It is possible thata spacetime singularity basically represents the incom-pleteness of the classical theory and when quantumeffects are combined with the gravitational force, theclassical singularity may be resolved.

Therefore, more than the existence of a naked sin-gularity, the important physical issue is whether theextreme gravity regions formed in the gravitational col-lapse of a massive star are visible to external observersin the universe. An affirmative answer here would meanthat such a collapse provides a good laboratory to studyquantum gravity effects in the cosmos, which may pos-sibly generate clues for an as yet unknown theory ofquantum gravity. Quantum gravity theories in the mak-ing, such as string theory or loop quantum gravity, infact, are badly in need of some kind of an observationalinput, without which it is nearly impossible to constrainthe plethora of possibilities.

We could say quite realistically that a laboratorysimilar to that provided by the early universe is createdin the collapse of a massive star. However, the big bang,which is also a naked singularity in that it is, in princi-ple, visible to all observers, occurred only once in thelife of the universe and is, therefore, a unique event.However, a naked singularity of gravitational collapsecould offer an opportunity to explore and observe thequantum gravity effects every time a massive star in theuniverse ends its life.

The important questions one could ask are the fol-lowing: If in realistic astrophysical situations the starterminates as a naked singularity, would there be anyobservable consequences which reflect the quantumgravity signatures in the ultra-strong gravity region? Donaked singularities have physical properties differentfrom those of a black hole? Such questions underlie ourstudy 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 avoid-ance of horizon formation as the star evolves collapsingunder gravity. This happens when collapse to a nakedsingularity takes place, where the horizon does not formearly enough or is avoided. In such a case, in the latestages of collapse if the star could radiate away mostof its mass, then this may offer a way out of the blackhole conundrum, while also resolving the singularityissue, because now there is no mass left to form the cur-vature singularity. The purpose is to resolve the blackhole paradoxes and avoid the singularity, either vis-

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Eventhorizon

Quantum gravityfuzzy region

Naked singularity

Eventhorizon

Black hole

Collapsingmatter

Fig. 20.9 Naked singularity may beresolved by quantum gravity effectsbut the ultra-strong gravity regionthat developed in gravitational col-lapse will still be visible to externalobservers in the universe

ible or within a black hole, which actually indicatesthe breakdown of physical theory. The current work ongravitational collapse suggests possibilities in this di-rection.

In this context, we considered a cloud that col-lapsed to a naked singularity final state, and in-troduced loop quantum gravity effects [20.54] (seealso [20.55]). It turned out that the quantum effectsgenerated an extremely powerful repulsive force withinthe cloud. Classically the cloud would have termi-nated into a naked singularity, but quantum effectscaused a burst-like emission of matter in the very lastphases of collapse, thus dispersing the star and dis-solving the naked singularity. The density remainedfinite and the spacetime singularity was eventuallyavoided. One could expect this to be a fundamen-tal feature of other quantum gravity theories as well,but more work would be required to confirm sucha conjecture.

For a realistic star, its final catastrophic collapsetakes place in a matter of seconds. A star that lived mil-lions of years thus collapses in only tens of seconds. Inthe very last fraction of a microsecond, almost a quarterof its total mass must be emitted due to quantum ef-fects, 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 signaturesof quantum effects taking place in such ultra-dense re-gions. In our case, these included a sudden dip in theintensity of emission just before the final burst-likeevaporation due to quantum gravity. The question is,whether such unique astrophysical signatures can be de-tected by modern experiments, and if so, what they tellus about quantum gravity, and if there are any new in-

sights into other aspects of cosmology and fundamentaltheories such as string theory. The key point is that be-cause the very final ultra-dense regions of the star areno longer hidden within a horizon as in the black holecase, the exciting possibility of observing these quan-tum effects now arises, independently of the quantumgravity theory used. An astrophysical connection to ex-treme high energy phenomena in the universe, such asthe gamma-ray bursts that have defied any explanationsso far, may not be ruled out.

Such a resolution of naked singularity throughquantum gravity could be a solution to some of theparadoxes mentioned above (see also [20.56]; for otherpossibilities on singularity resolution). Then, whenevera massive star undergoes a gravitational collapse, thismight create a laboratory for quantum gravity in theform of a quantum star (see, e.g., [20.53]), that wemay possibly be able to access. This would also sug-gest intriguing connections to high energy astrophysicalphenomena. The present situation poses one of the mostinteresting challenges that has emerged through the re-cent work on gravitational collapse.

We hope the considerations here have shown thatgravitational collapse, which is essentially the investi-gation of dynamical evolutions of matter fields underthe force of gravity in spacetime, provides one of themost exciting research frontiers in gravitation physicsand high energy astrophysics. In our view, there isscope, therefore, for both theoretical as well as numeri-cal investigations in these areas, which may have muchto tell for our quest on basic issues in quantum gravity,fundamental physics, and gravity theories, and towardsthe expanding frontiers of modern high energy astro-physical observations.

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20.20 Concluding RemarksWe have considered here several aspects of spacetimesingularities and the physical scenarios where thesemay be relevant because they play an interesting andintriguing role. We hope this creates a fairly good viewof the exciting new physics that the spacetime singular-ities are leading us to, presenting a whole spectrum ofnew possibilities towards our search of the universe.

After discussing their existence and certain key ba-sic properties, we discussed in some detail the gravita-tional collapse scenarios and the useful conclusions thathave emerged so far in this context. In the first place,singularities not covered fully by the event horizon oc-cur in several collapsing configurations from regularinitial data, with reasonable equations of state such asthose describing radiation, dust, or a perfect fluid witha nonzero pressure, or also for general forms of mat-ter. These naked singularities are physically significantin that densities and curvatures diverge powerfully nearthe same. Such results on the final fate of collapse,generated from the study of different physically rea-sonable collapse scenarios, may provide useful insightsinto black hole physics and may be of help for any pos-sible formulation of the cosmic censorship hypothesis.

An insight that seems to emerge is that the final stateof a collapsing star, in terms of either a black hole ora naked singularity, may not really depend on the formor equation of state of collapsing matter, but is actuallydetermined by the physical initial data in terms of theinitial density profiles and pressures.

As an example, for inhomogeneous dust collapse,the final fate could be a black hole or a naked sin-gularity depending on the values of initial parameters.The collapse ends in a naked singularity if the lead-ing nonvanishing derivative of density at the center iseither the first one or the second one. There is a tran-sition from the naked singularity phase to the blackhole phase as the initial density profile is made moreand more homogeneous near the center. As one pro-gresses towards more homogeneity, and hence towardsa stronger gravitational field, there first occurs a weaknaked singularity, then a strong naked singularity, andfinally a black hole.

The important question then is the genericity andstability of such naked singularities arising from regularinitial data. Will the initial data subspace giving rise tonaked singularity have zero measure in a suitable sense?In that case, one would be able to reformulate more suit-ably the censorship hypothesis, based on a criterion thatnaked singularities could form in collapse but may not

be generic. As we pointed out, the answer is far fromclear due to ambiguities in the definitions of measuresand the stability criteria.

One may try to evolve some kind of a physical for-mulation for cosmic censorship, where the availablestudies on various gravitational collapse scenarios suchas the above may provide a useful guide. The variousproperties of naked singularities may be collectivelystudied as they emerge from the studies so far, and onewould then argue that objects with such properties arenot physical. However, the way forward is again farfrom clear.

One could also invoke quantum effects and quantumgravity. While naked singularities may form in classicalgeneral relativity, quantum gravity presumably removesthem. The point is that even though the final singular-ity may be removed in this way, there would still bevery high density and curvature regions in the classicalregime which would be causally communicating withoutside 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 cen-sorship.

We hope the considerations here have shown thatgravitational collapse, which essentially is the investi-gation of dynamical evolutions of matter fields underthe force of gravity in spacetime, provides one of themost exciting research frontiers in gravitation physicsand high energy astrophysics. There are issues herewhich have deep relevance both for theory as well asobservational aspects in astrophysics and cosmology.Also these problems are of relevance for the basics ofgravitation theory and quantum gravity, and these in-spire a philosophical interest and inquiry into the natureand structure of spacetime, causality, and profound is-sues such as predictability in the universe, as we haveindicated here.

Research is already taking place in many of theseareas as the discussion here has pointed out. Someof the most interesting questions from the author’spersonal perspective are: genericity and stability of col-lapse outcomes, examination of the quantum gravityeffects near singularities, observational and astrophys-ical signatures of the collapse outcomes, and otherrelated issues. In particular, one of the most interest-ing questions would be, if naked singularities whichare hypothetical astrophysical objects, did actually formin nature, what distinct observational signatures wouldthey present? That is, how one distinguishes black holes

Spacetime Singularities References 435Part

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from naked singularities would be an important issue.There have been some efforts on this issue in recentyears, as we indicated above. The point is that there arealready very high energy astrophysical phenomena be-ing observed today, with several observational missionsworking both from ground and space. The black holesand naked singularities, which are logical consequencesof star collapse in general relativity, would appear tobe the leading candidates to explain these phenomena.The observational signatures that each of these would

present, and their astrophysical consequences would beof much interest for future theoretical and computa-tional research, and for their astrophysical applications.

In our view, there is, therefore, a scope for boththeoretical as well as numerical investigations in thesefrontier areas, which may have much to say for ourquest on basic issues in quantum gravity, fundamentalphysics, and gravity theories, and towards the expand-ing frontiers of modern high energy astrophysical ob-servations.

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