Spacetime Singularities and Cosmic Censorship
aa r X i v : . [ g r- q c ] O c t Spacetime Singularities and Cosmic Censorship
Pankaj S. Joshi
Tata Institute for Fundamental ResearchHomi Bhabha Road, Mumbai 400005, India
We present here a brief review and discussion on recent developments in the theory ofspacetime singularities. After mentioning some key motivations on the main ideas andconcepts involved, we take the approach that the singularities will be eventually resolved bythe quantum gravity effects. Some consequences are indicated when such singularities arevisible to far away observers in the universe.
I. INTRODUCTION
Physical phenomena in the universe take place in the arena of space, and evolve in time. Theknown laws of physics describe and govern these happenings that occur in nature. But then whatare space and time, what are their interconnections if any, and how the space and time themselvesoriginate, or whether each of these is actually infinite and endless? These are some of the mostprofound questions that have exercised greatest of minds in science and philosophy over past manycenturies.The best possible scientific theory that we have today, that governs the universe of space andtime is the general theory of relativity. In relativity, the space and time are no longer separateand independent of each other, but they are intertwined with each other and the actual physi-cal measurements of these quantities are always mutually related. While general relativity wasoriginally developed to describe the force of gravitation, it has provided us with some of the mostintriguing insights on the nature of space and time, their inter-connections, and the possible originsof the time and space itself. General relativity suggests not to view gravity as a ‘force’ in the usualsense, but describes it as the curvature and geometry of the ‘space-time continuum’, which is ouruniverse.General relativity describes the interply of the space-time curvature, and the matter within itthat generates such a curvature, just as a metal ball placed on a rubber sheet curves it. It is aclassical theory that governs the universe in its large scale structure. Given any physical systemgoverned mainly by gravity, such as a large collection of galaxies, or a massive star that it close tothe end of its evolution having burnt all its nuclear fuel, the Einstein equations govern the futureevolution of such a system in time. Thus we can ask the questions such as how the universe ofgalaxies that is continuously expanding will evolve in future, and whether it will continue to expandor will it start shrinking at some point in time in future. Or one could ask, what will be the finalend point of evolution of a massive star that has started contracting under the force of its owngravity when its internal fuel is exhausted.One of the most remarkable predictions of general relativity, developed during the 1960s andearly 1970s has been that, dynamical evolution of matter fields in a space-time generically producesa space-time singularity. Such a singularity is where the densities, space-time curvatures and allother physical quantities blow up and grow arbitrarily large, and thus all known physical laws nolonger hold there. In that sense, the singularity is the end (or beginning) of the space and timethemselves.The singularity theorems developed by Roger Penrose, Stephan Hawking and Robert Gerochshow that the evolution of matter fields in a spacetime generically yields such a singularity, pro-vided reasonable physical conditions are satisfied such as the causality ensuring that you do notreturn to your own past, a suitable energy condition ensuring the positivity of energy density, andformation of what are called ‘trapped surfaces’ in a space-time that indicate and characterize thatthe gravitational field is sufficiently strong. The space-time singularities develop in cosmology,where they signal the beginning of time, and in gravitational collapse of massive stars, which isan issue of great interest in gravitation physics today that has been investigated in much detail inrecent years in the Einstein theory.We outline here some aspects of space-time singularities that occur in cosmology and in grav-itational collapse. The singularity theorems predicting the occurrence of singularities allow thesingularities of gravitational collapse to be either visible to external observers or covered by anevent horizon of gravity. Some consequences of this fact are indicated. The role of space-timesingularities as an inevitable feature of Einstein’s theory of gravity has became clear now as sig-nalling the situations where the gravitational field becomes ultra-strong and grows without anyupper bound. Close to the singularity is the regime of strong gravity fields, where general relativitycomes into its own to imply most interesting physical consequences.
II. THE OCCURRENCE OF SINGULARITIES
We discuss now the occurrence of space-time singularities in some detail within a general space-time framework. The basic ideas involved in the singularity theorems are indicated, and what thesetheorems do not imply is pointed out.We observe the universe today to the very far depths in space and time through the telescopesthat observe the objects which are billions of light years away. One could look deep into space, letus say in diametrically opposite directions. Then the regions with extremely distant galaxies areseen in each of these directions. It is most interesting to observe that these regions have actuallyquite similar properties in terms of their appearance and there is a homogeneity seen in the spatialdistribution of the far away galaxies. The universe also looks similar in different directions, thusexhibiting an isotropy. These regions are, however, so far away from each other that they havehad no time to interact mutually. That is because, general relativity equations imply that ahomogeneous and isotropic universe had a finite age in the past when the energy density of matteris positive. The age of the universe since such a big bang that indicated the origin of both spaceand time has not been actually large enough for any such interactions to have taken place in thepast. Thus, within the big bang framework of cosmology, a very relevant question arises: How comethese regions have such similar properties? This is one of the major puzzles of modern cosmologytoday.This observed homogeneity and isotropy of the universe at large enough scales can be modelledby the so called Friedmann-Robertson-Walker geometry. The metric describing the geometry ofthe corresponding space-time universe is given by ds = − dt + R ( t ) (cid:2) dr (1 − kr ) + r d Ω (cid:3) . (1)Here d Ω = dθ + sin θdφ is the metric on a two dimensional sphere and the universe is assumedto be spherically symmetric here. The additional assumption here is that the matter content ofthe space-time is homogeneous and isotropic, to represent these observed features of the universe.The scale factor R ( t ) increases in time so as to model the observed expansion of the universe wheregalaxies recede from each other in space. Thus, the matter density is the same everywhere in theuniverse at any given epoch of time, and also the visual appearance of universe looks the same inall directions.The galaxies here are represented by point-like objects which form ‘dust particles’ of this uni-verse, which is the matter content of the space-time. Combining these geometrical features withthe Einstein equations and solving the same, one is led to the Friedmann solution yielding a de-scription of the dynamical evolution of the universe and the matter within it. The picture obtainedfrom such an evolution implies that the universe must have had a beginning at a finite time inthe past. This is the epoch of the so called big bang singularity. The matter density as well asthe curvatures of spacetime diverge in the limit of approaching this cosmological singularity. Thisis an epoch where all non-spacelike geodesics that represent the trajectories of the photons andthe material particles come to an end and these are ‘incomplete’ at a point in the past where thespace-time comes to an end.A similar occurrence and formation of a space-time singularity takes place when a massive starcollapses freely under the force of its own gravity when it has exhausted its internal fuel whichmade it shine earlier. If the mass of the star is small enough, it can stabilize as a white dwarf or aneutron star at the end of its life cycle, which would then be the natural endstate of its evolution.However, in case the mass is much larger, say of the order of tens of solar masses, a continualgravitational collapse of the star is inevitable once there are no internal pressures left to sustainthe star. This is because no known forces can then stabilize such a star. Such a scenario wasconsidered and modelled using general relativity by R. Oppenheimer and H. Snyder in 1939, andby B. Datt in 1938, when they considered a collapsing spherical cloud of dust. Again, according tothe equations of general relativity, a space-time singularity of infinite density and curvature formsat the center of the collapsing cloud. We shall discuss gravitational collapse in some more detailin the next section.Such space-time singularities were discovered in the context of specific models of universe orof a collapsing massive star, such as those discussed above, and after their discovery these weredebated extensively by the gravitation theorists in the 1940s and 1950s. An important key questionthat was persistently asked at this juncture in this connection was the following: Why shouldthese models be taken seriously at all, when they were so special because they assumed so manysymmetries of space-time? As a result, perhaps such a space-time singularity was arising due tothese special symmetry assumptions, and may be it could occur in such special circumstances onlyas described and assumed by these models, but possibly it would not actually develop in the actualphysical reality which is our universe. In other words, these singularities could be just some isolatedexamples occurring in some special models and manifestation of the symmetry assumptions made.After all, the Einstein equations R ab − Rg ab = 8 πT ab , (2)governing the ever present force of gravity, are a complex system of second-order, non-linear, partialdifferential equations, which admit an infinite space of solutions. The models discussed above areonly special cases and isolated examples in this full space of solutions.Therefore, the main issue was the absence of any general enough proof that such space-timesingularities would always occur in a general enough gravitational collapse depicting actual physicalsystems when a massive star dies, or in a generic enough cosmological scenario. In fact, there wasa widespread belief in the 1940s and 1950s that such singularities would be simply removed andgo away both from stellar collapse and from the cosmological considerations of the universe (whichare two very important physical situations), once assumptions such as the dust form of matter, thespherical symmetry of the model, and such others were relaxed and when more general solutionsto the Einstein equations were found and considered.This is where the work by A. K. Raychaudhuri in 1955, on the gravitational focusing of matterand light in a space-time universe became relevant. This was used by R. Penrose, S. W. Hawking,and R. Geroch, who analyzed the causal structure and global properties of a fully general spacetime,and who then combined it with the considerations on gravitational focussing effect as developed byRaychaudhuri. The culmination of these efforts was the singularity theorems in general relativity,which showed that space-time singularities, such as those depicted in the examples of gravitationalcollapse and cosmology we discussed above, in fact manifested themselves in a rather large class ofspace-time universes under quite general physical conditions.The Raychaudhuri equation played a central role in the analysis of space-time singularities ingeneral relativity. Prior to the use of this equation to analyse collapsing and cosmological situationsfor the occurrence of singularities [1] [4], most works on related issues had considered only ratherspecial cases with many symmetry conditions assumed on the underlying spacetime. But with thehelp of this equation these aspects could be discussed within the framework of a general spacetimewithout any symmetry conditions. This was in terms of the overall behaviour of the congruences oftrajectories of material particles and photons propagating and evolving dynamically. This analysisof the congruences of non-spacelike curves which represent either material particles or light rays,showed how gravitational focusing took place in the universe giving rise to what are called causticswhere nearby trajectories intersect due to gravitational focussing.Before general singularity theorems could be constructed, however, another important mathe-matical input was needed in addition to the Raychaudhuri equation. This was the analysis of thecausality structure and general global properties of a spacetime manifold. This particular devel-opment took place mainly in the late 1960s (for a detailed discussion, see e.g. [2]). The singularitytheorems then combined these two important features, namely the gravitational focusing effectsdue to matter and energy content of the space-time and the causal structure constraints whichfollowed from the global spacetime properties, to obtain the existence of space-time singularitiesin the form of geodesic incompleteness in the space-time.As we discussed earlier, the main question here was that of genericity of the space-time singu-larities, either in cosmology or in collapse situations. The singularity theorems, while proving theexistence of singularities in a generic manner, by themselves provide no information either on thestructure and properties of such singularities, or on the growth of curvature and densities in theirvicinity.There are several singularity theorems available which establish the non-spacelike geodesic in-completeness for a spacetime under different sets of physical conditions. Each of these may bemore relevant to one or the other specific physical situation, and may be applicable to differentphysical systems such as stellar collapse or the universe as a whole. However, the most general ofthese is the Hawking-Penrose theorem, which is applicable to both the collapse situation and thecosmological scenario. To outline briefly the basic idea and the chain of logic behind the same,firstly, using causal structure analysis it is shown that between certain pairs of events in space-timethere must exist timelike geodesics curves of maximal length. However, both from causal structureanalysis and from the global properties of a general space-time manifold (which is assumed tosatisfy a specific energy condition), it follows that a causal geodesic curve, which is complete inregard to both the future and past, must contain caustics where nearby null or timelike geodesicsmust intersect. One is then led to a contradiction, because the maximal geodesic curves mentionedabove are not allowed to contain any such conjugate points, the existence of which would be againsttheir maximality. Thus the space-time itself must have non-spacelike geodesic incompleteness.Such theorems do assume some physical reasonability conditions. First such condition assumedon the space-time is an energy condition. All classical fields have been observed to satisfy asuitable positivity of energy density requirement, and therefore this may be considered to be quitereasonable. The second condition is a statement that all non-spacelike trajectories do encountersome non-zero matter or stress-energy density somewhere during their entire path. This is calledthe genericity condition. The third is a global causality requirement to the effect that there areno closed timelike curves in the spacetime. Finally, there is a condition that relates to eithera gravitational collapse situation or to gravitational focussing within a cosmological framework,which considers how the congruences of non-spacelike curves in a space-time expand or converge.If these conditions are satisfied then the theorem goes through, proving the existence of space-timesingularities within a general space-time scenario without special symmetry conditions. III. GRAVITATIONAL COLLAPSE
Another important physical situation where space-time singularities do occur is the gravitationalcollapse of a massive star. In fact, it was pointed out by S. Chandrasekhar in 1935 that, ”..the lifehistory of a star of small mass must be essentially different from that of a star of large mass... Asmall mass star passes into White-dwarf stage... A star of large mass cannot pass into this stageand one is left speculating on other possibilities.”While we have pointed out above that space-time singularities must develop in gravitationalcollapse, the most important question that this situation gives rise to is:
What is the final fateof a massive star when it undergoes a continual gravitational collapse ? This has been one ofthe most important key problems in astronomy and astrophysics for past many decades. If thestar is sufficiently massive, beyond the white dwarf or neutron star mass limits, then a continuedgravitational collapse must ensue when the star has exhausted its nuclear fuel.What are then the possible end states of such a continued gravitational collapse is the issueto be resolved. To answer this question, one must study dynamical collapse scenarios withinthe framework of a gravitation theory such as the Einstein theory. We now outline some recentdevelopments in past couple of decades in this area on the final state of a gravitationally collapsingmassive matter cloud. We point out how the black hole and naked singularity end states arisenaturally and generically as spherical collapse final states. We see that it is the geometry oftrapped surfaces in the space-time that governs this phenomena.It was conjectured by Penrose in 1969, that the ultra-dense regions forming in gravitationalcollapse, that is the space-time singularities where the physical quantities such as densities andcurvatures are having extreme values, must be hidden within the event horizon of gravity. Thatis, the collapse must end in a black hole. This is called the ‘cosmic censorship conjecture’. Thereis, however, no proof or any suitable mathematical formulation available for the same as of todaydespite many attempts.If the gravitational collapse always produces a black hole, then that provides a very strongfoundation for the theory as well as the astrophysical applications of black holes. On the otherhand, if collapse produces visible ultra-strong gravity regions, or naked singularities, then thephysical processes in these super-strong gravity regions can propagate, in principle, to externalobservers in the universe, thus giving rise to very interesting physical consequences.Under the situation, very many researchers have made extensive studies of various dynamicalcollapse models, mainly spherically symmetric, over past couple of decades, to investigate the finaloutcome of a continual gravitational collapse. When no proof, or even a suitable mathematicalformulation of censorship conjecture is available, it is only such studies that can throw light on thisissue. The generic conclusion that follows is: Either a black hole or a naked singularity develops asend product of collapse, depending on the initial data for the collapsing matter cloud (for example,the initial density, pressures, and velocity profiles for the collapsing shells of matter), from whichthe collapse develops, and the nature of dynamical evolutions as permitted by Einstein equations.While extensive study is made of astrophysics of black holes, for the visible singularities wemay still want to inquire into questions such as: Are naked singularities of gravitational collapsegeneric, or What are the physical factors that cause a naked singularity, rather than a black holeforming as collapse end state. That is, one may wish to understand in a better way the nakedsingularity formation in gravitational collapse. Basically, it turns out that the black hole or nakedsingularity phases of collapse are determined by the geometry of the trapped surfaces that developas the collapse evolves.What governs the geometry of the trapped surfaces, or the formation or otherwise of the nakedsingularities in the spacetime? We can ask in other words, what is it that causes the nakedsingularity to develop rather than a black hole as collapse final state? It turns out that physicalagencies such as inhomogeneities in matter profiles play an important role to distort the trappedsurface geometry to delay the trapped surface formation during the collapse, thus giving rise to anaked singularity.When the collapsing dust matter is homogeneous, the final outcome of collapse is a black holeand the singularity is hidden within the horizon. But if the collapsing cloud has a density higherat the center, then the trapped surfaces are delayed and the outcome is a naked singularity fromwhich the light or matter particles can escape away (see Fig.1 and Fig.2).This, in a way, provides the physical understanding of the phenomena of black hole and visiblesingularities occurring as end states for gravitational collapse.What would be the outcome when the collapse is non-spherical? There are some examples whichindicate the outcome to be somewhat similar in nature, but the evidence in this case is limited sofar. The main difficulty is the complexity of the Einstein system of differential equations. It willbe necessary to understand non-spherical collapse better before we can decide on the genericityaspect of the visible singularities.There are several quite interesting questions which are under active investigation at the moment.For example, could naked singularities generate bursts of gravity waves? What kind of quantumeffects will take place near a visible singularity? Many of these issues would have interesting physical (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Spacetime singularityCollapsing matter cloudInitial surface at t = 0Event horizon
FIG. 1: The space-time singularity developing in collapse is hidden within an event horizon in the case ofgravitational collapse of a homogeneous density dust cloud. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Outgoing null geodesicsSpacetime singularityCollapsing matter cloudInitial surface at t = 0
FIG. 2: When the density of collapsing cloud is higher at the center, light rays or particle trajectories canreach the external observer from the vicinity of the singularity. implications. It appears likely from the current investigations that the astrophysical phenomenasuch as the Gamma Rays Bursts will have a strong connection to the physics and dynamics ofgravitational collapse of massive stars.The above discussion points to a wide variety of circumstances under which singularities developin general relativistic cosmologies and in many gravitational collapse processes. Singularity theo-0rems imply the existence of vast classes of solutions to the Einstein equations that must containspacetime singularities, as characterized by the conditions of these theorems, and of which the bigbang singularity is one example. These theorems therefore imply that singularities must occur inEinstein’s theory quite generically, that is, under rather general physically reasonable conditions onthe underlying spacetime. Historically, this implication considerably strengthened our confidencein the big bang model which is used extensively in cosmology today.While singularity theorems tell mainly on the existence part, what we really need is moreinformation on the structure of the singularities in terms of their visibility or otherwise, curvaturestrengths and other such aspects. What is therefore called for is a detailed investigation of thedynamics of gravitational collapse within the framework of Einstein’s theory.In such a context, discussion of the gravitational collapse for specific models in general relativitycan turn out to be of great help. One such model is that given by the Vaidya metric, which wasoriginally developed by P. C. Vaidya in 1941 in the context of modelling a radiating star in generalrelativity. One can use this metric to study collapse of radiation shells within a Vaidya geometry,and it provided a great deal of information on the black hole and naked singularity formation insuch collapse geometries [3], [4]).
IV. SINGULARITIES AND QUANTUM GRAVITY
Though general relativity deals with matter in the space-time as a purely classical entity thatgenerates the curvature of space-time, we know that actually the matter and particles, and theirinteractions, obey and are governed by the laws of quantum theory. On very large scale in theuniverse, and at relatively lower matter densities, it may be possible to ignore the intrinsic quantumnature of the matter. In that case, general relativity provides us with fairly accurate predictionson the evolution of the universe.The occurrence of singularities, however, offers us the regime where the matter densities, space-time curvatures, and gravity are all indeed extreme, and where the quantum gravity effects wouldbe certainly important. As of today, we do not yet have a combined theory governing in a uni-fied manner the forces operating within atom and at nuclear densities, and the force of gravity.Therefore a study of physical processes occurring in the vicinity of the space-time singularity wouldpossibly offer a unique opportunity to study the physics of the gravity and the quantum together,and could possibly lead to a unified theory of all forces of nature. This is the cherished dream ofthe physicists which is to create a quantum theory of gravity. It is for this reason that the study1 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Fuzzy region QG
FIG. 3: Eventhough the singularity may resolve by the quantum gravity effects, the physical processes inthe ultra-strong gravity regions may be seen by the observers far away in the universe. (Figure courtesy:Ref 3 below of the PSJ book.) of space-time singularities occupies such a central place in fundamental physics today.It is possible that such singularities represent the incompleteness of the theory of general rel-ativity itself. Further, they may be resolved or avoided when quantum effects near the same areincluded in a more complete theory of quantum gravity. Nevertheless, there is a key point here.Even if the final singularity is dissolved by quantum gravity, what is really important is the in-evitable occurrence of an ultra-strong gravity region, close and in the vicinity to the location of theclassical singularity, either in cosmology or in dynamical processes involved in gravitational col-lapse. Such processes must affect the physics of the universe. An example of such a situation is thebig bang singularity of cosmology. Even though such singularities may be possibly resolved througheither quantum gravity effects, or due to features such as chaotic initial conditions, the effects ofthe super ultra-dense region of gravity that existed near the big bang epoch profoundly influencethe physics and subsequent evolution of the universe. Similarly, we have in gravitational collapseof a massive star, the occurrence of singularities which are either visible to external observers orhidden behind the event horizon giving rise to a black hole. In either case again, the importantissue is how the inevitably occurring super ultra-dense region would influence the physics outside.We discussed here some aspects of space-time singularities. It is seen that in Einstein gravitythey occur generically, whether covered within event horizons or as visible to external observers.If a future quantum theory of gravity resolves the final singularity of collapse or the initial one incosmology, what is really interesting physically is the occurrence of regions of ultra-strong gravity2and space-time curvatures, that develop as the result of the dynamical gravitational processes.The following physical picture then emerges. Dynamical gravitational processes proceed andevolve to create ultra-strong gravity regions in the universe. Once these form, strong curvatureand quantum effects both come into their own in these regions. Quantum gravity then takes overand may resolve the final singularity. Particularly interesting is the case when the singularities ofcollapse are visible. In such a situation, quantum gravity effects, taking place in those ultra-stronggravity regions, will in principle be accessible and observable to external observers (see Fig.3). Theconsequences of such a scenario would be surely intriguing. [1] R. M. Wald,
General Relativity , University of Chicago Press, Chicago (1983).[2] R. Geroch, ‘The Structure of space-time’ , in
General Relativity- an Einstein Centenary Survey (eds. S.W. Hawking and W. Israel), Cambridge University Press, Cambridge (1979).[3] P. S. Joshi, ‘Naked singularities’ , in
Scientific American , February 2009.[4] P. S. Joshi,