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Naked Singularities

The black hole has a troublesome sibling, the naked singularity. Physicists have long thought—hoped—it could never exist. But could it?


In Brief

  • Conventional wisdom has it that a large star eventually collapses to a black hole, but some theoretical models suggest it might instead become a so-called naked singularity [see opposite page for definition]. Sorting out what happens is one of the most important unresolved problems in astrophysics.
  • The discovery of naked singularities would transform the search for a unified theory of physics, not least by providing direct observational tests of such a theory.

Modern science has introduced the world to plenty of strangeideas, but surely one of the strangest is the fate of a massive star that has reached the end of its life. Having exhausted the fuel that sustained it for millions of years, the star is no longer able to hold itself up under its own weight, and it starts collapsing catastrophically. Modest stars like the sun also collapse, but they stabilize again at a smaller size. Whereas if a star is massive enough, its gravity overwhelms all the forces that might halt the collapse. From a size of millions of kilometers across, the star crumples to a pinprick smaller than the dot on an “i.”

Most physicists and astronomers think the result is a black hole, a body with such intense gravity that nothing can escape from its immediate vicinity. A black hole has two parts. At its core is a singularity, the infinitesimal point into which all the matter of the star gets crushed. Surrounding the singularity is the region of space from which escape is impossible, the perimeter of which is called the event horizon. Once something enters the event horizon, it loses all hope of exiting. Whatever light the falling body gives off is trapped, too, so an outside observer never sees it again. It ultimately crashes into the singularity.

But is this picture really true? The known laws of physics are clear that a singularity forms, but they are hazy about the event horizon. Most physicists operate under the assumption that a horizon must indeed form, if only because the horizon is very appealing as a scientific fig leaf. Physicists have yet to figure out what exactly happens at a singularity: matter is crushed, but what becomes of it then? The event horizon, by hiding the singularity, isolates this gap in our knowledge. All kinds of processes unknown to science may occur at the singularity, yet they have no effect on the outside world. Astronomers plotting the orbits of planets and stars can safely ignore the uncertainties introduced by singularities and apply the standard laws of physics with confidence. Whatever happens in a black hole stays in a black hole.

Yet a growing body of research calls this working assumption into question. Researchers have found a wide variety of stellar collapse scenarios in which an event horizon does not in fact form, so that the singularity remains exposed to our view. Physicists call it a naked singularity. Matter and radiation can both fall in and come out. Whereas visiting the singularity inside a black hole would be a one-way trip, you could in principle come as close as you like to a naked singularity and return to tell the tale.

If naked singularities exist, the implications would be enormous and would touch on nearly every aspect of astrophysics and fundamental physics. The lack of horizons could mean that mysterious processes occurring near the singularities would impinge on the outside world. Naked singularities might account for unexplained high-energy phenomena that astronomers have seen, and they might offer a laboratory to explore the fabric of spacetime on its finest scales.

Cosmic Censor

Event horizons were supposed to have been the easy part about black holes. Singularities are clearly mysterious. They are places where the strength of gravity becomes infinite and the known laws of physics break down. According to physicists' current understanding of gravity, encapsulated in Einstein's general theory of relativity, singularities inevitably arise during the collapse of a giant star. General relativity does not account for the quantum effects that become important for microscopic objects, and those effects presumably intervene to prevent the strength of gravity from becoming truly infinite. But physicists are still struggling to develop the quantum theory of gravity they need to explain singularities.

By comparison, what happens to the region of spacetime around the singularity seems as though it should be rather straightforward. Stellar event horizons are many kilometers in size, far larger than the typical scale of quantum effects. Assuming that no new forces of nature intervene, horizons should be governed purely by general relativity, a theory that is based on well-understood principles and has passed 90 years of observational tests.

That said, applying the theory to stellar collapse is still a formidable task. Einstein's equations of gravity are notoriously complex, and solving them requires physicists to make simplifying assumptions. American physicists J. Robert Oppenheimer and Hartland S. Snyder—and, independently, Indian physicist B. Datt—made an initial attempt in the late 1930s. To simplify the equations, they considered only perfectly spherical stars, assumed the stars consisted of gas of a homogeneous (uniform) density and neglected gas pressure. They found that as this idealized star collapses, the gravity at its surface intensifies and eventually becomes strong enough to trap all light and matter, thereby forming an event horizon. The star becomes invisible to outside observers and soon thereafter collapses all the way down to a singularity.

Real stars, of course, are more complicated. Their density is inhomogeneous, the gas in them exerts pressure, and they can assume other shapes. Does every sufficiently massive collapsing star turn into a black hole? In 1969 University of Oxford physicist Roger Penrose suggested that the answer is yes. He conjectured that the formation of a singularity during stellar collapse necessarily entails the formation of an event horizon. Nature thus forbids us from ever seeing a singularity because a horizon always cloaks it. Penrose's conjecture is termed the cosmic censorship hypothesis. It is only a conjecture, but it underpins the modern study of black holes. Physicists hoped we would be able to prove it with the same mathematical rigor we used to show the inevitability of singularities.

Singularities Au Naturel

That has not happened. Instead of coming up with a direct proof of censorship that applies under all conditions, we have had to embark on the longer route of analyzing case studies of gravitational collapse one by one, gradually embellishing our theoretical models with the features that the initial efforts lacked. In 1973 German physicist Hans Jürgen Seifert and his colleagues considered inhomogeneity. Intriguingly, they found that layers of infalling matter could intersect to create momentary singularities that were not covered by horizons. But singularities come in various types, and these ones were fairly benign. Although the density at one location became infinite, the strength of gravity did not, so the singularity did not crush matter and infalling objects to an infinitesimal pinprick. Thus, general relativity never broke down, and matter continued to move through this location rather than meeting its end.

In 1979 Douglas M. Eardley of the University of California, Santa Barbara, and Larry Smarr, then at the University of Illinois at Urbana-Champaign, went a step further and performed a numerical simulation of a star with a realistic density profile: highest at its center and slowly decreasing toward the surface. An exact paper-and-pencil treatment of the same situation, undertaken by Demetrios Christodoulou of the Swiss Federal Institute of Technology in Zurich, followed in 1984. Both studies found that the star shrank to zero size and that a naked singularity resulted. But the model still neglected pressure, and Richard P.A.C. Newman, then at the University of York in England, showed that the singularity was again gravitationally weak.

Inspired by these findings, many researchers, including me, tried to formulate a rigorous theorem that naked singularities would always be weak. We were unsuccessful. The reason soon became clear: naked singularities are not always weak. We found scenarios of inhomogeneous collapse that led to singularities where gravity was strong—that is, genuine singularities that could crush matter into oblivion—yet remained visible to external observers. A general analysis of stellar collapse in the absence of gas pressure, developed in 1993 by Indresh Dwivedi, then at Agra University, and me, clarified and settled these points.

In the early 1990s physicists considered the effects of gas pressure. Amos Ori of the Technion-Israel Institute of Technology and Tsvi Piran of the Hebrew University of Jerusalem conducted numerical simulations, and my group solved the relevant equations exactly. Stars with a fully realistic relation between density and pressure could collapse to naked singularities. At about the same time, teams led by Giulio Magli of the Polytechnic University of Milan and by Kenichi Nakao of Osaka City University considered a form of pressure generated by rotation of particles within a collapsing star. They, too, showed that in a wide variety of situations, collapse ends in a naked singularity after all.

These studies analyzed perfectly spherical stars, which is not as severe a limitation as it might appear, because most stars in nature are very close to this shape. Moreover, spherical stars have, if anything, more favorable conditions for horizon formation than stars of other shapes do, so if cosmic censorship fails even for them, its prospects look questionable. That said, physicists have been exploring nonspherical collapse. In 1991 Stuart L. Shapiro of the University of Illinois and Saul A. Teukolsky of Cornell University presented numerical simulations in which oblong stars could collapse to a naked singularity. A few years later Andrzej Królak of the Polish Academy of Sciences and I studied nonspherical collapse and also found naked singularities. To be sure, both these studies neglected gas pressure.

Skeptics have wondered whether these situations are contrived. Would a slight change to the initial configuration of the star abruptly cause an event horizon to cover the singularity? If so, then the naked singularity might be an artifact of the approximations used in the calculations and would not truly arise in nature. Some scenarios involving unusual forms of matter are indeed very sensitive. But our results so far also show that most naked singularities are stable to small variations of the initial setup. Thus, these situations appear to be what physicists call generic—that is, they are not contrived.

How to Beat the Censor

These counterexamples to Penrose's conjecture suggest that cosmic censorship is not a general rule. Physicists cannot say, “Collapse of any massive star makes a black hole only,” or “Any physically realistic collapse ends in a black hole.” Some scenarios lead to a black hole and others to a naked singularity. In some models, the singularity is visible only temporarily, and an event horizon eventually forms to cloak it. In others, the singularity remains visible forever. Typically the naked singularity develops in the geometric center of collapse, but it does not always do so, and even when it does, it can also spread to other regions. Nakedness also comes in degrees: an event horizon might hide the singularity from the prying eyes of faraway observers, whereas observers who fell through the event horizon could see the singularity prior to hitting it. The variety of outcomes is bewildering.

My colleagues and I have isolated various features of these scenarios that cause an event horizon to arise or not. In particular, we have examined the role of inhomogeneities and gas pressure. According to Einstein's theory, gravity is a complex phenomenon involving not only a force of attraction but also effects such as shearing, in which different layers of material are shifted laterally in opposite directions. If the density of a collapsing star is very high—so high that by all rights it should trap light—but also inhomogeneous, those other effects may create escape routes. Shearing of material close to a singularity, for example, can set off powerful shock waves that eject matter and light—in essence, a gravitational typhoon that disrupts the formation of an event horizon.

To be specific, consider a homogeneous star, neglecting gas pressure. (Pressure alters the details but not the broad outlines of what happens.) As the star collapses, gravity increases in strength and bends the paths of moving objects ever more severely. Light rays, too, become bent, and there comes a time when the bending is so severe that light can no longer propagate away from the star. The region where light becomes trapped starts off small, grows and eventually reaches a stable size proportional to the star's mass. Meanwhile because the star's density is uniform in space and varies only in time, the entire star is crushed to a point simultaneously. The trapping of light occurs well before this moment, so the singularity remains hidden.

Now consider the same situation except that the density decreases with distance from the center. In effect, the star has an onionlike structure of concentric shells of matter. The strength of gravity acting on each shell depends on the average density of matter within that shell. Because the denser inner shells feel a stronger pull, they collapse faster than the outer ones. The entire star does not collapse to a singularity simultaneously. The innermost shells collapse first, and then the outer shells pile on, one by one.

The resulting delay can postpone the formation of an event horizon. If the horizon can form anywhere, it will form in the dense inner shells. But if density decreases with distance too rapidly, these shells may not constitute enough mass to trap light. The singularity, when it forms, will be naked. Thus, there is a threshold: if the degree of inhomogeneity is very small, below a critical limit, a black hole will form; with sufficient inhomogeneity, a naked singularity arises.

In other scenarios, the salient issue is the rapidity of collapse. This effect comes out very clearly in models where stellar gas has converted fully to radiation and, in effect, the star becomes a giant fireball—a scenario first considered by the late Indian physicist P. C. Vaidya in the 1940s. Again there is a threshold: slowly collapsing fireballs become black holes, but if a fireball collapses rapidly enough, light does not become trapped.

Slime and socks

One reason it has taken so long for physicists to accept the possibility of naked singularities is that they raise a number of conceptual puzzles. A commonly cited concern is that such singularities would make nature inherently unpredictable. Because general relativity breaks down at singularities, it cannot predict what those singularities will do. John Earman of the University of Pittsburgh memorably suggested that green slime and lost socks could emerge from them.

As long as singularities remain safely ensconced within event horizons, this randomness remains contained and general relativity is a fully predictive theory, at least outside the horizon. But if singularities can be naked, their unpredictability would infect the rest of the universe. For example, when physicists applied general relativity to Earth's orbit around the sun, they would in effect have to make allowance for the possibility that a singularity somewhere in the universe could emit a random gravitational pulse and send our planet flying off into deep space.

Yet this worry is misplaced. Unpredictability is actually common in general relativity—and not always directly related to censorship violation. The theory permits time travel, which could produce causal loops with unforeseeable outcomes, and even ordinary black holes can become unpredictable. For example, if we drop an electrical charge into an uncharged black hole, the shape of spacetime around the hole radically changes and is no longer predictable. A similar situation holds when the black hole is rotating. Specifically, what happens is that spacetime no longer neatly separates into space and time, so physicists cannot consider how the black hole evolves from some initial time into the future. Only the purest of pure black holes, with no charge or rotation at all, is fully predictable.

The loss of predictability and other problems with black holes actually stem from the occurrence of singularities; it does not matter whether they are hidden or not. The solution probably lies in a quantum theory of gravity, which will go beyond general relativity and offer a full explication of singularities. Within that theory, every singularity would prove to have a high but finite density. A naked singularity would be a “quantum star,” a hyperdense body governed by the rules of quantum gravity.

Another possibility is that singularities may really have an infinite density after all—that they are not things to be explained away by quantum gravity but to be accepted as they are. The breakdown of general relativity at such a location may not be a failure of the theory per se but a sign that space and time have an edge. The singularity marks the place where the physical world ends. We should think of it as an event rather than an object, a moment when collapsing matter reaches the edge and ceases to be, like the big bang in reverse.

In that case, questions such as what will come out of a naked singularity are not really meaningful; there is nothing to come out of, because the singularity is just a moment in time. What we see from a distance is not the singularity itself but the processes that occur in the extreme conditions of matter near this event, such as shock waves caused by inhomogeneities in this ultradense medium.

In addition to unpredictability, a second issue troubles many physicists. Having provisionally assumed that the censorship conjecture holds, they have spent the past several decades formulating laws that black holes should obey. But the laws are not free of major paradoxes. For example, they hold that a black hole swallows and destroys information—which appears to contradict the basic principles of quantum theory. This paradox and other predicaments stem from the presence of an event horizon. If the horizon goes away, these problems might go away, too. For instance, if the star could radiate away most of its mass in the late stages of collapse, it would destroy no information and leave behind no singularity. In that case, it would not take a quantum theory of gravity to explain singularities; general relativity might do the trick itself.

A Lab for Quantum Gravity

Far from considering naked singularities a problem, physicists can see them as an asset. If the singularities that formed in the gravitational collapse of a massive star are visible to external observers, they could provide a laboratory to study quantum-gravitational effects. Quantum gravity theories in the making, such as string theory and loop quantum gravity, are badly in need of some kind of observational input, without which it is nearly impossible to constrain the plethora of possibilities. Physicists commonly seek that input in the early universe, when conditions were so extreme that quantum-gravitational effects dominated. But the big bang was a unique event. If singularities could be naked, they would allow astronomers to observe the equivalent of a big bang every time a massive star in the universe ends its life.

To explore how naked singularities might provide a glimpse into otherwise unobservable phenomena, we recently simulated how a star collapses to a naked singularity, taking into account the effects predicted by loop quantum gravity. According to this theory, space consists of tiny atoms, which become conspicuous when matter becomes sufficiently dense; the result is an extremely powerful repulsive force that prevents the density from ever becoming infinite. In our model, such a repulsive force dispersed the star and dissolved the singularity. Nearly a quarter of the mass of the star was ejected within the final fraction of a microsecond. Just before it did so, a faraway observer would have seen a sudden dip in the intensity of radiation from the collapsing star—a direct result of quantum-gravitational effects.

The explosion would have unleashed high-energy gamma rays, cosmic rays and other particles such as neutrinos. Upcoming experiments such as the Extreme Universe Space Observatory—hosted by the Japanese Experiment Module of the International Space Station and expected to fly in 2017—may have the needed sensitivity to see this emission. Because the details of the outpouring depend on the specifics of the quantum gravity theory, observations would provide a way to discriminate among theories.

Either proving or disproving cosmic censorship would create a mini explosion of its own within physics because naked singularities touch on so many deep aspects of current theories. What comes out unambiguously from the theoretical work so far is that singularities can be naked if the conditions are suitable. The question remains whether these conditions could ever arise in nature.

MORE TO EXPLORE

Black Holes, Naked Singularities and Cosmic Censorship. Stuart L. Shapiro and Saul A. Teukolsky in American Scientist, Vol. 79, No. 4, pages 330–343; July/August 1991.

Black Holes and Time Warps. Kip S. Thorne. W. W. Norton, 1994.

Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. John Earman. Oxford University Press, 1995.

Quantum Evaporation of a Naked Singularity. Rituparno Goswami, Pankaj S. Joshi and Parampreet Singh in Physical Review Letters, Vol. 96, No. 3, Article No. 031302; January 27, 2006. http://arxiv.org/abs/gr-qc/0506129

Gravitational Collapse and Space-time Singularities. Pankaj S. Joshi. Cambridge University Press, 2007.

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