XII. Black Holes, Worm Holes, and Beyond

Black holes may form from stars, but they are vastly different from stars. One way to see this is to examine the intellectual frontiers to which research on black holes has led. There one finds mind-bending concepts of worm holes, time machines, multidimensional space, self-reproducing universes and radical new notions of how to think of time and space under conditions where neither can exist.

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1. Worm Holes

"Time is the fire in which we all burn." This quote from a recent Star Trek movie embodies the hold that time has on our imaginations. Time has been a frequent subject of science fiction, especially the fascinating and philosophically thorny issue of time travel. Although ultimate understanding of time and its manipulation remain beyond our grasp, physicists have brought the issues to focus in a remarkable story of the exploration of time that has been carried out for the last decade.

Time has always been a fundamental part of physics. The equations physicists use are chronologically symmetric -- in other words, they do not differentiate a forward from a backward direction in time. If you took a movie of dust particles floating in a sunbeam and then ran it backward, you would have great difficulty telling the difference. If, on the other hand, the projectionist ran an episode of Star Trek backward, there would be howls of protest from the audience. Where does the "arrow of time" arise? Why is it that we age from 18 to 38 to 58, but not the other way around? Is that progression immutable?

This chapter in physicists’ attack on time travel arose from a work of science fiction. In the original version of Carl Sagan's novel, Contact, he invoked a means of interstellar travel left by an ancient civilization for earthlings to discover. Effectively, he wanted his passageway to be a black hole, one where you flew into the event horizon and emerged -- elsewhere. Sagan sent the draft of the book to his friend and colleague, Kip Thorne. Thorne, a physicist at Caltech, is one of the world’s experts on black holes. He has written his personal version of this story in a recent book entitled Black Holes and Time Warps: Einstein's Outrageous Legacy. Thorne realized that Sagan's proposed method was flawed. The solution required two things: different physics and more imagination!

In the solution for Einstein's equations for a black hole, a passage can exist between two universes or between two parts of the same universe. Physicists call such a structure an Einstein-Rosen bridge. In more casual parlance, this bridge is called a worm hole, another phrase invented by the word-master physicist, John A. Wheeler. Black hole experts have known for decades that the apparent worm hole in the standard black hole solution represents only a single moment in time. Just before or just after that instant, there is no passage, only the terrible maw of the singularity, waiting to destroy anything that passed into the event horizon. Furthermore, if you tried to race through the worm hole in the instant it opened at anything less than the speed of light, the worm hole would snap shut. You would be trapped and summarily flung into the singularity. Abandon all hope, ye who enter here!

As he mused further, however, Thorne realized that there might be another approach. Suppose you were dealing with an arbitrarily advanced civilization, the citizens of which could engineer anything that was not absolutely forbidden by the laws of physics. What would be possible under those circumstances?

Thorne’s solution involved what he came to call exotic matter. Ordinary matter has a finite energy and exerts a finite pressure and creates a normal, pulling, gravitational field. One can envisage, however, matter that has a negative energy, that exerts a negative pressure, like the tension in a rubber band. For exotic matter, this tension is at such an extreme level that the tension energy is greater than the rest mass energy, E = Mc2, of the rubber band. Such material has the property one would label "anti-gravity." Whereas ordinary matter pushes outward with pressure and pulls inward with gravity, such exotic matter pulls inward with its tension and pushes outward with its gravity.

Remarkably, related stuff has become a prominent topic in cosmology, as mentioned in Chapter 11. Cosmologists describe an inflationary stage occurring in the split seconds after the Big Bang, in which the Universe underwent a rapid expansion that led to its current size and smoothness. The condition that is hypothesized to cause inflation is some form of negative energy field that would have a negative pressure that pushed against normal gravity resulting in rapid expansion. After a brief interval of hyper expansion, this field is presumed to decay away leaving what we regard today as the normal vacuum with its small but non-zero quantum vacuum energy. Another version of these ideas arises in the context of the Cosmological Constant discussed in Chapter 11. If the Universe is accelerating in its expansion there must be something involved other than the matter in it and irreducible quantum energy of the vacuum. One suggestion is yet another negative energy field, one invented to account for the acceleration of the Universe like the Cosmological Constant, but one that varies in time. This field has been termed quintessence. Thorne did not attempt to make the nature of exotic matter explicit. In the most general sense, however, the exotic matter needed to create worm holes would share some of the repulsive properties of the inflationary energy and "quintessence."

In his reply to Sagan, Thorne speculated that an advanced civilization could slather some of this exotic matter on a mortar board, pick up a trowel, and make something of it. What could you make? The answer was that, cleverly applied, you could manipulate an Einstein-Rosen bridge with it and hold the bridge open indefinitely, thanks to the repulsive nature of the gravity of your exotic cement! The way was open conceptually to traverse through hyperspace from one place in the Galaxy to a very distant one in a short time. The result would effectively be faster-than-light travel through a worm hole. That was enough for Sagan. He adopted that basic outline and described such a worm hole in the book that went to press. The movie was finally released in the summer of 1997, but, sadly, Carl Sagan succumbed to a rare disease and was not there to see how his vision of a worm hole translated to the screen.

This solution was not enough for Thorne, however. He continued thinking about the problem. He worked on it with students and together they published a number of papers showing that a proper arrangement of exotic matter could lead to a stable, permanent worm hole.

What would a worm hole look like? That would depend on how it was constructed. In its simplest form, it might appear spherical from the outside, that is, all approaches from all directions would look the same. The boundary of the worm hole would not necessarily look black, like a

black hole, even though the outer structure of their space-time geometries were similar. A black hole has an event horizon from within which nothing can escape. You can, in principle, see through a worm hole. If you traveled through one you would head straight toward the center of the spherical space. Without changing the direction of your propagation, you would eventually find yourself traveling away from the center, to emerge in another place.

From inside a spherical worm hole, the perspective would be tunnel-like. You would be able to see light coming in from the normal space at either end of the worm hole. The view sideways, however, would seem oddly constricted. Even in portions of the worm hole’s space-time that were vacuum -- empty space -- the space is highly curved. This would mean that light heading off in any direction "perpendicular" to the radius through the center of the worm hole would travel straight in the local space, but end up back where it started, like a line drawn around the surface of a sphere, only in three dimensional space. Hence, if you faced sideways in a worm hole you could, in principle, see the back of your head. In practice, the light might be distorted and your view very fuzzy. The effect might look like a halo of light around you that differentiated the "sideways" direction from that straight through the center of the worm hole. Figure 12.1 shows how it might look to you as you shined a flashlight on the interior of the worm hole.

FIGURE 12.1: A flashlight beamed into a wormhole would shine out the other end, but one aimed sideways would illuminate the back of your head.

The tunnel-like aspects of a worm hole are not to be confused with the funnel-like diagram that physicists use to make a two-dimensional representation of the real three-dimensional space around a black hole or worm hole. In a two-dimensional embedding diagram, a circle in two-dimensional space is the analog of a sphere in three-dimensional space. The real curved space around a three-dimensional worm hole is represented in two dimensions by a stretched two-dimensional space that resembles a funnel, just as it was for a black hole as discussed in Chapter 9. In this two-dimensional analog, you cannot travel through what we perceive to be the mouth of the funnel. That is a third-dimensional hyperspace in the two-dimensional analog. You have to imagine crawling, spider-like, along the surface of the two-dimensional space to get the true meaning of the nature of that space and some feeling for the three-dimensional reality. A version of this two-dimensional analog of a worm hole is shown in Figure 12.2. Figure 12.2 illustrates a worm hole connecting two different parts of an open, saddle-shaped, universe. One can also picture a worm hole cutting through a sphere in the two-dimensional analogy of a closed universe. It is more difficult to portray in an illustration, but worm holes can also provide such short cuts in flat space. If they are properly designed, worm holes can, in principle, yield an arbitrarily short path between arbitrarily distant reaches of normal space in any sort of universe.

FIGURE 12.2: A two-dimensional worm hole giving a "short cut" through an open "saddle-shaped" universe. In this representation, the three dimensional space surrounding the universe and threading the worm hole is a "hyperspace" that two-dimensional residents of the universe could not perceive. A two-dimensional denizen of the two-dimensional universe could approach this worm hole from any direction in 360 degrees and pass through the worm hole along the two-dimensional surface to emerge on the other side of the universe. An astronomer near the "mouth" of the worm hole could see a colleague within the worm hole, and vice versa. The astronomer within the worm hole could travel "straight" on a path at right angles to the way in or out and end up back where he started.

Movies portray worm holes in various ways. The most recent are based on some of these modern notions, but there may also be a tendency to confuse the actual tunnel-like nature with the two-dimensional funnel-like analog. In the first Star Trek movie, the Enterprise is captured in a worm hole when it jumps into warp drive too soon after leaving Earth. If you have a chance to watch the film again, notice that you can see stars through the sides of the worm hole. That is definitely wrong. Light from stars could come in the end of the worm hole you entered, or it could come in through the opposite end toward which you are headed, but inside the worm hole light is trapped by the severe curvature of the space. There is no literal tunnel wall and hence you cannot look out "sideways" through it.

In the TV series Babylon 5, there is a "constructed" worm hole, but its whirlpool-like nature may draw more on the two-dimensional analogy than the proper manifestation in real space. In another TV series, Deep Space 9, the worm hole can be approached from any direction and the tunnel-like interior is as close to reality as one can expect from graphic designers appealing to a TV audience.

The classic worm hole is that of the movie 2001: A Space Odyssey. There, the exterior of the worm hole looks like a flat, rectangular monolith. Matt Visser of Washington University of St. Louis has designed a worm hole that looks much like that, with the exotic matter confined to struts along the boundaries of the rectangular body.

In the movie version of Contact, the heroine is thrust into a worm hole by an alien-designed machine that opens the portal to the worm hole. The tunnel-like aspects are portrayed reasonably realistically and there is an attempt to invoke the other amazing property of worm holes, the distortion of time.

2. Time Machines

The modern story of worm holes does not stop with exotic matter and superluminal travel. As they thought more about the nature of worm holes, Thorne and his co-workers realized that worm holes would also function as time machines. In this phase, Thorne was joined by Igor Novikov, then of Moscow, now at the University of Copenhagen, and his colleagues. Special relativity has long since taught us the lesson of the Twin Paradox. A pair of twins moving at some velocity with respect to one another will each measure the other to be aging more slowly. What happens when one of the twins rockets out into space and then returns while the other remains at home? The motion is relative, but how can both be younger? The resolution is that the one that traveled will be younger. That traveler experienced a force, an acceleration upon turning around, and that makes all the difference. Thorne realized that you could do this experiment at the two ends of a worm hole. Grab one end (gravitationally), and rocket it out and back. It will be absolutely younger than the end that was not accelerated. Novikov argued that you can do the same thing by putting one end of a worm hole in empty space and the other near a gravitating body. General Relativity says that time will flow more slowly in the gravity well.

In either of these arrangements, you have a time machine. You can walk into one end of the worm hole and emerge at an earlier era, albeit not one before the worm hole time machine was created. If you walk to the other end of the worm hole though the exterior space, time passes and you age normally. You could meet your younger self emerging from the hole!

Is this possible? Do the laws of physics allow such things? What happens to causality? These questions lead to another classic paradox, one that has been explored thoroughly in science-fiction literature and continues to be a common theme of TV and movies: the "Grandfather Paradox." The notion is that a time traveler goes back in time and kills her grandfather before her mother (never mind herself) was born; thus, the paradox. Thorne argues that this example is too paternalistic and has the time traveler kill her mother, thus the "Matricide Paradox." Novikov argues for leaving out the middle-man. Kill your younger self in a time-contorted suicide. The result is the same. The time traveler could not have existed in the first place to commit the array of ansatz-testing crimes.

All these examples require people to make them graphic, but people raise the issue of consciousness and free will and those issues are messy for a physicist. Joe Polchinski, then of the University of Texas, now at the University of California at Santa Barbara, invented a simple mechanical paradox. Physicists are fond of referring to "pool-ball" physics, meaning reducing a problem to something as visceral as pool balls bouncing off one another so that the physics -- conservation of momentum, for instance -- can be easily visualized. Polchinski dipped into this metaphor to present the "pool-ball crisis." In this thought experiment, a pool ball rolls into one end of the worm hole time machine. It comes out the other end in the past. It smacks its earlier incarnation, deflecting it so that it does not enter the worm hole. So how does it "get there" in the future if it never entered in the past? Is this simple mechanical paradox valid?

POSSIBLE FIGURE - illustrate the "pool ball" paradox.

No, counter the time-machine explorers. The flaw in this argument, according to Novikov, is that one posits the original pool ball rolling unimpeded into the worm hole and only speaks about the collision when it emerges to collide with itself. That is manifestly self-inconsistent. The original pool ball must be involved in the collision as it first rolls toward the hole opening. Chronology notwithstanding, physics must be self-consistent. That is the only criterion. Novikov and his colleagues have carefully studied the pool ball crisis and have shown that it cannot arise. They have looked at every conceivable interaction. Pool balls can miss, or they can strike a glancing blow, but they simply can never undergo a hard collision that leads to a paradox. Novikov’s group even explored an exploding pool ball, one fragment of which manages to enter the worm hole, come back in time, and hit the exploding pool ball, causing it to blow up and rendering the whole experiment self-consistent. The notion that physics can incorporate time machines in this way is called, in some circles, the "Novikov Consistency Conjecture."

What, then, of the original paradoxes? What happens with people? According to the consistency conjecture, the complex interpersonal interactions must work themselves out, like the pool balls, so that there is no paradox. That is the resolution. This means, if taken literally, that if time machines exist there can be no free will. You simply cannot will yourself to kill your younger self if you travel back in time. You can co-exist, take yourself out for a beer, celebrate your birthday together, but somehow circumstances will dictate that you can take no behavior that will lead to a paradox in time. Novikov urges another viewpoint. Physics already restricts our free will every day. You may will yourself to fly, or to walk through a concrete wall, but gravity and condensed-matter physics dictate that you cannot.

Is the converse true? If personal free will exists then time machines cannot? That question is unresolved (free will is often up for debate!). The consistency conjecture does say that certain time-travel plots are allowed in stories and movies and others are not. In particular, the consistency conjecture would say that one cannot change the future, the basic notion behind both the Back to the Future and the Terminator movies. Loops in time are allowed, but according to the consistency conjecture, the future is as fixed as the past and cannot be affected by an act of will or any other physical act.

Another way to resolve these issues is to say time somehow "forks off" at the moment of a paradox and in one time prong a time traveler lives on, even having killed her younger self. In this view, her younger self lives in the old time prong, but not in the current one. It is not clear that this resolves the origin of the memories of the time traveler of having been younger and having later wielded the knife, in the cause of science of course, at the critical moment.

Philosophical questions aside, there are still issues involved in time-machine research that go right to the heart of modern physics. We have known since the advent of quantum mechanics that the vacuum does not have zero energy. Having a specific energy, even zero, would violate the Heisenberg Uncertainty principle. Rather, the vacuum is riven with fluctuations, particles of light, matter and anti-matter that constantly form and annihilate. The worm hole mouths, like the space near the event horizon of a black hole, will be endowed with these vacuum fluctuations. In the case of a black hole, these fluctuations lead to Hawking radiation and to the evaporation of the black hole. For a worm hole, the issue is, if anything, even deeper. The vacuum fluctuations can travel in normal space to the opposite mouth of the worm hole, zip inside, and emerge in the past just at the time they left. If that were to happen, there would be twice as much energy in vacuum fluctuations. The cycle might repeat indefinitely and build up an infinite energy density, thus sealing off the worm hole or preventing it from having existed in the first place.

To properly address this issue, physicists need a full theory of quantum gravity, a theory that incorporates both violently curved space time and all the probabilistic nature of the quantum theory. Such a theory remains the holy grail of modern physics, a theory needed to understand the singularity of the Big Bang and that inside a black hole. There are still great conceptual problems facing the development of such a theory of space-time that applies on scales where time and space themselves are uncertain in a quantum manner, where up and down and before and after lose their meaning. Only with the development of this ultimate Theory of Everything will we really know whether time machines are conceptually possible.

Or until we find one.

3. Quantum Gravity

The search for quantum gravity, a theory that unites both the aspects of uncertainty from the quantum theory and the aspects of curved space from General Relativity, a Theory of Everything, is the current frontier of physics. Black holes are at the center of the action. The current contender for this intellectual prize is what is called by physicists string theory. The basic notion is that the fundamental entities of the Universe are not particles, dots of matter, but strings of matter, entities with one dimensional extent.

That seems like a simple, maybe even unnecessary, generalization of our standard picture of elementary particles, electrons, neutrinos, protons, neutrons, quarks. The doors that have been opened by this change in viewpoint are, however, wondrous.

For perspective, let us go back to the theory of Newton. Newton gave a rigorous mathematical framework in which to understand gravity and much else of basic physics, how things move under the imposition of forces. Newton’s law of gravity was based on the concept of a force between two objects. It was encapsulated in a simple formula that said that the force of gravity was proportional to the mass of two gravitating objects and inversely proportional to the square of the distance between them. This prescription was immensely successful. It is still used with great effect in most of astronomy to predict the motions of stellar objects from asteroids to the swirling of majestic galaxies. It is used to guide man-made satellites and rockets. We know now, however, that Newton’s theory is wrong. It is wrong in concept and wrong in application.

A hint of the conceptual problem with Newton’s theory comes by examining the law of gravity. Newton’s version of this law tells of the dependence on the masses of the gravitating objects and the distance between them, but is mute on the dependence on time. To see this, consider the following question. Take two objects that are attracting one another with their mutual force of gravity. Then, suddenly, move one further away. How soon does the other object know that the gravity pulling on it is weaker? Newton’s formulation demands the answer: no time elapses. The effect is instantaneous. Newton knew that the speed of light was a speed limit, yet his theory demanded communication of information, the strength of gravity, at infinite speed. Another clue to problems with Newton’s theory is that if you reduce the distance between two objects to zero, the gravitational force between them is infinite. The lesson of modern physics has been that if your theory gives you an infinite result, in this case infinite speed or infinite force, there is an error in your physics. If one looks sufficiently closely at Newton, those errors exist. The ultimate test is comparison of theory with observation and experiment. Newton is exceedingly successful in many applications, but fails in some. The famous case of the shift of the orientation of the orbit of Mercury is one. The angle of deflection of light by a gravitating object is another. Newton’s theory gives the wrong answer to these subtle and carefully posed experimental situations.

Einstein’s theory of gravity, General Relativity, was based on an incredibly simple and elegant idea: that physics should behave the same independent of the motion of the experimenter. The earlier version of this idea, Einstein’s Special Theory of Relativity, arose from the young Einstein asking another simple question: what would an electromagnetic wave look like if an observer moved along with it at the speed of light? To answer that question, to show that the observer could not move at the speed of light, Einstein had to show that the speed of light was the same independent of the motion of the observer. This result, one of the deeply true aspects of physics, remains one of the most incredible of human insights. Einstein also proved with his Special Theory that the lengths and times measured by an observer depended on how the measured object was moving, not in an absolute sense, but moving with respect to the observer.

Einstein’s General Theory took another step and asked about observers not in uniform motion, the subject of Special Relativity, but observers in accelerated motion. He realized that an observer freely falling in a gravitational field would measure physical effects and find them identical to an observer moving at uniform speed far from any gravitating object, but that an observer in an accelerating frame would feel exactly the same as one feeling the effects of gravity. This notion has been enshrined as Einstein’s Equivalence Principle, that an acceleration gives the same effects as being at rest in a gravitational field. If you sat in a chair in a lecture hall that accelerated at a uniform rate, the floor would push on your feet and the seat would push on your rear end, exactly the same forces you feel sitting in your chair reading this book. The Equivalence Principle is elegantly simple to state. To put it into a self-consistent mathematical framework, Einstein found that he had to introduce the notions of curved space and a complex set of tensor equations to describe it. Our sense of the nature of space has never been the same.

Einstein’s theory of gravity has passed every test put to it. It gets the right answers for the shift of Mercury’s orbit and the deflection of light, and has passed numerous other tests to the limit of our current ability to devise those tests. This makes General Relativity a better theory of gravity than Newton’s. General Relativity also becomes identical to Newton’s theory, mathematically, and hence in its precise predictions, when gravity is weak, distances are large, and motion is small. It must do so in order to reproduce Newton’s manifest success of predictability in those regimes. To accomplish this great success, Einstein had to abandon, not just the mathematical structure adopted by Newton, but the fundamental concept behind gravity. Einstein abandoned the notion of a "force" of gravity, and replaced it with the notion of curved space and warped time. Space is curved, and that tells matter how to move, to orbit, to fall. Gravity is geometry, the geometry of curved space. The change in conception wrought by Einstein was deeply profound. General Relativity is, however, wrong.

So far we only know that General Relativity is wrong because of conceptual problems. We have not been able to devise a test sensitive enough to display the fact. The conceptual problem is in the prediction of the singularity. General Relativity predicts that right at the center of a black hole a region must form of infinitesimal size, with infinite space-time curvature and infinite tidal forces. Those predictions of infinity are the undoing of General Relativity. General Relativity is fine as a predictive theory in the regimes where it works, just as Newton’s theory was in its own regime. General Relativity not only does everything that Newton could do, but more, including predictions of black holes and event horizons.

A Theory of Everything must take its place in this hierarchy. It must incorporate everything that Newton accurately predicts. It must also incorporate everything that Einstein consumed so elegantly. Then it must also answer the question: what is this amazing thing called a singularity? The theory must tell us what happens to space and time under conditions where quantum uncertainty dictates that the very notions of "front," "back," "here," "there," "before," and "after" lose their meaning. There must be space without space as we know it and time without time as we know it. Is there any wonder that physicists since Einstein have labored against immense conceptual problems in attempting to cross this barrier?

4. String Theory

Work on string theories is beginning to penetrate these barriers. The previous summary of the history of this development gives some preparation for what is necessary. Whereas Einstein overthrew the concept of gravity as a force between two objects, the theory of everything is likely to bring with it entirely new ways to think about gravity and, indeed, about space and time. In the appropriate regime, one can still think of curved space as the origin of gravity, just as for weak gravity it is still useful to think of a force of gravity and to use Newton’s theory in appropriate circumstances. One of the steps that energized string theory was the understanding that within the full mathematics of the theory, a subset described exactly Einstein’s theory of General Relativity. Just as Einstein’s theory "contains" Newton’s theory of gravity in the limit of weak gravity, string theory "contains" Einstein’s theory. String theory, however, holds a lot more. The underlying concepts of a Theory of Everything may require a shift in conceptual basis as profound as that from a force of gravity to gravity as curved space. Recent developments point strongly to the conclusion that, at a sufficiently small scale, where neither Newton nor Einstein dared tread, neither space nor time exist.

To see how this idea has arisen, a sketch of string theory is necessary. The basic notion is that particles, mathematical points, are too simple to contain the wonders of nature. True point particles have no inner structure, no richness. A string, on the other hand, by adding only one more dimension to the structure, can vibrate in many modes. You can’t make music with four grains of sand, but with four violin strings you can have Mozart! In the view of string theory, different modes of vibrations of the string represent different particles, just as one string on a violin can give different notes depending on where the violinist’s finger is placed.

Unlike violin strings, the strings that represent the fundamental entities in this theory do not exist only in our ordinary three-dimensional space. To make a mathematically self-consistent picture, one free of infinities and other inconsistencies, the space through which the strings thread must be of much higher dimension. The currently most viable versions of the theory have 10 or 11 dimensions.

To illustrate black holes and curved space, we have had recourse to embedding diagrams that reduce the fullness of the curved three-dimensional space to two so that we, as three dimensional creatures, can view these warped spaces from our higher dimensional perspective. From this perspective, it is clear to us that while there is no two-dimensional outside to the two-dimensional space, there is a very natural "outside" to the two-dimensional space, the very three-dimensional "hyperspace" that we occupy. This naturally leads one to wonder whether there is a "real" fourth spatial dimension that we, as three-dimensional creatures, cannot perceive, into which our three-dimensional Universe curves. This hyperspace would be where worm holes go when they go.

Despite the intuitively natural sense that invokes this sort of higher dimension, this is not what the string theorists are talking about. Physicists can construct mathematical models of curved three-dimensional spaces and universes, even worm holes, completely within the confines of that three-dimensional space. There is no need, or means, to invoke any higher dimension, no way to measure it, no way to do physics with it. Not yet, anyway.

Rather, the higher dimensions invoked by string theorists are all "compact." To picture a compact space, start again with a two-dimensional analog, a sheet of paper. As shown in Figure 12.3, roll the paper up into a tight roll. From a distance, the resulting object looks like a straight line, a string of length of perceptible extent, but no width. Imagine rolling the paper up laterally so you have a tiny ball. Now from a distance, the whole original sheet of paper resembles a point, a particle of no extent. A string in that original sheet of paper could still exist and vibrate away in that compact space that we could not directly perceive. We could, however, deduce that the higher dimensions exist because the nature of particles in our Universe demands it!

FIGURE 12.3: A schematic example of how a space could be compact and still contain a "string" capable of vibrating. A two-dimensional sheet containing a one-dimensional string can, in principle, be rolled up compactly so that it would appear to have only one dimension, length. The one-dimensional string would still be there, just wound up in the compact space. If the space were rolled up again, it could, in principle, be compacted into a point, a one-dimensional space, yet still contain the string.

The last few years have seen some immense advances in string theory that have given great hope that it is the basis for the Theory of Everything. One step has been to prove that what looked like five or six different string theories are all versions of the same underlying theory, the full shape of which has not yet been elucidated. These connections were established by what physicists have called duality, a connection between the properties of the theories. In one version of the theory a parameter could be small, and as the parameter got large, the mathematics of the theory broke down. In another string theory, the dual to the first, there would be a parameter that was just the inverse of the first. In that second theory as the parameter got large, the inverse parameter got small and the mathematics in that theory was well behaved. The middle ground is unknown, but this "duality" yields a sign post for how to link the disparate theories and show that they are deeply connected; that they are aspects of the same thing. This grand string theory that is taking shape is called M theory, M for matrix.

One of the concepts that has emerged from string theory is that there are not only strings threading the 10 or 11 dimensions of the string theory hyperspace, but surfaces. These surfaces can be canted in hyperspace in just the same way a sheet of paper can be oriented in all sorts of ways in our ordinary three-dimensional space. A more general word for a surface is a membrane, a term that also connotes a certain elasticity, a property that these surfaces have. These membranes can vibrate just as the strings can vibrate and their modes of motion are also important to the behavior that emerges as ordinary physics in our ordinary space time. To classify the membranes in spaces of various dimensions, they are referred to as p-branes, where p is a symbol denoting the dimension in which the membrane is embedded, p = 2 for a two dimensional space, p = 3 for three dimensions, p = 10 for 10 dimensions. An important development of string theory in recent years has been the recognition of the critical nature of the interaction of strings with p-branes. The ends of the string can attach to the p-branes or snap off to form closed rings. Much of this work has been done by Joe Polchinski at the University of California at Santa Barbara, and his colleague, Gary Horowitz.

A striking feat that followed the development of the theory of p-branes and their interactions with strings has been the capacity to construct simple models of black holes. These black holes are not the creatures of the curved space time of Einstein, but simpler versions in two dimensions constructed from the entities of p-branes and strings. Nevertheless, since string theory contains Einstein’s theory, objects that exert gravitational pull and that have event horizons can be constructed. The difference is that string theorists can count the numbers of modes and vibrations of the strings within the black holes they have constructed and tell exactly what the temperature and entropy should be. They get precisely the same answer as Hawking did in predicting Hawking radiation (Chapter 9), even though the mathematics and, indeed, the conceptual framework they use, is completely different. This striking concordance is the sort of development that tells physicists that they are getting close to a universal truth and that string theory has deep lessons to reveal.

String theory has also brought new insight into another problem that arises from thinking about the nature of black holes. This is called the information crisis. Information, the bits and bytes of computers, is about as fundamental as you can get. The problem is that black holes seem to destroy information and that bugs physicists. The idea was already there in our previous discussions of the nature of black holes in Chapter 9 and captured in John Wheeler’s phrase "black holes have not hair." You can throw stars, cars, people, protons into a black hole and all the information that described that ordinary stuff vanishes inside the event horizon. The only properties of a black hole that can be measured from the outside are its mass, spin, and electrical charge. Now Stephen Hawking enters the game. Black holes can evaporate, giving off Hawking radiation. Given enough time, the black hole will just disappear, leaving pure radiation with very little information content, essentially pure randomness. This process conserves energy, the energy equivalent of all the stuff that went down the black hole eventually emerges as the energy in the radiation. What happened to the information that defined that stars, the cars, the people and

the protons that went down the hole? Physicists have been debating this fundamental problem since the implications of Hawking’s ideas of black hole evaporation were first assimilated.

One can sense a possible wrinkle in this argument. Hawking’s theory was designed to work for ordinary size black holes where the event horizon was well separated from the singularity at the center of the black hole. When a black hole evaporates down to the last of its essence one needs a theory that can simultaneously treat the event horizon and the singularity and that probably requires a quantum gravity, a Theory of Everything. In the absence of that theory, it is not clear that one can use Hawking’s original theory to account for the final moments. String theory gives a different possibility. It suggests that the black hole can not evaporate entirely, but that as the process runs away, one is left with a string vibrating intensely somewhere in its ten dimensional space time. In those vibrations is the epitaph of all that entered the black hole, all that original information, the size of the stars, the bumper stickers on the cars, the personalities of the people, the number of protons.

Einstein wrote down a full and self-consistent set of equations to describe gravity (in the absence of quantum effects) in 1916. Those equations have yet to be fully solved. String theory is like that, only more so. The full mathematical structure of string theory is very complex and only a few solutions have been wrested from it. Those solutions have been tremendously encouraging. Exactly what theory of space and time will emerge from string theory is thus not yet clear. One can see that since string theory is a theory of quantum fields and forces, the fundamental concept of gravity will again be a force, but a quantum force, not that of Newton. Away from any singularity, this "force" of gravity will act just as in Einstein’s theory. One will be able to speak in the language of curved space and time, and dream of the construction of worm hole time machines.

On the microscopic scale, however, the new concepts of string theory will lead to different pictures, pictures that are only just now beginning to take hazy conceptual form. One can see that gravity will be represented by the familiar terms of Einstein’s gravity plus "something else" that comes in ever more strongly as one approaches, intellectually at least, the singularity. At the singularity itself, Einstein’s theory will be completely inapplicable, as Newton’s is within the event horizon of a black hole. At the singularity, space and time as we think of them on our large, human scale will probably not exist.

Even with these encouraging clues and the startling successes of string theory, one can see that we have a long way to go in our intellectual quest to understand the Universe.

5. When the Singularity is not a Singularity

The singularity of Einstein’s theory cannot exist. Something else must happen to space and time "there." In the absence of the full development of a string theory or some other Theory of Everything, physicists are left to grope. When physicists grope, startling ideas emerge.

We know the scale on which Einstein’s theory must break down even if we do not fully understand what must replace it. This scale can be estimated from the simple idea of asking about the conditions where quantum uncertainty must be as important as the space-time curvature of gravity. The fundamental constants of quantum gravity are the strength of gravity as measured by Newton’s constant from the world of the large, the degree of quantum uncertainty as measured by Planck’s constant from the world of the small, and nature’s speed limit, the speed of light from the world of the very fast. With values for these constants of nature in some set of units, English or metric, it does not matter, one can estimate the scale where Einstein’s theory, and ordinary quantum theory, fail. This scale, of length, time, density, is called the Planck scale. Newton’s constant has units of length cubed, time squared, and the inverse of mass. Planck’s constant has units of mass, length squared and the inverse of time. The speed of light has units of length over time. There is only one way one can combine these three fundamental constants with their individual units to produce a quantity of only length, only one other way to produce a time, only one third way to produce a mass. This exercise is a simple one of sorting out units, but it has profound implications, because the building blocks are the fundamental constants that tell us how space curves, the degree of quantum uncertainty, and how fast things can move. Their combination implicitly tells us where space gets so curved that a quantum wave cannot exist and simultaneously where quantum uncertainty is so large that speaking of a given curvature makes no sense. We learn the conditions where the two great theories of twentieth century physics butt heads and contradict one another, the conditions that call for a new theory of physics.

The resulting value of the length, the Planck length, is about 10-33 centimeters. This is an incredibly small value, much smaller than the size of a proton, but it is not zero! This is roughly how large the "singularity" must be. At this level, space and time break down into something else and Einstein’s prediction of a singularity goes awry. The corresponding Planck time is about

10-43 seconds. This is again an incredibly short time, but not zero. Time as we know it probably does not exist at shorter intervals so that asking "what happened when the Universe was younger than 10-43 seconds?" or "what happened before the Big Bang?" may not make sense, at least not in the traditional way. The Planck mass is about 10-5 grams. This is a small number, but not incredibly small. It is vastly bigger than any elementary particle we know. One can also work out the Planck density, the Planck mass divided by the cube of the Planck length. The answer is about 1093 grams per cubic centimeter. This is a gigantic density, but it is not infinite. In some average way, this must be the density of a "singularity," the density from which our Universe expanded in the Big Bang.

One way to think about the singularity is as a bubbling sea of Planck masses, each a Planck length in extent winking in and out of existence for intervals of a Planck time. This quantum-bubbling mess has been called a "quantum foam," another bit of etymological brilliance from John A. Wheeler. This term is a picturesque name intended to describe something we do not understand, yet capture the flavor of the idea that it is not ordinary space and time. In the quantum foam, one could not speak of "front" and "back" because space itself would be so quantum uncertain that such concepts are invalid. Likewise for the ideas of "before" and "after" with time a quantum froth.

Even in the absence of a full theory, if we picture the singularity not as a point of zero size and infinite density, but a dollop of quantum foam, then other ideas begin to emerge. The Universe was not born from a point of infinite density, but emerged as a bubble of ordinary space and time from this quantum foam. This bubble was highly energetic, and expanded to become everything we see. Again, as we discussed in Chapter 11, the expansion is pictured in the sense that all points of space move away from all other points of space, not an explosion of stuff into a pre-existing space. Also, as three-dimensional physicists we do not have to address the issue of what the three-dimensional Universe is expanding into, as much as that question seems to intrude. That the Universe emerges from the quantum foam already gives some predictability to the nature of the Universe. There must have been quantum fluctuations in the density and temperature of the very young, hot Big Bang as it emerged from the quantum foam 10-33 centimeters across and 10-43 seconds old. These unavoidable fluctuations can be calculated from the quantum theory with some assumptions and they later cause the tiny irregularities in temperature that COBE detected (Chapter 11) and that later grow to form all the structure we see, stars, galaxies, clusters of galaxies.

The notion of a quantum foam also plays a role in the thinking about worm holes and shows again that we can not pursue the physics of worm holes without a theory of the quantum foam, a theory of the singularity, a quantum gravity Theory of Everything. One way to picture the quantum foam is quantum-connected fragments of space and time, connecting different places and different times willy-nilly in a probabilistic way. These connections, although dominated by quantum uncertainty, are essentially tiny quantum worm holes. One can imagine making a worm hole by taking a little quantum loop of space and time and blowing it up to become a worm home big enough to travel through.

Another way to imagine making a worm hole leads to similar issues of the quantum nature of space and time. If you start from ordinary space and want to make a black hole you have to stretch and distort the space, but you do not have to rip or tear it (at least not until you get to that nasty singularity). That is not true for a worm hole. To make a worm hole you have to tear and reconnect space. You have to change not just the curvature of space, but its connectedness, its topology. If you think about it, a tea cup with a nice handle and a donut are the same basic thing in terms of how they are connected. They are both solid objects with one hole through them. You could make both from the same lump of clay by just molding a side of the donut shape to be the cup and shape the clay around the hole to be the handle. You would not have to tear the clay or reattach it at any point. You can not, however, make a solid lump of clay into either a donut or tea cup without tearing a hole in the clay.

Think of how you could connect space on a large scale to make a worm hole. It helps to imagine this in two dimensions. Picture a balloon. Push two fingers inward from opposite sides until your fingers almost touch, separated only by the thin rubber of the balloon. You have almost made a worm hole. If the connection could be made there in the center of the balloon, there would be a way to travel on a short cut through the center of the balloon, rather than taking the long way around on the surface. The balloon serves as a two dimensional analog of our three-dimensional space, so all motion is confined to the rubber of the surface. Now think of what you need to do to make the connection between your fingers. You would have to cut the rubber and attach the ends of the two cones, but cutting the rubber is the analogy of cutting the very fabric of space. That would be the issue in our real three-dimensional space in order to make a three dimensional worm hole. The cutting and re-attaching of space would amount to, at least temporarily, introducing an end to space, a singularity, before the re-attachment is made. To make a worm hole or a worm hole time machine in this way, we have to bring in the operation of introducing a tear in space time, a tear in the quantum foam. We will not know whether such an operation even makes sense until we have a theory of quantum gravity that tells how space and time behave if such a rent is threatened. Once again, we cannot think constructively about worm holes or time machines without a theory of quantum gravity to guide us.

If the Universe were born not from a singularity of infinite density, but from a spot of quantum foam, then the inverse is true. When a star collapses to make a black hole, the matter of the star does not disappear into a singularity of zero volume, but is crushed into a froth of quantum foam of a Planck density. One of the most dramatic ideas to emerge in the last few years was to ask, if a black hole leads back to the quantum foam from which the Universe arose, why cannot the cycle repeat. This idea was first put forth by Andre Linde, a Russian physicist, now at Stanford University. Linde was striving for some new idea to present at a conference to which he had been invited. He was ill and contemplating skipping the meeting, when this notion came to him. He worked out the basic mathematical and physical picture and presented it at the meeting.

The idea is that the quantum foam that forms at the center of the black hole is identical to that from which the Big Bang, our whole Universe arose. This means, Linde argued, that a new Universe can arise from the quantum foam of the black hole. The dramatic implication is that the chain could be endless. A universe forms, it expands to form stars. Some of the stars collapse to make black holes. From the "singularities" of those black holes, new Universes can be born. Here, perhaps, is a way to answer the question of what came before, and what comes after the Big Bang, endless Universes forming endless black holes.

Like many grand ideas of physics, this one will have to be poked and pummeled and analyzed. How do you prove such a startling conjecture? We can not travel to other Universes to see how they work. We are stuck in this one, but empowered with our imaginations and our mathematics and physics. Physicists are already at work generalizing the old cosmologies to see how these ideas could fit in. The easiest way to picture a bubble being blown in the quantum foam to become our Universe is to picture a literal bubble being blown. Such a bubble, basically a sphere, is a two-dimensional analog, an embedding diagram, for a closed three-dimensional Universe. Such a Universe would have a finite lifetime and would have to recollapse. That is in conflict with the favored model of an "inflating" Universe, that only makes conceptual sense if the Universe were flat, the three-dimensional equivalent of an infinite, flat plane. The results reported in the Chapter 11 suggest that the Universe is not closed and "spherical." It might be flat, but accelerating or open and doomed to expand forever, a saddle shape in the two-dimensional analog. Physicists and cosmologists are working now to develop models of inflating Universes that are consistent with infinite expansion. Such universes, can, of course, make black holes as they expand, and that is enough to raise Linde’s conjecture of new universes being constantly created.

POSSIBLE FIGURE: show bubbles blown from quantum foam making black holes and new universes.

These ideas have been taken one more dramatic step by Lee Smolin of Penn State University in his book, The Life of the Cosmos. Smolin addresses the deepest issue that drives both physicists and theologians. Why are WE here? What is it about our Universe that gave rise to life, to us. Smolin may not have the answer, but he has put the issues in an especially thought-provoking way by combining these ideas from physics with the basic ideas of biology, the power of natural selection. Smolin notes the amazing coincidences of numbers and physical conditions that are required to give rise to life as we know it. What if, Smolin wonders, each new universe had different numbers, for instance different values of the fundamental constants, Newton’s constant of Gravity, Planck’s constant, the speed of light, and other physical constants of nature. Most of those universes would fail. Some would not get out of the quantum foam or would quickly fall back. Others would expand so rapidly that stars did not have a chance to form, so there would be no black holes. In either case, those Universes would be barren, unable to produce progeny, new universes with new properties. Smolin makes a natural selection argument that after countless trials, the universes that survive would be those that maximize the production of black holes so that maximum progeny are ensured. Smolin argues that physicists may have to give up on a purely reductionist approach to science wherein the constants of nature have set values that theory and experiment can reveal, and accept that our Universe has arisen from a process of trial and error, a result of probabilities, not certainty. To be fruitful, such a universe would have to expand about as fast as ours, make stars like ours, produce heavy elements like ours to control the heating and cooling of the interstellar gas to keep star formation going for billions of years. Such a universe, Smolin deduces, would have to have the properties of our Universe and such a universe naturally gives rise to life to contemplate and make sense of it. Now that is a grand vision.

For all its inventiveness, Smolin’s picture does not really address the fundamental issue. Given that there are infinite universes experimenting with all possible forms, how did it all arise in the first place? Was there a beginning to this process? Is there an end? James Gott of Princeton has put another wrinkle on the game by combining the self-reproduction of universes through black holes with the notions of time machines. If new universes emerge from the quantum foam of a black hole singularity, can they emerge in the past? If that were possible, Gott conjectures, then the universe that emerges from a black hole could be the one that made the black hole from which it emerged, or a universe somewhere back in the chain of universes that Linde and Smolin contemplate. Recall that the Novikov Consistency Conjecture does not rule out time travel, it only demands self-consistency. Could it be that the Universe or a complex web of universes gave rise to itself in a closed, but self-consistent time loop? Could it be that there is no "beginning" and no "end" but just an infinite closed loop? As Gott asks, could the Universe have created itself?

This is heady stuff. It is amazing that these ideas have emerged, not from science fiction, but from hard-nosed physicists wrestling to make sense of the Universe of our observations. Progress can be made by examining these ideas for self consistency and that enterprise will go forward with great energy. The real solution, or at least the one we can contemplate today is to develop the theory of quantum gravity, the Theory of Everything. Today the best bet for that appears to be string theory, M theory. So one can ask, what does string theory say about the quantum foam? Quantum foam was just a name, a place holder until some physics came along. What exactly does string theory say about the conditions at the Planck scale? Does string theory allow new universes to be born from the conditions predicted by string theory for "not time" and "not space" at the center of a black hole constructed from strings?

Other, more speculative questions also arise. What are these higher dimensions that are forced on the string theorists by mathematical self-consistency? Do they simply dictate the properties of particles that appear in the three-dimensional Universe of our space time, or can they be manipulated in some way? Does string theory allow worm holes and time machines? Prevent them?

It is somewhat old fashioned, but my guess is that even with a Theory of Everything in sight we are not about to see the end of physics.