by Brian Greene
For technical reasons, Witten first came upon the eleventh dimension in his studies of the strong coupling properties of the Type I IA string, and there the story is quite similar. As in the Heterotic-E example, there is an eleventh dimension whose size is controlled by the Type IIA coupling constant. When its value is increased, the new dimension grows. As it does, Witten argued, the Type IIA string, rather than stretching into a ribbon as in the Heterotic-E case, expands into an "inner tube," as illustrated in Figure 12.8. Once again, Witten argued that although theorists have always viewed Type IIA strings as one-dimensional objects, having only length but no thickness, this view is a reflection of the perturbative approximation scheme in which the string coupling constant is assumed to be small. If nature does require a small value of this coupling constant then it is a trustworthy approximation. Nevertheless, Witten's arguments and those of other physicists during the second superstring revolution do give strong evidence that the Type IIA and Heterotic-E "strings" are, fundamentally, two-dimensional membranes living in an eleven-dimensional universe.
But what is this eleven-dimensional theory? At low energies (low compared to the Planck energy), Witten and others argued, it is approximated by the long-neglected eleven-dimensional supergravity quantum field theory. But for higher energies, how can we describe this theory? This topic is currently under intense scrutiny. We know from Figures 12.7 and 12.8 that the eleven-dimensional theory contains two-dimensional extended objects—two-dimensional membranes. And as we shall soon discuss, extended objects of other dimensions play an important role as well. But beyond a hodgepodge of properties, no one knows what this eleven-dimensional theory is. Are membranes its fundamental ingredients? What are its defining properties? How does it purport to make contact with physics as we know it? If the respective coupling constants are small, our best current answers to these questions are described in previous chapters, since at small coupling constants we are led back to the theory of strings. But if the coupling constants are not small, no one currently knows the answers.
Whatever the eleven-dimensional theory is, Witten has provisionally named it M-theory. The name stands for as many things as people you poll. Some samples: Mystery Theory, Mother Theory (as in "Mother of all Theories"), Membrane Theory (since, whatever it is, membranes seem to be part of the story), Matrix Theory (after some recent work by Tom Banks of Rutgers University, Willy Fischler of the University of Texas at Austin, Stephen Shenker of Rutgers University, and Susskind that offers a novel interpretation of the theory). But even without having a firm grasp on its name or its properties, it is already clear that M-theory provides a unifying substrate for pulling together all five string theories.
M-Theory and the Web of Interconnections
There is an old proverb about three blind men and an elephant. The first blind man grabs hold of the elephant's ivory tusk and describes the smooth, hard surface that he feels. The second blind man grabs hold of one of the elephant's legs. He describes the tough, muscular girth that he feels. The third blind man grabs hold of the elephant's tail and describes the slender and sinewy appendage that he feels. Since their mutual descriptions are so different, and since none of the men can see the others, each thinks that he has grabbed hold of a different animal. For many years, physicists were as much in the dark as the blind men, thinking that the different string theories were very different. But now, through the insights of the second superstring revolution, physicists have realized that M-theory is the unifying pachyderm of the five string theories.
In this chapter we have discussed changes in our understanding of string theory that arise when we venture beyond the domain of the perturbative framework—a framework implicitly in use prior to this chapter. Figure 12.9 summarizes the interrelations we have found so far, with arrows to indicate dual theories. As you can see, we have a web of connections, but it is not yet complete. By also including the dualities of Chapter 10, we can finish the job.
Recall the large/small circular radius duality that interchanges a circular dimension of radius R with one whose radius is 1/R. Previously, we glossed over one aspect of this duality, which we now must clarify. In Chapter 10, we discussed the properties of strings in a universe with a circular dimension without carefully specifying which of the five string formulations we were working with. We argued that the interchange of winding and vibration modes of a string allows us to rephrase exactly the string theoretic description of a universe with a circular dimension of radius 1/R in terms of one in which the radius is R. The point we glossed over is that the Type IIA and Type IIB string theories actually get exchanged by this duality, as do the Heterotic-O and Heterotic-E strings. That is, the more precise statement of the large/small radius duality is this: The physics of the Type IIA string in a universe with a circular dimension of radius R is absolutely identical to the physics of the Type IIB string in a universe with a circular dimension of radius 1/R (a similar statement holds for the Heterotic-E and Heterotic-O strings). This refinement of the large/small radius duality has no significant effect on the conclusions of Chapter 10, but it does have an important impact on the present discussion.
The reason is that by providing a link between the Type IIA and Type IIB string theories, as well as between the Heterotic-O and Heterotic-E theories, the large/small radius duality completes the web of connections, as illustrated by the dotted lines in Figure 12.10. This figure shows that all five string theories, together with M-theory, are dual to one another. They are all sewn together into a single theoretical framework; they provide five different approaches to describing one and the same underlying physics. For some or other application, one phrasing may be far more effective than another. For instance, it's far easier to work with the weakly coupled Heterotic-O theory than it is to work with the strongly coupled Type I string. Nevertheless, they describe exactly the same physics.
The Overall Picture
We can now more fully understand the two figures—Figures 12.1 and 12.2—that we introduced in the beginning of this chapter to summarize the essential points. In Figure 12.1 we see that prior to 1995, without taking any dualities into account, we had five apparently distinct string theories. Various physicists worked on each, but without an understanding of the dualities they appeared to be different theories. Each of the theories had variable features such as the size of their coupling constant and the geometrical form and sizes of curled-up dimensions. The hope was (and still is) that these defining properties would be determined by the theory itself, but without the ability to determine them with the current approximate equations, physicists have naturally studied the physics that follows from a range of possibilities. This is represented in Figure 12.1 by the shaded regions—each point in such a region denotes one specific choice for the coupling constant and the curled-up geometry. Without invoking any dualities, we still have five disjointed (collections of) theories.
But now, if we apply all of the dualities we have discussed, then as we vary the coupling and geometric parameters, we can pass from any one theory to any other, so long as we also include the unifying central region of M-theory; this is shown in Figure 12.2. Even though we have only a scant understanding of M-theory, these indirect arguments lend strong support to the claim that it provides a unifying substrate for our five naively distinct string theories. Moreover, we have learned that M-theory is closely related to yet a sixth theory—eleven-dimensional supergravity—and this is recorded in Figure 12.11, a more precise version of Figure 12.2.13
Figure 12.11 illustrates that the fundamental ideas and equations of M-theory, although only partially understood at the moment, unify those of all of the formulations of string theory. M-theory is the theoretical elephant that has opened the eyes of string theorists to a far grander unifying framework.
A Surprising Feature of M-Theory: Democracy in Extension
When the string coupling constant is small in any of the upper five peninsular regions of the theory map in Figure 12.11, the fundamental ingredient of
the theory appears to be a one-dimensional string. We have, however, just gained a new perspective on this observation. If we start in either the Heterotic-E or Type IIA regions and turn the value of the respective string coupling constants up, we migrate toward the center of the map in Figure 12.11, and what appeared to be one-dimensional strings stretch into two-dimensional membranes. Moreover, through a more or less intricate sequence of duality relations involving both the string coupling constants and the detailed form of the curled-up spatial dimensions, we can smoothly and continuously move from any point in Figure 12.11 to any other. Since the two-dimensional membranes we have come upon from the Heterotic-E and Type IIA perspectives can be followed as we migrate to any of the three other string formulations in Figure 12.11, we learn that each of the five string formulations involves two-dimensional membranes as well.
This raises two questions. First, are two-dimensional membranes the true fundamental ingredient of string theory? And second, having made the bold leap in the 1970s and early 1980s from zero-dimensional point particles to one-dimensional strings, and having now seen that string theory actually involves two-dimensional membranes, might it be that there are even higher-dimensional ingredients in the theory as well? As of this writing, the answers to these questions are not fully known, but the situation appears to be the following.
We relied heavily on supersymmetry to give us some understanding of each formulation of string theory beyond the domain of validity of perturbative approximation methods. In particular, the properties of BPS states, their masses and their force charges, are uniquely determined by supersymmetry, and this allowed us to understand some of their strongly coupled characteristics without having to perform direct calculations of unimaginable difficulty. In fact, through the initial efforts of Horowitz and Strominger, and through subsequent groundbreaking work of Polchinski, we now know even more about these BPS states. In particular, not only do we know their masses and the force charges they carry, but we also have a clear understanding of what they look like. And the picture is, perhaps, the most surprising development of all. Some of the BPS states are one-dimensional strings. Others are two-dimensional membranes. By now, these shapes are familiar. But, the surprise is that yet others are three-dimensional, four-dimensional—in fact, the range of possibilities encompasses every spatial dimension up to and including nine. String theory or M-theory, or whatever it is finally called, actually contains extended objects of a whole slew of different spatial dimensions. Physicists have coined the term three-brane to describe extended objects with three spatial dimensions, four-brane for those with four spatial dimensions, and so on up to nine-branes (and, more generally, for an object with p space dimensions, where p represents a whole number, physicists have coined the far from euphonious terminology p-brane). Sometimes, using this terminology, strings are described as one-branes, and membranes as two-branes. The fact that all of these extended objects are actually part of the theory has led Paul Townsend to declare a "democracy of branes."
Notwithstanding brane democracy, strings—one-dimensional extended objects—are special for the following reason. Physicists have shown that the mass of the extended objects of every dimension except for one-dimensional strings is inversely proportional to the value of the associated string coupling constant when we are in any of the five string regions of Figure 12.11. This means that with weak string coupling, in any of the five formulations, all but the strings will be enormously massive—orders of magnitude heavier than the Planck mass. Because they are so heavy and, therefore, from E = mc2, require such unimaginably high energy to be produced, branes have only a small effect on much of physics (but not on all, as we shall see in the next chapter). However, when we venture outside the peninsular regions of Figure 12.11, the higher-dimensional branes become lighter and hence increasingly important.14
And so, the image you should have in mind is the following. In the central region of Figure 12.11, we have a theory whose fundamental ingredients are not just strings or membranes, but rather "branes" of a variety of dimensions, all more or less on equal footing. Currently, we do not have a firm grasp on many essential features of this full theory. But one thing we do know is that as we move from the central region to any of the peninsular regions, only the strings (or membranes curled up to look ever more like strings, as in Figures 12.7 and 12.8) are light enough to make contact with physics as we know it—the particles of Table 1.1 and the four forces through which they interact. The perturbative analyses string theorists have made use of for close to two decades have not been refined enough to discover even the existence of the super-massive extended objects of other dimensions; strings dominated the analyses and the theory was given the far-from-democratic name of string theory. Again, in these regions of Figure 12.11 we are justified, for most considerations, in ignoring all but the strings. In essence, this is what we have done so far in this book. We see now, though, that in actuality the theory is more rich than anyone previously imagined.
Does Any of This Solve the Unanswered Questions in String Theory?
Yes and no. We have managed to deepen our understanding by breaking free of certain conclusions that, in retrospect, were a consequence of perturbative approximate analyses rather than true string physics. But the current scope of our nonperturbative tools is quite limited. The discovery of the remarkable web of duality relations affords us far greater insight into string theory, but many issues remain unresolved. At present, for example, we do not know how to go beyond the approximate equations for the value of the string coupling constant—equations that, as we have seen, are too coarse to give us any useful information. Nor do we have any greater insight into why there are precisely three extended spatial dimensions, or how to choose the detailed form for the curled-up dimensions. These questions require more sharply honed nonperturbative methods than we currently possess.
What we do have is a far deeper understanding of the logical structure and theoretical reach of string theory. Prior to the realizations summarized in Figure 12.11, the strong coupling behavior of each string theory was a black box, a complete mystery. As on maps of old, the realm of strong coupling was uncharted territory, potentially filled with dragons and sea monsters. But now we see that although the journey to strong coupling may take us through unfamiliar regions of M-theory, it ultimately lands us back in the comfortable surrounds of weak coupling—albeit in the dual language of what was once thought to be a different string theory.
Duality and M-theory unite the five string theories and they suggest an important conclusion. It may well be that there aren't other surprises, on par with the ones just discussed, that are awaiting our discovery. Once a cartographer can fill in every region on a spherical globe of the earth, the map is done and geographical knowledge is complete. That's not to say explorations in Antarctica or on an isolated island in Micronesia are without scientific or cultural merit. It only means that the age of geographic discovery is over. The absence of blank spots on the globe ensures this. The "theory map" of Figure 12.11 plays a similar role for string theorists. It covers the entire range of theories that can be reached by setting sail from any one of the five string constructions. Although we are far from a full understanding of the terra incognita of M-theory, there are no blank regions on the map. Like the cartographer, the string theorist can now claim with guarded optimism that the spectrum of logically sound theories incorporating the essential discoveries of the past century—special and general relativity; quantum mechanics; gauge theories of the strong, weak, and electromagnetic forces; supersymmetry; extra dimensions of Kaluza and Klein—is fully mapped out by Figure 12.11.
The challenge to the string theorist—or perhaps we should say the M-theorist—is to show that some point on the theory map of Figure 12.11 actually describes our universe. To do this requires finding the full and exact equations whose solution will pick out this elusive point on the map, and then understanding the corresponding physics with sufficient precision to allow comparisons with
experiment. As Witten has said, "Understanding what M-theory really is—the physics it embodies—would transform our understanding of nature at least as radically as occurred in any of the major scientific upheavals of the past."15 This is the program for unification in the twenty-first century.
Chapter 13: Black Holes: A String/M-Theory Perspective:
The pre-string theory conflict between general relativity and quantum mechanics was an affront to our visceral sense that the laws of nature should fit together in a seamless, coherent whole. But this antagonism was more than a towering abstract disjunction. The extreme physical conditions that occurred at the moment of the big bang and that prevail within black holes cannot be understood without a quantum mechanical formulation of the gravitational force. With the discovery of string theory, we now have a hope of solving these deep mysteries. In this and the next chapter, we describe how far string theorists have gone toward understanding black holes and the origin of the universe.
Black Holes and Elementary Particles
At first sight it's hard to imagine any two things more radically different than black holes and elementary particles. We usually picture black holes as the most gargantuan of heavenly bodies, whereas elementary particles are the most minute specks of matter. But the research of a number of physicists during the late 1960s and early 1970s, including Demetrios Christodoulou, Werner Israel, Richard Price, Brandon Carter, Roy Kerr, David Robinson, Hawking, and Penrose, showed that black holes and elementary particles are perhaps not as different as one might think. These physicists found increasingly persuasive evidence for what John Wheeler has summarized by the statement "black holes have no hair." By this, Wheeler meant that except for a small number of distinguishing features, all black holes appear to be alike. The distinguishing features? One, of course, is the black hole's mass. What are the others? Research has revealed that they are the electric and certain other force charges a black hole can carry, as well as the rate at which it spins. And that's it. Any two black holes with the same mass, force charges, and spin are completely identical. Black holes do not have fancy "hairdos"—that is, other intrinsic traits—that distinguish one from another. This should ring a loud bell. Recall that it is precisely such properties—mass, force charges, and spin—that distinguish one elementary particle from another. The similarity of the defining traits has led a number of physicists over the years to the strange speculation that black holes might actually be gigantic elementary particles.
