The idea that is central to the Maldacena conjecture—that somehow all the physical information in a volume can be encoded on its surface—has thus become known as the holographic principle. I stress that while it has been applied in a variety of contexts by various theorists, the actual Maldacena conjecture itself involved two very specific spaces: a four-dimensional flat space with supersymmetry and quantum Yang-Mills fields, and a fivedimensional space with classical supergravity, along with a very weird specific source of gravity throughout the five dimensions (empty space full of negative energy—unlike anything we have measured in our own universe). Such a space is called an Anti-de Sitter space.
In any case, if Maldacena’s conjecture is correct—namely, that there is absolutely no physical difference other than appearance between these two spaces—then the physical distinction between different dimensions itself gets blurred. A host of questions naturally seems to arise. What is the utility of an extra hidden dimension if ultimately nothing is hidden except the existence of the extra dimension? And what is the practical meaning of extra dimensions if you can experience all there is to experience without actually moving into them?
Moreover, we may find ourselves somewhat like the holodeck characters in Star Trek, who have no sense that they may be mere projections. Are we merely a pale reflection of the real world behind the mirror? Or, if the surface contains everything that is inside, is it the extra dimension itself that is illusory? If the world of our experience is a hologram, where does the illusion end and reality begin? Ultimately, if Maldacena’s conjecture is correct, then it implies that these questions, as fascinating or troubling as they may seem, are moot. Reality is in the eye of the beholder. Both worlds are real, and identical, as different as they may seem.
If your head is now spinning, it should be. In one chapter, you have been treated, or perhaps subjected, to a menagerie of mathematical marvels associated with strings and D-branes in ten dimensions, M-theory in eleven dimensions, and holography in five dimensions. New dimensions have magically appeared and disappeared with more aplomb than the Cheshire Cat and with an uncanniness that might appear to make Alice’s voyage in Wonderland pale in comparison. Most importantly, you may be wondering what all of this wizardry has wrought? Are these imaginings of theoretical physicists any more real or of any more utility than those of Lewis Carroll?
These are good and valid questions. Remember what ostensibly caused all of this mathematical effort in the first place. String theory, or rather the Theory Formerly Known as String Theory, must, if it is to be useful to physicists, address some concrete physical problems and make concrete physical predictions. In its original form, it had simply failed to do so, all the hype surrounding it notwithstanding.
So, as mathematically remarkable as M-theory might be, or as useful as the Maldacena conjecture might be for trying to solve difficult mathematical problems associated with Yang-Mills theories, unless all of these ideas eventually help resolve fundamental physical questions, it is all just mathematics.
Thankfully, however, there has been some progress. In my mind it is not clear that it fully justifies the periodic hubris associated with string theory, but we shall see. It is at least an encouraging beginning. You will recall that a central problem in quantum gravity, which early work on string theory did not appear to address, was the “black hole information loss paradox.” Do black holes violate quantum mechanics? And if not, where does all the information that falls into black holes go?
A new approach to this problem did become possible once D-branes began to be explored. Recall that D-branes allow a new connection between the strongly interacting phase of some theories and the weakly interacting phase, where reliable calculations might be performed. It turns out that in certain limits one finds objects in string theory that resemble black holes, with highly curved geometries (in the extra dimensions). These are called black p-branes. Interestingly, if one explores a different limit of the same theory, where strings are weakly interacting, one can describe much of the physics in a calculable way using standard D-branes. One can hope, then, that the results of calculations one can explicitly perform in the one limit of the theory where such calculations are feasible might also be applicable in the other limit of the theory, where one cannot do direct calculations, and where the strongly gravitating black p-brane description applies. Now, if one examines a very special sort of five-dimensional p-brane, then in the weakly interacting limit of the theory, where D-brane calculations become reliable, it turns out that one can explicitly count the number of fundamental quantum states that could be occupied by an object that would, in the strong coupling limit of the theory, be associated with a black p-brane.
The result is striking. The number of quantum states turns out to be precisely the number of states needed to encode the information that was supposedly hidden behind the event horizon of a black hole—the so-called Hawking-Bekenstein entropy. This would suggest that the information is not, in fact, lost down the black hole, but is instead somehow preserved and if we had a way of accurately treating the quantum mechanics of realistic black holes (which, I remind you, are not to be confused with the very special five-dimensional black p-branes in this idealized calculation), we would uncover it.
Note that this result is far from a proof that black holes in string theory must behave like sensible quantum mechanical objects, nor does it provide any hint of what might actually happen to the information stored in a black hole’s interior as it evaporates. Moreover, the black p-branes in question are actually very finely tuned objects, which wouldn’t themselves even evaporate by Hawking processes because of their special configuration. However, this calculation is at least very encouraging. In the regime where D-branes, which are perfectly well-behaved quantum mechanical objects, are the appropriate description of string theory/M-theory, there are precisely the correct number of states to account for what one might hope a well-behaved quantum mechanical accounting of black holes might require. This was a real computational success in string theory, and it has generated tremendous enthusiasm.
Nevertheless, a host of caveats remain. As one increases the strength of the interaction needed to move from the D-brane to black p-brane picture, the physics could change, and information could be lost. Until one can calculate precisely where the information flows in the evaporation process of realistic black holes, extrapolating the apparent success of this aspect of the theory remains a conjecture.
Also, as I have mentioned, a few months before this writing Stephen Hawking made headlines throughout the world by retracting his claim that black hole evaporation destroys information. He has claimed that a new computation he has performed in the context of classical general relativity demonstrates explicitly how the information that falls into black holes gets preserved as they evaporate. He has spoken about this at several meetings. Many physicists are skeptical. However, when it comes to black holes, Stephen has a good track record.
If Hawking’s new claim is correct, then it will have a profound implication for the apparent success of string theory in potentially addressing the black hole information loss problem in classical general relativity, because the problem will have literally evaporated. This will not mean that the string accounting of p-brane states is incorrect, just that string theory would not have been needed to solve this fundamental problem that otherwise appeared to suggest the need to move beyond general relativity. String theorists will have to turn their attention to other problems the theory might more uniquely address. Which brings us back, finally, to Einstein’s revenge: the cosmological constant problem. This, after all, remains the key mystery in theoretical physics, and the clearest place where a theory of quantum gravity should shed some light. And it is the place where, I think it is fair to say, string theory had its biggest unmitigated lack of success. Nothing in all the work following the first string revolution, or even immediately following the discovery of the importance of D-branes and the emergence of M-theory, had shed any light
on the question of how the energy of empty space could be precisely zero.
So, when in 1998 cosmological observations led to the discovery that the energy of empty space isn’t precisely zero, just almost zero, everyone—including string theorists—stood up and took notice. Maybe, just maybe, this finding might provide a vital clue that could either vindicate the string revolution or help us move beyond it.
The result was a sudden new explosion of interest in—you guessed it—extra dimensions—but not the hypothetical, aetherial, and perhaps illusory extra dimensions that had so fixated the ten-or eleven-dimensional imaginations of string theorists. Rather, they were concrete and even potentially accessible extra dimensions that might literally be hiding behind the looking glass or on the other side of the wardrobe.
C H A P T E R 1 6
D IS FOR BRANEWORLD
The small man said to the other:
“Where does a wise man hide a pebble?”
And the tall man answered in a low voice:
“On the beach.”
—G. K. Chesterton
It is easy, in the midst of discussing such things as D-branes and supersymmetric state counting, to forget precisely what we are really talking about here. In order to understand what might otherwise be considered a somewhat esoteric corner of physical theory—the intersection of gravity and quantum mechanics—string theory or its successor, M-theory, suggests that we need to believe that the world of our experience is but a minor reflection of a higher-dimensional reality. The tragedies of human existence may be very poignant, and the evolution of our visible universe may be remarkable, but actually, they are all fundamentally a cosmic afterthought. Somehow the key to our existence lies in the poorly understood, but remarkably rich, possibilities available to a universe with perhaps seven extra dimensions, although one or more of these may not behave like any dimensions we have experienced. Moreover, the conventional wisdom, steeped in a tradition established by Kaluza and Klein almost a century ago, suggests these seven dimensions are “compactified,” bundled up for as-of-yet unknown reasons into regions so small that a pebble lying on a beach would be, by comparison, as large as our own galaxy is compared to the pebble.
At the same time deeper questions arise, some of which I have already considered. If the convolutions appropriate to extra-dimensional physics really do ultimately lead to a picture of the four-dimensional universe that accurately resembles the reality we experience, but if at the same time these dimensions remain forever hidden, ephemeral theoretical entities, inaccessible to our experiments, if not our imagination, then in what sense are these extra dimensions more than merely mathematical constructs?
What, in this case, does it mean to be real ?
There are times when I have wondered whether Michael Faraday, as he developed his fantastic mental images of hypothetical electric and magnetic fields permeating space in 1840 felt that they were so simple and beautiful that they had to exist. Or did he consider them to be merely a convenient crutch, so that someone like himself, unschooled in mathematics, could comprehend in an intuitive way some sliver of the physical world?
As I have mentioned, there is, of course, a noble tradition in physics of mathematical crutches turning out to have a physical reality. Faraday’s electromagnetic fields are just one example. Quarks, when they were first introduced, were also seen primarily as a mathematical classification scheme, rather than as real entities. So, too, were atoms, for that matter. Indeed, Ludwig Boltzmann committed suicide in part because he felt he could not convince his contemporaries that atoms had to be real. On the other hand, many mathematical models that have been proposed have thus far borne no relation to the real world—even mathematics that at one time or another seemed to show great promise. So the questions posed earlier remain relevant, and short of a theoretical breakthrough that unambiguously allows a prediction of unique laws of nature that match the ones we observe, the only way we may know if any of these higher-dimensional imaginings are correct is if somehow we can ultimately experimentally probe the extra dimensions, either directly or indirectly. Traditionally in string theory this has seemed like a colossal long shot. If the string scale is comparable with the Planck scale—about 10–33 cm, where quantum mechanical effects in gravity are presumed to become important—it is far removed from anything remotely accessible in the laboratory. Imagine you were looking at our galaxy through a distant telescope from another galaxy far, far away. Say your telescope could just barely resolve individual stars in the Milky Way, as the Hubble Space Telescope can in the nearby Andromeda galaxy, two million light-years away. The problem of measuring extra dimensions on the Planck scale is for us, then, similar to the problem of your trying to detect and probe individual atoms in that distant galaxy using your telescope!
The past decade has, however, produced some remarkable transformations in the way we think about fundamental physics, driven largely, I am happy to say, by the surprises nature has wrought.
Nature provided a cosmic wake-up call, in the form of dark energy, that not even those fully immersed in eleven-dimensional mathematics could ignore. In particular, the discovery that dark energy dominates the expansion of the universe is so shocking that it seems very likely that it is related to something fundamentally profound about the structure of space and time. And since string theory has taken as its mantra the revelation of profound new truths in these areas, the unexpected appearance of dark energy cried out for attention. Or, at the very least, it was irritating to the point of distraction.
The distraction was key, however. It stood as a stark reminder that, at the earliest moments of the big bang, what is now the visible universe was of a size comparable to the microscopically small scale of the purported extra dimensions. Thus perhaps the universe itself could provide the experiment that might ultimately reveal these extra dimensions for all to witness.
I remember David Gross’s telling me in 2002 why string theorists had suddenly become so interested in cosmology. The big bang, taken back to t = 0, inevitably leads to a singularity (a point of infinite density) at the beginning of time. There is clearly something physically implausible about such a state of infinitely high density. One of the main virtues of string theory, however, is its apparent ability to dispense with such infinite singularities, at least those that seemed to plague general relativity. Thus, string theory might be able to dispense with the big bang singularity, and perhaps in the process explain the mystery of dark energy. I confess that in a skeptical moment I responded to David by expressing the concern that string theory might instead do for observational cosmology what it has thus far done for experimental elementary particle physics: namely, nothing.
Sarcasm aside, however, in 1998 several theoretical breakthroughs transformed the way much of modern research is being performed, and have made the question of the possible reality of extra dimensions something of immediate and practical interest. They did not arise from cosmology, however, although they opened up, literally, a whole new universe of cosmological possibilities. Rather, they were inspired by a new consideration, reflected in the glow of D-branes, of the very same problem that first motivated many particle physicists to adopt supersymmetry as a useful guiding principle in nature: the hierarchy problem.
Recall that the hierarchy problem in particle physics relates to the question of why the GUT energy scale, where the three nongravitational forces in nature may be unified, could be fifteen orders of magnitude larger than the scale at which the weak and electromagnetic interactions are unified. Worse still, the Planck energy scale, where quantum gravity should become important, is seventeen orders of magnitude larger than this latter scale. Not only are these large discrepancies of scale inexplicable, but it turns out that formally, within the context of the standard model of particle physics without supersymmetry, this hierarchy is unstable. Namely, as I have described, the effects of high-energy virtual particles will tend to lead to intolerably large corrections in the low-energy theory.
In 1998 physicists Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali proposed a dramatic new way of avoiding the hierarchy between the Planck scale and the weak scale. They suggested that perhaps the Planck scale is not really where we think it is. The group was motivated by considering the possible existence of extra dimensions, and also indirectly by the development of D-branes in string theory. As I shall describe, their argument relied on the possibility that perhaps the extra dimensions, or at least one of them, might in fact not be microscopically small, but rather could be “almost” visible—perhaps, in fact, the size of a small pebble lying on a gravel road.
An immediate question that comes to mind when this possibility is raised is: If the extra dimensions are that big, why don’t we see them? A possible answer lies in the magic of D-branes. Remember that in string theory, open strings can end on D-branes, so that the charges on the ends of these strings, and the Yang-Mills fields and forces associated with these charges, might reside only on the D-branes. Remember, however, that gravitons, the particles associated with gravitational fields, are associated with closed string loops (i.e., objects without ends). These loops can also move about in the space between the branes, and thus gravitational fields are not restricted to exist only on the D-branes, but can also exist in the “bulk,” as the space between the branes is called.
Thus, imagine that the three spatial dimensions of our experience lie on a three-brane “surface” in a higher-dimensional space. If gravity is the only force that can exist outside of our three-brane, then only gravity could probe these extra dimensions.
How would gravity do so? Well, Newton’s theory of gravity tells us that the gravitational force between two objects falls off inversely as the square of the distance between them. This is, after all, precisely the same behavior that characterizes the electric force between charged objects. We now return at long last to Michael Faraday, whose brilliant idea of field lines helped to provide an intuitive understanding of why electric forces actually fell off as the inverse square of distance. Remember that if field lines move out in all directions from a charged particle, the number of field lines per unit area crossing any surface will fall off inversely with this area or, equivalently, inversely as the square of the distance from the source.
Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone) Page 23
