Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone)

by Lawrence M. Krauss

  Beyond this, what the models of Arkani-Hamed and coworkers and Randall-Sundrum have shown is that if our three-dimensional universe comprises a three-brane within a higher-dimensional space, then it might be possible to resolve at least one fundamental mystery in particle physics while providing a possible new set of signatures that could open up both the extra dimensions, and the complexities of string theory or M-theory to the bright light of experiment.

  That is the good news. Once again, however, just below the surface in these models lies a host of problems that suggest, that as far as the possible existence of extra dimensions are concerned, it is very difficult to have one’s cake and eat it, too.

  First, note that in the second Randall-Sundrum model, which switches branes around from their earlier compactified model, the whole extradimensional solution of the hierarchy problem disappears. In the first model the exponential falloff of gravity near our brane is sufficient to make gravity anomalously weak compared to the other forces in nature, while in their second model, unless one fine-tunes things, the four-dimensional Planck scale is identical to the fundamental curvature scale in the higher dimension, which is presumably related to the string scale in this higherdimensional space. All of the scales are vastly different than the electroweak scale. One must then find another mechanism to enforce the wide disparity between the strength of gravity and the other observed forces in our world. This is inconvenient and, frankly, reduces the motivation for introducing an extra dimension in the first place. Were it not for the fact that three-branes and extra dimensions arise within the context of string/M-theory, one might wonder what one would gain from this albeit fascinating mathematical construction. But there are other, more fundamental concerns. The key to all of these interesting recent results is the newly recognized possible existence within string theory of three-branes, onto which all nongravitational charges and fields could be constrained, and a higher-dimensional bulk space into which gravity can propagate. However, string theory appears to naturally incorporate branes of all dimensions up to perhaps ten dimensions itself, with all of these comprising all possible orientations within the context of the complicated and as-of-yet not understood ground state of string/M-theory. The notion that our world should lie completely within an isolated three-brane is quite frankly not suggested by anything that is known about string theory at the present time.

  From my point of view there is another more immediate issue that strongly diminishes the beauty of the proposed extra-dimensional solutions of the hierarchy problem. The one profoundly important experimental fact we know about the fundamental forces of nature on the scales that we can probe them is that as these scales become smaller and smaller, the strengths of the forces appear to approach a common value. It was this fact that provided one of the most direct pieces of evidence suggesting the existence of a possible grand unified theory in the first place, and that still provides one of the best reasons to believe in supersymmetry as a symmetry of nature. Recall that this symmetry is really the underpinning of all of modern string theory.

  But remember, too, that the scale at which this evidence suggests the forces may unify is unambiguously fourteen to fifteen orders of magnitude smaller than the scale at which the weak and electromagnetic interactions themselves unify, and within a few orders of magnitude of the Planck scale itself. Moreover, the other new great discovery of the past twenty-five years in particle physics is the remarkable fact that neutrinos, the ghostly particles that experience only the weak force, are not absolutely massless, but rather have a very small mass, more than a hundred thousand times smaller than that of the next lightest particle, the electron. Such masses are not explicable within the context of the standard model, which incorporates all known physics up to the electroweak scale. However, if one adds new physics at the grand unified scale, one can naturally arrive at neutrino masses in this range.

  Thus, if one makes the electroweak scale the fundamental scale in nature on which extra dimensions, gravity, and string phenomena arise, one might remove the hierarchy between our four-dimensional Planck scale and the electroweak scale, but in doing so one swims strongly against the tide of experimental evidence. This is not a good precedent for what is supposed to be an empirical science.

  It could be that the apparent unification of the strengths of the known nongravitational forces, and the existence of neutrino masses, are just coincidences with no fundamental explanation in terms of grand unification near the Planck scale. But here I paraphrase Einstein: Nature may be subtle, but she is not malicious. If the only evidence that nature seems to be providing us about fundamental scales turns out to be a red herring, this would break a tradition that has stood us in good stead for over four hundred years. Finally, we once again return to the Achilles heel of all theories of quantum gravity: Einstein’s cosmological constant, the energy of empty space. It turns out that in order for our three-dimensional space to exist as a flat three-brane within a warped higher-dimensional space, the vacuum energy associated with the higher-dimensional space would have to be very large, and negative. It would then have to be precisely cancelled on our brane by a contribution that is large and positive in order for the observed energy of three-dimensional space to be both very small and nonzero. In short, the biggest fine-tuning problem known in nature becomes even more significant in these models, which, after all, were motivated by a desire to solve a much less severe numerical issue. Still, I come here not to bury these new ideas about extra dimensions but to praise them. For all their potential weaknesses, they have revealed as at least experimentally allowable a whole host of possible extra dimensions that had hitherto been considered ruled out. And whenever new theoretical possibilities exist, there is always the chance that nature will actually take advantage of them.

  All the problems and challenges aside, the realization that the world of our experience could, in principle, be embedded in a larger space that could become directly experimentally accessible in the near future has caused a tremendous explosion of energy devoted to exploring all the potential consequences of (possibly large) extra dimensions and the branes that may exist within them.

  The sociology of physics is a strange and wonderful thing. The reaction to the Arkani-Hamed and coworkers and Randall-Sundrum papers was nothing short of phenomenal. Within six years, no fewer than 2,500 separate scientific papers appeared exploring their ramifications. Like a well-timed drama that somehow captures the public’s imagination, the notions of large extra dimensions and/or a low-energy string scale seemed to have everything going for them in the theoretical physics community. They were novel, sexy, and potentially testable.

  New phenomena associated with strings and extra dimensions that had previously been assumed to be forever inaccessible are, if these ideas are correct (and I remain dubious), possibly on the verge of being measured in the laboratory. Direct probes of gravity on scales smaller than one mm are being developed that might probe for a change from the inverse square law. Alternatively, if the Planck scale coincides with the electroweak scale, then because we can probe the latter scale with modern particle accelerators, perhaps we could directly use these devices to probe extradimensional quantum gravitational phenomena. But as interesting to physicists as these direct tests might be, the poetic and philosophical implications of potentially large extra dimensions lie elsewhere. Other branes could represent possibly infinitely large alternate universes that could exist, literally, less than a fingernail’s width away from our own. Each of these universes could have laws of physics that might be dramatically different from our own as well as a dramatically different life history. And so, even if possible extra dimensions continue to elude the able probes of direct laboratory experiments, it could be that observations associated with the origin and evolution of our entire universe may unlock the door to their discovery.

  Evidence for the existence or absence of extra dimensions is likely to come ultimately not from an attempt to understand the dynamics of objects within our uni
verse, but rather from an attempt to understand the dynamics of our universe itself and to address the ultimate questions that have beset science since it first emerged from the fog of history: How did the universe begin? How will it end?

  And it is here, as I earlier suggested, that string theory, too, must ultimately face the music. If it is really ever to provide an explanation of anything we see, much less everything we see, it must address the fundamental nature of that which we cannot see but which we know is there. It must explain the energy of nothing.

  C H A P T E R 1 7

  A THEORY OF NOTHING?

  Wherever you go, there you are.

  —The Adventures of Buckaroo Banzai across the Eighth Dimension

  What could be more romantic than the notion that extra dimensions might not be truly hidden, but that objects from our universe might cross over into this new realm? And since physics is a two-way street, with that possibility comes a more exciting or perhaps terrifying one: What if material or information from these extra dimensions can “leak” into our own world? What if, ultimately, the source of our own existence lies across that invisible boundary?

  As we have seen, these questions have been the fodder for speculation and belief for almost four centuries, since sixteenth-century theologians first speculated that spirits and angels emerge from the extra-dimensional universe. But now they have reemerged in a new scientific context that might actually be testable.

  For a literary mind, the science fiction possibilities of these concepts are endless, and Buckaroo Banzai’s adventures are merely one particularly wacky manifestation. So, too, for physicists and their graduate students, long starved of new calculations that might be performed and even tested, the possibility of large extra dimensions and the existence of other branes than our own have provided countless new opportunities for exploration and creative expression. These have become popularly known as “Braneworld” scenarios, which sounds like a science fiction movie title as much as anything ever did. Even Stephen Hawking has gotten into the act with a recent popular lecture entitled “Brane New World.”

  In some sense it is appropriate that this research area does sound like science fiction, because most of it probably is. What is too often underappreciated about science is that almost all of the ideas it proposes turn out to be wrong. If they weren’t, the line between science and science fiction would be much less firm. But the “present” can perhaps be defined as that time when we teeter on the edge of understanding, and where the line between speculative science and science fiction is most easily blurred. And that is precisely where we now are in this narrative of our ongoing love affair with extra dimensions. This is not to suggest, however, that all ideas are equally attractive. Over the past five years, hundreds, if not thousands, of scientific papers have been written considering cosmological possibilities that might be associated with Braneworld scenarios. One cannot do justice to all of them, but the greatest justice I could probably do to many of them is to not mention them here. Nevertheless, it is undeniable that the mere fact that we might live on a three-brane in a possibly infinite or large but compactified extradimensional space dramatically has broadened the scope of cosmological investigation. For example: What may have caused our three spatial dimensions to have become potentially so much larger than the other extra dimensions, and could the latter’s dynamic evolution have an impact upon the cosmological evolution of our visible universe? What about the possible existence of other nearby branes? In the earliest moments of our big bang expansion, when the scale of our presently observable universe was as small or smaller than the present size of any compactified extra dimension, how could the presence of significant other dimensions have affected both the origin and evolution of our universe? And, how might our brane evolve dynamically within the bulk space today, or, equivalently, how might the changing nature of gravity on small or large scales in extradimensional Braneworld scenarios have an impact upon current measurements in observational cosmology?

  The first question has been around in one form or another since Kaluza and Klein first wrote down their ideas involving compact extra dimensions, and, as I have argued earlier, it is fair to say that no very good answer has yet been provided. If one compactifies extra dimensions into some small radii, r, then the size of these radii leaves an imprint on the remaining large dimensions via the existence of new fields in nature, called moduli fields. String theory is replete with such moduli fields. One can explore the dynamics of these fields and it turns out that they tend to want to relax to a zero value, which, in the higher dimensional picture, corresponds to the radii of the extra dimensions going to infinity. To stop this runaway expansion of dimensions, one generally has to introduce ad hoc mechanisms, which is one of the reasons that the Randall-Sundrum warped-extra-dimensional scenario, with its infinitely large extra dimensions, was proposed. Nevertheless, there have been suggestions that somehow the expansion of our three dimensions might arise at a cost to the extra dimensions, with our dimensions expanding, while the others perhaps contract. While this notion has some aesthetic appeal, no otherwise attractive mechanism has been proposed to generate a workable model. The next question, regarding the possible existence of other branes, and their potential effects on our own, is more intriguing. One particularly inventive proposal in this regard actually explored the possibility that these “extra” branes might actually be our own.

  Shortly after the first Arkani-Hamed and colleagues (ADD) proposal for large extra dimensions, these authors, along with several others, proposed that our brane might actually be folded over on itself many times, with different sheets located less than a millimeter away in the extra compact dimension. Since electromagnetic radiation and all nongravitational fields propagate only along our brane and not out into this extra dimension, these other regions would be invisible to us as long as the “folds” in our brane occurred at distances along our brane so far away that light has not yet had sufficient time since the big bang to travel across such distances. Thus, the only effect of these extra sheets would be their gravitational effects on us, since gravity can cross into the bulk between them. But since these extra sheets are really part of our brane, the laws of physics on them are identical to our own. Thus, otherwise invisible gas, stars, and galaxies could exist superimposed “on top” of our space. The authors of papers on this topic have suggested that these invisible objects might somehow comprise the dark matter that we infer to dominate the mass of our galaxies, for example.

  While this might be plausible in a science fiction universe, it will not pass muster in the real universe. The question of why these invisible stars and galaxies should tend to cluster along with our own but why the material in them should nevertheless spread out in halos around visible galaxies, was unanswered. Indeed, there are a host of other issues that must be addressed, including what mechanism might fold our brane and keep it folded.

  The problems that beset this idea are typical of many Braneworld scenarios for cosmology. The freedom allowed by extra dimensions introduces lots of exotic possibilities, but almost every one of them involves a set of new cosmological problems that must be dealt with in order to agree with observations. Most important of all, though, is the fact that there are often very plausible non-Braneworld approaches that address many of the same cosmological issues these new scenarios propose to deal with. For example, elementary particle physics now offers many realistic candidates for dark matter along with natural mechanisms to explain how it might have survived the earliest moments of the hot big bang so that it might come to dominate the mass of the universe today. Morever, particle physics provides very elegant mechanisms for generating the density perturbations in the very early universe that might ultimately collapse to form galaxies of visible and dark matter. It is not clear that the additional intellectual overhead associated with branes and extra dimensions is needed to explain anything that we might otherwise explain without it.

  Another example involves an idea that has becom
e central to modern cosmology, inflation. Recall that in 1980 the physicist Alan Guth proposed that phase transitions in the early universe could lead to periods of rapid early expansion. What he also showed is that such periods would resolve two otherwise completely inexplicable but central features of our universe, including its remarkable isotropy (i.e., uniformity) on large scales and the fact that the universe does not appear to be curved on large scales. Moreover, it was subsequently demonstrated that quantum mechanical processes during inflation could generate density fluctuations that could in principle later gravitationally collapse to produce the observed distribution of galaxies in the universe. Recently the observation of small temperature fluctuations in the cosmic microwave background radiation appears to be completely consistent with this scenario. While such consistency cannot prove inflation actually happened in the early universe, it is strongly suggestive. Nevertheless, in spite of the beauty of the idea of inflation, no particle physics models have been developed that provide compelling or even particularly attractive mechanisms that might underlie it. One might wonder therefore whether, in this case, Braneworlds might come to the rescue. Within a year after Arkani-Hamed and colleagues’ article, Gia Dvali and his collaborator Henry Tye recognized that as two branes approach each other, the residual moduli field in our dimension that results from their separation in the extra dimension strongly resembles the kind of field that previously had been proposed, ad hoc, to result in an inflationary phase in the early universe. Furthermore, depending upon the net energy associated with empty space on each of the branes, there would be forces of either attraction or of repulsion between the branes that might produce a period of inflation that could in principle gently end as the two branes approached or diverged from each other. While this picture has the advantage of allowing an inflationary phase without the need to introduce additional elementary particles and fields in the early universe, it is not without its own weaknesses. The brane energies have to be carefully adjusted for the scenario to work. More than this, it is very difficult in these scenarios, once brane interaction energies are converted into the matter and radiation necessary to produce the early hot universe that was the precursor of the universe we now observe, to stop most of the produced energy from instead being transferred to invisible gravitational modes that would be radiated off into the bulk and not on our brane.