The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos

by Brian Greene

  GOAL: Experimental confirmation

  IS GOAL REQUIRED?: Yes.

  This is the only way to determine whether a theory is a correct description of nature.

  STATUS: Indeterminate; no predictions.

  This is the most important criterion; to date, string theory has not been tested on it. Optimists hope that experiments at the Large Hadron Collider and observations by satellite-borne telescopes have the capacity to bring string theory much closer to data. But there’s no guarantee that current technology is sufficiently refined to reach this goal.

  GOAL: Cure singularities

  IS GOAL REQUIRED?: Yes.

  A quantum theory of gravity should make sense of singularities arising in situations that are, even just in principle, physically realizable.

  STATUS: Excellent.

  Tremendous progress; many kinds of singularities have been resolved by string theory. The theory still needs to address black hole and big bang singularities.

  GOAL: Black hole entropy

  IS GOAL REQUIRED?: Yes.

  A black hole’s entropy provides a hallmark context in which general relativity and quantum mechanics interface.

  STATUS: Excellent.

  String theory has succeeded in explicitly calculating, and confirming, the entropy formula proposed in the 1970s.

  GOAL: Mathematical contributions

  IS GOAL REQUIRED?: No.

  There’s no requirement that correct theories of nature yield mathematical insights.

  STATUS: Excellent.

  Although mathematical insights aren’t necessary to validate string theory, significant ones have emerged from the theory, revealing the profound reach of its mathematical underpinnings.

  *If you’d like to know how string theory surmounts the problems that blocked earlier attempts to join gravity and quantum mechanics, see The Elegant Universe, Chapter 6; for a sketch, see note 8. For an even briefer summary, note that whereas a point particle exists at a single location, a string, because it has length, is slightly spread out. This spreading, in turn, dilutes the raucous short-distance quantum jitters that stymied previous attempts. By the late 1980s, there was strong evidence that string theory successfully melds general relativity and quantum mechanics; more recent developments (Chapter 9) make the case overwhelming.

  CHAPTER 5

  Hovering Universes in Nearby Dimensions

  The Brane and Cyclic Multiverses

  Late one night many years ago, I was in my office at Cornell University putting together the freshman physics final exam that would be given the following morning. Since this was the honors class, I wanted to enliven things a little by giving them one somewhat more challenging problem. But it was late and I was hungry, so rather than carefully working through various possibilities, I quickly modified a standard problem that most of them had already encountered, wrote it into the exam, and headed home. (The details hardly matter, but the problem had to do with predicting the motion of a ladder, leaning against a wall, as it loses its footing and falls. I modified the standard problem by having the density of the ladder vary along its length.) During the exam the next morning, I sat down to write the solutions, only to find that my seemingly modest modification to the problem had made it exceedingly difficult. The original problem took perhaps half a page to complete. This one took me six pages. I write big. But you get the point.

  This little episode represents the rule rather than the exception. Textbook problems are very special, being carefully designed so that they’re completely solvable with reasonable effort. But modify textbook problems just a bit, changing this assumption or dropping that simplification, and they can quickly become intricate or intractable. That is, they can quickly become as difficult as analyzing typical real-world situations.

  The fact is, the vast majority of phenomena, from the motion of planets to the interactions of particles, are just too complex to be described mathematically with complete precision. Instead, the task of the theoretical physicist is to figure out which complications in a given context can be discarded, yielding a manageable mathematical formulation that still captures essential details. In predicting the course of the earth you’d better include the effects of the sun’s gravity; if you include the moon’s too, all the better, but the mathematical complexity rises significantly. (In the nineteenth century, the French mathematician Charles-Eugène Delaunay published two 900-page volumes related to intricacies of the sun-earth-moon gravitational dance.) If you try to go further and account fully for the influence of all the other planets, the analysis becomes overwhelming. Luckily, for many applications, you can safely disregard all but the sun’s influence, since the effect of other bodies in the solar system on earth’s motion is nominal. Such approximations illustrate my earlier assertion that the art of physics lies in deciding what to ignore.

  But as practicing physicists know well, approximation is not just a potent means for progress; on occasion it also brings peril. Complications of minimal importance for answering one question can sometimes have a surprisingly significant impact in answering another. A single drop of rain will hardly affect the weight of a boulder. But if the boulder is teetering high on a cliff’s edge, that drop of rain could very well coax it to fall, initiating an avalanche. An approximation that disregards the raindrop would miss a crucial detail.

  In the mid-1990s, string theorists discovered something akin to a raindrop. They found that various mathematical approximations, widely used to analyze string theory, were overlooking some vital physics. As more precise mathematical methods were developed and applied, string theorists could finally step beyond the approximations; when they did, numerous unanticipated features of the theory came into focus. And among these were new types of parallel universes; one variety in particular may be the most experimentally accessible of all.

  Beyond Approximations

  Every major established discipline of theoretical physics—such as classical mechanics, electromagnetism, quantum mechanics, and general relativity—is defined by a central equation, or set of equations. (You don’t need to know these equations, but I’ve listed some of them in the notes.)1 The challenge is that in all but the simplest situations, the equations are extraordinarily difficult to solve. For this reason, physicists routinely use simplifications—like ignoring Pluto’s gravity or treating the sun as perfectly round—that make the mathematics easier and bring approximate solutions within reach.

  For a long time, research in string theory has faced even bigger challenges. Just finding the central equations proved so difficult that physicists could develop only approximate versions. And even the approximate equations were so intricate that physicists had to make simplifying assumptions to find solutions, thus basing research on approximations of approximations. During the 1990s, however, the situation vastly improved. In a series of advances, a number of string theorists showed how to go well beyond the approximations, offering unmatched clarity and insight.

  To get a feel for these breakthroughs, imagine that Ralph is planning to play the next two rounds of the weekly worldwide lottery, and he’s proudly worked out the odds of winning. He tells Alice that since he has a 1 in a billion chance each week, if he plays both rounds his chance of winning is 2 in a billion, .000000002. Alice smirks. “Well, that’s close, Ralph.” “Really, wise guy. What do you mean close?” “Well,” she says, “you’ve overestimated. Should you win the first round, playing a second time won’t increase your chances of winning; you would already have done so. If you win twice, we’ll have more money, sure, but since you’re working out the odds of winning at all, winning the second lottery after the first just doesn’t matter. So, to get the precise answer you’d need to subtract the odds of winning both rounds—1 in a billion times 1 in a billion, or .000000000000000001. That yields a final tally of .000000001999999999. Questions, Ralph?”

  Minus the smugness, Alice’s method is an example of what physicists call a perturbative approach. In doing a calculation, it’s often
easiest to make a first pass that incorporates only the most obvious contributions—that’s Ralph’s starting point—and then make a second pass that includes finer details, modifying or “perturbing” the first-pass answer, as in Alice’s contribution. The approach easily generalizes. If Ralph were planning to play the next ten weekly lotteries, the first-pass approach suggests that his chance of winning is about 10 in a billion, .00000001. But, as in the previous example, this approximation fails to account correctly for multiple wins. When Alice takes over, her second pass would properly account for instances in which Ralph wins twice—say, on the first and second lotteries, or the first and third, or the second and fourth. These corrections, as Alice pointed out above, are proportional to 1 in a billion times 1 in a billion. But there’s also an even tinier chance that Ralph wins three times; Alice’s third pass takes that, too, into account, producing modifications proportional to 1 in a billion multiplied by itself three times, .000000000000000000000000001. The fourth pass does the same for the even tinier chance of winning four rounds, and so on. Each new contribution is far smaller than the previous, so at some point Alice deems the answer sufficiently accurate and calls it a day.

  Calculations in physics, and in many other branches of science too, often proceed in an analogous fashion. If you are interested in how likely it is that two particles heading in opposite directions around the Large Hadron Collider will bang into each other, the first pass imagines they hit once and ricochet (where “hit” doesn’t mean they directly touch, but rather that a single force-carrying “bullet,” such as a photon, flies from one and is absorbed by the other). The second pass takes into account the chance that the particles hit each other twice (two photons are fired between them); the third pass modifies the previous two by accounting for the chance of the particles hitting each other three times; and so on (Figure 5.1). As with the lottery, this perturbative approach works well if the chance of an ever-greater number of particle interactions—like the chance of an ever-greater number of lottery wins—drops precipitously.

  For the lottery, the drop-off is determined by each successive win coming with a factor of 1 in a billion; in the physics example, it’s determined by each successive hit coming with a numerical factor, called a coupling constant, whose value captures the likelihood that one particle will fire a force-carrying bullet and that the second particle will receive it. For particles such as electrons, governed by the electromagnetic force, experimental measurements have determined that the coupling constant, associated with photon bullets, is about .0073.2 For neutrinos, governed by the weak nuclear force, the coupling constant is about 10–6. For quarks, components of protons, that are racing around the Large Hadron Collider and whose interactions are governed by the strong nuclear force, the coupling constant is somewhat less than 1. These numbers are not as small as the lottery’s .000000001, but if for example we multiply .0073 by itself the result quickly becomes minuscule. After one iteration it’s about .0000533, after two it’s about .000000389. This explains why theorists only rarely go to the trouble of accounting for electrons hitting each other numerous times. The calculations involving many hits are exceedingly intricate to carry out, and the resulting contributions are so terribly tiny that you can stop at just a few photons fired and still get an extraordinarily accurate answer.

  Figure 5.1 Two particles (represented by the two solid lines on the left in each diagram) interact by firing various “bullets” at each other (the “bullets” are force-carrying particles, represented by the squiggly lines), and then ricochet forward (the two solid lines on the right). Each diagram contributes to the overall likelihood that the particles bounce off each other. The contributions of processes with ever-more bullets are ever smaller.

  To be sure, physicists would love to have exact results. But for many calculations the mathematics proves too difficult, so the perturbative approach is the best we can do. Fortunately, for small enough coupling constants, the approximate calculations can yield predictions that agree extremely well with experiment.

  A similar perturbative approach has long been a mainstay of string theory research. The theory contains a number, called the string coupling constant (string coupling, for short), that governs the chance that one string bumps off another. If the theory proves correct, the string coupling may one day also be measured, much like the couplings enumerated above. But since such a measurement is at present purely hypothetical, the value of the string coupling is a complete unknown. Over the past few decades, with no guidance from experiment, string theorists have made the key assumption that the string coupling is a small number. To some extent, this has been like the drunkard looking for his keys under a lamppost, because a small string coupling allows physicists to shine the bright lights of perturbative analysis on their calculations. Since many successful approaches prior to string theory do have a small coupling, a more favorable version of the analogy notes that the drunkard has been justifiably emboldened by frequently finding his keys in the very location that’s illuminated. Either way, the assumption has made possible a vast collection of mathematical calculations that have not only clarified the basic processes of how one string interacts with another, but have also revealed much about the fundamental equations underlying the subject.

  If the string coupling is small, these approximate calculations are expected to accurately reflect the physics of string theory. But what if it isn’t? Unlike what we found with the lottery and with colliding electrons, a large string coupling would mean that successive refinements to first-pass approximations would yield ever-larger contributions, so you’d never be justified in stopping a calculation. The thousands of calculations that have used the perturbative scheme would be baseless; years of research would collapse. Adding to the concerns, even with a small yet moderate string coupling, you might also worry that your approximations, at least in some circumstances, were overlooking subtle yet vital physical phenomena, like the raindrop that hits the boulder.

  Through the early 1990s, not much could be said about these vexing questions. By the second half of that decade, the silence gave way to a clamor of insight. Researchers found new mathematical methods that could outflank the perturbative approximations by leveraging something called duality.

  Duality

  In the 1980s, theorists realized that there was not one string theory but rather five different versions, to which they gave the catchy names Type I, Type IIA, Type IIB, Heterotic-O, and Heterotic-E. I’ve not yet mentioned this complication because although calculations established that the theories differ in detail, all five include the same gross features—vibrating strings and extra spatial dimensions—on which we’ve so far focused. But we’re now at a point where the five variations on the string theory theme come to the fore.

  For many years, physicists had relied on perturbative methods to analyze each of the string theories. When working with the Type I string theory, they assumed its coupling was small, and pressed on with multi-pass calculations similar to what Ralph and Alice did in the lottery analysis. When working with the Heterotic-O, or any of the other string theories, they did the same. But outside of this restricted domain of small string couplings, researchers could do nothing more than shrug, throw up their hands, and admit that the math they were using was too feeble to provide any reliable insight.

  Until, that is, the spring of 1995, when Edward Witten rocked the string theory community with a series of stunning results. Drawing on the insights of scientists including Joe Polchinski, Michael Duff, Paul Townsend, Chris Hull, John Schwarz, Ashoke Sen, and many others, Witten provided strong evidence that string theorists could safely navigate beyond the shores of small couplings. The central idea was simple and powerful. Witten argued that when the coupling constant in any one formulation of string theory is dialed ever larger, the theory—remarkably—steadily morphs into something thoroughly familiar: one of the other formulations of string theory, but with a coupling constant that’s dialed ever smaller. For example, when
the Type I string coupling is large, it transforms into the Heterotic-O string theory with a coupling that’s small. Which means that the five string theories are not different after all. Each appears different when examined in a limited context—small values of its particular coupling constant—but when this restriction is lifted, each string theory transforms into the others.

  I recently encountered a splendid graphic that from close up looks like Albert Einstein, with a bit more distance becomes ambiguous, and from far away resolves into Marilyn Monroe (Figure 5.2). If you saw only the images that come into focus at the two extremes, you’d have every reason to think you were looking at two separate pictures. But if you steadily examine the image through the range of intermediate distances, you unexpectedly find that Einstein and Monroe are aspects of a single portrait. Similarly, an examination of two string theories, in the extreme case when each has a small coupling, reveals that they’re as different as Albert and Marilyn. If you stopped there, as for years string theorists did, you’d conclude that you were studying two separate theories. But if you examine the theories as their couplings are varied over the range of intermediate values, you find that, like Albert turning into Marilyn, each gradually morphs into the other.

  The morphing from Einstein to Monroe is amusing. The morphing of one string theory into another is transformative. It implies that if perturbative calculations in one string theory can’t be undertaken because that theory’s coupling is too large, the calculations can be faithfully translated into the language of another formulation of string theory, one in which a perturbative approach succeeds because the coupling is small. Physicists call the transition between naïvely distinct theories duality. It has become one of the most pervasive themes in modern string theory research. By providing two mathematical descriptions of one and the same physics, duality doubles our calculational arsenal. Calculations that are impossibly difficult from one perspective become perfectly doable from another.*