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

by Brian Greene

  12. There are some mild restrictions that quantum field theories place on their internal parameters. To avoid certain classes of unacceptable physical behavior (violations of critical conservation laws, violations of certain symmetry transformations, and so on), there can be constraints on the charges (electric and also nuclear) of the theory’s particles. Additionally, to ensure that in all physical processes, probabilities add to 1, there can also be constraints on particle masses. But even with these constraints, there is wide latitude in the allowed values of particle properties.

  13. Some researchers will note that even though neither quantum field nor our current understanding of string theory provides an explanation of the particle properties, the issue is more urgent in string theory. The point is a bit involved, but for the technically minded here’s the summary. In quantum field theory, the properties of particles—say their masses, to be definite—are controlled by numbers that are inserted into the theory’s equations. The fact that quantum field theory’s equations allow such numbers to be varied is the mathematical way of saying that quantum field theory does not determine particle masses but instead takes them as input. In string theory, the flexibility in the masses of particles has a similar mathematical origin—the equations allow particular numbers to vary freely—but the manifestation of this flexibility is more significant. The freely varying numbers—numbers, that is, that can be varied with no cost in energy—correspond to the existence of particles with no mass. (Using the language of potential energy curves introduced in Chapter 3, envision a potential energy curve that’s completely flat, a horizontal line. Just as walking on a perfectly flat terrain would have no impact on your potential energy, changing the value of such a field would have no cost in energy. Since a particle’s mass corresponds to the curvature of its quantum field’s potential energy curve around its minimum, the quanta of such fields are massless.) Excessive numbers of massless particles are a particularly awkward feature of any proposed theory since there are tight limits on such particles coming from both accelerator data and cosmological observations. For string theory to be viable it is imperative that these particles acquire mass. In recent years, various discoveries have revealed ways in which this might happen, having to do with fluxes that can thread through holes in the extra-dimensional Calabi-Yau shapes. I will discuss aspects of these developments in Chapter 5.

  14. It is not impossible for experiments to provide evidence that would strongly disfavor string theory. The structure of string theory ensures that certain basic principles should be respected by all physical phenomena. Among these are unitarity (the sum of all probabilities of all possible outcomes in a given experiment must be 1) and local Lorentz invariance (in a small enough domain the laws of special relativity hold), as well as more technical features such as analyticity and crossing symmetry (the result of particle collisions must depend on the particles’ momentum in a manner that respects a particular collection of mathematical criteria). Should evidence be found—perhaps at the Large Hadron Collider—that any of these principles are violated, it would be a challenge to reconcile those data with string theory. (It would also be a challenge to reconcile those data with the standard model of particle physics, which incorporates these principles too, but the underlying assumption is that the standard model must give way to some kind of new physics at a high enough energy scale since the theory does not incorporate gravity. Data conflicting with any of the principles enumerated would argue that the new physics is not string theory.)

  15. It is common to speak of the center of a black hole as if it were a position in space. But it’s not. It is a moment in time. When crossing the event horizon of a black hole, time and space (the radial direction) interchange roles. If you fall into a black hole, for example, your radial motion represents progress through time. You are thus pulled toward the black hole’s center in the same way you are pulled to the next moment in time. The center of the black hole is, in this sense, akin to a last moment in time.

  16. For many reasons, entropy is a key concept in physics. In the case discussed, entropy is being used as a diagnostic tool to determine if string theory is leaving out any essential physics in its description of black holes. If it was, the black hole disorder that the string mathematics is being used to calculate would be inaccurate. The fact that the answer agrees exactly with what Bekenstein and Hawking found using very different considerations is a sign that string theory has successfully captured the fundamental physical description. This is a very encouraging result. For more details, see The Elegant Universe, Chapter 13.

  17. The first hint of this pairing between Calabi-Yau shapes came from the work of Lance Dixon, as well as independently from Wolfgang Lerche, Nicholas Warner, and Cumrun Vafa. My work with Ronen Plesser found a method for producing the first concrete examples of such pairs, which we named mirror pairs, and the relationship between them mirror symmetry. Plesser and I also showed that difficult calculations on one member of a mirror pair, involving seemingly impenetrable details such as the number of spheres that can be packed into the shape, could be translated into far more manageable calculations on the mirror shape. This result was picked up by Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes and put into action—they developed techniques for explicitly evaluating the equality Plesser and I had established between the “difficult” and “easy” formulas. Using the easy formula, they then extracted information about its difficult partner, including the numbers associated with the sphere packing given in the text. In the years since, mirror symmetry has become its own field of research, with a great many important results being established. For a detailed history, see Shing-Tung Yau and Steve Nadis, The Shape of Inner Space (New York: Basic Books, 2010).

  18. String theory’s claim to have successfully melded quantum mechanics and general relativity rests on a wealth of supporting calculations, made yet more convincing by results we will cover in Chapter 9.

  Chapter 5: Hovering Universes in Nearby Dimensions

  1. Classical Mechanics: . Electromagnetism: d*F = *J;dF = 0. Quantum mechanics: . General relativity: .

  2. I am referring here to the fine structure constant, e2/hc, whose numerical value (at typical energies for electromagnetic processes) is about 1/137, which is roughly .0073.

  3. Witten argued that when the Type I string coupling is dialed large, the theory morphs into the Heterotic-O theory with a coupling that’s dialed small, and vice versa; the Type IIB at large coupling morphs into itself, the Type IIB theory but with small coupling. The cases of the Heterotic-E and Type IIA theories are a little more subtle (see The Elegant Universe, Chapter 12, for details), but the overall picture is that all five theories participate in a web of interrelations.

  4. For the mathematically inclined reader, the special thing about strings, one-dimensional ingredients, is that the physics describing their motion respects an infinite dimensional symmetry group. That is, as a string moves, it sweeps out a two-dimensional surface, and so the action functional from which its equations of motion are derived is a two-dimensional quantum field theory. Classically, such two-dimensional actions are conformally invariant (invariant under angle-preserving rescalings of the two-dimensional surface), and such symmetry can be preserved quantum mechanically by imposing various restrictions (such as on the number of spacetime dimensions through which the string moves—the dimension, that is, of spacetime). The conformal group of symmetry transformations is infinite-dimensional, and this proves essential to ensuring that the perturbative quantum analysis of a moving string is mathematically consistent. For example, the infinite number of excitations of a moving string that would otherwise have negative norm (arising from the negative signature of the time component of the spacetime metric) can be systematically “rotated” away using the infinite-dimensional symmetry group. For details, the reader can consult M. Green, J. Schwarz, and E. Witten, Superstring Theory, vol. 1 (Cambridge: Cambridge University Press, 1988).

  5. As with man
y major discoveries, credit deserves to be given to those whose insights laid its groundwork as well as to those whose work established its importance. Among those who played such a role for the discovery of branes in string theory are: Michael Duff, Paul Howe, Takeo Inami, Kelley Stelle, Eric Bergshoeff, Ergin Szegin, Paul Townsend, Chris Hull, Chris Pope, John Schwarz, Ashoke Sen, Andrew Strominger, Curtis Callan, Joe Polchinski, Petr Hořava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard DeWitt.

  6. The diligent reader might argue that the Inflationary Multiverse also entwines time in a fundamental way, since, after all, our bubble’s boundary marks the beginning of time in our universe; beyond our bubble is thus beyond our time. While true, my point here is meant more generally—the multiverses discussed so far all emerge from analyses that focus fundamentally on processes occurring throughout space. In the multiverse we will now discuss, time is central from the outset.

  7. Alexander Friedmann, The World as Space and Time, 1923, published in Russian, as referenced by H. Kragh, in “Continual Fascination: The Oscillating Universe in Modem Cosmology,” Science in Context 22, no. 4 (2009): 587–612.

  8. As an interesting point of detail, the authors of the braneworld cyclic model invoke an especially utilitarian application of dark energy (dark energy will be discussed fully in Chapter 6). In the last phase of each cycle, the presence of dark energy in the braneworlds ensures agreement with today’s observations of accelerated expansion; this accelerated expansion, in turn, dilutes the entropy density, setting the stage for the next cosmological cycle.

  9. Large flux values also tend to destabilize a given Calabi-Yau shape for the extra dimensions. That is, the fluxes tend to push the Calabi-Yau shape to grow large, quickly running into conflict with the criterion that extra dimensions not be visible.

  Chapter 6: New Thinking About an Old Constant

  1. George Gamow, My World Line (New York: Viking Adult, 1970); J. C. Pecker, Letter to the Editor, Physics Today, May 1990, p. 117.

  2. Albert Einstein, The Meaning of Relativity (Princeton: Princeton University Press, 2004), p. 127. Note that Einstein used the term “cosmologic member” for what we now call the “cosmological constant”; for clarity, I have made this substitution in the text.

  3. The Collected Papers of Albert Einstein, edited by Robert Schulmann et al. (Princeton: Princeton University Press, 1998), p. 316.

  4. Of course, some things do change. As pointed out in the notes to Chapter 3, galaxies generally have small velocities beyond the spatial swelling. Over the course of cosmological timescales, such additional motion can alter position relationships; such motion can also result in a variety of interesting astrophysical events such as galaxy collisions and mergers. For the purpose of explaining cosmic distances, however, these complications can be safely ignored.

  5. There is one complication that does not affect the essential idea I’ve explained but which does come into play when undertaking the scientific analyses described. As photons travel to us from a given supernova, their number density gets diluted in the manner I’ve described. However, there is another diminishment to which they are subject. In the next section, I’ll describe how the stretching of space causes the wavelength of photons to stretch too, and, correspondingly, their energy to decrease—an effect, as we will see, called redshift. As explained there, astronomers use redshift data to learn about the size of the universe when the photons were emitted—an important step toward determining how the expansion of space has varied through time. But the stretching of photons—the diminishment of their energy—has another effect: It accentuates the dimming of a distant source. And so, to properly determine the distance of a supernova by comparing its apparent and intrinsic brightness, astronomers must take account not just of the dilution of photon number density (as I’ve described in the text), but also the additional diminishment of energy coming from redshift. (More precisely still, this additional dilution factor must be applied twice; the second red shift factor accounts for the rate at which photons arrive being similarly stretched by the cosmic expansion.)

  6. Properly interpreted, the second proposed answer for the meaning of the distance being measured may also be construed as correct. In the example of earth’s expanding surface, New York, Austin, and Los Angeles all rush away from one another, yet each continues to occupy the same location on earth it always has. The cities separate because the surface swells, not because someone digs them up, puts them on a flatbed, and transports them to a new site. Similarly, because galaxies separate due to the cosmic swelling, they too occupy the same location in space they always have. You can think of them as being stitched to the spatial fabric. When the fabric stretches, the galaxies move apart, yet each remains tethered to the very same point it has always occupied. And so, even though the second and third answers appear different—the former focusing on the distance between us and the location a distant galaxy had eons ago, when the supernova emitted the light we now see; the latter focusing on the distance now between us and that galaxy’s current location—they’re not. The distant galaxy is now, and has been for billions of years, positioned at one and the same spatial location. Only if it moved through space rather than solely ride the wave of swelling space would its location change. In this sense, the second and third answers are actually the same.

  7. For the mathematically inclined reader, here is how you do the calculation of the distance—now, at time tnow—that light has traveled since being emitted at time temitted. We will work in the context of an example in which the spatial part of spacetime is flat, and so the metric can be written as ds2 = c2dt2 – a2(t)dx2, where a(t) is the scale factor of the universe at time t, and c is the speed of light. The coordinates we are using are called co-moving. In the language developed in this chapter, such coordinates can be thought of as labeling points on the static map; the scale factor supplies the information contained in the map’s legend.

  The special characteristic of the trajectory followed by light is that ds2 = 0 (equivalent to the speed of light always being c) along the path, which implies that , or, over a finite time interval such as that between . The left side of this equation gives the distance light travels across the static map between emission and now. To turn this into the distance through real space, we must rescale the formula by today’s scale factor; therefore, the total distance the light traveled equals . If space were not stretching, the total travel distance would be , as expected. When calculating the distance traveled in an expanding universe, we thus see that each segment of the light’s trajectory is multiplied by the factor , which is the amount by which that segment has stretched, since the moment the light traversed it, until today.

  8. More precisely, about 7.12 × 10–30 grams per cubic centimeter.

  9. The conversion is 7.12 × 10–30 grams/cubic centimeter = (7.12 × 10–30 grams/cubic centimeter) × (4.6 × 104 Planck mass/gram) × (1.62 × 10–33 centimeter/Planck length)3 = 1.38 × 10–123 Planck mass/cubic Planck volume.

  10. For inflation, the repulsive gravity we considered was intense and brief. This is explained by the enormous energy and negative pressure supplied by the inflaton field. However, by modifying a quantum field’s potential energy curve, the amount of energy and negative pressure it supplies can be diminished, thus yielding a mild accelerated expansion. Additionally, a suitable adjustment of the potential energy curve can prolong this period of accelerated expansion. A mild and prolonged period of accelerated expansion is what’s required to explain the supernova data. Nevertheless, the small non-zero value for the cosmological constant remains the most convincing explanation to have emerged in the more than ten years since the accelerated expansion was first observed.

  11. The mathematically inclined reader should note that each such jitter contributes an energy that’s inversely proportional to its wavelength, ensuring that the sum over all possible wavelengths yields an infinite energy.

  12. For the mathematically inclined reader, the cancellation occurs because supers
ymmetry pairs bosons (particles with an integral spin value) and fermions (particles with a half [odd] integral spin value). This results in bosons being described by commuting variables, fermions by anticommuting variables, and that is the source of the relative minus sign in their quantum fluctuations.

  13. While the assertion that changes to the physical features of our universe would be inhospitable to life as we know it is widely accepted in the scientific community, some have suggested that the range of features compatible with life might be larger than once thought. These issues have been widely written about. See, for example: John Barrow and Frank Tipler, The Anthropic Cosmological Principle (New York: Oxford University Press, 1986); John Barrow, The Constants of Nature (New York: Pantheon Books, 2003); Paul Davies, The Cosmic Jackpot (New York: Houghton Mifflin Harcourt, 2007); Victor Stenger, Has Science Found God? (Amherst, N.Y.: Prometheus Books, 2003); and references therein.

  14. Based on the material covered in earlier chapters, you might immediately think the answer is a resounding yes. Consider, you say, the Quilted Multiverse, whose infinite spatial expanse contains infinitely many universes. But you need to be careful. Even with infinitely many universes, the list of different cosmological constants represented might not be long. If, for example, the underlying laws don’t allow for many different cosmological constant values, then regardless of the number of universes, only the small collection of possible cosmological constants would be realized. So, the question we’re asking is whether (a) there are candidate laws of physics that give rise to a multiverse, (b) the multiverse so generated contains far more than 10124 different universes, and (c) the laws ensure that the cosmological constant’s value varies from universe to universe.