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

by Brian Greene

  5. In the approach to inflation I have just described, there is no fundamental explanation for why the inflaton field’s value would begin high up on the potential energy curve, nor why the potential energy curve would have the particular shape it has. These are assumptions the theory makes. Subsequent versions of inflation, most notably one developed by Andrei Linde called chaotic inflation, find that a more “ordinary” potential energy curve (a parabolic shape with no flat section that emerges from the simplest mathematical equations for the potential energy) can also yield inflationary expansion. To initiate the inflationary expansion, the inflaton field’s value needs to be high up on this potential energy curve too, but the enormously hot conditions expected in the early universe would naturally cause this to happen.

  6. For the diligent reader, let me note one additional detail. The rapid expansion of space in inflationary cosmology entails significant cooling (much as a rapid compression of space, or of most anything, causes a surge in temperature). But as inflation comes to a close, the inflaton field oscillates around the minimum of its potential energy curve, transferring its energy to a bath of particles. The process is called “re-heating” because the particles so produced will have kinetic energy and thus can be characterized by a temperature. As space then continues to undergo more ordinary (non-inflationary) big bang expansion, the temperature of the particle bath steadily decreases. The important point, though, is that the uniformity set down by inflation provides uniform conditions for these processes, and so results in uniform outcomes.

  7. Alan Guth was aware of the eternal nature of inflation; Paul Steinhardt wrote about its mathematical realization in certain contexts; Alexander Vilenkin brought it to light in the most general terms.

  8. The value of the inflaton field determines the amount of energy and negative pressure it suffuses through space. The larger the energy, the greater the expansion rate of space. The rapid expansion of space, in turn, has a back reaction on the inflaton field itself: the faster the expansion of space, the more violently the inflaton field’s value jitters.

  9. Let me address a question that may have occurred to you, one we will return to in Chapter 10. As space undergoes inflationary expansion, its overall energy increases: the greater the volume of space filled with an inflaton field, the greater the total energy (if space is infinitely large, energy is infinite too—in this case we should speak of the energy contained in a finite region of space as the region grows larger). Which naturally leads one to ask: What is the source of this energy? For the analogous situation with the champagne bottle, the source of additional energy in the bottle came from the force exerted by your muscles. What plays the role of your muscles in the expanding cosmos? The answer is gravity. Whereas your muscles were the agent that allowed the available space inside the bottle to expand (by pulling out the cork), gravity is the agent that allows the available space in the cosmos to expand. What’s vital to realize is that the gravitational field’s energy can be arbitrarily negative. Consider two particles falling toward each other under their mutual gravitational attraction. Gravity coaxes the particles to approach each other faster and faster, and as they do, their kinetic energy gets ever more positive. The gravitational field can supply the particles with such positive energy because gravity can draw down its own energy reserve, which becomes arbitrarily negative in the process: the closer the particles approach each other, the more negative the gravitational energy becomes (equivalently, the more positive the energy you’d need to inject to overcome the force of gravity and separate the particles once again). Gravity is thus like a bank that has a bottomless credit line and so can lend endless amounts of money; the gravitational field can supply endless amounts of energy because its own energy can become ever more negative. And that’s the energy source that inflationary expansion taps.

  10. I will use the term “bubble universe,” although the imagery of a “pocket universe” that opens up within the ambient inflaton-filled environment is a good one too (that term was coined by Alan Guth).

  11. For the mathematically inclined reader, a more precise description of the horizontal axis in Figure 3.5 is as follows: consider the two-dimensional sphere comprising the points in space at the time the cosmic microwave background photons began to stream freely. As with any two-sphere, a convenient set of coordinates on this locus are the angular coordinates from a spherical polar coordinate system. The temperature of the cosmic microwave background radiation can then be viewed as a function of these angular coordinates and, as such, can be decomposed in a Fourier series using as a basis the standard spherical harmonics, . The vertical axis in Figure 3.5 is related to the size of the coefficients for each mode in this expansion—farther to the right on the horizontal axis corresponds to smaller angular separation. For technical details, see for example Scott Dodelson’s excellent book Modern Cosmology (San Diego, Calif.: Academic Press, 2003).

  12. A little more precisely, it is not the strength of the gravitational field, per se, that determines the slowing of time, but rather the strength of the gravitational potential. For instance, if you were to hang out inside a spherical cavity at the center of a massive star, you wouldn’t feel a gravitational force at all, but because you were deep inside a gravitational-potential well, time for you would run slower than time for someone far outside the star.

  13. This result (and closely related ideas) was found by a number of researchers in different contexts, and was most explicitly articulated by Alexander Vilenkin and also by Sidney Coleman and Frank De Luccia.

  14. In our discussion of the Quilted Multiverse, you may recall that we assumed particle arrangements would vary randomly from patch to patch. The connection between the Quilted and Inflationary Mulitverses also allows us to make good on that assumption. A bubble universe forms in a given region when the inflaton field’s value drops; as it does, the energy the inflaton contained is converted into particles. The precise arrangement of these particles at any moment is determined by the precise value of the inflaton during the conversion process. But because the inflaton field is subject to quantum jitters, as its value drops it will be subject to random variations—the same random variations that give rise to the pattern of slightly hotter and slightly colder spots in Figure 3.4. When considered across the patches in a bubble universe, these jitters thus imply that the inflaton’s value will display random quantum variations. And this randomness ensures randomness of the resulting particle distributions. That’s why we expect any particle arrangement, such as the one responsible for all we see right now, to be replicated as often as any other.

  Chapter 4: Unifying Nature’s Laws

  1. I thank Walter Isaacson for personal communications on this and a number of other historical issues related to Einstein.

  2. In a little more detail, the insights of Glashow, Salam, and Weinberg suggested that the electromagnetic and weak forces were aspects of a combined electroweak force, a theory that was confirmed by accelerator experiments in the late 1970s and early 1980s. Glashow and Georgi went a step further and suggested that the electroweak and the strong forces were aspects of a yet more fundamental force, an approach that’s called grand unification. The simplest version of grand unification, however, was ruled out when scientists failed to observe one of its predictions—that protons should, every so often, decay. Nevertheless, there are many other versions of grand unification that remain experimentally viable since, for example, the rate of proton decay they predict is so slow that existing experiments would not yet have the sensitivity to detect it. However, even if grand unification is not borne out by data, it is already beyond doubt that the three nongravitational forces can be described using the same mathematical language of quantum field theory.

  3. The discovery of superstring theory spawned other, closely related, theoretical approaches seeking a unified theory of nature’s forces. In particular, supersymmetric quantum field theory, and its gravitational extension supergravity, have been vigorously pursued since the mid
-1970s. Supersymmetric quantum field theory and supergravity are based on the new principle of supersymmetry, which was discovered within superstring theory, but these approaches incorporate supersymmetry in conventional point-particle theories. We will briefly discuss supersymmetry later in the chapter, but for the mathematically inclined reader, I’ll note here that supersymmetry is the last available symmetry (beyond rotational symmetry, translational symmetry, Lorentz symmetry, and, more generally, Poincaré symmetry) of a nontrivial theory of elementary particles. It relates particles of different quantum mechanical spin, establishing a deep mathematical kinship between particles that communicate forces and the particles making up matter. Supergravity is an extension of supersymmetry that includes the gravitational force. In the early days of string theory research, scientists realized that the frameworks of supersymmetry and supergravity emerged from a low-energy analysis of string theory. At low energies, the extended nature of a string generally cannot be discerned, so it appears to be a point particle. Correspondingly, as we will discuss in this chapter, when applied to low energy processes, the mathematics of string theory transforms into that of quantum field theory. Scientists found that because both supersymmetry and gravity survive the transformation, low energy string theory gives rise to supersymmetric quantum field theory and to supergravity. In more recent times, as we will discuss in Chapter 9, the link between supersymmetric field theory and string theory has grown yet more profound.

  4. The informed reader may take exception to my statement that every field is associated to a particle. So, more precisely, the small fluctuations of a field about a local minimum of its potential are generally interpretable as particle excitations. That’s all we need for the discussion at hand. Additionally, the informed reader will note that localizing a particle at a point is itself an idealization, because it would take—from the uncertainty principle—infinite momentum and energy to do so. Again, the essence is that in quantum field theory there is, in principle, no limit to how finally localized a particle can be.

  5. Historically speaking, a mathematical technique known as renormalization was developed to grapple with the quantitative implications of severe, small-scale (high-energy) quantum field jitters. When applied to the quantum field theories of the three nongravitational forces, renormalization cured the infinite quantities that had emerged in various calculations, allowing physicists to generate fantastically accurate predictions. However, when renormalization was brought to bear on the quantum jitters of the gravitational field, it proved ineffective: the method failed to cure infinities that arose in performing quantum calculations involving gravity.

  From a more modern vantage point, these infinities are now viewed rather differently. Physicists have come to realize that en route to an ever-deeper understanding of nature’s laws, a sensible attitude to take is that any given proposal is provisional, and—if relevant at all—is likely capable of describing physics only down to some particular length scale (or only up to some particular energy scale). Beyond that are phenomena that lie outside the reach of the given proposal. Adopting this perspective, it would be foolhardy to extend the theory to distances smaller than those within its arena of applicability (or to energies above its arena of applicability). And with such inbuilt cutoffs (much as described in the main text), no infinities ever arise. Instead, calculations are undertaken within a theory whose range of applicability is circumscribed from the outset. This means that the ability to make predictions is limited to phenomena that lie within the theory’s limits—at very short distances (or at very high energies) the theory offers no insight. The ultimate goal of a complete theory of quantum gravity would be to lift the inbuilt limits, unleashing quantitative, predictive capacities on arbitrary scales.

  6. To get a feel for where these particular numbers come from, note that quantum mechanics (discussed in Chapter 8) associates a wave to a particle, with the heavier the particle the shorter its wavelength (the distance between successive wave crests). Einstein’s general relativity also associates a length to any object—the size to which the object would need to be squeezed to become a black hole. The heavier the object, the larger that size. Imagine, then, starting with a particle described by quantum mechanics and then slowly increasing its mass. As you do, the particle’s quantum wave gets shorter, while its “black hole size” gets larger. At some mass, the quantum wavelength and the black hole size will be equal—establishing a baseline mass and size at which quantum mechanical and general relativistic considerations are both important. When one makes this thought experiment quantitative, the mass and size are found to be those quoted in the text—the Planck mass and Planck length, respectively. To foreshadow later developments, in Chapter 9 I will discuss the holographic principle. This principle uses general relativity and black hole physics to argue for a very particular limit on the number of physical degrees of freedom that can reside in any volume of space (a more refined version of the discussion in Chapter 2 regarding the number of distinct particle arrangements within a volume of space; also mentioned in note 14 of Chapter 2). If this principle is correct, then the conflict between general relativity and quantum mechanics can arise before distances are small and curvatures large. A huge volume containing even a low density gas of particles would be predicted by quantum field theory to have many more degrees of freedom than the holographic principle (which relies on general relativity) would allow.

  7. Quantum mechanical spin is a subtle concept. Especially in quantum field theory, where particles are viewed as dots, it is hard to fathom what “spinning” would even mean. What really happens is that experiments show that particles can possess an intrinsic property that behaves much like an immutable quantity of angular momentum. Moreover, quantum theory shows, and experiments confirm, that particles will generally only have angular momentum that is an integer multiple of a fundamental quantity (Planck’s constant divided by 2). Since classical spinning objects possess an intrinsic angular momentum (one, however, that is not immutable—it changes as the object’s rotational speed changes), theoreticians have borrowed the name “spin” and applied it to this analogous quantum situation. Hence the name “spin angular momentum.” While “spinning like a top” provides a reasonable mental image, it’s more accurate to imagine that particles are defined not only by their mass, their electric charge, and their nuclear charges, but also by the intrinsic and immuatable spin angular momentum they possess. Just as we accept a particle’s electric charge as one of its fundamental defining features, experiments establish that the same is true of its spin angular momentum.

  8. Recall that the tension between general relativity and quantum mechanics arises from the powerful quantum jitters of the gravitational field that shake spacetime so violently that the traditional mathematical methods can’t cope. Quantum uncertainty tells us that these jitters become ever stronger when space is examined on ever-smaller distances (which is why we don’t see these jitters in everyday life). Specifically, the calculations show that it is the wildly energetic jitters over distances shorter than the Planck scale that make the math go haywire (the shorter the distance, the greater the jitters’ energy). Since quantum field theory describes particles as points with no spatial extent, the distances these particles probe can be arbitrarily small, and hence the quantum jitters they feel can be arbitrarily energetic. String theory changes this. Strings are not points—they have spatial extent. This implies that there is a limit to how small a distance can be accessed, even in principle, since a string can’t probe a distance smaller than its own size. In turn, a limit to how small a scale can be probed translates into a limit on how energetic the jitters can become. This limit proves sufficient to tame the unruly mathematics, allowing string theory to merge quantum mechanics and general relativity.

  9. If an object were truly one-dimensional, we wouldn’t be able to see it directly since it would offer no surface from which photons could reflect and would have no capacity to produce photons of its own through atomic
transitions. So, when I say “see” in the text, that’s a stand-in for any means of observation or experimentation you might use to seek evidence of an object’s spatial extent. The point, then, is that any spatial extent smaller than the resolving power of your experimental procedure will escape your experiment’s notice.

  10. “What Einstein Never Knew,” NOVA documentary, 1985.

  11. More precisely, the component of the universe most relevant to our existence would be completely different. Since the familiar particles and the objects they compose—stars, planets, people, etc.—amount to less than 5 percent of the mass of the universe, such a disruption would not affect the vast majority of the universe, at least as measured by mass. However, as measured by its effect on life as we know it, the change would be profound.