The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

  9. You could also travel to just outside the edge of a black hole, and remain there, engines firing away to avoid being pulled in. The black hole's strong gravitational field manifests itself as a severe warping of spacetime, and that results in your clock's ticking far slower than it would in a more ordinary location in the galaxy (as in a relatively empty spatial expanse). Again, the time duration measured by your clock is perfectly valid. But, as in zipping around at high speed, it is a completely individualistic perspective. When analyzing features of the universe as a whole, it is more useful to have a widely applicable and agreed upon notion of elapsed time, and that's what is provided by clocks that move along with the cosmic flow of spatial expansion and that are subject to a far more mild, far more average gravitational field.

  10. The mathematically inclined reader will note that light travels along null geodesics of the spacetime metric, which, for definiteness, we can take to be ds 2 = dt 2 — a 2 (t)(dx 2 ), where dx 2 = dx 1 2 + dx 2 2 + dx 3 2 , and the x i are comoving coordinates. Setting ds 2 = 0, as appropriate for a null geodesic, we can write t t0 (dt/a(t)) for the total comoving distance light emitted at time t can travel by time t 0 . If we multiply this by the value of scale factor a(t 0 ) at time t 0 , then we will have calculated the physical distance that the light has traveled in this time interval. This algorithm can be widely used to calculate how far light can travel in any given time interval, revealing whether two points in space, for example, are in causal contact. As you can see, for accelerated expansion, even for arbitrarily large t 0 , the integral is bounded, showing that the light will never reach arbitrarily distant comoving locations. Thus, in a universe with accelerated expansion, there are locations with which we can never communicate, and conversely, regions that can never communicate with us. Such regions are said to be beyond our cosmic horizon.

  11. When analyzing geometrical shapes, mathematicians and physicists use a quantitative approach to curvature developed in the nineteenth century, which today is part of a mathematical body of knowledge known as differential geometry. One nontechnical way of thinking about this measure of curvature is to study triangles drawn on or within the shape of interest. If the triangle's angles add up to 180 degrees, as they do when it is drawn on a flat tabletop, we say the shape is flat. But if the angles add up to more or less than 180 degrees, as they do when the triangle is drawn on the surface of a sphere (the outward bloating of a sphere causes the sum of the angles to exceed 180 degrees) or the surface of a saddle (the inward shrinking of a saddle's shape causes the sum of the angles to be less than 180 degrees), we say the shape is curved. This is illustrated in Figure 8.6.

  12. If you were to glue the opposite vertical edges of a torus together (which is reasonable to do, since they are identified—when you pass through one edge you immediately reappear on the other) you'd get a cylinder. And then, if you did the same for the upper and lower edges (which would now be in the shape of circles), you'd get a doughnut. Thus, a doughnut is another way of thinking about or representing a torus. One complication of this representation is that the doughnut no longer looks flat! However, it actually is. Using the notion of curvature given in the previous endnote, you'd find that all triangles drawn on the surface of the doughnut have angles that add up to 180 degrees. The fact that the doughnut looks curved is an artifact of how we've embedded a two-dimensional shape in our three-dimensional world. For this reason, in the current context it is more useful to use the manifestly uncurved representations of the two- and three-dimensional tori, as discussed in the text.

  13. Notice that we've been loose in distinguishing the concepts of shape and curvature. There are three types of curvatures for completely symmetric space: positive, zero, and negative. But two shapes can have the same curvature and yet not be identical, with the simplest example being the flat video screen and the flat infinite tabletop. Thus, symmetry allows us to narrow down the curvature of space to three possibilities, but there are somewhat more than three shapes for space (differing in what mathematicians call their global properties) that realize these three curvatures.

  14. So far, we've focused exclusively on the curvature of three-dimensional space— the curvature of the spatial slices in the spacetime loaf. However, although it's hard to picture, in all three cases of spatial curvature (positive, zero, negative), the whole four-dimensional spacetime is curved, with the degree of curvature becoming ever larger as we examine the universe ever closer to the big bang. In fact, near the moment of the big bang, the four-dimensional curvature of spacetime grows so large that Einstein's equations break down. We will discuss this further in later chapters.

  Chapter 9

  1. If you raised the temperature much higher, you'd find a fourth state of matter known as a plasma, in which atoms disintegrate into their component particles.

  2. There are curious substances, such as Rochelle salts, which become less ordered at high temperatures, and more ordered at low temperatures—the reverse of what we normally expect.

  3. One difference between force and matter fields is expressed by Wolfgang Pauli's exclusion principle. This principle shows that whereas a huge number of force particles (like photons) can combine to produce fields accessible to a prequantum physicist such as Maxwell, fields that you see every time you enter a dark room and turn on a light, matter particles are generally excluded by the laws of quantum physics from cooperating in such a coherent, organized manner. (More precisely, two particles of the same species, such as two electrons, are excluded from occupying the same state, whereas there is no such restriction for photons. Thus, matter fields do not generally have a macroscopic, classical-like manifestation.)

  4. In the framework of quantum field theory, every known particle is viewed as an excitation of an underlying field associated with the species of which that particle is a member. Photons are excitations of the photon field—that is, the electromagnetic field; an up-quark is an excitation of the up-quark field; an electron is an excitation of the electron field, and so on. In this way, all matter and all forces are described in a uniform quantum mechanical language. A key problem is that it has proved very difficult to describe all the quantum features of gravity in this language, an issue we will discuss in Chapter 12.

  5. Although the Higgs field is named after Peter Higgs, a number of other physicists—Thomas Kibble, Philip Anderson, R. Brout, and François Englert, among others— played a vital part in its introduction into physics and its theoretical development.

  6. Bear in mind that the field's value is given by its distance from the bowl's center, so even though the field has zero energy when its value is in the bowl's valley (since the height above the valley denotes the field's energy), its value is not zero.

  7. In the text's description, the value of the Higgs field is given by its distance from the bowl's center, and so you may be wondering how points on the bowl's circular valley— which are all the same distance from the bowl's center—give rise to any but the same Higgs value. The answer, for the mathematically inclined reader, is that different points in the valley represent Higgs field values with the same magnitude but different phases (the Higgs field value is a complex number).

  8. In principle, there are two concepts of mass that enter into physics. One is the concept described in the text: mass as that property of an object which resists acceleration. Sometimes, this notion of mass is called inertial mass. The second concept of mass is the one relevant for gravity: mass as that property of an object which determines how strongly it will be pulled by a gravitational field of a specified strength (such as the earth's). Sometimes this notion of mass is called gravitational mass. At first glance, the Higgs field is relevant only for an understanding of inertial mass. However, the equivalence principle of general relativity asserts that the force felt from accelerated motion and from a gravitational field are indistinguishable—they are equivalent. And that implies an equivalence between the concepts of inertial mass and gravitational mass. Thus, the Higgs field is relevan
t for both kinds of mass we've mentioned since, according to Einstein, they are the same.

  9. I thank Raphael Kasper for pointing out that this description is a variation on the prize-winning metaphor of Professor David Miller, submitted in response to British Science Minister William Waldegrave's challenge in 1993 to the British physics community to explain why taxpayer money should be spent on searching for the Higgs particle.

  10. The mathematically inclined reader should note that the photons and W and Z bosons are described in the electroweak theory as lying in the adjoint representation of the group SU(2) × U(1), and hence are interchanged by the action of this group. Moreover, the equations of the electroweak theory possess complete symmetry under this group action and it is in this sense that we describe the force particles as being interrelated. More precisely, in the electroweak theory, the photon is a particular mixture of the gauge boson of the manifest U(1) symmetry and the U(1) subgroup of SU(2); it is thus tightly related to the weak gauge bosons. However, because of the symmetry group's product structure, the four bosons (there are actually two W bosons with opposite electric charges) do not fully mix under its action. In a sense, then, the weak and electromagnetic interactions are part of a single mathematical framework, but one that is not as fully unified as it might be. When one includes the strong interactions, the group is augmented by including an SU(3) factor—"color" SU(3)—and this group's having three independent factors, SU(3) × SU(2) × U(1), only highlights further the lack of complete unity. This is part of the motivation for grand unification, discussed in the next section: grand unification seeks a single, semi-simple (Lie) group—a group with a single factor—that describes the forces at higher energy scales.

  11. The mathematically inclined reader should note that Georgi and Glashow's grand unified theory was based on the group SU(5), which includes SU(3), the group associated with the strong nuclear force, and also SU(2) × U(1), the group associated with the electroweak force. Since then, physicists have studied the implications of other potential grand unified groups, such as SO(10) and E 6 .

  Chapter 10

  1. As we've seen, the big bang's bang is not an explosion that took place at one location in a preexisting spatial expanse, and that's why we've not also asked where it banged. The playful description of the big bang's deficiency we've used is due to Alan Guth; see, for example, his The Inflationary Universe (Reading, Eng.: Perseus Books, 1997), p. xiii.

  2. The term "big bang" is sometimes used to denote the event that happened at time-zero itself, bringing the universe into existence. But since, as we'll discuss in the next chapter, the equations of general relativity break down at time-zero, no one has any understanding of what this event actually was. This omission is what we've meant by saying that the big bang theory leaves out the bang. In this chapter, we are restricting ourselves to realms in which the equations do not break down. Inflationary cosmology makes use of such well-behaved equations to reveal a brief explosive swelling of space that we naturally take to be the bang left out by the big bang theory. Certainly, though, this approach leaves unanswered the question of what happened at the initial moment of the universe's creation—if there actually was such a moment.

  3. Abraham Pais, Subtle Is the Lord (Oxford: Oxford University Press, 1982), p. 253.

  4. For the mathematically inclined reader: Einstein replaced the original equation G v = 8 T v by G v + g v = 8 T v where is a number denoting the size of the cosmological constant.

  5. When I refer to an object's mass in this context, I am referring to the sum total mass of its particulate constituents. If a cube, say, were composed of 1,000 gold atoms, I'd be referring to 1,000 times the mass of a single such atom. This definition jibes with Newton's perspective. Newton's laws say that such a cube would have a mass that is 1,000 times that of a single gold atom, and that it would weigh 1,000 times as much as a single gold atom. According to Einstein, though, the weight of the cube also depends on the kinetic energy of the atoms (as well as all other contributions to the energy of the cube). This follows from E=mc 2 : more energy (E), regardless of the source, translates into more mass (m). Thus, an equivalent way of expressing the point is that because Newton didn't know about E=mc 2 , his law of gravity uses a definition of mass that misses various contributions to energy, such as energy associated with motion.

  6. The discussion here is suggestive of the underlying physics but does not capture it fully. The pressure exerted by the compressed spring does indeed influence how strongly the box is pulled earthward. But this is because the compressed spring affects the total energy of the box and, as discussed in the previous paragraph, according to general relativity, the total energy is what's relevant. However, the point I'm explaining here is that pressure itself—not just through the contribution it makes to total energy—generates gravity, much as mass and energy do. According to general relativity, pressure gravitates. Also note that the repulsive gravity we are referring to is the internal gravitational field experienced within a region of space suffused by something that has negative rather than positive pressure. In such a situation, negative pressure will contribute a repulsive gravitational field acting within the region.

  7. Mathematically, the cosmological constant is represented by a number, usually denoted by (see note 4). Einstein found that his equations made perfect sense regardless of whether was chosen to be a positive or a negative number. The discussion in the text focuses on the case of particular interest to modern cosmology (and modern observations, as will be discussed) in which is positive, since this gives rise to negative pressure and repulsive gravity. A negative value for yields ordinary attractive gravity. Note, too, that since the pressure exerted by the cosmological constant is uniform, this pressure does not directly exert any force: only pressure differences, like what your ears feel when you're underwater, result in a pressure force. Instead, the force exerted by the cosmological constant is purely a gravitational force.

  8. Familiar magnets always have both a north and a south pole. By contrast, grand unified theories suggest that there may be particles that are like a purely north or purely south magnetic pole. Such particles are called monopoles and they could have a major impact on standard big bang cosmology. They have never been observed.

  9. Guth and Tye recognized that a supercooled Higgs field would act like a cosmological constant, a realization that had been made earlier by Martinus Veltman and others. In fact, Tye has told me that were it not for a page limit in Physical Review Letters, the journal to which he and Guth submitted their paper, they would not have struck a final sentence noting that their model would entail a period of exponential expansion. But Tye also notes that it was Guth's achievement to realize the important cosmological implications of a period of exponential expansion (to be discussed later in this and in the next chapter), and thereby put inflation front and center on cosmologists' maps.

  In the sometimes convoluted history of discovery, the Russian physicist Alexei Starobinsky had, a few years earlier, found a different means of generating what we now call inflationary expansion, work described in a paper that was not widely known among western scientists. However, Starobinsky did not emphasize that a period of such rapid expansion would solve key cosmological problems (such as the horizon and flatness problems, to be discussed shortly), which explains, in part, why his work did not generate the enthusiastic response that Guth's received. In 1981, the Japanese physicist Katsuhiko Sato also developed a version of inflationary cosmology, and even earlier (in 1978), Russian physicists Gennady Chibisov and Andrei Linde hit upon the idea of inflation, but they realized that—when studied in detail—it suffered from a key problem (discussed in note 11) and hence did not publish their work.

  The mathematically inclined reader should note that it is not difficult to see how accelerated expansion arises. One of Einstein's equations is d 2 a/dt 2 /a = -4/3( + 3p) where a, , and p are the scale factor of the universe (its "size"), the energy density, and the pressure density, respectively.
Notice that if the righthand side of this equation is positive, the scale factor will grow at an increasing rate: the universe's rate of growth will accelerate with time. For a Higgs field perched on a plateau, its pressure density turns out to equal the negative of its energy density (the same is true for a cosmological constant), and so the righthand side is indeed positive.

  10. The physics underlying these quantum jumps is the uncertainty principle, covered in Chapter 4. I will explicitly discuss the application of quantum uncertainty to fields in both Chapter 11 and Chapter 12, but to presage that material, briefly note the following. The value of a field at a given point in space, and the rate of change of the field's value at that point, play the same role for fields as position and velocity (momentum) play for a particle. Thus, just as we can't ever know both a definite position and a definite velocity for a particle, a field can't have a definite value and a definite rate of change of that value, at any given point in space. The more definite the field's value is at one moment, the more uncertain is the rate of change of that value—that is, the more likely it is that the field's value will change a moment later. And such change, induced by quantum uncertainty, is what I mean when referring to quantum jumps in the field's value.

  11. The contribution of Linde and of Albrecht and Steinhardt was absolutely crucial, because Guth's original model—now called old inflation— suffered from a pernicious flaw. Remember that the supercooled Higgs field (or, in the terminology we introduce shortly, the inflaton field) has a value that is perched on the bump in its energy bowl uniformly across space. And so, while I've described how quickly the supercooled inflaton field could take the jump to the lowest energy value, we need to ask whether this quantum-induced jump would happen everywhere in space at the same time. And the answer is that it wouldn't. Instead, as Guth argued, the relaxation of the inflaton field to a zero energy value takes place by a process called bubble nucleation: the inflaton drops to its zero energy value at one point in space, and this sparks an outward-spreading bubble, one whose walls move at light speed, in which the inflaton drops to the zero energy value with the passing of the bubble wall. Guth envisioned that many such bubbles, with random centers, would ultimately coalesce to give a universe with zero-energy inflaton field everywhere. The problem, though, as Guth himself realized, was that the space surrounding the bubbles was still infused with a non-zero-energy inflaton field, and so such regions would continue to undergo rapid inflationary expansion, driving the bubbles apart. Hence, there was no guarantee that the growing bubbles would find one another and coalesce into a large, homogeneous spatial expanse. Moreover, Guth argued that the inflaton field energy was not lost as it relaxed to zero energy, but was converted to ordinary particles of matter and radiation inhabiting the universe. To achieve a model compatible with observations, though, this conversion would have to yield a uniform distribution of matter and energy throughout space. In the mechanism Guth proposed, this conversion would happen through the collision of bubble walls, but calculations—carried out by Guth and Erick Weinberg of Columbia University, and also by Stephen Hawking, Ian Moss, and John Steward of Cambridge University—revealed that the resulting distribution of matter and energy was not uniform. Thus, Guth's original inflationary model ran into significant problems of detail.