46 Another subtle point needs to be explained. The expansion rate of the universe is measured by the Hubble constant, which relates distance to redshift. It’s not really a “constant”; in the early universe the expansion was much faster, and what we might call the Hubble “parameter” was a lot larger than our current Hubble constant. We might expect that the phrase the universe is accelerating means “the Hubble parameter is increasing,” but that’s not true—it just means “it’s not decreasing very fast.” The “acceleration” refers to an increase in the apparent velocity of any particular galaxy over time. But that velocity is equal to the Hubble parameter times the distance, and the distance is increasing as the universe expands. So an accelerating universe is not necessarily one in which the Hubble parameter is increasing, just one in which the product of the Hubble parameter with the distance to any particular galaxy is increasing. It turns out that, even with a cosmological constant, the Hubble parameter never actually increases; it decreases more slowly as the universe expands and dilutes, until it approaches a fixed constant value after all the matter has gone away and there’s nothing left but cosmological constant.
47 We’re being careful to distinguish between two forms of energy that are important for the evolution of the contemporary universe: “matter,” made of slowly moving particles that dilute away as the universe expands, and “dark energy,” some mysterious stuff that doesn’t dilute away at all, but maintains a constant energy density. But matter itself comes in different forms: “ordinary matter,” including all of the kinds of particles we have ever discovered in experiments here on Earth, and “dark matter,” some other kind of particle that can’t be anything we’ve yet directly seen. The mass (and therefore energy) in ordinary matter is mostly in the form of atomic nuclei—protons and neutrons—but electrons also contribute. So ordinary matter includes you, me, the Earth, the Sun, stars, and all the gas and dust and rocks in space. We know how much of that stuff there is, and it’s not nearly enough to account for the gravitational fields observed in galaxies and clusters. So there must be dark matter, and we’ve ruled out all known particles as candidates; theorists have invented an impressive menu of possibilities, including “axions” and “neutralinos” and “Kaluza-Klein particles.” All told, ordinary matter makes up about 4 percent of the energy in the universe, dark matter makes up about 22 percent, and dark energy makes up about 74 percent. Trying to create or detect dark matter directly is a major goal of modern experimental physics. See Hooper (2007), Carroll (2007), or Gates (2009) for more details.
48 So how much energy is there in the dark energy, anyway? It’s about one calorie per cubic centimeter. Note that the “calories” used to measure the energy content of food are actually kilocalories (1,000 standard calories). So if we took all of the cubic centimeters within the volume of Lake Michigan, their total dark energy content is roughly equal to the nutritional energy content of one Big Mac. Seen another way, if we converted all of the dark energy in all the cubic centimeters within the volume of the Earth into electricity, it would be roughly equal to the electricity usage of an average American over one year. The point is, there’s not all that much dark energy per cubic centimeter—it’s spread very thinly throughout the universe. Of course, we cannot convert dark energy into useful energy of this form—dark energy is completely useless. (Why? Because it’s in a high-entropy state.)
49 Planck wasn’t really doing quantum gravity. In 1899, in attempting to understand some mysteries of blackbody radiation, he had hit upon the need for a new fundamental constant of nature, now known as “Planck ’s constant,” ħ. Taking that new quantity and multiplying and dividing in appropriate ways by the speed of light c and Newton’s constant of gravitation G, Planck invented a system of fundamental units that we now think of as characteristic of quantum gravity: the Planck length LP = 1.6 × 10-35 meters, the Planck time tP = 5.4 × 10-44 seconds, and the Planck mass MP = 2.2 × 10-8 kilograms, along with the Planck energy. Interestingly, Planck’s first thought was that the universal nature of these quantities—based in physics, rather than determined by human convention—could someday help us communicate with extraterrestrial civilizations.
50 Fred Adams and Greg Laughlin devoted an entire book to the subject, well worth reading (Adams and Laughlin, 1999).
51 Huw Price has diagnosed this tendency very convincingly (Price, 1996). He accuses cosmologists of an implicit double standard, applying criteria of naturalness to the early universe that they would never apply to the late universe, and vice versa. Price suggests that a consistent cosmology governed by time-symmetric laws should have time-symmetric evolution. Given that the Big Bang has a low entropy, this implies that the future should feature eventual re-collapse to a Big Crunch that also has low entropy—the Gold universe, first contemplated by Thomas Gold (of Steady State fame). In such a universe, the arrow of time would reverse when the universe reached its maximum size, and entropy would begin to decrease toward the Crunch. This kind of scenario seems less likely now that we have discovered dark energy. (The way we will meet Price’s challenge in this book is to imagine that the universe is indeed time-symmetric on large scales, with high entropy toward both the far past and the far future, which can obviously be achieved only if the Big Bang is not really the beginning.)
52 The universe is not actually going to collapse into one big black hole. As discussed, it’s going to empty out. Remarkably, however, in the presence of dark energy even empty space has entropy, and we obtain the same number (10120) for the maximum entropy of the observable universe. Note that 10120 was also the discrepancy between the theoretical estimate of the vacuum energy and its observed value. This apparent coincidence of two different numbers is actually the same coincidence as that between the current density of matter (which is related to the maximum entropy) and the energy density in a vacuum. In both cases, the numbers work out to be given by taking the size of the observable universe—roughly 10 billion light years—dividing by the Planck length, and squaring the result.
4. TIME IS PERSONAL
53 On the other hand, the achievements for which Paris Hilton is famous are also pretty mysterious.
54 Einstein’s “miraculous year” was 1905, when he published a handful of papers that individually would have capped the career of almost any other scientist: the definitive formulation of special relativity, the explanation of the photoelectric effect (implying the existence of photons and laying the groundwork for quantum mechanics), proposing a theory of Brownian motion in terms of random collisions at the atomic level, and uncovering the equivalence between mass and energy. For most of the next decade he concentrated on the theory of gravity; his ultimate answer, the general theory of relativity, was completed in 1915, when Einstein was thirty-six years old. He died in 1955 at the age of seventy-six.
55 We should also mention Dutch physicist Hendrik Antoon Lorentz, who beginning in 1892 developed the idea that times and distances were affected when objects moved near the speed of light, and derived the “Lorentz transformations,” relating measurements obtained by observers moving with respect to each other. To Lorentz, velocities were measured with respect to a background of aether; Einstein was the one who first realized that the aether was an unnecessary fiction.
56 Galison (2003). One gets the impression from Galison’s book that he finds the case of Poincaré to actually be more interesting than that of Einstein. However, when an author has a chance to put Einstein in a book title, his name will generally go first. Einstein is box office.
57 George Johnson (2008), in reviewing Leonard Susskind’s book The Black Hole Wars (2008), laments the fate of the modern reader of popular physics books.
I was eager to learn how, in the end, Susskind and company showed that Hawking was probably wrong—that information is indeed conserved. But first I had to get through a sixy-six-page crash course on relativity and quantum mechanics. Every book about contemporary physics seems to begin this way, which can be frustrating to anyone who
reads more than one. (Imagine if every account of the 2008 presidential campaign had to begin with the roots of Athenian democracy and the heritage of the French Enlightenment.)
The solution is obvious: The basics of relativity and quantum mechanics should be a regular part of secondary education, just like the roots of Athenian democracy and the heritage of the French Enlightenment. In the meantime, this chapter will be part of the inevitable crash course, but by concentrating in particular on the role of “time” we’ll hopefully be able to avoid the most shopworn ways of explaining things.
58 Science fiction movies and television shows tend to flagrantly disregard this feature of reality, mostly for the practical reason that it’s very hard to fake weightlessness. (Star Trek: Enterprise did feature one amusing scene in which the ship “lost its gravity” while Captain Archer was taking a shower.) The artificial gravity you need to make the captain and crew stride purposefully about the ship’s bridge doesn’t seem compatible with the laws of physics as we know them. If you’re not accelerating, the only way to make that much gravity is to carry around a small planet’s worth of mass, which doesn’t seem practical.
59 Velocity is just the rate of change of position, and acceleration is the rate of change of velocity. In terms of calculus, velocity is the first derivative of the position, and acceleration is the second derivative. It is a deep feature of classical mechanics that the information one can specify about the state of a particle is its position and velocity; the acceleration is then determined by the local conditions and the appropriate laws of physics.
60 Left as exercises for the reader: Can we imagine a world in which absolute orientation in space were observable? What about a world in which position, velocity, and acceleration were all unobservable, but the rate of change of acceleration were observable?
61 Don’t get lost in the hypotheticals here. Today we strongly believe that there is not any medium pervading space, with respect to which we could measure our velocity. But they did believe that in the late nineteenth century; that’s the aether we’ll be talking about. On the other hand, we do believe that there are fields defined at every point in space, and some of those fields (such as the hypothetical Higgs field) might even have nonzero values in empty space. We now believe that waves, electromagnetic and otherwise, are propagating oscillations in these fields. But a field doesn’t really count as a “medium,” both because it can have a zero value, and because we can’t measure our velocity with respect to it.
On the third hand, it’s possible that we don’t know everything, and some imaginative theoretical physicists have been wondering whether there actually might be fields that do define a rest frame, and with respect to which we could imagine measuring our velocities (see, for example, Mattingly, 2005). Such fields have been whimsically dubbed “aether,” but they are not really the kind of aether that was being proposed in the nineteenth century. In particular, they have nothing to do with the propagation of electromagnetic waves, and are perfectly consistent with the underlying principles of relativity.
62 For some of the historical background, see Miller (1981). Many of the original papers concerning relativity are reprinted in Einstein (1923).
63 To actually experience length contraction or time dilation, we need either to have incredibly exquisite measuring devices, or to be moving at velocities close to the speed of light. Neither such devices nor such velocities are part of our everyday lives, which is why special relativity seems so counterintuitive to us. Of course, the fact that most objects around us have relative velocities that are small compared to the speed of light is an interesting fact about the world, which a complete theory of the universe should try to explain.
64 You might be suspicious that this argument doesn’t really demonstrate the impossibility of moving faster than light, only the impossibility of taking something moving slower than light and accelerating it to move faster than light. We might imagine that there exist objects that are always moving faster than light, so they don’t have to be accelerated. And that certainly is a logical possibility; such hypothetical particles are known as “tachyons.” But as far as we know, tachyons do not exist in the real world, and it’s a good thing, too; the ability to send signals faster than light would entail the ability to send signals backward in time, and that would wreak havoc with our notions of causality.
65 You will sometimes hear that special relativity is unable to deal with accelerating bodies, and you need general relativity to take acceleration into account. That is complete rubbish. General relativity is required when (and only when) gravity becomes important and spacetime is curved. Far away from any gravitational fields, where spacetime is flat, special relativity applies, no matter what is going on—including accelerating bodies. It’s true that freely falling (unaccelerated) trajectories have a special status in special relativity, as they are all created equal. But it is entirely incorrect to leap from there to the idea that accelerated trajectories cannot even be described within the language of special relativity.
66 Apologies for the sloppy lapse into temporal chauvinism (by presuming that one moves forward in time), not to mention giving in to the metaphor of “moving” through time. Rather than saying “Every object moves through spacetime,” it would be less prejudicial to say “The history of every object describes a world line that extends through spacetime.” But sometimes it’s just too tedious to be so pedantically precise all the time.
67 One way of relating relativity to Newtonian spacetime is to imagine “letting the speed of light get infinitely large.” Then the light cones we draw would become wider and wider, and the spacelike region would be squeezed down to a single surface, just as in the Newtonian setup. This is a suggestive picture but not terribly respectable. For one thing, we can always choose units in which the speed of light is unity; just measure time in years, and distance in light-years. So what we would actually try to do is change all of the constants of nature so that other velocities diminished with respect to the speed of light. Even if we did that, the process is highly non-unique; we have make an arbitrary choice about how to take the limit so that the light cones converge to some particular surfaces of constant time.
68 That is, at least three dimensions of space. It is quite possible, and taken for granted in certain corners of the theoretical-physics community, that there exist additional dimensions of space that for some reason are invisible to us, at least at the low energies to which we have ready access. There are a number of ways in which extra spatial dimensions could be hidden; see Greene (2000), or Randall (2005). Extra hidden timelike dimensions are considered much less likely, but you never know.
69 Both are reprinted in Einstein (1923).
5. TIME IS FLEXIBLE
70 Special relativity grew out of the incompatibility of Newtonian mechanics and Maxwellian electrodynamics, while general relativity grew out of the incompatibility of special relativity and Newtonian gravity. Right now, physics faces another troublesome incompatibility: general relativity and quantum mechanics. We are all hopeful that someday they will be united into a theory of quantum gravity. String theory is the leading candidate at present, but matters are not yet settled.
71 It might seem crazy that tension, which pulls things together, is responsible for the acceleration of the universe, which pushes things apart. The point is that the tension from dark energy is equal at every point throughout space, and precisely cancels, so there is no direct pulling. Instead, we are left with the indirect effect of the dark energy on the curvature of spacetime. That effect is to impart a perpetual push to the universe, because the dark energy density does not dilute away.
72 Here is another way of thinking about it. The fact that energy is conserved in Newtonian mechanics is a reflection of an underlying symmetry of the theory: time-translation invariance. The background spacetime in which particles move is fixed once and for all. But in general relativity that’s no longer true; the background is dynamical, pushing things around, changin
g their energies.
73 See Michell (1784); Laplace’s essay is reprinted as an appendix in Hawking and Ellis (1974). It is occasionally pointed out, with great raising of eyebrows and meaningful murmurs, that the radius of a “black star” as calculated according to Newtonian gravity is precisely the same size as the predicted Schwarzschild radius of a black hole in general relativity (2GM/c2, where G is Newton’s constant of gravitation, M is the mass of the object, and c is the speed of light). This coincidence is completely accidental, due primarily to the fact that there aren’t many ways you can create a quantity with units of length out of G, M, and c.
74 For purposes of this chapter, we are assuming the validity of classical general relativity, even though we know that it must be replaced by a better theory when it comes to singularities. For more on these issues, see Hawking (1988) or Thorne (1994).
75 Feel free to construct your own moral lessons.
6. LOOPING THROUGH TIME
76 Referring, of course, to the time machines in George Pal’s 1960 movie version of H. G. Wells’s The Time Machine; Robert Zemeckis’s 1985 film Back to the Future; and the long-running BBC serial Doctor Who, respectively.
77 In the interest in getting on with our story, we’re not being completely fair to the subject of tachyons. Allowing objects that travel faster than light opens the door to paradoxes, but that doesn’t necessarily force us to walk through the door. We might imagine models that allowed for tachyons, but only in self-consistent ways. For some discussion see Feinberg (1967) or Nahin (1999). To make things more confusing, in quantum field theory the word “tachyon” often simply refers to a momentarily unstable configuration of a field, where nothing is actually traveling faster than light.
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