More informally, logarithms have the nice property that they take large numbers and whittle them down to manageable sizes. When we take the logarithm of an unwieldy number like a trillion, we get a nice number like 9. The logarithm is a monotonic function—it always increases as we increase the number we’re taking the logarithm of. So the logarithm gives a specific measure of how big a number is, but it collapses huge numbers down to a reasonable size, which is very helpful in fields like cosmology, statistical mechanics, or even economics.
One final crucial detail is that, just like exponentials, logarithms can come in different bases. The “log base b” of a number x is the number to which we would have to raise b in order to get x. That is:
log2(2x) = x,
log12(12x) = x,
and so on. Whenever we don’t write the base explicitly, we take it to be equal to 10, because that’s how many fingers most human beings have. But scientists and mathematicians often like to make a seemingly odd choice: they use the natural logarithm , often written ln(x), in which the base is taken to be Euler’s constant:
ln(x) = loge(x),
e = 2.7182818284 . . .
Euler’s constant is an irrational number, like pi or the square root of two, so its explicit form above would go on forever. At first glance that seems like a truly perverse choice to use as a base for one’s logarithms. But in fact e has a lot of nice properties, once you get deeper into the math; in calculus, for example, the function ex is the only one (aside from the trivial function equal to zero everywhere) that is equal to its own derivative, as well as its own integral. In this book all of our logarithms have used base 10, but if you launch yourself into physics and math at a higher level, it will be natural logarithms all the way.
NOTES
PROLOGUE
1 Wikipedia contributors (2009).
2 Let’s emphasize the directions here, because they are easily confused: Entropy measures disorder, not order, and it increases with time, not decreases. We informally think “things wind down,” but the careful way of saying that is “entropy goes up.”
1. THE PAST IS PRESENT MEMORY
3 In an effort not to be too abstract, we will occasionally lapse into a kind of language that assumes the directionality of time—“time passes,” we “move into the future,” stuff like that. Strictly speaking, part of our job is to explain why that language seems so natural, as opposed to phrasings along the lines of “there is the present, and there is also the future,” which seems stilted. But it’s less stressful to occasionally give into the “tensed” way of speaking, and question the assumptions behind it more carefully later on.
4 Because the planets orbit in ellipses rather than perfect circles, their velocity around the Sun is not strictly constant, and the actual angle that the Earth describes in its orbit every time Mars completes a single revolution will depend on the time of year. These are details that are easy to take care of when we actually sit down to carefully define units of time.
5 The number of vibrations per second is fixed by the size and shape of the crystal. In a watch, the crystal is tuned to vibrate 32,768 times per second, which happens to be equal to 2 to the 15th power. That number is chosen so that it’s easy for the watch’s inner workings to divide successively by 2 to obtain a frequency of exactly once per second, appropriate for driving the second hand of a watch.
6 Alan Lightman’s imaginative novel Einstein’s Dreams presents a series of vignettes that explore what the world would be like if time worked very differently than it does in the real world.
7 See for example Barbour (1999) or Rovelli (2008).
8 There is a famous joke, attributed to Einstein: “When a man sits with a pretty girl for an hour, it seems like a minute. But let him sit on a hot stove for a minute and it’s longer than any hour. That’s relativity.” I don’t know whether Einstein actually ever said those words. But I do know that’s not relativity.
9 Here is a possible escape clause, if we were really committed to restoring the scientific integrity of Baker’s fantasy: Perhaps time in the rest of the world didn’t completely stop, but just slowed down by a tremendous factor, and still ticked along at a sufficient rate that light could travel from the objects Arno was looking at to his eyes. Close, but no cigar. Even if that happened, the fact that the light was slowed down would lead to an enormous redshift—what looked like visible light in the ordinary world would appear to Arno as radio waves, which his poor eyes wouldn’t be able to see. Perhaps X-rays would be redshifted down to visible wavelengths, but X-ray flashlights are hard to come by. (It does, admittedly, provoke one into thinking how interesting a more realistic version of this scenario might be.)
10 Temporal: of or pertaining to time. It’s a great word that we’ll be using frequently. Sadly, an alternative meaning is “pertaining to the present life or this world”—and we’ll be roaming very far away from that meaning.
11 As a matter of historical accuracy, while Einstein played a central role in the formulation of special relativity, it was legitimately a collaborative effort involving the work of a number of physicists and mathematicians, including George FitzGerald, Hendrik Lorentz, and Henri Poincaré. It was eventually Hermann Minkowski who took Einstein’s final theory and showed that it could be understood in terms of a four-dimensional spacetime, which is often now called “Minkowski space.” His famous 1909 quote was “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality” (Minkowski, 1909).
12 Pirsig (1974), 375.
13 Price (1996), 3.
14 Vonnegut (1969), 34. Quoted in Lebowitz (2008).
15 Augustine (1998), 235.
16 Good discussions of these issues can be found in Callender (2005), Lockwood (2005), and Davies (1995).
17 Philosophers often discuss different conceptions of time in terms laid out by J. M. E. McTaggart in his famous paper “The Unreality of Time” (1908). There, McTaggart distinguished between three different notions of time, which he labeled as different “series” (see also Lockwood, 2005). The A-series is a series of events measured relative to now, that move through time—“one year ago” doesn’t denote a fixed moment, but one that changes as time passes. The B-series is the sequence of events with permanent temporal labels, such as “October 12, 2009.” And the C-series is simply an ordered list of events—“x happens before y but after z”—without any time stamps at all. McTaggart argued—very roughly—that the B-series and C-series are fixed arrays, lacking the crucial element of change, and therefore insufficient to describe time. But the A-series itself is incoherent, as any specific event will be classified simultaneously as “past,” “present,” and “future,” from the point of view of different moments in time. (The moment of your birth is in the past to you now but was in the future to your parents when they first met.) Therefore, he concludes, time doesn’t exist.
If you get the feeling that this purported contradiction seems more like a problem with language than one with the nature of time, you are on the right track. To a physicist, there seems to be no contradiction between stepping outside the universe and thinking of all of spacetime at once, and admitting that from the point of view of any individual inside the universe time seems to flow.
2. THE HEAVY HAND OF ENTROPY
18 Amis (1991), 11.
19 Fitzgerald (1922).
20 Carroll, L. (2000), 175.
21 Obviously.
22 Diedrick (1995) lists a number of stories that feature time reversals in one form or another, in addition to the ones mentioned here: Lewis Carroll’s Sylvie and Bruno, Jean Cocteau’s Le Testament d’Orphee, Brian Aldiss’s An Age, and Philip K. Dick’s Counter-Clock World. In T. H. White’s The Once and Future King, the character of Merlyn experiences time backward, a
lthough White doesn’t try very hard to consistently maintain the conceit. More recently, the technique has been used by Dan Simmons in Hyperion, and serves as a major theme in Andrew Sean Greer’s The Confessions of Max Tivoli and in Greg Egan’s short story “The Hundred-Year Diary.” Vonnegut’s Slaughterhouse-Five includes a brief description of the firebombing of Dresden in reversed order, which Amis credits in the Afterword to Time’s Arrow.
23 Stoppard (1999), 12.
24 In addition to the First Law of Thermodynamics (“the total energy remains constant in any physical process”) and the Second Law (“the entropy of a closed system never decreases”), there is also a Third Law: As the temperature of a system is lowered, there is a minimum value (absolute zero) for which the entropy is also a minimum. These three laws have been colorfully translated as: “You can’t win; you can’t break even; and you can’t even get out of the game.” There is also a Zeroth Law: If two systems are both in thermal equilibrium with a third system, they are in thermal equilibrium with each other. Feel free to invent your own whimsical sporting analogies.
25 Eddington (1927), 74.
26 Snow (1998), 15.
27 In fact, it would be fair to credit Sadi Carnot’s father, French mathematician and military officer Lazare Carnot, with the first glimmerings of this concept of entropy and the Second Law. In 1784, Lazare Carnot wrote a treatise on mechanics in which he argued that perpetual motion was impossible, because any realistic machine would dissipate useful energy through the rattling and shaking of its component parts. He later became a successful leader of the French Revolutionary Army.
28 Not strictly true, actually. Einstein’s general theory of relativity, which explains gravitation in terms of the curvature of spacetime, implies that what we ordinarily call “energy” is not really conserved, for example, in an expanding universe. We’ll talk about that in Chapter Five. But for the purposes of most combustion engines, the expansion of the universe can be neglected, and energy really is conserved.
29 Specifically, by “measures the number of ways we can rearrange the individual parts,” we mean “is proportional to the logarithm of the number of ways we can rearrange the individual parts.” See the Appendix for a discussion of logarithms, and Chapter Nine for a detailed discussion of the statistical definition of entropy.
30 The temperature of the surface of the Sun is approximately 5,800 Kelvin. (One Kelvin is the same as one degree Celsius, except that zero Kelvin corresponds to -273 degrees C—absolute zero, the lowest possible temperature.) Room temperature is approximately 300 Kelvin. Space—or, more properly, the cosmic background radiation that suffuses space—is at about 3 Kelvin. There is a nice discussion of the role of the Sun as a hot spot in a cold sky in Penrose (1989).
31 You will sometimes hear claims by creationists to the effect that evolution according to Darwinian natural selection is incompatible with the growth of entropy, since the history of life on Earth has involved increasingly complex organisms purportedly descending from less complex forms. This is crazy on several levels. The most basic level is simply: The Second Law refers to closed systems, and an organism (or a species, or the biosphere) is not a closed system. We’ll discuss this a bit more in Chapter Nine, but that’s basically all there is to it.
32 Thompson (1862).
33 Pynchon (1984), 88.
3. THE BEGINNING AND END OF TIME
34 In fact there was a literal debate—the “Great Debate” between astronomers Harlow Shapley and Heber Curtis was held in 1920 at the Smithsonian in Washington, D.C. Shapley defended the position that the Milky Way was the entirety of the universe, while Curtis argued that the nebulae (or at least some of them, and in particular the Andromeda nebula M31) were galaxies like our own. Although Shapley ended up on the losing side of the big question, he did correctly understand that the Sun was not at the center of the Milky Way.
35 That’s a bit of poetic license. As we will explain later, the cosmological redshift is conceptually distinct from the Doppler effect, despite their close similarity. The former arises from the expansion of space through which the light is traveling, while the latter arises from the motion of the sources through space.
36 After decades of heroic effort, modern astronomers have finally been able to pin down the actual value of this all-important cosmological parameter: 72 kilometers per second per Megaparsec (Freedman et al., 2001). That is, for every million parsecs of distance between us and some galaxy, we will observe an apparent recession velocity of 72 km/sec. For comparison, the current size of the observable universe is about 28 billion parsecs across. A parsec is about 3.26 light years, or 30 trillion kilometers.
37 Strictly speaking, we should say “every sufficiently distant galaxy . . .” Nearby galaxies could be bound into pairs or groups or clusters under the influence of their mutual gravitational attraction. Such groups, like any bound systems, do not expand along with the universe; we say that they have “broken away from the Hubble flow.”
38 Admittedly, it’s a bit subtle. Just two footnotes prior, we said the observable universe was “28 billion parsecs” across. It’s been 14 billion years since the Big Bang, so you might think there are 14 billion light-years from here to the edge of the observable universe, which we can multiply by two to get the total diameter—28 billion light years, or about 9 billion parsecs, right? Was there a typo, or how can these be reconciled? The point is that distances are complicated by the fact that the universe is expanding, and in particular because it is being accelerated by dark energy. The physical distance today to the most distant galaxies within our observable universe is actually larger than 14 billion light-years. If you go through the math, the farthest point that was ever within our observable patch of universe is now 46 billion light-years, or 14 billion parsecs, distant.
39 The idea that particles aren’t created out of empty space should be clearly labeled as an assumption, although it seems to be a pretty good one—at least, within the current universe. (Later we’ll see that particles can very rarely appear from the vacuum in an accelerating universe, in a process analogous to Hawking radiation around black holes.) The old Steady State theory explicitly assumed the opposite, but had to invoke new kinds of physical processes to make it work (and it never really did).
40 To be careful about it, the phrase Big Bang is used in two different ways. One way is as we’ve just defined it—the hypothetical moment of infinite density at the beginning of the universe, or at least conditions in the universe very, very close to that moment in time. But we also speak of the “Big Bang model,” which is simply the general framework of a universe that expands from a hot, dense state according to the rules of general relativity; and sometimes we drop the model. So you might read newspaper stories about cosmologists “testing the predictions of the Big Bang.” You can’t test the predictions of some moment in time; you can only test predictions of a model. Indeed, the two concepts are fairly independent—we will be arguing later in the book that a complete theory of the universe will have to replace the conventional Big Bang singularity by something better, but the Big Bang model of the evolution of the universe over the last 14 billion years is well established and not going anywhere.
41 The microwave background has a messy history. George Gamow, Ralph Alpher, and Robert Herman wrote a series of papers in the late 1940s and early 1950s that clearly predicted the existence of relic microwave radiation from the Big Bang, but their work was subsequently largely forgotten. In the 1960s, Robert Dicke at Princeton and A. G. Doroshkevich and Igor Novikov in the Soviet Union independently recognized the existence and detectability of the radiation. Dicke went so far as to assemble a talented group of young cosmologists (including David Wilkinson and P. J. E. Peebles, who would go on to become leaders in the field) to build an antenna and search for the microwave background themselves. They were scooped by Penzias and Wilson, just a few miles away, who were completely unaware of their work. Gamow passed away in 1968, but it remains mysterious why Alpher
and Herman never won the Nobel Prize for their predictions. They told their side of the story in a book, Genesis of the Big Bang (Alpher and Herman, 2001). In 2006, John Mather and George Smoot were awarded the Prize for their measurements of the blackbody spectrum and temperature anisotropies in the microwave background, using NASA’s Cosmic Background Explorer (COBE) satellite.
42 The full story is told by Farrell (2006).
43 Bondi and Gold (1948); Hoyle (1948).
44 See for example Wright (2008).
45 Needless to say, that’s making a long story very short. Type Ia supernovae are believed to be the result of the catastrophic gravitational collapse of white dwarf stars. A white dwarf is a star that has used up all of its nuclear fuel and just sits there quietly, supported by the basic fact that electrons take up space. But some white dwarfs have companion stars, from which matter can slowly dribble onto the dwarf. Eventually the white dwarf hits a point—the Chandrasekhar Limit, named after Subrah manyan Chandrasekhar—where the outward pressure due to electrons cannot compete with the gravitational pull, and the star collapses into a neutron star, ejecting its outer layers as a supernova. Because the Chandrasekhar Limit is approximately the same for every white dwarf in the universe, the brightness of the resulting explosions is approximately the same for every Type Ia supernova. (There are other types of supernovae, which don’t involve white dwarfs at all.) But astronomers have learned how to correct for the differences in brightness by using the empirical fact that brighter supernovae take longer to decline in brightness after the peak luminosity. The story of how astronomers search for such supernovae, and how they eventually discovered the acceleration of the universe, is told in Goldsmith (2000), Kirshner (2004), and Gates (2009); the original papers are Riess et al. (1998) and Perlmutter et al. (1999).
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