Regardless of how the details play out, we are left with the same long-term picture. Other galaxies move away from us and disappear; our own galaxy will evolve through various stages, but the end result is a thin gruel of particles dissipating into the void. In the very far future, the universe becomes once again a very simple place: It will be completely empty, as empty as space can be. That’s the diametric opposite of the hot, dense state in which the universe began; a vivid cosmological manifestation of the arrow of time.
THE ENTROPY OF THE UNIVERSE
An impressive number of brain-hours on the part of theoretical physicists have been devoted to the question of why the universe evolved in this particular fashion, rather than in some other way. It’s certainly possible that this question simply has no answer; perhaps the universe is what it is, and the best we can do is to accept it. But we are hopeful, not without reason, that we can do more than accept it—we can explain it.
Given perfect knowledge of the laws of physics, the question “Why has the universe evolved in the fashion it has?” is equivalent to “Why were the initial conditions of the universe arranged in the way they were?” But that latter formulation is already sneaking in an implicit notion of time asymmetry, by privileging past conditions over future conditions. If our understanding of the fundamental, microscopic laws of nature is correct, we can specify the state of the universe at any time, and from there derive both the past and the future. It would be better to characterize our task as that of understanding what would count as a natural history of the universe as a whole.51
There is some irony in the fact that cosmologists have underappreciated the importance of the arrow of time, since it is arguably the single most blatant fact about the evolution of the universe. Boltzmann was able to argue (correctly) for the need for a low-entropy boundary condition in the past, without knowing anything about general relativity, quantum mechanics, or even the existence of other galaxies. Taking the problem of entropy seriously helps us look at cosmology in a new light, which might suggest some resolutions to long-standing puzzles.
But first, we need to be a little more clear about what exactly we mean about “the entropy of the universe.” In Chapter Thirteen we will discuss the evolution of entropy in our observable universe in great detail, but the basic story goes as follows:
1. In the early universe, before structure forms, gravity has little effect on the entropy. The universe is similar to a box full of gas, and we can use the conventional formulas of thermodynamics to calculate its entropy. The total entropy within the space corresponding to our observable universe turns out to be about 1088 at early times.
2. By the time we reach our current stage of evolution, gravity has become very important. In this regime we don’t have an ironclad formula, but we can make a good estimate of the total entropy just by adding up the contributions from black holes (which carry an enormous amount of entropy). A single supermassive black hole has an entropy of order 1090, and there are approximately 1011 such black holes in the observable universe; our total entropy today is therefore something like 10101.
3. But there is a long way to go. If we took all of the matter in the observable universe and collected it into a single black hole, it would have an entropy of 10120. That can be thought of as the maximum possible entropy obtainable by rearranging the matter in the universe, and that’s the direction in which we’re evolving.52
Our challenge is to explain this history. In particular, why was the early entropy, 1088, so much lower than the maximum possible entropy, 10120? Note that the former number is much, much, much smaller than the latter; appearances to the contrary are due to the miracle of compact notation.
The good news is, at least the Big Bang model provides a context in which we can sensibly address this question. In Boltzmann’s time, before we knew about general relativity or the expansion of the universe, the puzzle of entropy was even harder, simply because there was no such event as “the beginning of the universe” (or even “the beginning of the observable universe”). In contrast, we are able to pinpoint exactly when the entropy was small, and the particular form that low-entropy state took; that’s a crucial step in trying to explain why it was like that.
It’s possible, of course, that the fundamental laws of physics simply aren’t reversible (although we’ll give arguments against that later on). But if they are, the low entropy of our universe near the Big Bang leaves us with two basic possibilities:
1. The Big Bang was truly the beginning of the universe, the moment when time began. That may be because the true laws of physics allow spacetime to have a boundary, or because what we call “time” is just an approximation, and that approximation ceases to be valid near the Big Bang. In either case, the universe began in a low-entropy state, for reasons over and above the dynamical laws of nature—we need a new, independent principle to explain the initial state.
2. There is no such thing as an initial state, because time is eternal. In this case, we are imagining that the Big Bang isn’t the beginning of the entire universe, although it’s obviously an important event in the history of our local region. Somehow our observable patch of spacetime must fit into a bigger picture. And the way it fits must explain why the entropy was small at one end of time, without imposing any special conditions on the larger framework.
As to which of these is the correct description of the real world, the only answer is that we don’t know. I will confess to a personal preference for Option 2, as I think it would be more elegant if the world were described as a nearly inevitable result of a set of dynamical laws, without needing an extra principle to explain why it appears precisely this way. Turning this vague scenario into an honest cosmological model will require that we actually take advantage of the mysterious vacuum energy that dominates our universe. Getting there from here requires a deeper understanding of curved spacetime and relativity, to which we now turn.
PART TWO
TIME IN EINSTEIN’S UNIVERSE
4
TIME IS PERSONAL
Time travels in divers paces with divers persons.
—William Shakespeare, As You Like It
When most people hear “scientist,” they think “Einstein.” Albert Einstein is an iconic figure; not many theoretical physicists attain a level of celebrity in which their likeness appears regularly on T-shirts. But it’s an intimidating, distant celebrity. Unlike, say, Tiger Woods, the precise achievements Einstein is actually famous for remain somewhat mysterious to many people who would easily recognize his name.53 His image as the rumpled, absentminded professor, with unruly hair and baggy sweaters, contributes to the impression of someone who embodied the life of the mind, disdainful of the mundane realities around him. And to the extent that the substance of his contributions is understood—equivalence of mass and energy, warping of space and time, a search for the ultimate theory—it seems to be the pinnacle of abstraction, far removed from everyday concerns.
The real Einstein is more interesting than the icon. For one thing, the rumpled look with the Don King hair attained in his later years bore little resemblance to the sharply dressed, well-groomed young man with the penetrating stare who was responsible for overturning physics more than once in the early decades of the twentieth century.54 For another, the origins of the theory of relativity go beyond armchair speculations about the nature of space and time; they can be traced to resolutely practical concerns of getting persons and cargo to the right place at the right time.
Figure 10: Albert Einstein in 1912. His “miraculous year” was 1905, while his work on general relativity came to fruition in 1915.
Special relativity, which explains how the speed of light can have the same value for all observers, was put together by a number of researchers over the early years of the twentieth century. (Its successor, general relativity, which interpreted gravity as an effect of the curvature of spacetime, was due almost exclusively to Einstein.) One of the major contributors to special relativity was the
French mathematician and physicist Henri Poincaré. While Einstein was the one who took the final bold leap into asserting that the “time” as measured by any moving observer was as good as the “time” measured by any other, both he and Poincaré developed very similar formalisms in their research on relativity.55
Historian Peter Galison, in his book Einstein’s Clocks, Poincaré’s Maps: Empires of Time, makes the case that Einstein and Poincaré were as influenced by their earthbound day jobs as they were by esoteric considerations of the architecture of physics.56 Einstein was working at the time as a patent clerk in Bern, Switzer land, where a major concern was the construction of accurate clocks. Railroads had begun to connect cities across Europe, and the problem of synchronizing time across great distances was of pressing commercial interest. The more senior Poincaré, meanwhile, was serving as president of France’s Bureau of Longitude. The growth of sea traffic and trade led to a demand for more accurate methods of determining longitude while at sea, both for the navigation of individual ships and for the construction of more accurate maps.
And there you have it: maps and clocks. Space and time. In particular, an appreciation that what matters is not questions of the form “Where are you really?” or “What time is it actually?” but “Where are you with respect to other things?” and “What time does your clock measure?” The rigid, absolute space and time of Newtonian mechanics accords pretty well with our intuitive understanding of the world; relativity, in contrast, requires a certain leap into abstraction. Physicists at the turn of the century were able to replace the former with the latter only by understanding that we should not impose structures on the world because they suit our intuition, but that we should take seriously what can be measured by real devices.
Special relativity and general relativity form the basic framework for our modern understanding of space and time, and in this part of the book we’re going to see what the implications of “spacetime” are for the concept of “time.”57 We’ll be putting aside, to a large extent, worries about entropy and the Second Law and the arrow of time, and taking refuge in the clean, precise world of fundamentally reversible laws of physics. But the ramifications of relativity and spacetime will turn out to be crucial to our program of providing an explanation for the arrow of time.
LOST IN SPACE
Zen Buddhism teaches the concept of “beginner’s mind”: a state in which one is free of all preconceptions, ready to apprehend the world on its own terms. One could debate how realistic the ambition of attaining such a state might be, but the concept is certainly appropriate when it comes to thinking about relativity. So let’s put aside what we think we know about how time works in the universe, and turn to some thought experiments (for which we know the answers from real experiments) to figure out what relativity has to say about time.
To that end, imagine we are isolated in a sealed spaceship, floating freely in space, far away from the influence of any stars or planets. We have all of the food and air and basic necessities we might wish, and some high school-level science equipment in the form of pulleys and scales and so forth. What we’re not able to do is to look outside at things far away. As we go, we’ll consider what we can learn from various sensors aboard or outside the ship.
But first, let’s see what we can learn just inside the spaceship. We have access to the ship’s controls; we can rotate the vessel around any axis we choose, and we can fire our engines to move in whatever direction we like. So we idle away the hours by alternating between moving the ship around in various ways, not really knowing or caring where we are going, and playing a bit with our experiments.
Figure 11: An isolated spaceship. From left to right: freely falling, accelerating, and spinning.
What do we learn? Most obviously, we can tell when we’re accelerating the ship. When we’re not accelerating, a fork from our dinner table would float freely in front of us, weightless; when we fire the rockets, it falls down, where “down” is defined as “away from the direction in which the ship is accelerating.”58 If we play a bit more, we might figure out that we can also tell when the ship is spinning around some axis. In that case, a piece of cutlery perfectly positioned on the rotational axis could remain there, freely floating; but anything at the periphery would be “pulled” to the hull of the ship and stay there.
So there are some things about the state of our ship we can determine observa tionally, just by doing simple experiments inside. But there are also things that we can’t determine. For example, we don’t know where we are. Say we do a bunch of experiments at one location in our unaccelerated, non-spinning ship. Then we fire the rockets for a bit, zip off somewhere else, kill the rockets so that we are once again unaccelerated and non-spinning, and do the same experiments again. If we have any skill at all as experimental physicists, we’re going to get the same results. Had we been very good record keepers about the amount and duration of our acceleration, we could possibly calculate the distance we had traveled; but just by doing local experiments, there doesn’t seem to be any way to distinguish one location from another.
Likewise, we can’t seem to distinguish one velocity from another. Once we turn off the rockets, we are once again freely floating, no matter what velocity we have attained; there is no need to decelerate in the opposite direction. Nor can we distinguish any particular orientation of the ship from any other orientation, here in the lonely reaches of interstellar space. We can tell whether we are spinning or not spinning; but if we fire the appropriate guidance rockets (or manipulate some onboard gyroscopes) to stop whatever spin we gave the ship, there is no local experiment we can do that would reveal the angle by which the ship had rotated.
These simple conclusions reflect deep features of how reality works. Whenever we can do something to our apparatus without changing any experimental outcomes—shift its position, rotate it, set it moving at a constant velocity—this reflects a symmetry of the laws of nature. Principles of symmetry are extraordinarily powerful in physics, as they place stringent restrictions on what form the laws of nature can take, and what kind of experimental results can be obtained.
Naturally, there are names for the symmetries we have uncovered. Changing one’s location in space is known as a “translation”; changing one’s orientation in space is known as a “rotation”; and changing one’s velocity through space is known as a “boost.” In the context of special relativity, the collection of rotations and boosts are known as “Lorentz transformations,” while the entire set including translations are known as “Poincaré transformations.”
The basic idea behind these symmetries far predates special relativity. Galileo himself was the first to argue that the laws of nature should be invariant under what we now call translations, rotations, and boosts. Even without relativity, if Galileo and Newton had turned out to be right about mechanics, we would not be able to determine our position, orientation, or velocity if we were floating freely in an isolated spaceship. The difference between relativity and the Galilean perspective resides in what actually happens when we switch to the reference frame of a moving observer. The miracle of relativity, in fact, is that changes in velocity are seen to be close relatives of changes in spatial orientation; a boost is simply the spacetime version of a rotation.
Before getting there, let’s pause to ask whether things could have been different. For example, we claimed that one’s absolute position is unobservable, and one’s absolute velocity is unobservable, but one’s absolute acceleration can be measured. 59 Can we imagine a world, a set of laws of physics, in which absolute position is unobservable, but absolute velocity can be objectively measured?60
Sure we can. Just imagine moving through a stationary medium, such as air or water. If we lived in an infinitely big pool of water, our position would be irrelevant, but it would be straightforward to measure our velocity with respect to the water. And it wouldn’t be crazy to think that there is such a medium pervading space.61 After all, ever since the work of Maxwell on el
ectromagnetism we have known that light is just a kind of wave. And if you have a wave, it’s natural to think that there must be something doing the waving. For example, sound needs air to propagate; in space, no one can hear you scream. But light can travel through empty space, so (according to this logic, which will turn out not to be right) there must be some medium through which it is traveling.
So physicists in the late nineteenth century postulated that electromagnetic waves propagated through an invisible but all-important medium, which they called the “aether.” And experimentalists set out to actually detect the stuff. But they didn’t succeed—and that failure set the stage for special relativity.
THE KEY TO RELATIVITY
Imagine we’re back out in space, but this time we’ve brought along some more sophisticated experimental apparatus. In particular, we have an impressive-looking contraption, complete with state-of-the-art laser technology, that measures the speed of light. While we are freely falling (no acceleration), to calibrate the thing we check that we get the same answer for the speed of light no matter how we orient our experiment. And indeed we do. Rotational invariance is a property of the propagation of light, just as we suspected.
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