From Eternity to Here

by Sean Carroll

If everything in the universe evolves toward increasing disorder, it must have started out in an exquisitely ordered arrangement.

Time comes in three different aspects, all of which are going to be important to us. 1. Time labels moments in the universe. Time is a coordinate; it helps us locate things. 2. Time measures the duration elapsed between events. Time is what clocks measure. 3. Time is a medium through which we move. Time is the agent of change. We move through it, or—equivalently—time flows past us, from the past, through the present, toward the future.

John Archibald Wheeler, an influential American physicist who coined the term black hole, was once asked how he would define “time.” After thinking for a while, he came up with this: “Time is Nature’s way of keeping everything from happening at once.”

What we call the “universe” is just the set of all events—every point in space, at every moment of time. So we need four numbers—three coordinates of space, and one of time—to uniquely pick out an event. That’s why we say that the universe is four-dimensional. This is such a useful concept that we will often treat the whole collection, every point in space at every moment of time, as a single entity called spacetime.

Time not only labels and orders different moments; it also measures the distance between them.

The passage of time can be completely recast in terms of certain things happening together, in synchrony. “The program lasts one hour” is equivalent to “there will be 117,964,800 oscillations of the quartz crystal in my watch between the beginning and end of the program” (give or take a few commercials). If you really wanted to, you could reinvent the entire superstructure of physics in a way that completely eliminated the concept of “time,” by replacing it with elaborate specifications of how certain things happen in coincidence with certain other things.

Our internal rhythms are not as reliable as a pendulum or a quartz crystal; they can be affected by external conditions or our emotional states, leading to the impression that time is passing more quickly or more slowly.

But in 1905, along comes Einstein with his special theory of relativity.11 The central conceptual breakthrough of special relativity is that our two aspects of time, “time labels different moments” and “time is what clocks measure,” are not equivalent, or even interchangeable. In particular, the scheme of setting up a time coordinate by sending clocks throughout the universe would not work: two clocks, leaving the same event and arriving at the same event but taking different paths to get there, will generally experience different durations along the journey, slipping out of synchronization. That’s not because we haven’t been careful enough to pick “good clocks,” as defined above. It’s because the duration elapsed along two trajectories connecting two events in spacetime need not be the same.

In the case of time, it’s not that we live in the spatial vicinity of an influential object; it’s that we live in the temporal vicinity of an influential event: the birth of the universe. The beginning of our observable universe, the hot dense state known as the Big Bang, had a very low entropy. The influence of that event orients us in time, just as the presence of the Earth orients us in space.

The principle underlying irreversible processes is summed up in the Second Law of Thermodynamics: The entropy of an isolated system either remains constant or increases with time.

Entropy is a measure of the number of particular microscopic arrangements of atoms that appear indistinguishable from a macroscopic perspective.29

In an isolated system entropy tends to increase, because there are more ways to be high entropy than to be low entropy.

Before Boltzmann, the Second Law was absolute—an ironclad law of nature. But the definition of entropy in terms of atoms comes with a stark implication: entropy doesn’t necessarily increase, even in a closed system; it is simply likely to increase. (Overwhelmingly likely, as we shall see, but still.)

In other words, part of the distinction we draw between “effects” and “causes” is that “effects” generally involve an increase in entropy. If two billiard balls collide and go their separate ways, the entropy remains constant, and neither ball deserves to be singled out as the cause of the interaction. But if you hit the cue ball into a stationary collection of racked balls on the break (provoking a noticeable increase in entropy), you and I would say “the cue ball caused the break ”—even though the laws of physics treat all of the balls perfectly equally.

These days we think of space not as some fixed and absolute stage through which matter moves, but as a dynamical and lively entity in its own right, according to Einstein’s general theory of relativity. When we say space is expanding, we mean that more space is coming into existence in between galaxies.

The rules of general relativity are unambiguous: Given certain kinds of stuff in the universe, there must have been a singularity in the past. But that’s not really an internally consistent conclusion. The singularity itself would be a moment when the curvature of spacetime and the density of matter were infinite, and the rules of general relativity simply would not apply. The correct deduction is not that general relativity predicts a singularity, but that general relativity predicts that the universe evolves into a configuration where general relativity itself breaks down. The theory cannot be considered to be complete; something happens where general relativity predicts singularities, but we don’t know what.

Imagine that we lived in a universe much like our current one, with the same kind of distribution of galaxies and clusters, but that was contracting rather than expanding. Would we expect that the galaxies would smooth out toward the future as the universe contracted, creating a homogeneous plasma such as we see in the past of our real (expanding) universe? Not at all. We would expect the contrast knob to continue to be turned up, even as the universe contracted—black holes and other massive objects would gather matter from the surrounding regions. Growth of structure is an irreversible process that naturally happens toward the future, whether the universe is expanding or contracting: It represents an increase in entropy. So the relative smoothness of the early universe, illustrated in the image of the cosmic microwave background, reflects the very low entropy of those early times.

It’s possible, of course, that general relativity is not the correct theory of gravity on cosmological scales, and that possibility is one that physicists take very seriously. It seems more likely, however, that general relativity is correct, and the observations are telling us that most of the energy in the universe is not in the form of “matter” at all, but rather in the form of some stubbornly persistent stuff that sticks around even as space expands. We’ve dubbed that mysterious stuff “dark energy,” and the nature of the dark energy is very much a favorite research topic for modern cosmologists, both theorists and observers. We don’t know much about dark energy, but we do know two very crucial things: It’s nearly constant throughout space (the same amount of energy from place to place), and also nearly constant in density through time (the same amount of energy per cubic centimeter at different times).

The answer, according to special relativity, is that it’s not the speed of light that depends on your reference frame—it’s your notion of a “kilometer” and a “second.”

Let’s try to flesh out some of the details. Examining the clocks on our probe ships closely, we realize that all of the traveling clocks are different in a similar way: They read shorter times than the one that was stationary. That is striking, as we were comforting ourselves with the idea that time is kind of like space, and the clocks were reflecting a distance traveled through spacetime. But in the case of good old ordinary space, moving around willy-nilly always makes a journey longer; a straight line is the shortest distance between two points in space. If our clocks are telling us the truth (and they are), it would appear that unaccelerated motion—a straight line through spacetime, if you like—is the path of longest time between two events.

Extraneous motion decreases the time elapsed between two events in spacetime, whereas it increases the distance traveled between two points in space.

The whole point of spacetime according to relativity is that it is not fundamentally divided up into “time” and “space.” The light cones, demarcating the accessible past and future of each event, are not added on top of the straightforward Newtonian decomposition of spacetime into time and space; they replace that structure entirely. Time can be measured along each individual world line, but it’s not a built-in feature of the entire spacetime.

It’s not the force of gravity that you feel when you are sitting in a chair; it’s the force of the chair pushing up on your posterior. According to general relativity, free fall is the natural, unforced state of motion, and it’s only the push from the surface of the Earth that deflects us from our appointed path.

Following the insight that an unaccelerated trajectory yields the greatest possible time a clock could measure between two events, a straight line through spacetime is one that does its best to maximize the time on a clock, just like a straight line through space does its best to minimize the distance read by an odometer.

According to the viewpoint of general relativity, the orbiting clock is not accelerating; it’s in free fall, doing its best to move in a straight line through spacetime. The tower clock, meanwhile, is accelerating—it’s being prevented from freely falling by the force of the tower keeping it up. Therefore, the orbiting clock will experience more elapsed time per orbit than the tower clock—compared to the accelerated clock on the tower, the freely falling one in orbit appears to run more quickly.

It would be wrong to think of the singularity as residing at the “center” of the black hole. If we look carefully at the representation of spacetime near a black hole shown in Figure 19, we see that the future light cones inside the event horizon keep tipping toward the singularity. But that light cone defines what the observer at that event would call “the future.” Like the Big Bang singularity in the past, the black hole singularity in the future is a moment of time, not a place in space. Once you are inside the event horizon, you have absolutely no choice but to continue on to the grim destiny of the singularity, because it lies ahead of you in time, not in some direction in space. You can no more avoid hitting the singularity than you can avoid hitting tomorrow.

General relativity is complicated. Not just conceptually, but technically; the equations governing the curvature of spacetime are enormously difficult to solve in any real-world situation. What we know about the exact predictions of the theory comes mostly from highly idealized cases with a great deal of symmetry, such as a static star or a completely smooth universe. Determining the spacetime curvature caused by two black holes passing by each other near the speed of light is beyond our current capabilities (although the state of the art is advancing rapidly).

A “state” of a physical system is “all of the information about the system, at some fixed moment in time, that you need to specify its future evolution,111 given the laws of physics.”

Instead of thinking of the evolution through time of all those atoms moving through three-dimensional space with their individual momenta, we can equally well think of the evolution of the entire system as the motion of a single point (the state) through a giant-dimensional space of states.

When you dig deeply into the guts of how particle physics works, it turns out that there are three different kinds of possible symmetries that involve “inverting” a physical property, each of which is denoted by a capital letter. We have time reversal T, which exchanges past and future. We also have parity P, which exchanges right and left. We discussed parity in the context of our checkerboard worlds, but it’s just as relevant to three-dimensional space in the real world. Finally, we have “charge conjugation” C, which is a fancy name for the process of exchanging particles with their antiparticles. The transformations C, P, and T all have the property that when you repeat them twice in a row you simply return to the state you started with.

As far as any experiment yet performed can tell, CPT is a perfectly good symmetry of Nature.

Our ability to successfully define “time reversal” so that some laws of physics are invariant under it depends on one other crucial assumption: conservation of information. This is simply the idea that two different states in the past always evolve into two distinct states in the future—they never evolve into the same state. If that’s true, we say that “information is conserved,” because knowledge of the future state is sufficient to figure out what the appropriate state in the past must have been. If that feature is respected by some laws of physics, the laws are reversible , and there will exist some (possibly complicated) transformations we can do to the states so that time-reversal invariance is respected.

When we say that a physical process is irreversible, we mean that we cannot construct the past from knowledge of the current state, and this checkerboard is a perfect example of that.

Boltzmann’s formula for the entropy, which is traditionally denoted by S (you wouldn’t have wanted to call it E, which usually stands for energy), states that it is equal to some constant k, cleverly called “Boltzmann’s constant,” times the logarithm of W, the number of microscopic arrangements of a system that are macroscopically indistinguishable.126 That is: S=k log W.

Entropy measures the uselessness of a configuration of energy.

The process of dividing up the space of microstates of some particular physical system (gas in a box, a glass of water, the universe) into sets that we label “macroscopically indistinguishable” is known as coarse-graining.

The total number of low-entropy states that evolve to high entropy is equal, as Loschmidt argued, to the total number of high-entropy states that evolve to low entropy.

Boltzmann has told us a compelling story about why entropy increases: There are more ways to be high entropy than low entropy, so most microstates in a low-entropy macrostate will evolve toward higher-entropy macrostates. But that argument makes no reference to the direction of time. Following that logic, most microstates within some macrostate will increase in entropy toward the future but will also have evolved from a higher-entropy condition in the past.

When all is said and done, if we think of the “message” as a specification of which macrostate a system is in, the relationship between entropy and information is very simple: The information is the difference between the maximum possible entropy and the actual entropy of the macrostate.

So the largest conceivable change in entropy that would be required to take a completely disordered collection of molecules the size of our biomass and turn them into absolutely any configuration at all—including the actual ecosystem we currently have—is 1044. If the evolution of life is consistent with the Second Law, it must be the case that the Earth has generated more entropy over the course of life’s evolution by converting high-energy photons into low-energy ones than it has decreased entropy by creating life. The number 1044 is certainly an overly generous estimate—we don’t have to generate nearly that much entropy, but if we can generate that much, the Second Law is in good shape. How long does it take to generate that much entropy by converting useful solar energy into useless radiated heat? The answer, once again plugging in the temperature of the Sun and so forth, is: about 1 year. Every year, if we were really efficient, we could take an undifferentiated mass as large as the entire biosphere and arrange it in a configuration with as small an entropy as we can imagine. In reality, life has evolved over billions of years, and the total entropy of the “Sun + Earth (including life) + escaping radiation” system has increased by quite a bit. So the Second Law is perfectly consistent with life as we know it—not that you were ever in doubt.

Perhaps the most remarkable thing about the book is Schrödinger’s deduction that the stability of genetic information over time is best explained by positing the existence of some sort of “aperiodic crystal” that stored the information in its chemical structure. This insight helped inspire Francis Crick to leave physics in favor of molecular biology, eventually leading to his discovery with James Watson of the double-helix structure of DNA.

According to quantum mechanics, what we can observe about the world is only a tiny subset of what actually exists.

But we don’t. That’s not what we see, in this idealized thought-experiment world where our cat is a truly quantum object. What we see when we choose not to observe whether she goes via the food bowl or the scratching post is that she ends up on the sofa 100 percent of the time! We never find her under the table—the final wave function assigns an amplitude of zero to that possible outcome. Apparently, if all this is to be believed, the very presence of our spy cameras changed her wave function in some dramatic way. The possibilities are summarized in the table.

This isn’t just a thought experiment; it’s been done. Not with real cats, who are unmistakably macroscopic and well described by the classical limit, but with individual photons, in what is known as the “double slit experiment.” A photon passes through two possible slits, and if we don’t watch which slit it goes through, we get one final wave function, but if we do, we get a completely different one, no matter how unobtrusive our measurements were.

In quantum mechanics, no matter how many individual pieces make up the system you are thinking about, there is only one wave function . Even if we consider the entire universe and everything inside it, there is still only one wave function, sometimes redundantly known as the “wave function of the universe.” People don’t always like to talk that way, for fear of sounding excessively grandiose, but at bottom that’s simply the way quantum mechanics works.

In order for a black hole to be hotter than the microwave background is today, it would have to be less than about 1014 kilograms—about the mass of Mt. Everest, and much smaller than any known black hole.

But if the entropy of the black hole is proportional to the area of its event horizon, that means there is a maximum amount of entropy you can possibly fit into a region of some fixed size, which is achieved by a black hole of that size.

The upshot is simple: Quantum gravity doesn’t obey the principle of locality. In quantum gravity, what goes on over here is not completely independent from what goes on over there. The number of things that can possibly go on (the number of possible microstates in a region) isn’t proportional to the volume of the region; it’s proportional to the area of a surface we can draw that encloses the region. The real world, described by quantum gravity, allows for much less information to be squeezed into a region than we would naïvely have imagined if we weren’t taking gravity into account.

Here is why holography is important: It means that spacetime is not fundamental. When we typically think about what goes on in the universe, we implicitly assume something like locality; we describe what happens at this location, and at that location, and give separate specifications for every possible location in space. Holography says that we can’t really do that, in principle—there are subtle correlations between things that happen at different locations, which cut down on our freedom to specify a configuration of stuff through space.

Hawking’s formula for the entropy of a black hole seems to be telling us that there are a very large number of microstates corresponding to any particular macroscopic black hole. What are those microstates? They are not apparent in classical general relativity. Ultimately, they must be states of quantum gravity. There’s good news and bad news here. The bad news is that we don’t understand quantum gravity very well in the real world, so we are unable to simply list all of the different microstates corresponding to a macroscopic black hole. The good news is that we can use Hawking’s formula as a clue, to test our ideas of how quantum gravity might work.