Figure 32: An example of checkerboard world, featuring a simple pattern within each vertical column.
In Figure 32 we have a particularly simple example of the game, labeled “checkerboard A.” Clearly there is some pattern going on here, which should be pretty evident. One way of putting it would be, “every square in any particular column is in the same state.” We should be careful to check that there aren’t any other patterns lurking around—if someone else finds more patterns than we do, we lose the game, and they go collect the checkerboard Nobel Prize. From the looks of checkerboard A, there doesn’t seem to be any obvious pattern as we go across a row that would allow us to make further simplifications, so it looks like we’re done.
As simple as it is, there are some features of checkerboard A that are extremely relevant for the real world. For one thing, notice that the pattern distinguishes between “time,” running vertically up the columns, and “space,” running horizontally across the rows. The difference is that anything can happen within a row—as far as we can tell, knowing the state of one particular square tells us nothing about the state of nearby squares. In the real world, analogously, we believe that we can start with any configuration of matter in space that we like. But once that configuration is chosen, the “laws of physics” tell us exactly what must happen through time. If there is a cat sitting on our lap, we can be fairly sure that the same cat will be nearby a moment later; but knowing there’s a cat doesn’t tell us very much about what else is in the room.
Starting completely from scratch in inventing the universe, it’s not at all obvious that there must be a distinction of this form between time and space. We could imagine a world in which things changed just as sharply and unpredictably from moment to moment in time as they do from place to place in space. But the real universe in which we live does seem to feature such a distinction. The idea of “time,” through which things in the universe evolve, isn’t a logically necessary part of the world; it’s an idea that happens to be extremely useful when thinking about the reality in which we actually find ourselves.
We characterized the rule exhibited by checkerboard A as “every square in a column is in the same state.” That’s a global description, referring to the entire column all at once. But we could rephrase the rule in another way, more local in character, which works by starting from some particular row (a “moment in time”) and working up or down. We can express the rule as “given the state of any particular square, the square immediately above it must be in the same state.” In other words, we can express the patterns we see in terms of evolution through time—starting with whatever state we find at some particular moment, we can march forward (or backward), one row at a time. That’s a standard way of thinking about the laws of physics, as illustrated schematically in Figure 33. You tell me what is going on in the world (say, the position and velocity of every single particle in the universe) at one moment of time, and the laws of physics are a black box that tells us what the world will evolve into just one moment later.108 By repeating the process, we can build up the entire future. What about the past?
Figure 33: The laws of physics can be thought of as a machine that tells us, given what the world is like right now, what it will evolve into a moment later.
FLIPPING TIME UPSIDE DOWN
A checkerboard is a bit sterile and limiting, as imaginary worlds go. It would be hard to imagine these little squares throwing a party, or writing epic poetry. Nevertheless, if there were physicists living in the checkerboards, they would find interesting things to talk about once they were finished formulating the laws of time evolution.
For example, the physics of checkerboard A seems to have a certain degree of symmetry. One such symmetry is time-translation invariance—the simple idea that the laws of physics don’t change from moment to moment. We can shift our point of view forward or backward in time (up or down the columns) and the rule “the square immediately above this one must be in the same state” remains true.109 Symmetries are always like that: If you do a certain thing, it doesn’t matter; the rules still work in the same way. As we’ve discussed, the real world is also invariant under time shifts; the laws of physics don’t seem to be changing as time passes.
Another kind of symmetry is lurking in checkerboard A: time-reversal invariance . The idea behind time reversal is relatively straightforward—just make time run backward. If the result “looks the same”—that is, looks like it’s obeying the same laws of physics as the original setup—then we say that the rules are time-reversal invariant. To apply this to a checkerboard, just pick some particular row of the board, and reflect the squares vertically around that row. As long as the rules of the checkerboard are also invariant under time shifts, it doesn’t matter which row we choose, since all rows are created equal. If the rules that described the original pattern also describe the new pattern, the checkerboard is said to be time-reversal invariant. Example A, featuring straight vertical columns of the same color squares, is clearly invariant under time reversal—not only does the reflected pattern satisfy the same rules; it is precisely the same as the original pattern.
Let’s look at a more interesting example to get a better feeling for this idea. In Figure 34 we show another checkerboard world, labeled “B.” Now there are two different kinds of patterns of gray squares running from bottom to top—diagonal series of squares running in either direction. (They kind of look like light cones, don’t they?) Once again, we can express this pattern in terms of evolution from moment to moment in time, with one extra thing to keep in mind: Along any single row, it’s not enough to keep track of whether a particular square is white or gray. We also need to keep track of what kinds of diagonal lines of gray squares, if any, are passing through that point. We could choose to label each square in one of four different states: “white,” “diagonal line of grays going up and to the right,” “diagonal line of grays going up and to the left,” or “diagonal lines of grays going in both directions.” If, on any particular row, we simply listed a bunch of 0’s and 1’s, that wouldn’t be enough to figure out what the next row up should look like.110 It’s as if we had discovered that there were two different kinds of “particles” in this universe, one always moving left and one always moving right, but that they didn’t interact or interfere with each other in any way.
Figure 34: Checkerboard B, on the left, has slightly more elaborate dynamics than checkerboard A, with diagonal lines of gray squares in both directions. Checkerboard B‘, on the right, is what happens when we reverse the direction of time by reflecting B about the middle row.
What happens to checkerboard B under time reversal? When we reverse the direction of time in this example, the result looks similar in form, but the actual configuration of white and black squares has certainly changed (in contrast to checkerboard A, where flipping time just gave us precisely the set of whites and grays we started with). The second panel in Figure 34, labeled B‘, shows the results of reflecting about some row in checkerboard B. In particular, the diagonal lines that were extending from lower left to upper right now extend from upper left to lower right, and vice versa.
Is the checkerboard world portrayed in example B invariant under time reversal? Yes, it is. It doesn’t matter that the individual distribution of white and gray squares is altered when we reflect time around some particular row; what matters is that the “laws of physics,” the rules obeyed by the patterns of squares, are unaltered. In the original example B, before reversing time, the rules were that there were two kinds of diagonal lines of gray squares, going in either direction; the same is true in example B‘. The fact that the two kinds of lines switched identities doesn’t change the fact that the same two kinds of lines could be found before and after. So imaginary physicists living in the world of checkerboard B would certainly proclaim that the laws of nature were time-reversal invariant.
THROUGH THE LOOKING GLASS
Well, then, what about checkerboard C, shown in Figure 35? Once again
, the rules seem to be pretty simple: We see nothing but diagonal lines going from lower left to upper right. If we want to think about this rule in terms of one-step-at-a-time evolution, it could be expressed as “given the state of any particular square, the square one step above and one step to the right must be in the same state.” It is certainly invariant under shifts in time, since that rule doesn’t care about what row you start from.
Figure 35: Checkerboard world C only has diagonal lines of gray squares running from lower left to upper right. If we reverse the direction of time to obtain C‘, we only have lines running from bottom right to top left. Strictly speaking, checkerboard C is not time-reversal invariant, but it is invariant under simultaneous reflection in space and reversal in time.
If we reverse the direction of time in checkerboard C, we get something like the checkerboard C’ shown in the figure. Clearly this is a different situation than before. The rules obeyed in C’ are not those obeyed in C—diagonal lines stretching from lower left to upper right have been replaced by diagonal lines stretching the other way. Physicists who lived in one of these checkerboards would say that time reversal was not a symmetry of their observed laws of nature. We can tell the difference between “forward in time” and “backward in time”—forward is the direction in which the diagonal lines move to the right. It is completely up to us which direction we choose to label “the future,” but once we make that choice it’s unambiguous.
However, that’s surely not the end of the story. While checkerboard C might not be, strictly speaking, invariant under time reversal as we have defined it, there does seem to be something “reversible” about it. Let’s see if we can’t put our fingers on it.
In addition to time reversal, we could also consider “space reversal,” which would be obtained by flipping the checkerboard horizontally around some given column. In the real world, that’s the kind of thing we get by looking at something in a mirror; we can think of space reversal as just taking the mirror image of something. In physics, it usually goes by the name of “parity,” which (when we have three dimensions of space rather than just the one of the checkerboard) can be obtained by simultaneously inverting every spatial direction. Let’s call it parity, so that we can sound like physicists when the occasion demands it.
Our original checkerboard A clearly had parity as a symmetry—the rules of behavior we uncovered would still be respected if we flipped right with left. For checkerboard C, meanwhile, we face a situation similar to the one we encountered when considering time reversal—the rules are not parity symmetric, since a world with only up-and-to-the-right diagonals turns into one with only up-and-to-the-left diagonals once we switch right and left, just as it did when we reversed time.
Nevertheless, it looks like you could take checkerboard C and do both a reversal in time and a parity inversion in space, and you would end up with the same set of rules you started with. Reversing time takes one kind of diagonal to the other, and reflecting space takes them back again. That’s exactly right, and it illustrates an important feature of time reversal in fundamental physics: It is often the case that a certain theory of physics would not be invariant under “naïve time reversal,” which reverses the direction of time but does nothing else, and yet the theory is invariant under an appropriately generalized symmetry transformation that reverses the direction of time and also does some other things. The way this works in the real world is a tiny bit subtle and becomes enormously more confusing the way it is often discussed in physics books. So let’s leave the blocky world of checkerboards and take a look at the actual universe.
THE STATE-OF-THE-SYSTEM ADDRESS
The theories that physicists often use to describe the real world share the underlying framework of a “state” that “evolves with time.” That’s true for classical mechanics as put together by Newton, or for general relativity, or for quantum mechanics, all the way up to quantum field theory and the Standard Model of particle physics. On one of our checkerboards, a state is a horizontal row of squares, each of which is either white or gray (with perhaps some additional information). In different approaches to real-world physics, what counts as a “state” will be different. But in each case we can ask an analogous set of questions about time reversal and other possible symmetries.
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.” In particular, we have in mind isolated systems—those that aren’t subject to unpredictable external forces. (If there are predictable external forces, we can simply count those as part of the “laws of physics” relevant to that system.) So we might be thinking of the whole universe, which is isolated by hypothesis, or some spaceship far away from any planets or stars.
First consider classical mechanics—the world of Sir Isaac Newton.112 What information do we need to predict the future evolution of a system in Newtonian mechanics? I’ve already alluded to the answer: the position and velocity of every component of the system. But let’s creep up on it gradually.
When someone brings up “Newtonian mechanics,” you know sooner or later you’ll be playing with billiard balls.113 But let’s imagine a game that is not precisely like conventional eight ball; it’s a unique, hypothetical setup, which we might call “physicist’s billiards.” In our eagerness to strip away complications and get to the essence of a thing, physicists imagine games of billiards in which there is no noise or friction, so that perfectly round spheres roll along the table and bounce off one another without losing any energy. Real billiard balls don’t quite behave this way—there is some dissipation and sound as they knock into one another and roll along the felt. That’s the arrow of time at work, as noise and friction create entropy—so we’re putting those complications aside for the moment.
Start by considering a single billiard ball moving alone on a table. (Generalization to many billiard balls is not very hard.) We imagine that it never loses energy, and bounces cleanly off of the bumper any time it hits. For purposes of this problem, “bounces cleanly off the bumper” is part of the “laws of physics” of our closed system, the billiard ball. So what counts as the state of that single ball?
You might guess that the state of the ball at any one moment of time is simply its position on the table. That is, after all, what would show up if we took a picture of the table—you would see where the ball was. But we defined the state to consist of all the information you would need to predict the future evolution of the system, and just specifying the position clearly isn’t enough. If I tell you that the ball is precisely in the middle of the table (and nothing else), and ask you to predict where it will be one second later, you would be at a loss, since you wouldn’t know whether the ball is moving.
Of course, to predict the motion of the ball from information defined at a single moment in time, you need to know both the position and the velocity of the ball. When we say “the state of the ball,” we mean the position and the velocity, and—crucially—nothing else. We don’t need to know (for example) the acceleration of the ball, the time of day, what the ball had for breakfast that morning, or any other pieces of information.
We often characterize the motion of particles in classical mechanics in terms of momentum rather than velocity. The concept of “momentum” goes all the way back to the great Persian thinker Ibn Sina (often Latinized as Avicenna) around the year 1000. He proposed a theory of motion in which “inclination”—weight times velocity—remained constant in the absence of outside influences. The momentum tells us how much oomph an object has, and the direction in which it is moving114; in Newtonian mechanics it is equal to mass times velocity, and in relativity the formula is slightly modified so that the momentum goes to infinity as the velocity approaches the speed of light. For any object with a fixed mass, when you know the momentum you know the velocity, and vice versa. We can therefore specify the state of a single particle by giving its position and i
ts momentum.
Figure 36: A lone billiard ball, moving on a table, without friction. Three different moments in time are shown. The arrows denote the momentum of the ball; it remains constant until the ball rebounds off a wall.
Once you know the position and momentum of the billiard ball, you can predict the entire trajectory as it rattles around the table. When the ball is moving freely without hitting any walls, the momentum stays the same, while the position changes with a constant velocity along a straight line. When the ball does hit a wall, the momentum is suddenly reflected with respect to the line defined by the wall, after which the ball continues on at constant velocity. That is to say, it bounces. I’m making simple things sound complicated, but there’s a method behind the madness.
All of Newtonian mechanics is like that. If you have many billiard balls on the same table, the complete state of the system is simply a list of the positions and momenta of each ball. If it’s the Solar System you are interested in, the state is the position and momentum of each planet, as well as of the Sun. Or, if we want to be even more comprehensive and realistic, we can admit that the state is really the position and momentum of every single particle constituting these objects. If it’s your boyfriend or girlfriend you are interested in, all you need to do is precisely specify the position and momentum of every atom in his or her body. The rules of classical mechanics give unambiguous predictions for how the system will involve, using only the information of its current state. Once you specify that list, Laplace’s Demon takes over, and the rest of history is determined. You are not as smart as Laplace’s Demon, nor do you have access to the same amount of information, so boyfriends and girlfriends are going to remain mysterious. Besides, they are open systems, so you would have to know about the rest of the world as well.
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 16