From Eternity to Here (23 page)

Reversing time turns out to be a surprisingly subtle concept for something that would appear at first glance to be relatively straightforward. (Just run the movie backward, right?) Blithely reversing the direction of time is
not
a symmetry of the laws of nature—we have to dress up what we really mean by “reversing time” in order to correctly pinpoint the underlying symmetry. So we’ll approach the topic somewhat circuitously, through simplified toy models. Ultimately I’ll argue that the important concept isn’t “time reversal” at all, but the similar-sounding notion of “reversibility”—our ability to reconstruct the past from the present, as Laplace’s Demon is purportedly able to do, even if it’s more complicated than simply reversing time. And the key concept that ensures reversibility is
conservation of information—
if the information needed to specify the state of the world is preserved as time passes, we will always be able to run the clock backward and recover any previous state. That’s where the real puzzle concerning the arrow of time will arise.

CHECKERBOARD WORLD

Let’s play a game. It’s called “checkerboard world,” and the rules are extremely simple. You are shown an array of squares—the checkerboard—with some filled in white, and some filled in gray. In computer-speak, each square is a “bit”—we could label the white squares with the number “0,” and the gray squares with “1.” The checkerboard stretches infinitely far in every direction, but we get to see only some finite part of it at a time.

The point of the game is to
guess the pattern
. Given the array of squares before you, your job is to discern patterns or rules in the arrangements of whites and grays. Your guesses are then judged by revealing more checkerboard than was originally shown, and comparing the predictions implied by your guess to the actual checkerboard. That last step is known in the parlance of the game as “testing the hypothesis.”

Of course, there is another name to this game: It’s called “science.” All we’ve done is describe what real scientists do to understand nature, albeit in a highly idealized context. In the case of physics, a good theory has three ingredients: a specification of the
stuff
that makes up the universe, the
arena
through which the stuff is distributed, and a set of
rules
that the stuff obeys. For example, the stuff might be elementary particles or quantum fields, the arena might be four-dimensional spacetime, and the rules might be the laws of physics. Checkerboard world is a model for exactly that: The stuff is a set of bits (0’s and 1’s, white and gray squares), the arena through which they are distributed is the checkerboard itself, and the rules—the laws of nature in this toy world—are the patterns we discern in the behavior of the squares. When we play this game, we’re putting ourselves in the position of imaginary physicists who live in one of these imaginary checkerboard worlds, and who spend their time looking for patterns in the appearance of the squares as they attempt to formulate laws of nature.
¹⁰⁷

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.