Figure 29: A wormhole in three-dimensional space, constructed by identifying two spheres whose interiors have been removed. Anything that enters one sphere instantly appears on the opposite side of the other.
The wormhole is reminiscent of our previous gate-into-yesterday example. If you look through one end of the wormhole, you don’t see swirling colors or flashing lights; you see whatever is around the other end of the wormhole, just as if you were looking through some sort of periscope (or at a video monitor, where the camera is at the other end). The only difference is that you could just as easily put your hand through, or (if the wormhole were big enough), jump right through yourself.
This sort of wormhole is clearly a shortcut through spacetime, connecting two distant regions in no time at all. It performs exactly the trick that Sagan needed for his novel, and on Thorne’s advice he rewrote the relevant section. (In the movie version, sadly, there were swirling colors and flashing lights.) But Sagan’s question set off a chain of ideas that led to innovative scientific research, not just a more accurate story.
TIME MACHINE CONSTRUCTION MADE EASY
A wormhole is a shortcut through space; it allows you to get from one place to another much faster than you would if you took a direct route through the bulk of spacetime. You are never moving faster than light from your local point of view, but you get to your destination sooner than light would be able to if the wormhole weren’t there. We know that faster-than-light travel can be used to go backward in time; travel through a wormhole isn’t literally that, but certainly bears a family resemblance. Eventually Thorne, working with Michael Morris, Ulvi Yurtsever, and others, figured out how to manipulate a wormhole to create closed timelike curves.99
The secret is the following: When we toss around a statement like, “the wormhole connects two distant regions of space,” we need to take seriously the fact that it really connects two sets of events in spacetime. Let’s imagine that spacetime is perfectly flat, apart from the wormhole, and that we have defined a “background time” in some rest frame. When we identify two spheres to make a wormhole, we do so “at the same time” with respect to this particular background time coordinate. In some other frame, they wouldn’t be at the same time.
Now let’s make a powerful assumption: We can pick up and move each mouth of the wormhole independently of the other. There is a certain amount of hand-waving justification that goes into this assumption, but for the purposes of our thought experiment it’s perfectly okay. Next, we let one mouth sit quietly on an unaccelerated trajectory, while we move the other one out and back at very high speed.
To see what happens, imagine that we attach a clock to each wormhole mouth. The clock on the stationary mouth keeps time along with the background time coordinate. But the clock on the out-and-back wormhole mouth experiences less time along its path, just like any other moving object in relativity. So when the two mouths are brought back next to each other, the clock that moved now seems to be behind the clock that stayed still.
Now consider exactly the same situation, but think of it from the point of view that you would get by looking through the wormhole. Remember, you don’t see anything spooky when you look through a wormhole mouth; you just see whatever view is available to the other mouth. If we compare the two clocks as seen through the wormhole mouth, they don’t move with respect to each other. That’s because the length of the wormhole throat doesn’t change (in our simplified example it’s exactly zero), even when the mouth moves. Viewed through the wormhole, there are just two clocks that are sitting nearby each other, completely stationary. So they remain in synchrony, keeping perfect time as far as they are each concerned.
How can the two clocks continue to agree with each other, when we previously said that the clock that moved and came back would have experienced less elapsed time? Easy—the clocks appear to differ when we look at them as an external observer, but they appear to match when we look at them through the wormhole. This puzzling phenomenon has a simple explanation: Once the two wormhole mouths move on different paths through spacetime, the identification between them is no longer at the same time from the background point of view. The sphere representing one mouth is still identified with the sphere representing the other mouth, but now they are identified at different times. By passing through one, you move into the past, as far as the background time is concerned; by passing through in the opposite direction, you move into the future.
Figure 30: A wormhole time machine. Double-sided arrows represent identifications between the spherical wormhole mouths. The two mouths start nearby, identified at equal background times. One remains stationary, while the other moves away and returns near the speed of light, so that they become identified at very different background times.
This kind of wormhole, therefore, is exactly like the gate into yesterday. By manipulating the ends of a wormhole with a short throat, we have connected two different regions of spacetime with very different times. Once we’ve done that, it’s easy enough to travel through the wormhole in such a way as to describe a closed timelike curve, and all of the previous worries about paradoxes apply. This procedure, if it could be carried out in the real world, would unambiguously count as “building a time machine” by the standards of our earlier discussion.
PROTECTION AGAINST TIME MACHINES
The wormhole time machines make it sound somewhat plausible that closed timelike curves could exist in the real world. The problem seemingly becomes one of technological ability, rather than restrictions placed by the laws of physics; all we need is to find a wormhole, keep it open, move one of the mouths in the right way . . . Well, perhaps it’s not completely plausible after all. As one might suspect, there turn out to be a number of good reasons to believe that wormholes don’t provide a very practical route to building time machines.
First, wormholes don’t grow on trees. In 1967, theoretical physicist Robert Geroch investigated the question of wormhole construction, and he showed that you actually could create a wormhole by twisting spacetime in the appropriate way—but only if, as an intermediate step in the process, you created a closed timelike curve. In other words, the first step to building a time machine by manipulating a wormhole is to build a time machine so that you can make a wormhole.100 But even if you were lucky enough to stumble across a wormhole, you’d be faced with the problem of keeping it open. Indeed, this difficulty is recognized as the single biggest obstacle to the plausibility of the wormhole time machine idea.
The problem is that keeping a wormhole open requires negative energies. Gravity is attractive: The gravitational field caused by an ordinary positive-energy object works to pull things together. But look back at Figure 29 and see what the wormhole does to a collection of particles that pass through it—it “defocuses” them, taking particles that were initially coming together and now pushing them apart. That’s the opposite of gravity’s conventional behavior, and a sign that negative energies must be involved.
Do negative energies exist in Nature? Probably not, at least not in the ways necessary to sustain a macroscopic wormhole—but we can’t say for sure. Some people have proposed ideas for using quantum mechanics to create pockets of negative energy, but they’re not on a very firm footing. A big hurdle is that the question necessarily involves both gravity and quantum mechanics, and the intersection of those two theories is not very well understood.
As if that weren’t enough to worry about, even if we found a wormhole and knew a way to keep it open, chances are that it would be unstable—the slightest disturbance would send it collapsing into a black hole. This is another question for which it’s hard to find a clear-cut answer, but the basic idea is that any tiny ripple in energy can zoom around a closed timelike curve an arbitrarily large number of times. Our best current thinking is that this kind of repeat journey is inevitable, at least for some small fluctuations. So the wormhole doesn’t just feel the mass of a single speck of dust passing through—it feels that effect over and over again, c
reating an enormous gravitational field, enough to ultimately destroy our would-be time machine.
Nature, it seems, tries very hard to stop us from building a time machine. The accumulated circumstantial evidence prompted Stephen Hawking to propose what he calls the “Chronology Protection Conjecture”: The laws of physics (whatever they may be) prohibit the creation of closed timelike curves.101 We have a lot of evidence that something along those lines is true, even if we fall short of a definitive proof.
Time machines fascinate us, in part because they seem to open the door to paradoxes and challenge our notions of free will. But it’s likely that they don’t exist, so the problems they present aren’t the most pressing (unless you’re a Hollywood screenwriter). The arrow of time, on the other hand, is indisputably a feature of the real world, and the problems it presents demand an explanation. The two phenomena are related; there can be a consistent arrow of time throughout the observable universe only because there are no closed timelike curves, and many of the disconcerting properties of closed timelike curves arise from their incompatibility with the arrow of time. The absence of time machines is necessary for a consistent arrow of time, but it’s by no means sufficient to explain it. Having laid sufficient groundwork, it’s time to confront the mystery of time’s direction head-on.
PART THREE
ENTROPY AND TIME’S ARROW
7
RUNNING TIME BACKWARD
This is what I mean when I say I would like to swim against the stream of time: I would like to erase the consequences of certain events and restore an initial condition.
—Italo Calvino, If on a Winter’s Night a Traveler
Pierre-Simon Laplace was a social climber at a time when social climbing was a risky endeavor.102 When the French Revolution broke out, Laplace had established himself as one of the greatest mathematical minds in Europe, as he would frequently remind his colleagues at the Académie des Sciences. In 1793 the Reign of Terror suppressed the Académie; Laplace proclaimed his Republican sympathies, but he also moved out of Paris just to be safe. (Not without reason; his colleague Antoine Lavoisier, the father of modern chemistry, was sent to the guillotine in 1794.) He converted to Bonapartism when Napoleon took power, and dedicated his Théorie Analytique des Probabilités to the emperor. Napoleon gave Laplace a position as minister of the interior, but he didn’t last very long—something about being too abstract-minded. After the restoration of the Bourbons, Laplace became a Royalist, and omitted the dedication to Napoleon from future editions of his book. He was named a marquis in 1817.
Social ambitions notwithstanding, Laplace could be impolitic when it came to his science. A famous anecdote concerns his meeting with Napoleon, after he had asked the emperor to accept a copy of his Méchanique Céleste—a five-volume treatise on the motions of the planets. It seems unlikely that Napoleon read the whole thing (or any of it), but someone at court did let him know that the name of God was entirely absent. Napoleon took the opportunity to mischievously ask, “M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator.” To which Laplace answered stubbornly, “I had no need of that hypothesis.”103
Figure 31: Pierre-Simon Laplace, mathematician, physicist, swerving politician, and unswerving determinist.
One of the central tenets of Laplace’s philosophy was determinism. It was Laplace who truly appreciated the implications of Newtonian mechanics for the relationship between the present and the future: Namely, if you understood everything about the present, the future would be absolutely determined. As he put it in the introduction to his essay on probability:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.104
These days we would probably say that a sufficiently powerful computer could, given all there was to know about the present universe, predict the future (and retrodict the past) with perfect accuracy. Laplace didn’t know about computers, so he imagined a vast intellect. His later biographers found this a bit dry, so they attached a label to this hypothetical intellect: Laplace’s Demon.
Laplace never called it a demon, of course; presumably he had no need to hypothesize demons any more than gods. But the idea captures some of the menace lurking within the pristine mathematics of Newtonian physics. The future is not something that has yet to be determined; our fate is encoded in the details of the current universe. Every moment of the past and future is fixed by the present. It’s just that we don’t have the resources to perform the calculation.105
There is a deep-seated impulse within all of us to resist the implications of Laplace’s Demon. We don’t want to believe that the future is determined, even if someone out there did have access to the complete state of the universe. Tom Stoppard’s Arcadia once again expresses this anxiety in vivid terms.
VALENTINE: Yes. There was someone, forget his name, 1820s, who pointed out that from Newton’s laws you could predict everything to come—I mean, you’d need a computer as big as the universe but the formula would exist.
CHLOË: But it doesn’t work, does it?
VALENTINE: No. It turns out the maths is different.
CHLOË: No, it’s all because of sex.
VALENTINE: Really?
CHLOË: That’s what I think. The universe is deterministic all right, just like Newton said, I mean it’s trying to be, but the only thing going wrong is people fancying other people who aren’t supposed to be in that part of the plan.
VALENTINE: Ah. The attraction Newton left out. All the way back to the apple in the garden. Yes. (Pause.) Yes, I think you’re the first person to think of this.106
We won’t be exploring whether sexual attraction helps us wriggle free of the iron grip of determinism. Our concern is with why the past seems so demonstrably different from the future. But that wouldn’t be nearly the puzzle it appears to be if it weren’t for the fact that the underlying laws of physics seem perfectly reversible; as far as Laplace’s Demon is concerned, there’s no difference between reconstructing the past and predicting the future.
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
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 15