From Eternity to Here: The Quest for the Ultimate Theory of Time - Sean Carroll (2010)

Part II. TIME IN EINSTEIN’S UNIVERSE

Chapter 6. LOOPING THROUGH TIME

You see, my son, here time changes into space.

—Richard Wagner, Parsifal

Everyone knows what a time machine looks like: something like a steampunk sled with a red velvet chair, flashing lights, and a giant spinning wheel on the back. For those of a younger generation, a souped-up stainless-steel sports car is an acceptable substitute; our British readers might think of a 1950s-style London police box.76 Details of operation vary from model to model, but when one actually travels in time, the machine ostentatiously dematerializes, presumably to be re-formed many millennia in the past or future.

That’s not how it would really work. And not because time travel is impossible and the whole thing is just silly; whether or not time travel is possible is more of an open question than you might suspect. I’ve emphasized that time is kind of like space. It follows that, if you did stumble across a working time machine in the laboratory of some mad inventor, it would simply look like a “space machine”—an ordinary vehicle of some sort, designed to move you from one place to another. If you want to visualize a time machine, think of launching a rocket ship, not disappearing in a puff of smoke.

So what is actually entailed in traveling through time? There are two cases of possible interest: traveling to the future and traveling to the past. To the future is easy: Just keep sitting in your chair. Every hour, you will move one hour into the future. “But,” you say, “that’s boring. I want to move far into the future, really quickly, a lot faster than one hour per hour. I want to visit the twenty-fourth century before lunchtime.” But we know it’s impossible to move faster than one hour per hour, relative to a clock that travels along with you. You might be able to trick yourself, by going to sleep or entering suspended animation, but time will still be passing.

On the other hand, you could distort the total amount of time experienced along your world line as compared to the world lines of other people. In a Newtonian universe even that wouldn’t be possible, as time is universal and every world line connecting the same two events experiences the same elapsed time. But in special relativity we can affect the elapsed time by moving through space. Unaccelerated motion gives us the longest time between events; so if you want to get to the future quickly (from your perspective), you just have to move on a highly non-straight path through spacetime. You could zip out on a rocket near the speed of light and then return; or, given sufficient fuel, you could just fly around in circles at ultra-high speed, never straying very far from your starting point in space. When you stopped and exited your spaceship, besides being dizzy you would have “traveled into the future”; more accurately, you would have experienced less time along your world line than was experienced by anyone who stayed behind. Traveling to the future is easy, and getting there more quickly is just a technology problem, not something that conflicts with the fundamental laws of physics.

But you might want to come back, and that’s where the challenge lies. The problems with traveling through time arise when we contemplate traveling to the past.

CHEATING SPACETIME

Lessons learned from watching Superman movies notwithstanding, traveling backward in time is not a matter of reversing the rotation of the Earth. Spacetime itself has to cooperate. Unless, of course, you cheat, by moving faster than the speed of light.

In a Newtonian universe, traveling backward in time is simply out of the question. World lines extend through a spacetime that is uniquely divided into three-dimensional moments of equal time, and the one unbreakable rule is that they must never double back and return to the past. In special relativity, things aren’t much better. Defining “moments of equal time” across the universe is highly arbitrary, but at every event we are faced with the restrictions enforced by light cones. If we are made of ordinary stuff, confined to move from each event forward into the interior of its light cone, there is no hope of traveling backward in time; in a spacetime diagram, we are doomed to march relentlessly upward.

Things would be a little bit more interesting if we were made of non-ordinary stuff. In particular, if we were made of tachyons—particles that always move faster than light. Sadly, we are not made of tachyons, and there are good reasons to believe that tachyons don’t even exist. Unlike ordinary particles, tachyons are forced to always travel outside the light cone. In special relativity, whenever we move outside the light cone, from somebody’s point of view we’re also moving backward in time. More important, the light cones are the only structure defined on spacetime in relativity; there is no separate notion of “space at one moment of time.” So if a particle can start at some event where you are momentarily located and move outside your light cone (faster than light), it can necessarily move into the past from your point of view. There’s nothing to stop it.

Tachyons, therefore, can apparently do something scary and unpredictable: “start” from an event on the world line of some ordinary (slower-than-light) object, defined by some position in space and some moment in time, and travel on a path that takes them to a previous point on the same world line. If you had a flashlight that emitted tachyons, you could (in principle) construct an elaborate series of mirrors by which you could send signals in Morse code to yourself in the past. You could warn your earlier self not to eat the shrimp in that restaurant that one time, or to go on that date with the weirdo from the office, or to sink your life savings into Pets.com stock.

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Figure 22: If tachyons could exist, they could be emitted by ordinary objects and zip around to be absorbed in the past. At every event along its trajectory, the tachyon moves outside the light cone.

Clearly, the possibility of travel backward in time raises the possibility of paradoxes, which is unsettling. There is a cheap way out: Notice that tachyons don’t seem to exist, and declare that they are simply incompatible with the laws of physics.77 That is both fruitful and accurate, at least as far as special relativity is concerned. When curved spacetime gets into the game, things become more interesting.

CIRCLES IN TIME

For those of us who are not made of tachyons, our trajectories through spacetime are limited by the speed of light. From whatever event defines our current location, we necessarily move “forward in time,” toward some other event inside our light cone—in technical jargon, we move on a timelike path through spacetime. This is a local requirement, referring only to features of our neighborhood of the universe. But in general relativity, spacetime is curved. That means light cones in our neighborhood may be tilted with respect to those far away. It’s that kind of tilting that leads to black holes.

But imagine that, instead of light cones tilting inward toward a singularity and creating a black hole, we had a spacetime in which light cones tilted around a circle, as shown in Figure 23. Clearly this would require some sort of extremely powerful gravitational field, but we’re letting our imaginations roam. If spacetime were curved in that way, a remarkable thing would happen: We could travel on a timelike path, always moving forward into our future light cone, and yet eventually meet ourselves at some moment in our past. That is, our world line could describe a closed loop in space, so that it intersected itself, bringing us at one moment in our lives face-to-face with ourselves at some other moment.

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Figure 23: In curved spacetime, we can imagine light cones tilting around in a circle, creating closed timelike curves.

Such a world line—always moving forward in time from a local perspective, but managing to intersect itself in the past—is a closed timelike curve, or CTC. That’s what we really mean when we talk about a “time machine” in general relativity. To actually move along the closed timelike curve would involve ordinary travel through spacetime, on a spaceship or something even more mundane—perhaps even sitting “motionless” in your chair. It’s the curvature of spacetime itself that brings you into contact with your own past. This is a central feature of general relativity that will become important later on when we return to the origin of the universe and the problem of entropy: Spacetime is not stuck there once and for all, but can change (and even come into or pop out of existence) in response to the effects of matter and energy.

It’s not difficult to find spacetimes in general relativity that include closed timelike curves. All the way back in 1949, mathematician and logician Kurt Gödel found a solution to Einstein’s equation that described a “spinning” universe, which contains closed timelike curves passing through every event. Gödel was friends with Einstein in his later years at the Institute for Advanced Study in Princeton, and the idea for his solution arose in part from conversations between the two men.78 In 1963, New Zealand mathematician Roy Kerr found the exact solution for a rotating black hole; interestingly, the singularity takes the form of a rapidly spinning ring, the vicinity of which is covered by closed timelike curves.79 And in 1974, Frank Tipler showed that an infinitely long, rotating cylinder of matter would create closed timelike curves around it, if the cylinder were sufficiently dense and rotating sufficiently rapidly.80

But you don’t have to work nearly so hard to construct a spacetime with closed timelike curves. Consider good old garden-variety flat spacetime, familiar from special relativity. But now imagine that the timelike direction (as defined by some particular unaccelerated observer) is a circle, rather than stretching on forever. In such a universe, something that moved forward in time would find itself coming back to the same moment in the universe’s history over and over. In the Harold Ramis movie Groundhog Day, Bill Murray’s character keeps waking up every morning to experience the exact same situations he had experienced the day before. The circular-time universe we’re imagining here is something like that, with two important exceptions: First, it would be truly identical every day, including the actions of the protagonist himself. And second, there would not be any escape. In particular, winning the love of Andie MacDowell would not save you.

The circular-time universe isn’t just an amusing playground for filmmak ers; it’s an exact solution to Einstein’s equation. We know that, by choosing some unaccelerated reference frame, we can “slice” four-dimensional flat spacetime into three-dimensional moments of equal time. Take two such slices: say, midnight February 2, and midnight February 3—two moments of time that extend throughout the universe (in this special case of flat spacetime, in this particular reference frame). Now take just the one-day’s-worth of spacetime that is between those slices, and throw everything else away. Finally, identify the beginning time with the final time. That is, make a rule that says whenever a world line hits a particular point in space on February 3, it instantly reemerges at the same point in space back on February 2. At heart, it’s nothing more than rolling up a piece of paper and taping together opposite sides to make a cylinder. At every event, even at midnight when we’ve identified different slices, everything looks perfectly smooth and spacetime is flat—time is a circle, and no point on a circle is any different than any other point. This spacetime is rife with closed timelike curves, as illustrated in Figure 24. It might not be a realistic universe, but it demonstrates that the rules of general relativity alone do not prohibit the existence of closed timelike curves.

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Figure 24: A circular-time universe, constructed by identifying two different moments in flat spacetime. Two closed timelike curves are shown: one that loops through only once before closing, from (a) to (a‘), and another that loops twice, from (b) to (b’) to (b“) to (b’ ”).

THE GATE INTO YESTERDAY

There are two major reasons why most people who have given the matter a moment’s thought would file the possibility of time travel under “Science Fiction,” not “Serious Research.” First, it’s hard to see how to actually create a closed timelike curve, although we’ll see that some people have ideas. But second, and more fundamentally, it’s hard to see how the notion could make sense. Once we grant the possibility of traveling into our own past, it’s just too easy to invent nonsensical or paradoxical situations.

To fix our ideas, consider the following simple example of a time machine: the gate into yesterday. (“Gate into tomorrow” would be equally accurate—just go the other way.) We imagine there is a magical gate, standing outside in a field. It’s a perfectly ordinary gate in every way, with one major exception: When you walk through from what we’ll call “the front,” you emerge in the same field on the other side, but one day earlier—at least, from the point of view of the “background time” measured by outside observers who don’t ever step through the gate. (Imagine fixed clocks standing in the field, never passing through the gate, synchronized in the rest frame of the field itself.) Correspondingly, when you walk through the back side of the gate, you emerge through the front one day later than when you left.

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Figure 25: The gate into yesterday, showing one possible world line. A traveler walks through the front of the gate from the right (a) and appears out the back one day in the past (a‘). The person spends half a day walking around the side of the gate to enter from the front again (b) and reappears one day earlier (b’). Then the person waits a day and enters the back side of the gate (c), emerging from the front one day in the future (c‘).

It sounds magical and wondrous, but all we’ve done is describe a particular sort of unusual spacetime; we’ve identified a set of points in space at unequal times. Nobody is disappearing in any puffs of smoke; from the point of view of any particular observer, their own world line marches uninterruptedly into the future, one second per second. When you look through the gate from the front, you don’t see inky blackness, or swirling psychedelic colors; you see the field on the other side, just as you would if you looked through any other door. The only difference is, you see what it looked like yesterday. If you peer around the side of the gate, you see the field today, while peering through the gate gives you a view of one day before. Likewise, if you move around the other side and peer through the gate from the back, you see just the other side of the field, but you see what it will look like tomorrow. There is nothing to stop you from walking through the gate and immediately back again, any number of times you like, or for that matter from standing with one foot on either side of the threshold. You wouldn’t feel any strange tingling sensations if you did so; everything would seem completely normal, except that an accurate fixed clock on one side of you would read one day later than a fixed clock on the other side.

The gate-into-yesterday spacetime clearly contains closed timelike curves. All you have to do is walk through the front side of the gate to go back in time by one day, then walk around the side of the gate back to the front, and wait there patiently. After one day has passed, you will find yourself at the same place and time in spacetime as you were one day earlier (by your personal reckoning)—and of course, you should see your previous self there. If you like, you can exchange pleasantries with your former self, chatting about what the last day was like. That’s a closed timelike curve.

This is where the paradoxes come in. For whatever reason, physicists love to make their thought experiments as violent and deadly as possible; think of Schrödinger and his poor cat.81 When it comes to time travel, the standard scenario is to imagine going back into the past and killing your grandfather before he met your grandmother, ultimately preventing your own birth. The paradox itself is then obvious: If your grandparents never met, how did you come into existence in order to go back and kill one of them?82

We don’t need to be so dramatic. Here’s a simpler and friendlier version of the paradox. You walk up to the gate into yesterday, and as you approach you see a version of yourself waiting for you there, looking about one day older than you presently are. Since you know about the closed timelike curves, you are not too surprised; obviously you lingered around after passing through the gate, looking forward to the opportunity to shake hands with a previous version of yourself. So the two versions of you exchange pleasantries, and then you leave your other self behind as you walk through the front of the gate into yesterday. But after passing through, out of sheer perverseness, you decide not to go along with the program. Rather than hanging around to meet up with your younger self, you wander off, catching a taxi to the airport and hopping on a flight to the Bahamas. You never do meet up with the version of yourself that went through the gate in the first place. But that version of yourself did meet with a future version of itself—indeed, you still carry the memory of the meeting. What is going on?

ONE SIMPLE RULE

There is a simple rule that resolves all possible time travel paradoxes.83 Here it is:

• Paradoxes do not happen.

It doesn’t get much simpler than that.

At the moment, scientists don’t really know enough about the laws of physics to say whether they permit the existence of macroscopic closed timelike curves. If they don’t, there’s obviously no need to worry about paradoxes. The more interesting question is, do closed timelike curves necessarily lead to paradoxes? If they do, then they can’t exist, simple as that.

But maybe they don’t. We all agree that logical contradictions cannot occur. More specifically, in the classical (as opposed to quantum mechanical84) setup we are now considering, there is only one correct answer to the question “What happened at the vicinity of this particular event in spacetime?” In every part of spacetime, something happens—you walk through a gate, you are all by yourself, you meet someone else, you somehow never showed up, whatever it may be. And that something is whatever it is, and was whatever it was, and will be whatever it will be, now and forever. If, at a certain event, your grandfather and grandmother were getting it on, that’s what happened at that event. There is nothing you can do to change it, because it happened. You can no more change events in your past in a spacetime with closed timelike curves than you can change events that already happened in an ordinary, no-closed-timelike-curves spacetime.85

It should be clear that consistent stories are possible, even in spacetimes with closed timelike curves. Figure 25 depicts the world line of one intrepid adventurer who jumps back in time twice, then gets bored and jumps forward once, before walking away. There’s nothing paradoxical about that. And we can certainly imagine a non-paradoxical version of the scenario from the end of the previous section. You approach the gate, where you see an older version of yourself waiting for you there; you exchange pleasantries, and then you leave your other self behind as you walk through the front of the gate into yesterday. But instead of obstinately wandering off, you wait around a day to meet up with the younger version of yourself, with whom you exchange pleasantries before going on your way. Everyone’s version of every event would be completely consistent.

We can have much more dramatic stories that are nevertheless consistent. Imagine that we have been appointed Guardian of the Gate, and our job is to keep vigilant watch over who passes through. One day, as we are standing off to the side, we see a stranger emerge from the rear side of the gate. That’s no surprise; it just means that the stranger will enter (“has entered”?—our language doesn’t have the tenses to deal with time travel) the front side of the gate tomorrow. But as you keep vigilant watch, you see that the stranger who emerged simply loiters around for one day, and when precisely twenty-four hours have passed, walks calmly through the front of the gate. Nobody ever approached from elsewhere—the entering and exiting strangers formed a closed loop, and that twenty-four hours constituted the stranger’s entire life span. That may strike you as weird or unlikely, but there is nothing paradoxical or logically inconsistent about it.86

The real question is, what happens if we try to cause trouble? That is, what if we choose not to go along with the plan? In the story where you meet a slightly older version of yourself just before you cross through the front of the gate and jump backward in time, the crucial point is that you seem to have a choice once you pass through. You can obediently fulfill your apparent destiny, or you can cause trouble by wandering off. If that’s the choice you make, what is to stop you? That is where the paradoxes seem to get serious.

We know what the answer is: That can’t happen. If you met up with an older version of yourself, we know with absolute metaphysical certainty that once you age into that older self, you will be there to meet with your younger self. Imagine that we remove messy human beings from the problem by just considering simple inanimate objects, like series of billiard balls passing through the gate. There may be more than one consistent set of things that could happen at the various events in spacetime—but one and only one set of things will actually occur.87 Consistent stories happen; inconsistent ones do not.

ENTROPY AND TIME MACHINES

The issue that troubles us, when we get right down to it, isn’t anything about the laws of physics; it’s about free will. We have a strong feeling that we can’t be predestined to do something we choose not to do; that becomes a difficult feeling to sustain, if we’ve already seen ourselves doing it.

There are times when our free will must be subjugated to the laws of physics. If we get thrown out of a window on the top floor of a skyscraper, we expect to hurtle to the ground, no matter how much we would rather fly away and land safely elsewhere. That kind of predestination we’re willing to accept. But the much more detailed kind implied by closed timelike curves, where it seems that the working out of a consistent history through spacetime simply forbids us from making free choices that would otherwise be possible, is bothersome. Sure, we could be committed determinists and imagine that all of the atoms in our bodies and in the external world, following the unbending dictates of Newton’s laws of motion, will conspire to force us to behave in precisely the required way in order to avoid paradoxes, but it seems somewhat at variance with the way we think about ourselves.88

The nub of the problem is that you can’t have a consistent arrow of time in the presence of closed timelike curves. In general relativity, the statement “We remember the past and not the future” becomes “We remember what happened within our past light cone, but not within our future light cone.” But on a closed timelike curve, there are spacetime events that are both in our past light cone and in our future light cone, since those overlap. So do we remember such events or not? We might be able to guarantee that events along a closed timelike curve are consistent with the microscopic laws of physics, but in general they cannot be compatible with an uninterrupted increase of entropy along the curve.

To emphasize this point, think about the hypothetical stranger who emerges from the gate, only to enter it from the other side one day later, so that their entire life story is a one-day loop repeated ad infinitum. Take a moment to contemplate the exquisite level of precision required to pull this off, if we were to think about the loop as “starting” at one point. The stranger would have to ensure that, one day later, every single atom in his body was in precisely the right place to join up smoothly with his past self. He would have to make sure, for example, that his clothes didn’t accumulate a single extra speck of dust that wasn’t there one day earlier in his life, that the contents of his digestive tract was precisely the same, and that his hair and toenails were precisely the same length. This seems incompatible with our experience of how entropy increases—to put it mildly—even if it’s not strictly a violation of the Second Law (since the stranger is not a closed system). If we merely shook hands with our former selves, rather than joining up with them, the required precision doesn’t seem quite so dramatic; but in either case the insistence that we be in the right place at the right time puts a very stringent constraint on our possible future actions.

Our concept of free will is intimately related to the idea that the past may be set in stone, but the future is up for grabs. Even if we believe that the laws of physics in principle determine the future evolution of some particular state of the universe with perfect fidelity, we don’t know what that state is, and in the real world the increase of entropy is consistent with any number of possible futures. The kind of predestination seemingly implied by consistent evolution in the presence of closed timelike curves is precisely the same we would get into if there really were a low-entropy future boundary condition in the universe, just on a more local scale.

In other words: If closed timelike curves were to exist, consistent evolution in their presence would seem just as strange and unnatural to us as a movie played backward, or any other example of evolution that decreases entropy. It’s not impossible; it’s just highly unlikely. So either closed timelike curves can’t exist, or big macroscopic things can’t travel on truly closed paths through spacetime—or everything we think we know about thermodynamics is wrong.

PREDICTIONS AND WHIMSY

Life on a closed timelike curve seems depressingly predestined: If a system moves on a closed loop along such a curve, it is required to come back to precisely the state in which it started. But from the point of view of an observer standing outside, closed timelike curves also raise what is seemingly the opposite problem: What happens along such a curve cannot be uniquely predicted from the prior state of the universe. That is, we have the very strong constraint that evolution along a closed timelike curve must be consistent, but there can be a large number of consistent evolutions that are possible, and the laws of physics seem powerless to predict which one will actually come to pass.89

We’ve talked about the contrast between a presentist view of the universe, holding that only the current moment is real, and an eternalist or block-universe view, in which the entire history of the universe is equally real. There is an interesting philosophical debate over which is the more fruitful version of reality; to a physicist, however, they are pretty much indistinguishable. In the usual way of thinking, the laws of physics function as a computer: You give as input the present state, and the laws return as output what the state will be one instant later (or earlier, if we wish). By repeating this process multiple times, we can build up the entire predicted history of the universe from start to finish. In that sense, complete knowledge of the present implies complete knowledge of all of history.

Closed timelike curves make that program impossible, as a simple thought experiment reveals. Hearken back to the stranger who appeared out of the gate into yesterday, then jumped back in the other side a day later to form a closed loop. There would be no way to predict the existence of such a stranger from the state of the universe at an earlier time. Let’s say that we start in a universe that, at some particular moment, has no closed timelike curves. The laws of physics purportedly allow us to predict what happens in the future of that moment. But if someone creates closed timelike curves, that ability vanishes. Once the closed timelike curves are established, mysterious strangers and other random objects can consistently appear and travel around them—or not. There is no way to predict what will happen, just from knowing the complete state of the universe at a previous time.

We can insist all we like, in other words, that what happens in the presence of closed timelike curves be consistent—there are no paradoxes. But that’s not enough to make it predictable, with the future determined by the laws of physics and the state of the universe at one moment in time. Indeed, closed timelike curves can make it impossible to define “the universe at one moment in time.” In our previous discussions of spacetime, it was crucially important that we were allowed to “slice” our four-dimensional universe into three-dimensional “moments of time,” the complete set of which was labeled with different values of the time coordinate. But in the presence of closed timelike curves, we generally won’t be able to slice spacetime that way.90 Locally—in the near vicinity of any particular event—the division of spacetime into “past” and “future” as defined by light cones is perfectly normal. Globally, we can’t divide the universe consistently into moments of time.

In the presence of closed timelike curves, therefore, we have to abandon the concept of “determinism”—the idea that the state of the universe at any one time determines the state at all other times. Do we value determinism so highly that this conflict means we should reject the possibility of closed timelike curves entirely? Not necessarily. We could imagine a different way in which the laws of physics could be formulated—not as a computer that calculates the next moment from the present moment, but as some set of conditions that are imposed on the history of the universe as a whole. It’s not clear what such conditions might be, but we have no way of excluding the idea on the basis of pure thought.

All this vacillation might come off as unseemly, but it reflects an important lesson. Some of our understanding of time is based on logic and the known laws of physics, but some of it is based purely on convenience and reasonable-sounding assumptions. We think that the ability to uniquely determine the future from knowledge of our present state is important, but the real world might end up having other ideas. If closed timelike curves could exist, we would have a definitive answer to the debate between eternalism and presentism: The eternalist block universe would win hands down, for the straightforward reason that the universe can’t be nicely divided into a series of “presents” if there are closed timelike curves lurking around.

The ultimate answer to the puzzles raised by closed timelike curves is probably that they simply don’t (and can’t) exist. But if that’s true, it’s because the laws of physics won’t let you warp spacetime enough to create them, not because they let you kill your ancestors. So it’s to the laws of physics we should turn.

FLATLAND

Closed timelike curves offer an interesting thought-experiment laboratory in which to explore the nature of time. But if we’re going to take them seriously, we need to ask whether or not they could exist in the real world, at least according to the rules of general relativity.

I’ve already mentioned a handful of solutions to Einstein’s equation that feature closed timelike curves—the circular-time universe, the Gödel universe, the inner region near the singularity of a rotating black hole, and an infinite spinning cylinder. But these all fall short of our idea of what it would mean to “build” a time machine—to create closed timelike curves where there weren’t any already. In the case of the circular-time universe, the Gödel universe, and the rotating cylinder, the closed timelike curves are built into the universe from the start.91 The real question is, can we make closed timelike curves in a local region of spacetime?

Glancing all the way back at Figure 23, it’s easy to see why all of these solutions feature some kind of rotation—it’s not enough to tilt light cones; we want them to tilt around in a circle. So if we were to sit down and guess how to make a closed timelike curve in spacetime, we might think to start something rotating—if not an infinite cylinder or a black hole, then perhaps a pretty long cylinder, or just a very massive star. We might be able to get even more juice by starting with two giant masses, and shooting them by each other at an enormous relative speed. And then, if we got lucky, the gravitational pull of those masses would distort the light cones around them enough to create a closed timelike curve.

That all sounds a bit loosey-goosey, and indeed we’re faced with an immediate problem: 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).

In this spirit of dramatic simplification, we can ask, what would happen if two massive objects passed by each other at high relative velocity, but in a universe with only three dimensions of spacetime? That is, instead of the three dimensions of space and one dimension of time in our real four-dimensional spacetime, let’s pretend that there are only two dimensions of space, to make three spacetime dimensions in total.

Throwing away a dimension of space in the interest of simplicity is a venerable move. Edwin A. Abbott, in his book Flatland, conceived of beings who lived in a two-dimensional space as a way of introducing the idea that there could be more than three dimensions, while at the same time taking the opportunity to satirize Victorian culture.92 We will borrow Abbott’s terminology, and refer to a universe with two spatial dimensions and one time dimension as “Flatland,” even if it’s not really flat—we care about cases where spacetime is curved, and light cones can tip, and timelike curves can be closed.

STUDYING TIME MACHINES IN FLATLAND (AND IN CAMBRIDGE)

Consider the situation portrayed in Figure 26, where two massive objects in Flatland are zooming past each other at high velocity. The marvelous feature of a three-dimensional universe is that Einstein’s equation simplifies enormously, and what would have been an impossibly complicated problem in the real four-dimensional world can now be solved exactly. In 1991, astrophysicist Richard Gott rolled up his sleeves and calculated the spacetime curvature for this situation. Remarkably, he found that heavy objects moving by each other in Flatland do lead to closed timelike curves, if they are moving fast enough. For any particular value of the mass of the two bodies, Gott calculated a speed at which they would have to be moving in order to tilt the surrounding light cones sufficiently to open up the possibility of time travel.93

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Figure 26: A Gott time machine in Flatland. If two objects pass by each other with sufficiently high relative velocity, the dashed loop will be a closed timelike curve. Note that the plane illustrated here is truly two-dimensional, not a projection of three-dimensional space.

This is an intriguing result, but it doesn’t quite count as “building” a time machine. In Gott’s spacetime, the objects start out far apart, pass by each other, and then zip back out to infinity again. Ultimately, the closed timelike curves were destined to occur; there is no point in the evolution where their formation could have been avoided. So the question still lingers—can we build a Gott time machine? For example, we could imagine starting with two massive objects in Flatland that were at rest with respect to each other, and hooking up rocket engines to each of them. (Keep telling yourself: “thought experiment.”) Could we accelerate them fast enough to create closed timelike curves? That would really count as “building a time machine,” albeit in somewhat unrealistic circumstances.

The answer is fascinating, and I was lucky enough to be in on the ground floor when it was worked out.94 When Gott’s paper appeared in 1991, I was a graduate student at Harvard, working mostly with my advisor, George Field. But like many Harvard students, I frequently took the Red Line subway down to MIT to take courses that weren’t offered at my home institution. (Plenty of MIT students came the other way for similar reasons.) Among these were excellent courses on theoretical particle physics from Edward (“Eddie”) Farhi, and on early-universe cosmology from Alan Guth. Eddie was a younger guy with a Bronx accent and a fairly no-nonsense attitude toward physics, at least for someone who wrote papers like “Is it Possible to Create a Universe in the Laboratory by Quantum Tunneling?”95 Alan was an exceptionally clear-minded physicist who was world-famous as the inventor of the inflationary universe scenario. They were both also friendly and engaged human beings, guys with whom you’d be happy to socialize with, even without interesting physics to talk about.

So I was thrilled and honored when the two of them pulled me into a collaboration to tackle the question of whether it was possible to build a Gott time machine. Another team of theorists—Stanley Deser, Roman Jackiw, and Nobel laureate Gerard ’t Hooft—were also working on the problem, and they had uncovered a curious feature of the two moving bodies in Gott’s universe: Even though each object by itself moved slower than the speed of light, when taken together the total system had a momentum equivalent to that of a tachyon. It was as if two perfectly normal particles combined to create a single particle moving faster than light. In special relativity, where there is no gravity and spacetime is perfectly flat, that would be impossible; the combined momentum of any number of slower-than-light particles would add up to give a nice slower-than-light total momentum. It is only because of the peculiarities of curved spacetime that the velocities of the two objects could add together in that funny way. But to us, it wasn’t quite the final word; who is to say that the peculiarities of curved spacetime didn’t allow you to make tachyons?

We tackled the rocket-ship version of the problem: Could you start with slowly moving objects and accelerate them fast enough to make a time machine? When put that way, it’s hard to see what could possibly go wrong—with a big enough rocket, what’s to stop you from accelerating the heavy objects to whatever speed you like?

The answer is, there’s not enough energy in the universe. We started by assuming an “open universe”—the plane in Flatland through which our particles were moving extended out to infinity. But it is a peculiar feature of gravity in Flatland that there is an absolute upper limit on the total amount of energy that you can fit in an open universe. Try to fit more, and the spacetime curvature becomes too much, so that the universe closes in on itself.96 In four-dimensional spacetime, you can fit as much energy in the universe as you like; each bit of energy curves spacetime nearby, but the effect dilutes away as you go far from the source. In three-dimensional spacetime, by contrast, the effect of gravity doesn’t dilute away; it just builds up. In an open three-dimensional universe, therefore, there is a maximum amount of energy you can possibly have—and it is not enough to make a Gott time machine if you don’t have one to start with.

That’s an interesting way for Nature to manage to avoid creating a time machine. We wrote two papers, one by the three of us that gave reasonable-sounding arguments for our result, and another with Ken Olum that proved it in greater generality. But along the way we noticed something interesting. There’s an upper limit to how much energy you can have in an open Flatland universe, but what about a closed universe? If you try to stick too much energy into an open universe, the problem is that it closes in on itself. But turn that bug into a feature by considering closed universes, where space looks something like a sphere instead of like a plane.97 Then there is precisely one value of the total amount of allowed energy—there is no wriggle room; the total curvature of space has to add up to be exactly that of a sphere—and that value is twice as large as the most you can fit in an open universe.

When we compared the total amount of energy in a closed Flatland universe to the amount you would need to create a Gott time machine, we found there was enough. This was after we had already submitted our first paper and it had been accepted for publication in Physical Review Letters, the leading journal in the field. But journals allow you to insert small notes “added in proof” to your papers before they are published, so we tacked on a couple of sentences mentioning that we thought you could make a time machine in a closed Flatland universe, even if it were impossible in an open universe.

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Figure 27: Particles moving in a closed Flatland universe, with the topology of a sphere. Think of ants crawling over the surface of a beach ball.

We goofed. (The single best thing about working with famous senior collaborators as a young scientist is that, when you goof, you can think to yourself, “Well if even those guys didn’t catch this one, how dumb could it have been?”) It did seem a little funny to us that Nature had been so incredibly clever in avoiding Gott time machines in open universes but didn’t seem to have any problem with them in closed universes. But there was certainly enough energy to accelerate the objects to sufficient velocity, so again—what could possibly go wrong?

Very soon thereafter, Gerard ’t Hooft figured out what could go wrong. A closed universe, unlike an open universe, has a finite total volume—really a “finite total area,” since we have only two spatial dimensions, but you get the idea. What ’t Hooft showed was that, if you set some particles moving in a closed Flatland universe in an attempt to make a Gott time machine, that volume starts to rapidly decrease. Basically, the universe starts to head toward a Big Crunch. Once that possibility occurs to you, it’s easy to see how spacetime avoids making a time machine—it crunches to zero volume before the closed timelike curves are created. The equations don’t lie, and Eddie and Alan and I acknowledged our mistake, submitting an erratum to Physical Review Letters . The progress of science marched on, seemingly little worse for the wear.

Between our result about open universes and ’t Hooft’s result about closed universes, it was clear that you couldn’t make a Gott time machine in Flatland by starting from a situation where such a time machine wasn’t already there. It may seem that much of the reasoning used to derive these results is applicable only to the unrealistic case of three-dimensional spacetime, and you would be right. But it was very clear that general relativity was trying to tell us something: It doesn’t like closed timelike curves. You can try to make them, but something always seems to go wrong. We would certainly like to ask how far you could push that conclusion into the real world of four-dimensional spacetime.

WORMHOLES

In the spring of 1985, Carl Sagan was writing a novel—Contact, in which astrophysicist Ellie Arroway (later to be played by Jodie Foster in the movie version) makes first contact with an alien civilization.98 Sagan was looking for a way to move Ellie quickly over interstellar distances, but he didn’t want to take the science fiction writer’s lazy way out and invoke warp drive to move her faster than light. So he did what any self-respecting author would do: He threw his heroine into a black hole, hoping that she would pop out unharmed twenty-six light-years away.

Not likely. Poor Ellie would have been “spaghettified”—stretched to pieces by the tidal forces near the singularity of the black hole, and not spit out anywhere at all. Sagan wasn’t ignorant of black-hole physics; he was thinking about rotating black holes, where the light cones don’t actually force you to smack into the singularity, at least according to the exact solution that had been found by Roy Kerr back in the sixties. But he recognized that he wasn’t the world’s expert, either, and he wanted to be careful about the science in his novel. Happily, he was friends with the person who was the world’s expert: Kip Thorne, a theoretical physicist at Caltech who is one of the foremost authorities on general relativity.

Thorne was happy to read Sagan’s manuscript, and noticed the problem: Modern research indicates that black holes in the real world aren’t quite as well behaved as the pristine Kerr solution. An actual black hole that might have been created by physical processes in our universe, whether spinning or not, would chew up an intrepid astronaut and never spit her out. But there might be an alternative idea: a wormhole.

Unlike black holes, which almost certainly exist in the real world and for which we have a great deal of genuine observational evidence, wormholes are entirely conjectural playthings of theorists. The idea is more or less what it sounds like: Take advantage of the dynamical nature of spacetime in general relativity to imagine a “bridge” connecting two different regions of space.

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Figure 28: A wormhole connecting two distant parts of space. Although it can’t be accurately represented in a picture, the physical distance through the wormhole could be much shorter than the ordinary distance between the wormhole mouths.

A typical representation of a wormhole is depicted in Figure 28. The plane represents three-dimensional space, and there is a sort of tunnel that provides a shortcut between two distant regions; the places where the wormhole connects with the external space are the “mouths” of the wormhole, and the tube connecting them is called the “throat.” It doesn’t look like a shortcut—in fact, from the picture, you might think it would take longer to pass through the wormhole than to simply travel from one mouth to the other through the rest of space. But that’s just a limitation on our ability to draw interesting curved spaces by embedding them in our boring local region of three-dimensional space. We are certainly welcome to contemplate a geometry that is basically of the form shown in the previous figure, but in which the distance through the wormhole is anything we like—including much shorter than the distance through ordinary space.

In fact, there is a much more intuitive way of representing a wormhole. Just imagine ordinary three-dimensional space, and “cut out” two spherical regions of equal size. Then identify the surface of one sphere with the other. That is, proclaim that anything that enters one sphere immediately emerges out of the opposite side of the other. What we end up with is portrayed in Figure 29; each sphere is one of the mouths of a wormhole. This is a wormhole of precisely zero length; if you enter one sphere, you instantly emerge out of the other. (The word instantly in that sentence should set off alarm bells—instantly to whom?)

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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.

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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, creating 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.