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.
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.
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 poss
ible, 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. I
n 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.
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 13
