People all have their own ideas of what a time machine would look like. If you are a fan of the 1960 movie version of H. G. Wells’s classic novel, it would be a steampunk sled with a red velvet chair, flashing lights, and a giant spinning wheel on the back. For those whose notions of time travel were formed in the 1980s, it would be a souped-up stainless steel sports car. Details of operation vary from model to model, but they all have one thing in common: When someone actually travels through time, the machine ostentatiously dematerializes, only to reappear many years in the past or future. And most people could tell you that such a time machine would never work, even if it looked like a DeLorean.
They would be half right: That is not how time travel might work, but time travel in some other form is not necessarily off the table. Since time is kind of like space (the four dimensions go hand in hand), a working time machine would zoom off like a rocket rather than disappearing in a puff of smoke. Einstein described our universe in four dimensions: the three dimensions of space and one of time. So traveling back in time is nothing more or less than the fourth-dimensional version of walking in a circle. All you would have to do is use an extremely strong gravitational field, like that of a black hole, to bend space-time. From this point of view, time travel seems quite difficult but not obviously impossible.
These days, most people feel comfortable with the notion of curved space-time. What they trip up on is actually a more difficult conceptual problem, the time travel paradox. This is the worry that someone could go back in time and change the course of history. What would happen if you traveled into the past, to a time before you were born, and murdered your parents? Put more broadly, how do we avoid changing the past as we think we have already experienced it? At the moment, scientists don’t know enough about the laws of physics to say whether these laws would permit the time equivalent of walking in a circle — or, in the parlance of time travelers, a “closed timelike curve.” If they don’t permit it, there is obviously no need to worry about paradoxes. If physics is not an obstacle, however, the problem could still be constrained by logic. Do closed timelike curves necessarily lead to paradoxes?
If they do, then they cannot exist, simple as that. Logical contradictions cannot occur. More specifically, there is only one correct answer to the question “What happened at the vicinity of this particular event in space-time?” Something happens: You walk through a door, 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, once 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 space-time with closed timelike curves than you can change events that already happened in ordinary space-time, with no closed timelike curves.
As we will see, the time travel paradox — the possibility of changing our past — seems intractable only because it conflicts with our notion of ourselves as beings with free will. Consistent stories are possible, even in space-times with closed timelike curves.
To illustrate this point, imagine that you stumble upon a time machine in the form of a gate. When you pass through it in one direction, it takes you exactly one day into the past; if you pass through in the other direction, it takes you exactly one day into the future. You walk up to the gate, where you see an older version of yourself waiting for you. The two of you exchange pleasantries. Then you leave your other self behind as you walk through the gate into yesterday. But instead of obstinately wandering off, you wait around a day to meet up with the younger version of yourself (you have now aged into the older version you saw the day before) 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 person walk out of the rear side of the gate, emerging from one day in the future. That’s no surprise; it just means that you will see that person enter the front side of the gate tomorrow. But as you keep watch, you notice that he simply loiters around for one day, and when precisely 24 hours have passed, the traveler walks calmly through the front of the gate. Nobody ever approached from elsewhere. That 24-hour period constitutes the entire life span of this time traveler. He experiences the same thing over and over again, although he doesn’t realize it himself, since he does not accumulate new memories along the way. Every trip through the gate is precisely the same to him. That may strike you as weird or unlikely, but there is nothing paradoxical or logically inconsistent about it.
The real question is this: What happens if we try to cause trouble? That is, what if we choose not to go along with the plan? Let’s say you meet a day-older version of yourself just before you cross through the front of the gate and jump backward in time, as if you will hang around for a day to greet yourself in the past. But once you actually do jump backward in time, you still seem to have a choice about what to do next. You can obediently fulfill your apparent destiny, or you can cause trouble by wandering off. What is to stop you from deciding to wander? That seems like it would create a paradox. Your younger self bumped into your older self, but your older self decides not to cooperate, apparently violating the consistency of the story.
We know what the answer is: That cannot happen. If you met up with an older version of yourself, we know with absolute certainty that once you age into that older self, you will be there to meet your younger self. That is because, from your personal point of view, that meet-up happened, and there is no way to make it un-happen, any more than we can change the past without any time travel complications. There may be more than one consistent set of things that could happen at the various events in space-time, but one and only one set of things actually does occur. Consistent stories happen; inconsistent ones do not. The vexing part is understanding what forces us to play along.
The issue that troubles us, when you get down to it, is free will. We have a strong feeling that we cannot be predestined to do something we choose not to do. That becomes a difficult feeling to sustain if we have already seen ourselves doing it.
Of course, there are some kinds of predestination we are willing to accept. 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. The much more detailed kind of predestination implied by closed timelike curves, where it seems that we simply cannot make certain choices (like walking away after meeting a future version of ourselves), is bothersome.
The nub of the problem is that you cannot have a consistent “arrow of time” in the presence of closed timelike curves. The arrow of time is simply the distinction between the past and the future. We can turn an egg into an omelet, but not an omelet into an egg; we remember yesterday, but not tomorrow; we are born, grow older, and die, never the reverse. Scientists explain all of these manifestations of the arrow of time in terms of entropy — loosely, the “disorderliness” of a system. A neatly stacked collection of papers has a low entropy, while the same collection scattered across a desktop has a high entropy. The entropy of any system left to its own devices will either increase with time or stay constant; that is the celebrated second law of thermodynamics. The arrow of time comes down to the fact that entropy increases toward the future and was lower in the past.
A statement like “We remember the past and not the future” makes perfect sense to us under ordinary circumstances. But in the presence of closed timelike curves, some events are in our past and also in our future. So do we remember such events or not? In general, events along a closed timelike curve cannot be compatible with an uninterrupted increase of entropy along the curve. That’s a puzzle: On a closed curve, the entropy has to finish exactly where it started, but the arrow of time says that entropy tends to increase and never decrease. Something has to give.
To emphasize this point, think about the hypothetical traveler who emerges from the gate, only to enter it from the other side one day later, so that his 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 think about the loop as “starting” at one point. The traveler 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 did not accumulate a single extra speck of dust that was not there one day earlier. This seems incompatible with our experience of how entropy increases. If we merely shook hands with our former selves, rather than joining up with them, the required precision doesn’t seem quite so dramatic. In either case, though, 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 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. A closed timelike curve seems to imply predestination: We know what is going to happen to us in the future because we witnessed it in our past.
Closed timelike curves, in other words, make the future resemble the past. It is set in stone, not up for grabs at all. The reason we think the past is fixed once and for all is that there is a boundary condition at the beginning of time. The entropy of the universe started very small (at the time of the Big Bang) and has been growing ever since. Ordinarily we do not imagine that there is any analogous boundary condition in the future — entropy continues to grow, but we cannot use that information to draw any conclusions. If we use a closed timelike curve to observe something about our future actions, those actions become predestined. That’s extra information about the history of the universe, over and above what we normally glean from the laws of physics, and it makes us uncomfortable.
If closed timelike curves exist, ensuring that all events are consistent is 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 cannot exist, or big, macroscopic things cannot travel on truly closed paths through space-time — unless everything we think we know about entropy and the arrow of time is wrong.
Life on a closed timelike curve seems pretty drab. Once you start moving along such a curve, you are required to come back to precisely the point at which you started. An observer standing outside, however, has what is seemingly the opposite problem: What happens along such a curve cannot be uniquely predicted from the prior state of the universe. We have the strong constraint that evolution along a closed timelike curve must be consistent, but there will always 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.
In the usual way of thinking, the laws of physics function like 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 many times, we can build up the entire 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 would make such a program impossible, as a simple thought experiment reveals. Hark 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 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. This ability vanishes as soon as someone builds a time machine and creates a closed timelike curve. Mysterious strangers and other random objects can then appear out of thin air and disappear just as quickly.
We can insist all we like that what happens in the presence of closed timelike curves be consistent. But that requirement is not enough to make the events 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 de-fine “the universe at one moment in time.” Ordinarily we can imagine “slicing” our four-dimensional universe into three-dimensional “moments of time.” In the presence of closed timelike curves, though, we generally will not be able to slice space-time that way. Locally — in the near vicinity of any particular point in space-time — we can always divide events into the “past” and the “future.” But we might not be able to do this throughout the universe. The warping associated with the closed timelike curve could cause our slice to twist back on itself, making it impossible to divide all of space-time into distinct moments.
We would therefore 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. We would also have to abandon free will — because witnessing part of our future history implies some amount of predestination.
Do we value determinism so highly that 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 a set of conditions that are imposed on the history of the universe as a whole. It is not clear what such conditions might be, but we have no way of excluding the idea on the basis of pure thought.
All this may sound like vacillation, but it provides 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 physicists discover that closed timelike curves really can exist, we will have to dramatically rethink the way we understand time. In that case, the universe could not be nicely divided into a series of separate “moments” of time.
The ultimate answer to the puzzles raised by closed timelike curves is probably that they simply cannot exist. If that is true, though, it is because the laws of physics do not let you warp space-time enough to create them — not because they let you kill your grandfather before you are born.
This piece is adapted from Cosmic Variance blogger Sean Carroll’s latest book, From Eternity to Here: The Quest for the Ultimate Theory of Time, which was published last month by Dutton.
