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
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.
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.
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 t
he 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?)
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 14
