From Eternity to Here (20 page)

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

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.