From Eternity to Here (21 page)

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 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
?)

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.