Arlie Petters. Photo by Michael Zirkle, Photographed at the Nasher Museum of Art
Arlie Petters. Photo by Michael Zirkle, Photographed at the Nasher Museum of Art

Ripple Effect

A mathematician's search for evidence of tiny black holes could disprove Einstein's general theory of relativity—and open up a whole new dimension.
January 31, 2008
Put your right hand on your right temple, and your left on your left temple. Now gently squeeze; don't let up.

Okay. Now you are ready for a conversation with Arlie Petters, the energetic, broadly smiling man in the striped short-sleeve shirt and comfortable brown slacks who has come to meet you at a Barnes & Noble café. Many topics will come up. He will wonder whether amoebas could join the conversation. He will suggest searching the solar system for tiny black holes as a practical business enterprise. And he will say things like, "I don't see why reality should contain only three spatial dimensions." Again with the beaming, cheerful smile. "Do you?"

Now that he mentions it, maybe there are no good reasons for limiting reality to three spatial dimensions, but you are rather used to it that way, and … it's a good thing you didn't order coffee, because you could not pick up the cup: You need both hands to hold your skull together.

Perhaps Petters, a Duke professor of mathematics and physics, smiles so much because once he decided that he may have an entire extra physical dimension to work with, the usual limitations that make the rest of us so grumpy stopped pinching quite so tight. An additional dimension might enable him to be two places at once—saving money on daycare, perhaps, or at least making it easier to pick a child up there; or it might offer limitless extra time, or … space, or … something.

But maybe Petters is so cheerful because, if everything goes just right with a NASA satellite scheduled to launch any day now, it's possible that his name will be forever linked to the physical evidence that disproves Einstein's general theory of relativity. Or not exactly disproves Einstein's brilliant conflating of space, time, energy, and matter. "I would say that he missed something," Petters says. Something that, if it turns out to be true, "would give us a complete philosophical shift in our understanding of the physical world,"by proving that the physical universe has four, not three, spatial dimensions, making ours, when you include time, a five-dimensional universe. "It's a very exciting idea," Petters says.

He opens a clasp envelope and produces a sheaf of scrawled notes on lined paper, including a simple graph: an X- and a Y-axis, on which a straight dashed line angles down from left to right, with a sine wave superimposed. "The telltale wiggle," he calls it.

Here's the idea. The Gamma-ray  Large Area Space Telescope (GLAST), scheduled to be launched into Earth orbit by NASA in February, will spend its time looking at gamma rays, the most energetic form of light there is—billions of times more energetic than the waves our eyes can perceive; millions of times stronger than even X-rays. The result should be new information about things like pulsars and supernovae, the kind of unimaginably massive energy sources that emit gamma rays and exist at the very edge of our current understanding of physics.

But with the new telescope, Petters and his colleague in this project, Charles Keeton, a Rutgers University astronomer, saw an opportunity to go even further. That graph that Petters produced, which he calls a "back of the envelope calculation," resulted from a flurry of e-mail messages between the two a couple of years ago when they heard about the telescope. The graph represents how gamma rays would bend—the "wiggle" in the graph—if they happened to pass a tiny but massive object called a braneworld black hole.

A black hole—not the braneworld kind, but the kind that most of us have heard of, even if we still don't quite comprehend what it is—is a massive object like a star or many stars that has collapsed into an unimaginably small and dense space with a gravitational pull so strong that even light cannot escape it. Einstein's theories predicted the existence of black holes, since verified by scientists. A braneworld black hole is a special kind of black hole. It's tiny, the size of an atomic nucleus or smaller, but has the mass of an asteroid. For now, its existence is theoretical. Proof will come only if a specific variant of the string theory of gravitation, which disagrees with Einstein's theory, turns out to be true.

But ignore that for a moment. For now, just keep in mind that Petters and Keeton want to look for the wiggle they predict they will find in the gamma-ray graph if the gamma rays happen to pass by one of those braneworld black holes. These wiggles are the subject of Petters' research.

They're caused by gravitational lensing, a process by which light (of any electromagnetic wavelength) is bent in the warped conditions of space and time that occur near massive objects like planets, stars, or black holes. Einstein predicted this phenomenon, and it was first observed during a 1919 total eclipse of the sun, when background stars viewed directly past the darkened sun appeared slightly out of position. The sun's mass had actually bent the rays of light from those distant stars. The phenomenon was regarded as a brilliant proof of the warping of space and time described in Einstein's general theory of relativity. Such lensing, now better understood, thanks in part to Petters, can produce not only bent but also multiple images of distant objects. What's more, objects with certain masses affect light of specific wavelengths according to specific signatures. Subjected to mathematical analysis, these signature bends yield secrets about the objects that cause them.

"Imagine dropping a pebble into a pond," Petters says. The pebble generates waves, with peaks and valleys: Big rock, big waves; tiny pebble, smaller waves. That is, an object massive enough to be a gravitational lens leaves a signature pattern affecting a specific wavelength of light. And those tiny black holes predicted by the braneworld theory would produce a wiggle in the specific electromagnetic range that the GLAST will be measuring, once it's in orbit.

Bend it like Petters: He and his collaborator, Charles Keeton, aim to prove the existence of tiny black holes originating from an extra dimension by capturing the interference patterns created when gravity bends light, in the same way radio waves, green, above, are bent by the sun's gravity in a 2002 experiment

Bend it like Petters: He and his collaborator, Charles Keeton, aim to prove the existence of tiny black holes originating from an extra dimension by capturing the interference patterns created when gravity bends light, in the same way radio waves, green, above, are bent by the sun's gravity in a 2002 experiment. NASA

Okay, braneworld black holes. These currently exist only on the blackboards of scientists who believe in a certain variation of string theory. String theory is a cosmological theory that considers the tiniest building blocks of matter to be something like vibrating strings rather than particles. The mathematics of this complex model end up requiring additional dimensions for everything to work out, though in most versions, extra dimensions are treated as little more than convenient theoretical constructs.

But a variant of string theory developed by Lisa Randall of Harvard University and Raman Sundrum of the Johns Hopkins University posits the universe we perceive as a sort of three-dimensional membrane (hence "braneworld") floating in a multidimensional universe. Petters loves the pragmatic elegance: "In braneworld, what I like is that this fourth dimension extends to infinity. In other string-theory models, it's this tiny, curled-up dimension," kind of stuffed into an inexpressibly small space like the end of a fiddlehead fern. In those models, the fourth dimension doesn't affect anything except strings. "They're not letting it loose," he says.

As he tries to explain the added spatial dimension, he runs his fingers along the café tabletop. "Imagine we are beings that live only on this desk," he says. That is, we're two-dimensional beings, inhabiting this flat, two-dimensional space. "That's not a limit of our eyes or ears, that's a limit of our physical existence." His eyes grow large: "You can't get off this table." He cites the famous Victorian satire Flatland, a book about two-dimensional creatures who receive a visit from a sphere and are simply unable to comprehend its three-dimensionality. He thinks a minute, then takes an intellectual step backward, to first principles.

"There are two acts of faith that go into science," he says. The first is that "the physical world is understandable to the human mind." The second, that "you can model it mathematically—quantitatively."

Then we're back to life on the tabletop. "That first postulate, that physical reality is accessible, is not quite right—not all of it. Think of an amoeba," he says. "It's a tiny entity that's wiggling around on this desk," in this essentially two-dimensional space. "Now think of our conversation. It completely transcends that amoeba, because of its wiring." That is, it completely lacks the capacity to perceive us: Living on its tabletop, it's going to think the universe has the limits of its perceptions.

"Who are we to think we are any different than this amoeba in the full spectrum of reality?" Petters says. Just because we can't think of where that fourth spatial dimension would be, and lack the capacity to perceive it, doesn't mean it's not there.

Telltale wiggle: X-Y graph predicting how gamma rays would be affected when close enough to be bent by a braneworld black hole. Dashed line shows gamma rays unaffected by the black holes. Diagram courtesy of Arlie Petters and Charles Keeton

Telltale wiggle: X-Y graph predicting how gamma rays would be affected when close enough to be bent by a braneworld black hole. Dashed line shows gamma rays unaffected by the black holes. Diagram courtesy of Arlie Petters and Charles Keeton


Charles Keeton, Petters' partner in the paper they published about that telltale wiggle, cites a common way to try to imagine this extra dimension: "People often draw a piece of paper standing vertically," he says. "The third dimension," poking outward from both sides of the two-dimensional paper, "is perpendicular to both dimensions. Braneworld would have a fourth dimension perpendicular to all three dimensions" that we now perceive.

That's about as good a description as anybody can come up with—we seem to be like those tabletop amoebas, doomed to our limited understanding of reality. Still, take for comfort these words by perhaps the most famous journalist of our generation, uttered when she, too, was trying to comprehend a five-dimensional reality: "It had height, length, breadth," she said, "and a couple of other things." Those are the words of Lois Lane, describing Superman's nemesis Mr. Mxyzptlk, who came from the fifth dimension, wore a derby hat, and could be forced to return to his five-dimensional space only if tricked by Superman into saying his own name backwards. Looking at Mr. Mxyzptlk's five-dimensional manifestation, Lane said, "made my head hurt."

But hard as it may be for two-dimensional comic-book characters, or even us three-dimensional types, to wrap our minds around spatial four-dimensionality, the point is that the braneworld model puts it there, and that Petters and Keeton have found a way to look for its signature through the data gathered by the GLAST satellite.

That takes us back to the tiny black holes of braneworld. In an Einsteinian universe, black holes of that size could be created only in the conditions present at the dawn of the universe, and any created then would have evaporated by now. But according to the braneworld universe, they would not have evaporated and so would still exist to put their signature juju on that gamma-ray vibration.

If you find the interference pattern in the gamma rays, you've found a tiny braneworld black hole. That tiny black hole doesn't fit in Einstein's equations. Ergo, if you find the braneworld black hole, you demonstrate that the braneworld theory, not Einstein's, is correct. And scientists will probably start spending a lot of time looking for ways to investigate a fourth spatial dimension.

Petters leans back in his chair. "The way I look at this is, remember when we thought the world was flat?" He shakes his head. "Think of the poor elementary-school kids who will have to learn geometry into the fourth dimension." He imagines possible consequences of a fourth spatial dimension, wondering, for example, whether people will instantly try to develop weapons based on the incredibly high energy with which particles will move between dimensions. On the other hand, he recalls that in Flatland, a three-dimensional being—like you—could put a finger right into the middle of a two-dimensional being—like a square—without penetrating its boundaries: You could poke its insides without going through its "skin." Similarly, a being from four spatial dimensions might be able to poke you in, say, the spleen—from the inside. "But that's like spooks, right?" Petters asks. "That's like ghosts. You can run with this metaphor." He smiles. "It's almost like science fiction."

Yes, like science fiction. But tiny black holes the size of atomic particles, four spatial dimensions, gravity bending light? That isn't like science fiction. That stuff is all in a day's work.

Up, up, and away: GLAST telescope will observe highly energetic light rays, or gamma rays, that could prove the existence of a fifth dimension

Up, up, and away: GLAST telescope will observe highly energetic light rays, or gamma rays, that could prove the existence of a fifth dimension. NASA

Actually, Petters calls that stuff mathematical thinking, and it's what he brings to the table with Keeton. A native of Belize, he emigrated to the U.S. as a teenager and in high school recognized that mathematics had "a beauty of its own." His course was set: Hunter College, then MIT for his Ph.D. He describes mathematical thinking as abstract thinking that takes the elegance of pure thought and creates structures that eventually can be applied to reality. "You're dealing with very general structures that you don't give any physical meaning to," he says. "You just look at how these structures interact logically." So you consider the relationship between speed and energy or between mass and light or among the sides of a triangle, and you work out equations to represent those relationships.

"Then, in physical reasoning, you're looking at a special case," he says. "It has mass, it moves, it's alive in that sense." As you investigate it, "sometimes you realize this is a structure you've already studied in your mathematical reasoning, and so you import all that structure and give it physical meaning."

He considers this connection something of a miracle. "So in physical reasoning, because it fits with the real world, you imagine a braneworld black hole traveling across the solar system." But then you take your mathematical structures, and you start calculating, and you learn remarkable things. You take the belief that these tiny black holes constitute, say, one percent or so of the mysterious dark matter in the solar system (a reasonable assumption), you do the math, and you discover that there ought to be some 3,000 of these braneworld black holes in our solar system—sixteen or so in our immediate neighborhood, inside the orbit of Mars.

"There's something mysterious that happens," Petters says. "You have all these mathematical structures. If something has a four-dimensional geography, you should see this wiggle. And then you do the calculation, and … you get a prediction of 200 mega-electron volts!" And then you can go looking for a wiggle in that neighborhood, and maybe you find one, and if you do, you change the philosophical foundation of the perception of the world.

Cosmic events: supermassive black hole in the galaxy Centaurus A, above: in Hubble telescope view, below, light rays from a distant galaxy (blue halo) bend around a closer galaxy (white center) directly in front of it

Cosmic events: supermassive black hole in the galaxy Centaurus A, above: in Hubble telescope view, below, light rays from a distant galaxy (blue halo) bend around a closer galaxy (white center) directly in front of it. NASA Marshall Space Flight Center

Though it seems counterintuitive, abstract mathematical thinking keeps Petters tethered to reality. One reason the braneworld model attracts him is that it seems to work in the universe he perceives. Apart from their scrunching of the fourth spatial dimension, other string-theory models don't satisfy him because they would require as proof reactions that take place at energies so high that we likely won't have the capacity to test them for thousands of years. "I'm drawn to models that at least have a chance of [being] observable in this astrophysical realm," he says. "You have one life to live. I'm digging for gold in the mountainside, and I'd rather dig where there's a good chance you'll find it."

In fact, that's what drew his gaze and Keeton's to the sky, rather than to, say, the Large Hadron Collider, now being built near Geneva, Switzerland, in a circular tunnel twenty-seven kilometers in circumference. The collider will accelerate particles to nearly the speed of light. Trying to create and observe a tiny black hole there would cost billions of dollars, and the result would be so small that it would instantly disappear. "So we took a different approach," Petters says. "We said, You know, nature is supposed to create these things." They did that back-of-the-envelope math. "We said, Here's the signature, that's the wiggle, and the GLAST satellite doesn't cost billions of dollars." He laughs. "So it's a practical business decision."

How practical is hard to say. Three thousand of those tiny black holes floating around in our solar system seems like a lot, but remember: Each one is about the size of an atomic nucleus, and the GLAST satellite will be looking all over interstellar space. (And don't worry—if one of those tiny black holes comes near, it'll just float through the Earth, or you. "You have a lot of space," Petters says blithely, "between particles.")

Petters and Keeton now can do little more than wait and hope one of those black holes happens to float through the satellite's field of vision while it's looking at a gamma ray source—a staggering long shot at best. "Even if the theory is true," says Keeton, "it's very possible that we will not see it, because the black hole is not in the right place." Petters dares to hope it might happen "in my lifetime," but he's not making bets. Even if you dig where you think the gold is, there's a lot of mountain and not much gold. Einstein predicted gravitational lensing in 1915, and it was proved in 1919; but in 1936, he predicted lensing that caused double images, and that wasn't observed until 1979.

But Petters draws inspiration from predictions made, and proved, before Einstein's. "I go back to the way Neptune was discovered," he says—in 1846. Trying to understand irregularities in the orbit of Uranus, mathematicians figured out not only that another planet would explain them but where that planet ought to be. They told astronomers. Mathematical thinking created a structure; physical observation went looking for it; and there, just where the mathematicians told the astronomers it ought to be, was Neptune. Maybe that's why Arlie Petters smiles so much: He's engaged, full time, in teasing secrets out of the universe. That's what mathematicians do.

And, of course, even beyond the difficulty of hoping to stumble across impossibly small objects in the vastness of space, if in its search for braneworld black holes the GLAST comes up empty, there may be another explanation. "It could be," Petters says, "that Einstein was right."

If so, he'll be happy to explain it to you. Fortunately, there's nothing difficult to understand in Einstein.

  • Scott Huler is the senior writer at Duke Magazine.