Duke University Alumni Magazine


Photo: Les Todd

A mathematician explores gravitational lensing, a method of tracing the intricate pathways of starlight.

s a boy, Arlie Petters lulled himself to sleep by peering out his window at the swath of stars across the black sky above his small hometown in Belize. The inquisitive youngster constantly peppered his indulgent elders in the Central American village with questions about those alluring, distant lights: "How did they get there? What holds them up? Why do they glimmer so?"

As Petters grew up, he kept that same eager sense of wonderment; today it has led him to revelations about the heavens more stunning than he once dreamed possible. His

curiosity has also propelled him into a brilliant career as a mathematician--with a Ph.D. from M.I.T., a post at Duke as the William and Sue Gross Associate Professor of Mathematics, and a mission to bring the wonder of astrophysics to students.

Petters still ponders the heavens, but he now concentrates his talents on a cosmic phenomenon that sounds like the most outlandish science fiction--gravitational lensing. Astronomers first observed gravitational lensing in 1979, when the startled scientists discovered that the image from a distant cosmic object appeared to split into multiple images due to the effect of gravitational force exerted by massive intervening objects between them and Earth. The effect begins, of course, with light streaming from a distant astronomical object, such as a star like the sun, or even an ultrabright quasar in the farthest reaches of the universe. Quasars--intergalactic beacons that blaze with the light of a trillion suns--are violent young galaxies with black holes lurking at their center that gobble stars like poppy seeds, crunching them into nothingness, and spewing out intense radiation.

As the light from a quasar speeds on its billion-year journey toward Earth, its path can be diverted when it slaloms past objects such as a whirling galaxy of billions of stars or a single neutron star--a collapsed, dead star so dense that a single teaspoon of its matter weighs 100 tons. Or the streaming light might be deflected by a "stellar black hole," the corpse of a giant star so immense that its gravity has caused it to collapse down to a pinpoint of infinitely dense matter. So powerful is the gravity of such objects that they warp space and time around them, bending the passing light rays like immense space-time lenses. These lenses--depending on the positions of Earth, the incoming light, and the intervening objects--may split a single image of a star or quasar into multiple images, or even magnify such images.

Gravitational lenses delighted astronomers because the phenomenon offered them the chance to "candle the universe"--using the distortion of light from distant galaxies, black holes, and stars to understand the structure of those objects. The lenses could even be used to detect planets around other stars and even intergalactic concentrations of "dark matter"--immense invisible masses of gas, dust, and dead stars that astrophysicists believe might profoundly affect the evolution and structure of the universe.

Unfortunately for astronomers, the universe seldom cooperates by producing simple cosmic phenomena. For one thing, the zigging and zagging of wayward celestial light can be extremely complex because of the gravity of many intervening bodies--like a golf ball putted across the frustrating, undulating green of the seventh hole at the Duke University Golf Course (its toughest green, says golf pro Ed Ibarguen). So, despite the scientific juiciness of gravitational lenses, no mathematician had dared tackle the incredibly thorny problem of creating a general mathematical theory to explain the lens' properties--the theory that would give astrophysicists the right tool to help them analyze the intricacies of wayward cosmic light and deduce what gravitational adventures it had experienced on its way to Earth.

Even Einstein, who first suggested that such lenses might exist, hadn't gone beyond figuring out how a lens would split the light of a single star into two images. Later theorists had gone through laborious calculations to yield only the result that light passing two stars would produce five images. Not until an affable, young, star-struck mathematician from Dangriga, Belize, tackled the problem would there arise the first promising general theory to sort out gravity-warped starlight streaming to Earth.

But that theory and, in fact, Arlie Petters' brilliant career almost didn't happen at all. In the first place, his initial boyhood enthusiasms were for his Methodist religion and art. "I didn't play a lot as a kid," he recalls. "After school I'd come home, do my homework, and then I'd sit at the dining table and draw. That's what I did all the time--draw, draw, draw. Initially, I drew a lot of pictures of nature and people. And then I began drawing more about how I felt, expressing my philosophical feelings by drawing distant horizons and quiet places and mysterious, even mystical settings." His religion and his art, he realized, were connected. "There's a certain joy and peace you get from a belief in God and trying to lead a Christian way of life. And I realized that the good feeling I had in church was similar to this good feeling I got when I look at a beautiful painting or sketch."

The young Arlie did have some inkling that a similar beauty might lie in the world of mathematics. "I had a cousin at home who used to explain a lot of things to me, so I used to bug him a lot. I remember he had books that had these strange symbols that I loved looking at." But when Petters emigrated to the U.S. as a teenager, he fully intended to become a preacher. "Although I did very well in science and math, it was just a side thing," he recalls. As he continued to explore math, though, he experienced the dawning of an intellectual passion. "At first, I didn't know what mathematical thinking was all about. I thought math dealt only with calculations, but as I began looking deeper into math, toward my junior year of high school, I began realizing it has a beauty of its own. And I discovered that the same joy I felt when I did art is present in the abstractions of mathematics."

Still, the starry sky of his boyhood beckoned: He entered Hunter College planning to study astrophysics. Again, he almost missed a promising mathematical career when family problems nearly forced him to drop out of college and return to Belize. Fortunately for Petters and for science, he received a Minority Access to Research Careers Fellowship. The support that came with the fellowship proved to be just the intellectual spark his career needed. When Petters told his mentor that he was interested in Einstein's theory of relativity, he was sent to an expert on the subject, Professor Edward Tryon. "Ed took me under his wing and voluntarily taught me relativity theory from scratch."

Relativity theory proved a stunning personal and intellectual revelation for the young Petters. "I grew up thinking of space and time as these independent things that have always existed. And what struck me about relativity was how Einstein had to alter completely his world view. He had to let go of tradition and immerse himself in this new world, where space and time are relative and can be warped. That turned my world view upside down. To me, it took so much bravery. He must have had a deep faith that relativity is right, because when you go through a whole shift in your world view like that, it's a frightening experience."

Taking to heart the lessons of both Einstein's courage and his mathematics, Petters himself became a sort of intellectual quasar, gobbling up every math and physics course at Hunter and proceeding to devour the scientific course offerings at City College of New York. There, he met the famed physicist Michio Kaku, author of such popular books as Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Kaku took Petters into an entirely different realm, from the vastness of the universe down to the infinitesimal world of the subatomic. He introduced Petters to "string theory," which holds that all matter is composed of infinitesimal mathematical entities resembling vibrating strings.

Elegant science: Mathematician Arlie Petters' theory of gravitational lensing can calculate how the gravitational fields of intervening objects will intricately sculpt the light from a distant cosmic object as it passes them on its way to Earth. These three computer diagrams plot the "caustics" produced by different arrays of such intervening objects. Caustics are positions in space where a distant object's image would be gravitationally focused by an array of intervening objects, down to intensely bright points called singularities.
Images courtesy of Arlie Petters

The adventure confirmed Petters' fascination with mathematics, and his brilliance as a student brought him yet another golden opportunity: a Bell Labs Fellowship to study math at one of the world's meccas of math and astronomy, M.I.T. There, he learned both math and humility. "I left Hunter College as a star, but when I arrived at M.I.T., I was just average," he says with a chuckle. The sobering experience led him to serious soul-searching to assess his talents realistically and seek a specialization to match them. "I discovered that my mathematical thinking is much stronger than my physical thinking. And so I anchored myself in math and looked out across physics."

Following his instinct, he spent his last two years of graduate work at Princeton, initially exploring string theory further, and it was there--working under renowned astrophysicist David Spergel--that he first learned about the startling observations of gravitational lenses. "It sounded exciting to me, like getting a chance to be on the frontier, clearing undeveloped land," he says. So, Petters plunged into the physics and mathematics of the lenses, seeking to stake his own claim in this new intellectual landscape. "Initially, it was tough trying to get through some of the actual physics literature because they looked at gravitational lensing strictly from a physics perspective. And I was trying to dig out the underlying mathematics of the subject."

Then, on a train ride back to M.I.T. to see his math adviser Bertram Kostant--as he watched the New England landscape slide past the train window and mulled, he says, over the complexities of gravitational lenses--he experienced the conceptual Eureka! that would propel his professional career. "The problem of how to describe gravitational lenses mathematically had stuck in the back of my head," he says. He knew that other attempts to create theories about them had involved "a lot of algebra, a little calculus, and a whole lot of messy, messy calculations." Suddenly, he says, he realized that the tools for creating such a theory already existed, in a sophisticated kind of mathematics called singularity theory. Ironically, a special case of this theory, called Morse theory, had existed even in Einstein's day, and the legendary scientist could have used it to create a lensing theory, had he thought of it.

Inspired by his train-ride flash of inight,

Petters launched himself on an effort to use singularity theory to build the first general mathematical theory of gravitational lensing. The payoff was immediate. His work revealed key details of how gravity from not just one or two objects but multiple objects at various cosmic distances will split passing light into images, including their number and magnification. It also allowed him to begin to map an optical "halo" phenomenon that occurs, for example, when a distant star lies directly behind an intervening object. In such rare instances, gravitational lensing causes the star's image to appear as a ring of intense brightness surrounding the object. Astrophysicists use the term "caustics" to describe the unique positions of distant stars or quasars that produce these infinitely bright points.

Advancing his work far beyond a single source and single lens, Petters has now used singularity theory to predict the caustics that result when the gravity of many objects--galaxies, black holes, and stars--sculpt starlight on its cosmic journey. "It turns out that when you have more than one intervening star, the caustics are no longer a point," he explains. "They form curves."

Remarkably beautiful curves, in fact. With a powerful computer, astrophysicists and mathematicians have used Petters' theory to generate intricate maps of such caustics, caused not only by the gravity of multiple objects such as stars but also by objects in different planes--each intensifying or attenuating the light refracted from the other. "My dream has been to isolate those properties of gravitational lensing that are generic and stable--features that are robust and independent of the simplifications used in most models of real lens systems," he says. "Singularity theory gives me a rigorous framework for accomplishing this goal, yielding insights where physical intuition can hardly penetrate."

Gravity lenses: The above diagrams show what happens when light speeding toward Earth from a distant cosmic object such as a quasar passes through the intense gravitational field of an intervening body such as a galaxy. When the quasar and the galaxy are in a line with Earth (top), the galaxy's gravity shapes the quasar's light to create an intensely bright ring around the galaxy. But when the quasar is slightly off the axis (bottom), the galaxy splits the quasar's image into two sepaate images, as seen from Earth.
Illustrations by Jerry Schoendorf

The scientific community will get its first comprehensive overview of Petters' theories with the publication of his book, Singularity Theory and Gravitational Lensing (Birkhauser, Boston), with co-authors Harold Levine and Joachim Wambsganss. Besides helping astrophysicists understand the properties of lensing, Petters hopes his theories will influence the entire direction of the field. "I think of these theories as being a first step toward mathematical astrophysics. Most of the twentieth century has seen little communication between mathematicians and astrophysicists. However, with gravitational lensing, we now have a bridge."

Inspired by his university mentors, and Bell Labs scientist Bill Massey, who gave him crucial help in developing his own career, Petters has also dedicated himself to helping students. Because of this commitment and his exceptional record of research, he has been named a Fellow in the Bass Program for Excellence in Undergraduate Education, launched in 1996 by a $10-million challenge gift from Anne and Robert Bass. In his teaching, Petters plans to use students' natural fascination with all things cosmic (witness the popularity of Star Wars) to pique their interest in math.

"I am hoping that my new course, 'The Mathematics of Light Deflection in the Universe,' based on our book, can show students how basic ideas in regular calculus can be extended and applied to real problems in gravitational lensing," he says. In the course, Petters will challenge students to develop their own projects using his mathematical tools to analyze different cases of gravitational lensing. What's more, he says, the students can more broadly apply the mathematics they learn in his course to other science and engineering problems, such as structural mechanics, aerodynamics, shock waves, laser physics, and climate studies.

Petters' goal will be to inspire in his students that same sense of wonder and aestheticism that he still enjoys himself. He still loves to draw, taking his kit of pencils and paper on afternoon walks in Sarah P. Duke Gardens to sketch and use the peace and quiet to ponder his latest mathematical problem. The same small boy who sat at his kitchen table industriously drawing pictures of nature has now become a renowned artist. But his medium is starlight and his canvas, the universe.

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