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The Math of Physics

A visit with Adam Helfer, Professor of Mathematics, Adjunct Professor of Physics & Astronomy

By Darcy Holtgrave
Published: - Topics: black holes gravitational waves physics mathematics
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Black holes loom large in the public imagination. Mathematical physicist Adam Helfer offers a definition: “Roughly speaking, a black hole is a region from which nothing can ever escape.” In other words, in its most simple definition (one uncomplicated, for the moment, by the nuances of scientific inquiry), it is the Alcatraz of the cosmos.

During our recent visit, Dr. Helfer, Professor of Mathematics and Adjunct Professor of Physics & Astronomy, cautions that black holes are only a portion of what he studies, but in general, he enjoys working on problems with real-world applications. Regardless of his topic, Dr. Helfer is a prime example of a scholar whose interests lie at the intersection of different fields—in this case, mathematics and physics.

Dr. Helfer completed his undergraduate degree in math and physics at Washington University in St. Louis. As an undergraduate, he became interested in black holes and what he characterizes as “problems of energy” in the description of them. According to the physicist Stephen Hawking, the negative energy within the vicinity of black holes means that black holes should give off light. While that theory is not uncontested—there are certainly, as Dr. Helfer states, unresolved “difficulties”—it is an exciting contribution to the field.

After he completed his undergraduate degree, Dr. Helfer went on to do a rigorous 3½-year PhD program at Oxford University. His PhD topic was in twistor theory and working out solutions to Einstein’s equations on gravitation; one way a twistor can be thought of is as “representing a particle moving at the speed of light,” he told us.

More recently he works with one Einstein’s major theories, gravitational waves. This theory posits that, in the wake of a collision between two cosmic entities like black holes or white dwarfs, ripples of gravity can be perceived in the fabric of space-time.

Dr. Helfer tells us that although scientists have good indirect evidence for gravitational waves, they are still looking for more direct confirmation. Research centers such as the Laser Interferometer Gravitational-Wave Observatory use techniques to attempt, from Earth, to measure gravitational waves, but their travel from distant sources weakens them, making them difficult if not impossible to measure. Other scientists have suggested looking at the waves’ effect on light that passes through them; the signature of the gravitational waves is “stamped on” the light.

For Dr. Helfer’s research, scientific inquiry includes questioning assumptions that have been fundamental in conceptualizing the problem in the first place. In the case of detecting gravitational waves, Dr. Helfer questioned the initial assumption about the source of the light being measured; instead of the light source being far behind the waves, what if the light source were closer? This question suggests that the effects of gravitational waves would be larger and therefore more readily measurable—or, as he qualified, with the extremely cautious optimism of a seasoned scientist, the predicted effects were getting to be large enough that “it wasn’t totally out of the question that we might be able to detect them.” He continues, “We would still have to get lucky, I think,” to find evidence to support this theory, “but it’s not hopeless.”

What gives Dr. Helfer an edge in reaching this theory of gravitational waves is his background in mathematics, particularly geometry. He enjoys the ability to “flip in and out” of theoretical exercises, bringing a mathematical perspective to physics. “You can spend a lot of time proving mathematical theorems…but even if you have an ironclad argument that the mathematics behaves in a certain way, it’s not necessarily speaking directly to the physical issues.“ The topics of mathematical physics seem enormous—and, indeed, on a cosmological scale, they are—but Dr. Helfer tells us, “a lot of the art of research is to find small pieces of it that you can make progress on, you have some hope of doing something with.”

In addition to his research, he teaches graduate and undergraduate mathematics courses. He is particularly enthusiastic about a history of mathematics course, which examines its various functions in culture, like economics and geometry, as well as the works of particular mathematicians, like Newton and Galileo, and finally the way mathematics is used to describe our world.

To return to the theory of black holes, no sooner did Dr. Helfer offer us a definition—a region from which nothing, not even light, escapes—then immediately he complicated it. The practical considerations for someone like the astronomer looking for them are important. “Are they supposed to wait forever to see if nothing escapes? How do they know that nothing could escape? Maybe there just was nothing trying to escape,” and so on; in the seeming impertinence, “these are actually rather technical points.” The Alcatraz of the cosmos remains an enigmatic entity, but piece by piece, the scientific community continues to chip away its surrounding mystery.