Black holes exist in our universe. Today it is widely accepted. Physicists have detected the X-rays that power black holes, analyzed gravitational waves from colliding black holes, and even imaged two of these giants.
But mathematician Elena Giorgi of Columbia University studies black holes in a different way. “Black holes are mathematical solutions to Einstein’s equation,” Giorgi says, the “master equation” that underlies general relativity.
She and other mathematicians aim to prove theorems about these solutions and otherwise explore the mathematics of general relativity. Their goal: to reveal the unknown truth about black holes or to confirm existing suspicions.
Within general relativity, “one can understand pure mathematical statements and study those statements and they can give an unambiguous answer within that theory,” says Christoph Kehle, a mathematician at ETH Zurich’s Institute for Theoretical Studies. Mathematicians can solve equations related to the nature of the formation, evolution and stability of black holes.
Last year, in a paper published on arXiv.org, Giorgi and his colleagues decided an ancient mathematical question on the stability of a black hole. Mathematically speaking, a stable black hole is one that, if pushed, nudged, or otherwise disturbed, will eventually turn back into a black hole. Like a rubber band that has been stretched and then released, a black hole does not snap, explode, or cease to exist, but returns to its original state.
Black holes seem to be physically stable—otherwise they wouldn’t be able to survive in the universe—but proving it mathematically is another matter entirely.
And a feat is necessary, says George. If black holes are stable, as the researchers assume, then the mathematics that describe them should better reflect that stability. If not, something is wrong with the underlying theory.
“Most of my work,” says Georgi, “is to prove what we already expect to be true.”
Mathematics has a history of major contributions to the realm of black holes. In 1916, Carl Schwarzschild published a solution to Einstein’s equations for general relativity near a single spherical mass. The math revealed a limit to how small mass can be compressed, an early sign of black holes. British mathematician Roger Penrose recently won the 2020 Nobel Prize in Physics for his calculations that showed black holes to be real predictions of general relativity. In a landmark paper published in 1965, Penrose described how matter could collapse to form a black hole with a singularity at its center.
A few years earlier, in 1963, the New Zealand mathematician Roy Kerr found a solution to Einstein’s equation for a rotating black hole. This was “a turning point for black holes,” Giorgi noted in a public lecture delivered at the virtual International Congress of Mathematicians 2022. Spinning black holes were much more realistic astrophysical objects than the non-spinning black holes for which Carl Schwarzschild solved the equations.
“Physicists have really believed for decades that the region of a black hole is a symmetry artifact that appears in the mathematical design of this object, but not in the real world,” Giorgi said during the talk. Kerr’s solution helped establish the existence of black holes.
In a nearly 1,000-page paper, Giorgi and his colleagues used a type of “proof from the contrary” to show that Kerr black holes, which rotate slowly (meaning they have a small angular momentum relative to their mass) are mathematically stable. The technique involves assuming the opposite statement to be proved and then finding the inconsistency. This shows that the assumption is wrong. The work is currently being reviewed. “It’s a long document, so it will take some time,” says Giorgi.
The result does not yet extend to rapidly rotating Kerr black holes, which are also known to exist in the universe.
Although the result is unlikely to change our understanding of black holes, such mathematical journeys can provide new insights.
This was true in Giorgi’s study of electrically charged black holes, which are also solutions to Einstein’s equations. She investigated what happens to these black holes during perturbations that contain both electromagnetic radiation and gravitational waves. These waves can surround black holes, fall inside them, or interact with them at a distance, she says. Thanks to this work, she found a new mathematical definition of electromagnetic radiation that could be used in additional studies of charged black holes.
Giorgi has been doing physics and mathematics since school, when she realized that “if I know mathematics, I can also do physics.” Her abiding interest in physics and attraction to differential geometry, which deals with the geometry of smooth spaces, made general relativity a natural fit. But her divisiveness led some colleagues to misunderstand her work.
Some physicists believe that black hole mathematicians are proving things “more rigorously than they’ve already supposedly proven what they believe,” says Georgi. At the same time, some mathematicians consider her work “more physics than mathematics” – until they see the scope of her complete mathematical proofs.
Georgie likes the freedom she has found in research. “You can work on anything,” she says. “You have to find your own problems.”