Albert Einstein’s theory of general relativity has a math problem, several problems actually. They have to do with black holes and are extremely complex and difficult to solve—just the kind of problems that have always fascinated Jonathan Luk.
This fascination led the Stanford mathematician to a long collaboration with Princeton’s Mihalis Dafermos that disproved the “strong cosmic censorship conjecture,” the hypothesis that sought to save general relativity from the loss of determinism. What Luk and Dafermos showed is that determinism, or the idea that the future is always predicated by past data, does not always hold true within certain types of black holes.
Their achievement was recently recognized with a 2026 Bôcher Memorial Prize, the American Mathematical Society’s top prize for mathematical analysis. Their work also pokes a hole in determinism, leaving open the possibility of unpredictable futures—at least deep within some black holes. This has troubling implications, but for Luk, it just means there is more work to be done.
“I think the important thing is to first understand what happens inside black holes; then we can try to understand what it means,” said Luk, professor of mathematics in Stanford’s School of Humanities and Sciences.
The search for solutions
When Einstein wrote the theory of general relativity in 1915, he included a set of equations describing how the gravity of large objects curves space-time—and physicists and mathematicians have been trying to find solutions to those equations ever since.
The equations suggested the existence of black holes, areas where gravity becomes so strong not even light can escape. Evidence of black holes was gathered in the ensuing 100 years, but it wasn’t until 2019 that astrophysicists captured the image of one.
Using the Event Horizon Telescope, scientists obtained the first picture of a black hole in 2019. Photo by Event Horizon Telescope Collaboration
Yet the search for solutions to Einstein’s equations also found some problems: namely that inside some rotating black holes, there is a location, called the Cauchy horizon, after which determinism breaks down. In other words, beyond that horizon the past doesn’t predict the future —an idea that challenges our understanding of the universe.
To save determinism, the physicist Roger Penrose proposed the strong cosmic censorship conjecture in 1979. He argued that the Cauchy horizon was unstable and any gravitational wave that made it past would just cause a crushing singularity, a point at which matter is condensed into an infinite density. This singularity would end space-time and protect the theory from producing unpredictable futures.
Through their work, Luk and Dafermos found that there is no crushing singularity, as Penrose proposed, even when space-time is “perturbed” or distorted by gravitational waves. That means that space-time could still be unpredictable beyond the Cauchy horizon within a black hole. The mathematicians essentially found a new aspect of Einstein’s equations.
“Luk and Dafermos really achieved something that was very unexpected,” said Rafe Mazzeo, the Cassius Lamb Kirk Professor of the Natural Sciences and professor of mathematics in H&S. “In terms of the mathematical understanding of relativity, it’s really a huge advance.”
The intersection of physics and mathematics
This discovery, like a lot of things in mathematics or science, took many years. Luk describes solving complex problems like this as a process that has many obstacles.
“In mathematics, I think you are always stuck,” he said.
That frustration doesn’t deter Luk. In fact, the more difficult a problem is, the more fascinating he finds it. This has been true ever since he faced his first challenging, nonstandard math problem as a fourth grader.