At Long Last, Mathematical Proof That Black Holes Are Stable

In 1963, the mathematician Roy Kerr found a solution to Einstein’s equations that precisely described the space-time outside what we now call a rotating black hole. (The term wouldn’t be coined for a few more years.) In the nearly six decades since his achievement, researchers have tried to show that these so-called Kerr black holes are stable. What that means, explained Jérémie Szeftel, a mathematician at Sorbonne University, “is that if I start with something that looks like a Kerr black hole and give it a little bump” — by throwing some gravitational waves at it, for instance — “what you expect, far into the future, is that everything will settle down, and it will once again look exactly like a Kerr solution.”

The opposite situation — a mathematical instability — “would have posed a deep conundrum to theoretical physicists and would have suggested the need to modify, at some fundamental level, Einstein’s theory of gravitation,” said Thibault Damour, a physicist at the Institute of Advanced Scientific Studies in France.

In a 912-page paper posted online on May 30, Szeftel, Elena Giorgi of Columbia University and Sergiu Klainerman of Princeton University have proved that slowly rotating Kerr black holes are indeed stable. The work is the product of a multiyear effort. The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, plus three background papers that established various mathematical tools — totals roughly 2,100 pages in all.

The new result “does indeed constitute a milestone in the mathematical development of general relativity,” said Demetrios Christodoulou, a mathematician at the Swiss Federal Institute of Technology Zurich.

Shing-Tung Yau, an emeritus professor at Harvard University who recently moved to Tsinghua University, was similarly laudatory, calling the proof “the first major breakthrough” in this area of general relativity since the early 1990s. “It is a very tough problem,” he said. He did stress, however, that the new paper has not yet undergone peer review. But he called the 2021 paper, which has been approved for publication, both “complete and exciting.”

One reason the question of stability has remained open for so long is that most explicit solutions to Einstein’s equations, such as the one found by Kerr, are stationary, Giorgi said. “These formulas apply to black holes that are just sitting there and never change; those aren’t the black holes we see in nature.” To assess stability, researchers need to subject black holes to minor disturbances and then see what happens to the solutions that describe these objects as time moves forward.

For example, imagine sound waves hitting a wineglass. Almost always, the waves shake the glass a little bit, and then the system settles down. But if someone sings loudly enough and at a pitch that exactly matches the glass’s resonant frequency, the glass could shatter. Giorgi, Klainerman and Szeftel wondered whether a similar resonance-type phenomenon could happen when a black hole is struck by gravitational waves.

They considered several possible outcomes. A gravitational wave might, for instance, cross the event horizon of a Kerr black hole and enter the interior. The black hole’s mass and rotation could be slightly altered, but the object would still be a black hole characterized by Kerr’s equations. Or the gravitational waves could swirl around the black hole before dissipating in the same way that most sound waves dissipate after encountering a wineglass.

Or they could combine to create havoc or, as Giorgi put it, “God knows what.” The gravitational waves might congregate outside a black hole’s event horizon and concentrate their energy to such an extent that a separate singularity would form. The space-time outside the black hole would then be so severely distorted that the Kerr solution would no longer prevail. This would be a dramatic sign of instability.

The three mathematicians relied on a strategy — called proof by contradiction — that had been previously employed in related work. The argument goes roughly like this: First, the researchers assume the opposite of what they’re trying to prove, namely that the solution does not exist forever — that there is, instead, a maximum time after which the Kerr solution breaks down. They then use some “mathematical trickery,” said Giorgi — an analysis of partial differential equations, which lie at the heart of general relativity — to extend the solution beyond the purported maximum time. In other words, they show that no matter what value is chosen for the maximum time, it can always be extended. Their initial assumption is thus contradicted, implying that the conjecture itself must be true.

Klainerman emphasized that he and his colleagues have built on the work of others. “There have been four serious attempts,” he said, “and we happen to be the lucky ones.” He considers the latest paper a collective achievement, and he’d like the new contribution to be viewed as “a triumph for the whole field.”

So far, stability has only been proved for slowly rotating black holes — where the ratio of the black hole’s angular momentum to its mass is much less than 1. It has not yet been demonstrated that rapidly rotating black holes are also stable. In addition, the researchers did not determine precisely how small the ratio of angular momentum to mass has to be in order to ensure stability.

Given that only one step in their long proof rests on the assumption of low angular momentum, Klainerman said he would “not be surprised at all if, by the end of the decade, we will have a full resolution of the Kerr [stability] conjecture.”

Giorgi is not quite so sanguine. “It is true that the assumption applies to just one case, but it is a very important case.” Getting past that restriction will require quite a bit of work, she said; she is not sure who will take it on or when they might succeed.

Looming beyond this problem is a much bigger one called the final state conjecture, which basically holds that if we wait long enough, the universe will evolve into a finite number of Kerr black holes moving away from each other. The final state conjecture depends on Kerr stability and on other sub-conjectures that are extremely challenging in themselves. “We have absolutely no idea how to prove this,” Giorgi admitted. To some, that statement might sound pessimistic. Yet it also illustrates an essential truth about Kerr black holes: They are destined to command the attention of mathematicians for years, if not decades, to come.