Finally a mathematical proof that black holes are stable

1963 became the Mathematician Roy Kerr found a solution to Einstein’s equations that accurately described spacetime outside of what we now call a rotating black hole. (The term wouldn’t be coined for a couple of years.) In the nearly six decades since its achievement, researchers have attempted to show that these so-called Kerr black holes are stable. What that means, explained Jérémie Szeftel, a mathematician at Sorbonne University, “is if I start with what looks like a Kerr black hole and give it a little bump” — by throwing some gravitational waves at it, for example — ” what you expect things to settle down far into the future and it looks exactly like a Kerr solution again.”

The opposite situation – a mathematical instability – “would have deeply puzzled theoretical physicists and suggested the need to modify Einstein’s theory of gravity at a fundamental level,” said Thibault Damour, a physicist at the Institute of Advanced Scientific Studies in France.

In a 912-page article posted online May 30, Szeftel, Columbia University’s Elena Giorgi, and Princeton University’s Sergiu Klainerman proved that slowly rotating Kerr black holes are indeed stable. The work is the result of many years of work. The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, and three background papers establishing various mathematical tools — totals about 2,100 pages.

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

Shing-Tung Yau, a Harvard University professor emeritus who recently moved to Tsinghua University, was similarly laudable, calling the evidence “the first major breakthrough” in this area of ​​general relativity since the early 1990s. “It’s a very difficult problem,” he said. However, he stressed that the new paper has not yet been peer-reviewed. But he called the 2021 paper, which was approved for publication, both “complete and exciting.”

One reason the question of stability has lingered for so long is that most explicit solutions to Einstein’s equations, like the one found by Kerr, are stationary, Giorgi said. “These formulas apply to black holes that just sit there and never change; these are not the black holes we see in nature.” To assess stability, researchers need to subject black holes to minor perturbations and then see what happens to the solutions that describe those objects over time.

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

They considered several possible outcomes. For example, a gravitational wave could cross the event horizon of a Kerr black hole and enter the interior. The black hole’s mass and rotation could be changed slightly, but the object would still be a black hole characterized by the Kerr equations. Or the gravitational waves could swirl around the black hole before dissipating in the same way most sound waves dissipate after hitting a wine glass.

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

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