At Long Last, Mathematical Proof That Black Holes Are Stable

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In 1963, the Mathematician Roy Kerr found a solution to Einstein’s equations that accurately described the spacetime outside of what we now call a spinning black hole. (The term wouldn’t be coined for several more years.) In the nearly six decades since its success, researchers have tried to show that these so-called Kerr black holes are stable. What this means, explained Jérémie Szeftel, a mathematician at the Sorbonne University, “is that if I start with something that looks like a Kerr black hole and give it a bit of a bump”—by throwing gravitational waves at it, for example-, “what do you expect, far into the future, that everything will be fixed and look exactly like a Kerr solution again.”

The opposite situation, a mathematical instability, “would have posed a profound conundrum for 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 Science. Studies in France.

In a 912-page paper published online May 30, Szeftel, Elena Giorgi of Columbia University, and Sergiu Klainerman of Princeton University have shown that slowly rotating Kerr black holes are indeed stable. The work is the result of a multi-year 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.

The new result “in fact constitutes 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 professor emeritus at Harvard University who recently moved to Tsinghua University, was equally complimentary, calling the test “the first major breakthrough” in this area of ​​general relativity since the beginning of the nineties “It’s a very tough problem,” he said. However, he stressed that the new paper has not yet undergone peer review. But he called the 2021 paper, which has been approved for publication, “comprehensive and exciting.”

One reason the stability question has remained open 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 are sitting 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 these objects as time goes on.

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

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

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

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