Maths Prove Slowly Rotating Black Holes Are Stable

Black holes. They’re massive, they’re mysterious, and they sound really eerie.

And up until now, we didn’t know how stable they might be. Fortunately, 2022 has seen a monster achievement in mathematical physics. Quanta Magazine‘s Steve Nadis sets the intergalactic stage for a remarkable mathematical proof:

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.

Would Einstein be proven wrong? As a wise man once said, “bet on Big Al and give the points”. In this case, Team Einstein has chalked up a new win in the form of a mathematical proof that such slowly rotating black holes are stable. And all it took was a team of three mathematical physicists working for years to invent the needed mathematical tools before finally producing a single 912 page paper to prove it.

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.

We may be understating the amount of work that resulted in the proof by calling it a monster achievement. What we found particularly interesting was the path by which they reached the proof:

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.

Proof by contradiction is one of the oldest tools in maths. The mathematician Euclid used it to prove there are an infinite number of prime numbers roughly 2,300 years ago. Because it’s a great introduction into how the tool of proof by contradiction works, here is a seven minute video of Trefor Bazett working through the basic steps Euclid followed to prove there are an infinite number of primes in Book IX of Elements:

One could reasonably argue that the multiyear effort to prove that slowly rotating Kerr black holes are stable really took hundreds, if not thousands, of years to reach.

We are however putting the cart before the horse at this point. These are preprint papers that are now going through peer review phase, so until it passes muster, the Giorgi-Klainerman-Szeftel proof is under challenge. That will almost certainly add some additional years to the total assuming it all comes together and the proof is confirmed.

References

Elena Giorgi, Sergiu Klainerman, Jeremie Szeftel. Kerr stability for small angular momentum. Arxiv. [PDF Document]. DOI: 10.48550/arXiv.2205.14808. 30 May 2022.

Sergiu Klainerman, Jeremie Szeftel. Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes. Arxiv. [PDF Document]. DOI: 10.48550/arXiv.2104.11857. 30 May 2022.

Image credit: Stable Diffusion Demo: “A rotating black hole in space”. Generated 28 August 2022.