New Math Proves That a Special Kind of Space-Time Is Unstable

Einstein’s equations describe three canonical configurations of space-time. Now one of these three — important in the study of quantum gravity — has been shown to be inherently unstable.

Four years ago, while still a graduate student at Princeton University, Georgios Moschidis took on a problem that likely wasn’t going to pan out. His adviser asked him to mathematically prove that a certain configuration of space-time is unstable — to show, in other words, that any small change to it would ultimately lead to a breakdown in the space-time itself.

His adviser, the mathematician Mihalis Dafermos, knew how difficult the task would be. “You could spend a lot of time banging your head against the wall without getting anywhere,” said Dafermos, who (along with Gustav Holzegel) posed the instability conjecture in 2006. “I didn’t think it could ever be proven.” But he encouraged Moschidis, now a postdoc at the University of California, Berkeley, to take a look at it anyway. Moschidis had already done enough work to earn a doctorate, so why not try for something big?

Dafermos’ faith in Moschidis was well placed. In a series of advances that began in 2017 and continue to this day, Moschidis has shown that a certain canonical configuration of Einsteinian space-time called anti-de Sitter (AdS) space is unstable. Throw a tiny bit of matter into AdS space, and eventually a black hole will emerge.

The Stanford mathematician Jonathan Luk describes Moschidis’ work as “amazing. … What he’s discovered is a rather general instability mechanism” — one that could apply to other settings, unrelated to AdS, in which matter or energy is cooped up within a physical system that has no escape hatches. Dafermos calls his former student’s work “spectacular” and “certainly the most original thing I’ve seen in the mathematics of general relativity in the last few years.”

And although we do not live in an anti-de Sitter universe (thank goodness for that, as we wouldn’t exist), the work also has implications for our understanding of everything from turbulence to the mysterious connections between theories of gravity and quantum mechanics.

The Swell of Gravity

The instability conjecture — and indeed the whole school of thought from which it sprang — goes back to Einstein’s equations of general relativity, which spell out exactly how mass and energy affect the curvature of space-time. In a vacuum, where there’s no matter at all, space-time can still be curved and gravity can still be present, due to the energy density of the vacuum itself, described by a “cosmological constant.” Empty space, it turns out, is not really empty at all.