mathematical physics

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.

The proof involves injecting a small bit of matter into the space-time, akin to dropping a stone into a pond. Waves ripple out and back, interacting in a way that eventually creates a black hole.

DVDP for Quanta Magazine

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.

The three simplest solutions to Einstein’s vacuum equations are the most symmetric ones — those in which the curvature of space-time is the same everywhere. In Minkowski space-time, where the cosmological constant is zero, the universe is perfectly flat. In de Sitter space-time, where the cosmological constant has a positive value, the universe is shaped like a sphere. And when the cosmological constant is negative, you get AdS space-time, which has a saddle shape. In the early days of cosmology, scientists wondered which one of these three space-times describes our universe.

Mathematicians, on the other hand, tended to wonder if these space-times were really, truly stable. That is, if you disturbed a vacuum space-time in any way — say, by injecting some matter into the system or sending in some gravitational waves — would it eventually settle down into something close to the original state? Or would it evolve into something wildly different? It’s the cosmic equivalent of dropping a rock into a pond: Will the waves gradually diminish, or will they build into a tsunami?

In 1986, a mathematician proved that de Sitter space-time is stable. A pair of mathematicians did the same for Minkowski space-time in 1993. The AdS problem has taken longer. The general consensus was that AdS, unlike the other two configurations, is unstable, which meant that mathematicians would have to take an entirely new approach. “A lot of tools in mathematics have been developed for stability problems,” said Dafermos. “But instability is a completely different arena — especially this type of instability,” which is nonlinear in nature, making for an inherently complicated situation, with correspondingly tricky calculations.

Researchers suspected that AdS space-time might be unstable because they believed its boundary would be reflective, thereby causing it “to act like a mirror so that any waves hitting it will come back,” Dafermos explained.

“Reflection at the boundary makes sense from a physical point of view,” said Juan Maldacena, a physicist at the Institute for Advanced Study in Princeton, New Jersey. This is partly due to the curvature of AdS space, but there’s an even simpler explanation: The premise upholds the principle of energy conservation.

If the boundary is, in fact, reflective, nothing can leak out of AdS space-time. So any matter or energy put into the system could potentially get concentrated — perhaps to such an extent that a black hole would form. The question was: Would that really happen, and, if so, what mechanism would cause matter and energy to cluster together to such a degree rather than staying spread out?

Moschidis imagined standing in the middle of AdS space-time, which would be like standing inside a giant ball whose edge or boundary lies at infinity. If you sent a light signal from there, it would travel out and reach the boundary in a finite amount of time. That kind of travel is only possible because of a well-known relativistic effect: Although the spatial distance to the boundary is indeed infinite, time slows down for a wave or object traveling at or near the speed of light. So an observer standing in the middle of AdS space-time would see a light ray reach the boundary in a finite amount of time (though some patience would be required).

Instead of using a light ray, Moschidis dropped into AdS space a form of matter that is commonly used in general relativity models — so-called Einstein-Vlasov particles. These particles create concentric waves of matter in space-time, similar to the water waves that appear in a pond.

Of the many concentric waves created when matter is suddenly plopped into this space-time, the first two will be the biggest. Since they contain the most matter and energy, we’ll focus on them. The first wave — call it wave 1 — will expand outward until it hits the boundary, bounces back, and contracts as it retreats toward the center. The second wave, wave 2, will follow.

When wave 1 rebounds off the boundary and starts contracting toward the center, it will hit wave 2, which is still expanding. One consequence of the Einstein equations, Moschidis determined, is that in an interaction like this, the expanding wave (wave 2 in this case) will always transfer energy to the contracting wave (wave 1).

After wave 1 reaches the center, it will start expanding again, meeting up with wave 2, which is now contracting. This time, wave 1 will impart energy to wave 2. This cycle can repeat many, many times.

Moschidis realized something else: Near the center, the waves occupy less space, and the energy they carry is more concentrated. Because of this, the waves exchange more energy during the interaction near the center than during the interaction near the boundary. The net result is that wave 1 gives more energy to wave 2 at the center than wave 2 gives to wave 1 at the boundary.

Over numerous iterations, wave 2 grows larger and larger, taking energy from wave 1. Consequently, the energy density of wave 2 continues to build. At some point, as wave 2 contracts toward the center, its energy will become so concentrated that a black hole will form.

Here’s the proof of instability: Moschidis showed that when he adds even a minuscule amount of matter to an AdS space-time, a black hole (or black holes) will inexorably form. However, AdS space-time has, by definition, uniform curvature everywhere, which means it cannot harbor space-contorting objects like black holes. “If you perturb AdS space-time and wait a sufficient time,” Moschidis said, “you’ll end up with a different geometry — one that contains black holes — and it’s no longer AdS. That’s what we mean by unstable.”

Moschidis has recently proved AdS instability for a different kind of matter perturbation — a so-called massless scalar field — presenting this not-yet-published work in several academic talks. “Because the waves generated by a scalar field are a proxy for gravitational waves,” Dafermos said, this brings Moschidis one step closer to the ultimate goal — proving AdS instability in a true vacuum, where the space-time is perturbed strictly by gravity without the introduction of any matter.

The Turbulent Future of AdS Space

The instability of AdS space-time has major implications for how we understand our own universe. First, since AdS space-time is unstable, it’s “something you will not see in nature,” Moschidis said.

But “even though AdS is not real,” he said, “it can still lead us to the discovery and study of real phenomena.”

For example, turbulence arises when energy gets concentrated from large scales to small scales — something that Moschidis showed can occur when AdS space-time is perturbed. But turbulence is a widespread (and poorly understood) phenomenon that appears in all kinds of fluid systems. AdS space-time — at least to someone with Moschidis’ skills and inclinations — is a “clean” and relatively simple system to work with, which is why he considers it “a good theoretical test bed” for studying turbulence. In the AdS setting, turbulence is caused by gravity, but Moschidis believes that the mathematical tools he is developing could aid the analysis of the turbulence that crops up in fluid mechanics as well.

AdS also features prominently in the so-called AdS/CFT correspondence — a key clue for how to unite quantum mechanics with gravity in an all-encompassing theory of quantum gravity. The correspondence states that a gravitational system in AdS space can be equivalent to a nongravitational quantum system in one fewer dimension. “We can take a quantum mechanical system that does not contain gravity and describe it instead by a theory of gravity — not a theory of gravity in our universe but a theory of gravity in an AdS universe,” said Maldacena, who uncovered the correspondence in 1997. He further noted that the instability of AdS — as recently proved by Moschidis — does not affect the validity of the correspondence.

Moschidis’ work, when combined with the AdS/CFT correspondence, could also help illuminate the more familiar domain of interacting particles. For example, Moschidis used small perturbations of AdS space-time to create black holes. This process correlates, via the correspondence, to the process of thermalization whereby quantum systems achieve equilibrium — an almost ubiquitous real-world phenomenon.

“Proving that AdS is unstable,” Moschidis concluded, “doesn’t mean it is uninteresting.”

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