Kunzinger and Sämann wanted to use their new way of estimating curvature to determine whether these singularity theorems would still be valid if they no longer assumed space-time is smooth. Would singularities persist even in rougher, more realistic-looking spaces? It’s important to find out if the smoothness condition can be waived, Sämann said, because doing so would bring the theorems closer to physical reality. After all, he added, “we believe non-smoothness is an inescapable part of the natural world.”
In 2019, together with Stephanie Alexander of the University of Illinois (who died in 2023) and Melanie Graf, now at the University of Hamburg, the mathematicians proved a special case of Hawking’s singularity theorem. For simpler models of space-time — which were not smooth, but had a special structure — they showed that if you traced the paths of particles or light rays backward in time, then those paths would have to be finite.
In other words, a singularity would inevitably arise at some point in the past.
“It’s a proof of concept that with our approach, we can prove singularity theorems that had been in more restricted, smooth domains,” Sämann said. Their triangle comparison method wasn’t just for show; it could help tell them something useful about the universe, about the presence of singularities in various kinds of space-times.
But the technique could only give them estimates of sectional curvature. And sectional curvature provides more detailed information about the curvature of space-time than Penrose’s and Hawking’s theorems had needed. By basing their argument on sectional curvature, Kunzinger, Sämann and their colleagues proved their result under a more limited set of conditions than they would have preferred to. To re-prove the singularity theorem in its full generality — as Hawking and Penrose had done — the mathematicians would instead need to base their arguments on less detailed information about curvature. They’d need to use Ricci curvature, not sectional curvature.
To achieve that, they needed some new players to join the effort.
A Napoleonic Notion
In 2018, while Kunzinger and Sämann were developing their techniques for sectional curvature, Robert McCann of the University of Toronto decided to approach the problem using tools from an entirely different area of math. In particular, he hoped to make use of a method called optimal transport.
In the late 18th century, Gaspard Monge found a way to efficiently transport soil to build fortifications for Napoleon’s army. Mathematicians have continued to develop his “optimal transport” technique to solve other optimization problems. Henri-Joseph Hesse via Wikimedia Commons
The idea dates back to the late 18th century, when Napoleon tasked the French geometer Gaspard Monge with transporting large quantities of soil to construct fortifications. Monge used his mathematical skills to figure out the most cost-efficient way to divvy the materials up and send them to their destinations.
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