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.
More than two centuries later, McCann found a way to use Monge’s technique to estimate Ricci curvature. Whereas sectional curvature tells you precisely how two-dimensional slices of a space bend in different directions, Ricci curvature gives a more average sense of that bending. It essentially measures how the volume of an object will change as it moves through regions of space-time with varying curvature. And optimal transport, McCann realized, could give you information about these changes in volume.
To get a sense of how this works, let’s consider a simpler example. Say you have a pile of sand at the Earth’s North Pole, and you want to transport it to the South Pole. You can use optimal transport techniques to study how grains of sand will move between the two poles, and how their volume will change along the way. As they travel over the surface of the Earth, following the most direct possible paths toward the equator, they spread out, encompassing a bigger volume, before contracting again. The way their volume changes reflects the curvature of the Earth.
McCann used the connection between optimal transport and curvature to develop a method for estimating the Ricci curvature of space-time without calculus. But the approach only worked when space-time was smooth.
Then, a few months later, two mathematicians — Andrea Mondino of the University of Oxford and Stefan Suhr of Ruhr University Bochum in Germany — figured out how to adapt optimal transport techniques (using insights from Kunzinger and Sämann’s research) to work in non-smooth settings. In 2020, Mondino and Fabio Cavalletti of the University of Milan showed that Hawking’s singularity theorem still held up in those settings. In fact, they were able to get it to work for more general models of space-time than Kunzinger and Sämann had. And their method for estimating Ricci curvature allowed them to prove the theorem without making the same limiting assumptions that Kunzinger and Sämann had to.
The proof not only showcases the power of their method, but also provides an even firmer mathematical basis for the idea of a Big Bang singularity.
“It shows that the singularity theorems are even more fundamental” than mathematicians and physicists had ever been able to show, according to Eric Ling of the University of Copenhagen, who was not involved in the research. Hawking’s and Penrose’s singularities don’t require a smooth space-time. Even in a rougher environment — one with corners or edges or other strange geometric features — they’ll inevitably arise.
“Major results in general relativity actually extend to a much weaker setting where a smooth underlying space-time is not necessary,” said Eric Woolgar, a mathematician at the University of Alberta. “The ideas involved are quite remarkable.”
A New Calculus
The ideas are still coming. Last year, McCann, Sämann and six colleagues started to develop ways to extend techniques from calculus to non-smooth settings. “We can’t do full-on calculus yet,” Sämann said, but “this should expand the toolbox a lot.” Mathematicians are already using those techniques to prove other singularity theorems and related statements.
And last month, Cavalletti and Mondino, along with Davide Manini of the International School for Advanced Studies in Italy, became the first mathematicians to re-prove Penrose’s singularity theorem about black holes in non-smooth space-times.
Financial support has come too. Last year, Steinbauer, Kunzinger, Sämann and their colleagues received a grant of 7 million euros from the Austrian Science Fund to continue their work. They’ve been recruiting more researchers to the team, who are now working on several projects — all aimed at developing novel mathematics to expand the reach of general relativity.
Steinbauer is excited by the possibility that this program might one day help establish a mathematical foundation for a theory of quantum gravity: a long-sought way to unify the laws of general relativity with those of the submicroscopic world of quantum physics. “There are many approaches to quantum gravity which predict that, on a fundamental level, space-time is discrete,” he said. “You have isolated points in space rather than a space-time continuum. And our framework can still speak about curvature in these discrete situations.” And if it can speak about curvature, then perhaps it can speak about gravity.
Sämann can’t wait to see what this collective enterprise will turn up next. “People are still arriving,” he said. “This project is really just starting.”
Correction: July 16, 2025
An earlier version of this article misstated when Napoleon tasked Gaspard Monge with finding an efficient way to transport materials for fortifications. He did so in the 1790s, not 1781.