In other words, a singularity would inevitably arise at some point in the past.<\/p>\n
\u201cIt\u2019s a proof of concept that with our approach, we can prove singularity theorems that had been in more restricted, smooth domains,\u201d S\u00e4mann said. Their triangle comparison method wasn\u2019t just for show; it could help tell them something useful about the universe, about the presence of singularities in various kinds of space-times.<\/p>\n
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\u2019s and Hawking\u2019s theorems had needed. By basing their argument on sectional curvature, Kunzinger, S\u00e4mann 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 \u2014 as Hawking and Penrose had done \u2014 the mathematicians would instead need to base their arguments on less detailed information about curvature. They\u2019d need to use Ricci curvature, not sectional curvature.<\/p>\n
To achieve that, they needed some new players to join the effort.<\/p>\n
A Napoleonic Notion<\/p>\n
In 2018, while Kunzinger and S\u00e4mann were developing their techniques for sectional curvature, Robert McCann<\/a> 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.<\/p>\n In the late 18th century, Gaspard Monge found a way to efficiently transport soil to build fortifications for Napoleon\u2019s army. Mathematicians have continued to develop his \u201coptimal transport\u201d technique to solve other optimization problems.<\/p>\n Henri-Joseph Hesse via Wikimedia Commons<\/p>\n The idea dates back to 1781, 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.<\/p>\n More than two centuries later, McCann found a way to use Monge\u2019s 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.<\/p>\n To get a sense of how this works, let\u2019s consider a simpler example. Say you have a pile of sand at the Earth\u2019s 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.<\/p>\n McCann used the connection between optimal transport and curvature to develop a method for estimating the Ricci curvature of space-time<\/a> without calculus. But the approach only worked when space-time was smooth.<\/p>\n Then, a few months later, two mathematicians \u2014 Andrea Mondino<\/a> of the University of Oxford and Stefan Suhr<\/a> of Ruhr University Bochum in Germany \u2014 figured out how to adapt optimal transport techniques (using insights from Kunzinger and S\u00e4mann\u2019s research) to work in non-smooth settings<\/a>. In 2020, Mondino and Fabio Cavalletti<\/a> of the University of Milan showed that Hawking\u2019s singularity theorem still held up in those settings<\/a>. In fact, they were able to get it to work for more general models of space-time than Kunzinger and S\u00e4mann had. And their method for estimating Ricci curvature allowed them to prove the theorem without making the same limiting assumptions that Kunzinger and S\u00e4mann had to.<\/p>\n 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.<\/p>\n \u201cIt shows that the singularity theorems are even more fundamental\u201d than mathematicians and physicists had ever been able to show, according to Eric Ling<\/a> of the University of Copenhagen, who was not involved in the research. Hawking\u2019s and Penrose\u2019s singularities don\u2019t require a smooth space-time. Even in a rougher environment \u2014 one with corners or edges or other strange geometric features \u2014 they\u2019ll inevitably arise.<\/p>\n \u201cMajor results in general relativity actually extend to a much weaker setting where a smooth underlying space-time is not necessary,\u201d said Eric Woolgar<\/a>, a mathematician at the University of Alberta. \u201cThe ideas involved are quite remarkable.\u201d<\/p>\n A New Calculus<\/p>\n The ideas are still coming. Last year, McCann, S\u00e4mann and six colleagues started to develop ways to extend techniques from calculus<\/a> to non-smooth settings. \u201cWe can\u2019t do full-on calculus yet,\u201d S\u00e4mann said, but \u201cthis should expand the toolbox a lot.\u201d Mathematicians are already using those techniques<\/a> to prove other singularity theorems<\/a> and related statements.<\/p>\n And last month, Cavalletti and Mondino, along with Davide Manini<\/a> of the International School for Advanced Studies in Italy, became the first mathematicians to re-prove Penrose\u2019s singularity theorem<\/a> about black holes in non-smooth space-times.<\/p>\n Financial support has come too. Last year, Steinbauer, Kunzinger, S\u00e4mann and their colleagues received a grant of 7 million euros from the Austrian Science Fund to continue their work. They\u2019ve been recruiting more researchers to the team, who are now working on several projects \u2014 all aimed at developing novel mathematics to expand the reach of general relativity.<\/p>\n 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. \u201cThere are many approaches to quantum gravity which predict that, on a fundamental level, space-time is discrete,\u201d he said. \u201cYou have isolated points in space rather than a space-time continuum. And our framework can still speak about curvature in these discrete situations.\u201d And if it can speak about curvature, then perhaps it can speak about gravity.<\/p>\n S\u00e4mann can\u2019t wait to see what this collective enterprise will turn up next. \u201cPeople are still arriving,\u201d he said. \u201cThis project is really just starting.\u201d<\/p>\n","protected":false},"excerpt":{"rendered":"Kunzinger and S\u00e4mann wanted to use their new way of estimating curvature to determine whether these singularity theorems…\n","protected":false},"author":2,"featured_media":506,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24],"tags":[49,48,314,66],"class_list":{"0":"post-505","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-ca","9":"tag-canada","10":"tag-physics","11":"tag-science"},"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/505","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/comments?post=505"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/505\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media\/506"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media?parent=505"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/categories?post=505"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/tags?post=505"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"An](\"https:\/\/www.newsbeep.com\/ca\/wp-content\/uploads\/2025\/07\/GaspardMonge-crPublicDomain-colored.webp.webp\")
<\/p>\n