wp-plugin-hostgator domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114ol-scrapes domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114Source:https:\/\/www.quantamagazine.org\/a-century-later-new-math-smooths-out-general-relativity-20231130\/#comments<\/a><\/br> Dong and Song proved a conjecture<\/a> that was formulated in 2001 by the mathematicians Gerhard Huisken<\/a> and Tom Ilmanen<\/a>. The conjecture states that as the mass of a space approaches zero, so too must its curvature. Huisken and Ilmanen recognized, however, that this scenario is complicated by the presence of bubbles and spikes (which are mathematically distinct from one another). They hypothesized that the bubbles and spikes could be cut off in such a way that the boundary area left behind on the surface of the space by each excision was small. They suggested, but couldn\u2019t prove, that the space that remained after these troublesome appendages had been removed would be close to flat. They also weren\u2019t sure how such cuts should be made.<\/p>\n \u201cThese questions were difficult, and I was not expecting to see a solution to the Huisken-Ilmanen conjecture,\u201d Lee said.<\/p>\n At the heart of the conjecture is\u00a0a measurement of curvature. Space\u00a0can curve in different ways, different amounts, and different directions \u2014 like a saddle (in two dimensions) that curves up going forward and back, but down going to the left and right.\u00a0Dong and Song ignore those details. They use a concept called scalar curvature, which\u00a0represents the curvature as a single number that summarizes the full curvature in all directions.<\/p>\n Dong and Song\u2019s new work, said Daniel Stern<\/a>\u00a0of Cornell University, is \u201cone of the strongest results we have so far that shows us how scalar curvature controls [the] geometry\u201d of the space as a whole. Their paper illustrates that \u201cif we have nonnegative scalar curvature and small mass, we understand the structure of space very well.\u201d<\/p>\n The Huisken-Ilmanen conjecture concerns the geometry of spaces with steadily decreasing mass. It prescribes a specific method for saying how close a space with small mass is to flat space. That measure is called Gromov-Hausdorff distance, named after the mathematicians Mikhael Gromov<\/a> and Felix Hausdorff. Calculating the Gromov-Hausdorff distance is a two-step process.<\/p>\n The first step is to find the Hausdorff distance. Suppose that you have two circles, A and B. Start with any point on A and figure out how far it is to the closest point on B.<\/p>\n Repeat this for every point on A. The largest distance you find is the Hausdorff distance between the circles.<\/p>\n Once you have the Hausdorff distance, you can calculate the Gromov-Hausdorff distance. To do that, place your objects in a bigger space so as to minimize the Hausdorff distance between them. In the case of two identical circles, since you could put them literally on top of each other, the Gromov-Hausdorff distance between them is zero. Geometrically identical objects like these are called \u201cisometric.\u201d<\/p>\n Measuring distance is more difficult, of course, when the objects or spaces being compared are alike but not the same. The Gromov-Hausdorff distance provides a precise measure of the similarities (or differences) between the shapes of two objects that initially lie in different spaces. \u201cGromov-Hausdorff distance is one of the best ways we have of saying that two spaces are almost isometric, and it gives a number to that \u2018almost,\u2019\u201d Stern said.<\/p>\n Before Dong and Song could draw comparisons between a space with a small mass and a space that is perfectly flat, they had to snip off the pesky protuberances \u2014 the narrow spikes where matter is tightly packed and even denser bubbles that may harbor tiny black holes. \u201cWe cut them so that the boundary area [where the slice was made] is small,\u201d Song said, \u201cand we showed that the area gets smaller as the mass goes down.\u201d<\/p>\n Although that tactic might sound like a cheat, Stern said it\u2019s permissible in proving the conjecture to do a sort of pre-processing by cutting out bubbles and spikes whose area shrinks to zero as the mass decreases.<\/p>\n As a proxy for a space with small mass, he suggested, we might imagine a crumpled sheet of paper that, after being smoothed out again, still has sharp creases and folds. You could use a hole punch to remove the most prominent irregularities, leaving a slightly uneven piece of paper with some holes in it. As the size of those holes shrinks, so will the unevenness of the paper\u2019s terrain. At the limit, you might say, the holes would shrink to zero, the mounds and ridges would disappear, and you\u2019d be left with a uniformly smooth piece of paper \u2014 a genuine stand-in for flat space.<\/p>\n That\u2019s what Dong and Song sought to prove. The next step was to see how these denuded spaces \u2014 shorn of their rough features \u2014 stacked up against the standard of utter flatness. The strategy they pursued made use of a special kind of map, which is a way of comparing two spaces by associating points in one space with points in another. The map they used was developed in a paper<\/a> written by Stern and three colleagues \u2014 Hubert Bray, Demetre Kazaras and Marcus Khuri. This procedure can spell out exactly how close two spaces are.<\/p>\n To simplify their task, Dong and Song adopted another mathematical trick from Stern and his co-authors, which showed that a three-dimensional space can be divided into infinitely many two-dimensional slices called level sets, much as a hard-boiled egg can be segmented into narrow sheets by the taut wires of an egg slicer.<\/p>\n The level sets inherit the curvature of the three-dimensional space they comprise. By focusing their attention on level sets rather than on the bigger three-dimensional space, Dong and Song were able to reduce the dimensionality of the problem from three to two. That\u2019s very beneficial, Song said, because \u201cwe know a lot about two-dimensional objects \u2026 and we have a lot of tools to study them.\u201d<\/p>\n If they could successfully show that each level set is \u201ckind of flat,\u201d Song said, this would allow them to attain their overall goal of showing that a three-dimensional space with little mass is close to flat. Fortunately, this strategy panned out.<\/p>\n Looking ahead, Song said one of the field\u2019s next challenges is to make the proof more explicit by laying out a precise procedure for getting rid of bubbles and spikes and better describing the regions that have been cut away. But for now, he admitted, \u201cwe don\u2019t have a clear strategy to achieve that.\u201d<\/p>\n \u00a0Another promising avenue, Song said, would be to explore a separate conjecture<\/a> that was formulated in 2011 by Lee and Christina Sormani<\/a>, a mathematician at the City University of New York. The Lee-Sormani conjecture asks a similar question to the one posed by Huisken and Ilmanen, but it relies on a different way of measuring the difference between shapes. Instead of considering the maximum distance between two shapes, as the Gromov-Hausdorff distance does, the Lee-Sormani approach asks about the volume of the space<\/a> between them. The smaller that volume, the closer they are.<\/p>\n Song, meanwhile, hopes to look into basic questions about scalar curvature that are not motivated by physics. \u201cIn general relativity,\u201d he said, \u201cwe deal with very special spaces that are almost flat at infinity, but in geometry we care about all kinds of spaces.\u201d<\/p>\n \u201cThere is hope that these techniques could be of value in other settings\u201d unrelated to general relativity, Stern said. \u201cThere is a large family of related problems,\u201d he said, that\u2019s waiting to be explored.<\/p>\n Quanta\u00a0is conducting a series of surveys to better serve our audience. Take our\u00a0<\/i>mathematics reader survey<\/i><\/a>\u00a0and you will be entered to win free\u00a0<\/i>Quanta\u00a0merch.<\/i><\/p>\n<\/div>\n <\/br><\/br><\/br><\/p>\n
\nA Century Later, New Math Smooths Out General Relativity<\/br>
\n2023-12-04 21:58:07<\/br><\/p>\nThe Proof<\/strong><\/h2>\n
Next Steps<\/strong><\/h2>\n