Jul 19, 2025 10:31 PM
Yet another adaptive enhancement of GR that will probably never be heard from again in a couple of years.
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A new geometry for Einstein’s theory of relativity
https://www.quantamagazine.org/a-new-geo...-20250716/
INTRO: In October 2015, a young mathematician named Clemens Sämann was flying home to Austria from a conference in Turin, Italy, when he had a chance encounter. He found himself seated beside Michael Kunzinger, another conference attendee. Kunzinger was a math professor at the University of Vienna, where Sämann had just started his postdoctoral research. They soon got to talking, landing on a subject Sämann had started thinking about in graduate school — whether there was a mathematical way to get around the limitations of Albert Einstein’s general theory of relativity.
Einstein’s theory defines gravity as the curvature of space-time caused by the presence of matter and energy. Since its formulation in 1915, it has held up remarkably well. Consisting of 10 interconnected differential equations, the theory describes how objects fall, how light bends, and how planets, stars and galaxies move. It tells us that the universe is expanding, and it predicted the existence of both black holes and gravitational waves a century before they were definitively observed.
But in spite of these successes, Einstein’s theory also has shortcomings. Its equations can only describe how matter curves space-time when the geometry of that space-time is smooth — with no sharp corners or cusps, no regions where it suddenly becomes jagged. Picture space-time as a flat rubber sheet, and matter as a bowling ball placed on that sheet, causing it to bend. If space-time is smooth, then this bending will be gradual.
But physicists know that’s not always true. A black hole, for instance, warps space-time more violently, causing the sheet to bend sharply until, at the black hole’s center (a so-called singularity), the curvature “blows up,” becoming infinite. Some physicists even posit that space-time doesn’t become non-smooth only at isolated singularities, but at every point. On the smallest scales, space-time might be “discrete,” or pixelated — broken up into tiny, disconnected bits in the same way that a fluid, while appearing to be a single uniform entity, is actually made up of distinct atoms and molecules.
In these situations, general relativity hits an impasse. Whenever space-time isn’t sufficiently smooth, Einstein’s equations stop working. They can no longer tell us how matter curves space-time, or how curved space-time influences matter.
That’s because the equations rely on a technique from calculus called differentiation that measures how quickly functions change, and differentiation is no longer possible in settings that are far from smooth. And so, on their flight back to Austria, Kunzinger and Sämann wondered whether they could devise alternative methods — ones that would still work in the inhospitable environments where the usual tools of calculus broke down.
The pair wouldn’t start working on the problem in earnest for another year. But since then, they’ve made significant advances toward their goal. They’ve found new ways to estimate curvature and other geometric properties without relying on smoothness and differentiation. In collaboration with other researchers, they’ve used their methods to rederive (and sometimes strengthen) core theorems about the universe without depending on Einstein’s equations, putting those theorems on even firmer mathematical footing.
And they’re now part of an ambitious new program — launched last year under the direction of Roland Steinbauer, another University of Vienna mathematician — that aims to provide “a new geometry for Einstein’s theory of relativity and beyond.”
“Standard general relativity talks about geometric objects, namely space-times, but only if they behave nicely enough,” Steinbauer said. “With this new framework, we can go beyond that. We can handle very edgy objects, very badly behaved objects.” (MORE - details)
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A new geometry for Einstein’s theory of relativity
https://www.quantamagazine.org/a-new-geo...-20250716/
INTRO: In October 2015, a young mathematician named Clemens Sämann was flying home to Austria from a conference in Turin, Italy, when he had a chance encounter. He found himself seated beside Michael Kunzinger, another conference attendee. Kunzinger was a math professor at the University of Vienna, where Sämann had just started his postdoctoral research. They soon got to talking, landing on a subject Sämann had started thinking about in graduate school — whether there was a mathematical way to get around the limitations of Albert Einstein’s general theory of relativity.
Einstein’s theory defines gravity as the curvature of space-time caused by the presence of matter and energy. Since its formulation in 1915, it has held up remarkably well. Consisting of 10 interconnected differential equations, the theory describes how objects fall, how light bends, and how planets, stars and galaxies move. It tells us that the universe is expanding, and it predicted the existence of both black holes and gravitational waves a century before they were definitively observed.
But in spite of these successes, Einstein’s theory also has shortcomings. Its equations can only describe how matter curves space-time when the geometry of that space-time is smooth — with no sharp corners or cusps, no regions where it suddenly becomes jagged. Picture space-time as a flat rubber sheet, and matter as a bowling ball placed on that sheet, causing it to bend. If space-time is smooth, then this bending will be gradual.
But physicists know that’s not always true. A black hole, for instance, warps space-time more violently, causing the sheet to bend sharply until, at the black hole’s center (a so-called singularity), the curvature “blows up,” becoming infinite. Some physicists even posit that space-time doesn’t become non-smooth only at isolated singularities, but at every point. On the smallest scales, space-time might be “discrete,” or pixelated — broken up into tiny, disconnected bits in the same way that a fluid, while appearing to be a single uniform entity, is actually made up of distinct atoms and molecules.
In these situations, general relativity hits an impasse. Whenever space-time isn’t sufficiently smooth, Einstein’s equations stop working. They can no longer tell us how matter curves space-time, or how curved space-time influences matter.
That’s because the equations rely on a technique from calculus called differentiation that measures how quickly functions change, and differentiation is no longer possible in settings that are far from smooth. And so, on their flight back to Austria, Kunzinger and Sämann wondered whether they could devise alternative methods — ones that would still work in the inhospitable environments where the usual tools of calculus broke down.
The pair wouldn’t start working on the problem in earnest for another year. But since then, they’ve made significant advances toward their goal. They’ve found new ways to estimate curvature and other geometric properties without relying on smoothness and differentiation. In collaboration with other researchers, they’ve used their methods to rederive (and sometimes strengthen) core theorems about the universe without depending on Einstein’s equations, putting those theorems on even firmer mathematical footing.
And they’re now part of an ambitious new program — launched last year under the direction of Roland Steinbauer, another University of Vienna mathematician — that aims to provide “a new geometry for Einstein’s theory of relativity and beyond.”
“Standard general relativity talks about geometric objects, namely space-times, but only if they behave nicely enough,” Steinbauer said. “With this new framework, we can go beyond that. We can handle very edgy objects, very badly behaved objects.” (MORE - details)