Math proof draws new frontiers on black hole formation


TThe modern idea of ​​a black hole has been around since February 1916, three months after Albert Einstein unveiled his theory of gravity. That’s when physicist Karl Schwarzschild, in the midst of fighting in the German army during World War I, published a paper with startling implications: If enough mass was confined within a perfectly spherical region (bounded by the “Schwarzschild radius”), nothing could. Escape the intense gravity of such an object, not even the light itself. At the center of this sphere is a singularity where density approaches infinity and known physics derails.

And in the more than 100 years since then, physicists and mathematicians have explored the properties of these enigmatic objects from the perspective of theory and experiment. So it might come as a surprise to hear that “if you take a region of space with a bunch of matter scattered in it and you ask a physicist if that region is going to collapse into a black hole, we don’t have the tools to answer yet.” He said this question Marcus Khourya mathematician at Stony Brook University.

Do not give up. Khoury and three of his colleaguesSven Hirsch at the Institute for Advanced Study, Demeter Casaras at Michigan State University, and Yiwu Chang at University of California, Irvine – They released a new version paper This brings us closer to determining the existence of black holes based solely on matter concentration. In addition, their research mathematically proves that black holes of higher dimensions – those with four, five, six or seven spatial dimensions – can exist, something that could not be said with confidence before.

At a singularity, the density approaches infinity and derails known physics.

To put the recent research in context, it might be helpful to go back to 1964, the year Roger Penrose began to advance the singularity theories that earned him a share of the research. Nobel Prize in Physics 2020. Penrose demonstrated that if space-time contains a so-called enclosed trapped surface—a surface whose curvature is so intense that light going outward gets wrapped around it and turned inward—then it must also contain a singularity.

It was a tremendous result, in part because Penrose brought powerful new tools from geometry and topology to the study of black holes and other phenomena in Einstein’s theory. But Penrose’s work didn’t show what it takes to create a closed confined surface in the first place.

And in 1972, physicist Kip Thorne took a step in this direction by formulating the ring conjecture. Thorne realized that knowing whether a non-spherical object—an object lacking the symmetry assumed in Schwarzschild’s pioneering efforts—would collapse into a black hole would be “much more difficult to calculate (and) actually far beyond my talents”. (Thorn would go on to win Nobel Prize in Physics 2017.) He felt, however, that his guess might make the problem more manageable. The basic idea is to first determine the mass of a given object, and then calculate the critical radius of the hoop within which the object must fit—regardless of how the hoop is oriented—to make the formation of a black hole inevitable. It would be like showing that a hula hoop that fits around your waist can also – if rotated 360 degrees – fit around your fully stretched body, including your feet and head. If the object is suitable, it will collapse into a black hole.

in body image
Hope Dreams: Kip Thorne in 1972, the same year he introduced The Ring Hunch. Credit: Wikimedia Commons.

“The girth guess is not well defined,” Cazaras commented. Thorne deliberately used vague wording in hopes that others would make a more accurate statement.

In 1983, mathematicians Richard Schoen and Cheng Tong Yao, Proof important version of Guess the RingWhich is later referred to as the theory of the existence of a black hole. In a well-defined mathematical argument, Shuen and Yao show how much matter must be crammed into a given volume to induce the space-time curvature needed to create a closed confined surface.

Cazaras praised Xwen Yao’s work for its originality and generality. Their technology can reveal whether any form of matter, regardless of symmetry considerations, is destined to become a black hole. But their approach had a major flaw. Cazaras said the way they measured the size of a given region of space — by determining the radius of the largest torus, or donut, that could fit inside — was for many observers “cumbersome and counterintuitive,” and therefore impractical.

The last paper offers an alternative. One of Shuen and Yao’s major innovations was the realization that the equation devised by physicist Bong Soo Jang, which originally had nothing to do with black holes, could “explode” – go to infinity – at certain points in space. Surprisingly, the place of its explosion coincides with the location of the closed trapping surface. So, if you want to find such a surface, first find out where Jang’s equation goes to infinity. “In high school, we often try to solve an equation when the solution is zero,” explained the mathematician. Mo Tao Wang from Columbia University. “In this case, we’re trying to solve (Jang’s) equation so that the solution is infinite.”

Hirsch, Casaras, Khoury, and Zhang also rely on Jang’s equation. But in addition to the torus, they use a cube, a cube that can be seriously deformed. The approach, Khoury said, is “similar to Thorne’s idea, using square hoops instead of the traditional round hoops.” It is based on the “cubic inequality” developed by mathematician Mikhail Gromov. This relationship relates the volume of a cube to the curvature of space in and around it.

The new research shows that if you can find a cube somewhere in space such that the concentration of matter is large compared to the volume of the cube, a confined surface will form. “It is much easier to verify this measurement,” he said, than to measure torso Bingzi Miaoa mathematician at the University of Miami, “because all you need to calculate is the distance between the two nearest opposite sides of the cube.”

Mathematicians can also construct cakes (torii) and cubes in higher dimensions. In order to extend their evidence for the existence of black holes to these spaces, Hirsch and his colleagues relied on engineering insights developed in the four decades since Schwen and Yao’s research in 1983. The team could not go beyond the seven spatial dimensions because singularities began to appear in their results. “Getting around those singularities is a common sticking point in engineering,” Khouri said.

He said the next logical step would be to prove the existence of a black hole on the basis of “sublocal mass,” which includes energy coming from both matter and gravitational radiation, not matter alone. This is no simple task, in part because there is no universally agreed upon definition of a quasi-local mass.

Meanwhile, another question looms: To create a black hole with three spatial dimensions, would the object have to be compressed in all three directions, as Thorne insisted, or could compression in two or even one direction be sufficient? Khoury said that all evidence points to the validity of Thorne’s statement, although it has not yet been proven. In fact, this is just one of many open questions about black holes that still linger after they first appeared more than a century ago in a German soldier’s notebook.

This article was Originally published On the Quantitative abstractions Blog.

Main image: The 51-year-old speculates that if matter is compressed into a hoop of a certain size, a black hole is sure to form. Alison Lee / Quanta Magazine.

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