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Math Proof Draws New Boundaries Around Black Hole Formation

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Summary

In this article, Steve Nadis examines the 51-year-old "hoop conjecture" which states that if matter is squeezed into a hoop of a certain size, a black hole will form. He discusses the work of Roger Penrose, Kip Thorne, Richard Schoen, and Shing-Tung Yau in the development of this conjecture and how their work has been used to make a new paper, developed by Marcus Khuri, Sven Hirsch, Demetre Kazaras, and Yiyue Zhang. The paper uses the Jang equation, developed by Pong Soo Jang, to measure the size of a given region of space and determine if it will eventually collapse into a black hole. It also uses the "cube inequality" developed by Mikhail Gromov to connect the size of a cube to the curvature of space. This paper extends the proof of black hole existence to higher dimensions and is the first step in answering the question of whether an object with certain mass will eventually become a black hole.

Q&As

What is the hoop conjecture and how was it formulated?
The hoop conjecture is an idea formulated by the physicist Kip Thorne in 1972 that states that if enough mass is confined within a perfectly spherical region (bounded by the “Schwarzschild radius”), then a black hole is sure to form.

What does the Jang equation measure and how does it relate to the formation of black holes?
The Jang equation measures the size of a given region of space by determining the radius of the fattest torus, or doughnut, that could fit inside. It can "blow up" — go to infinity — at certain points in space, which coincides with the location of a closed trapped surface.

How does the new paper published by four mathematicians help to determine the presence of black holes?
The new paper published by four mathematicians helps to determine the presence of black holes by showing that if you can find a cube somewhere in space such that the matter concentration is large compared to the size of the cube, then a trapped surface will form.

How did the mathematicians Richard Schoen and Shing-Tung Yau prove an important version of the hoop conjecture?
The mathematicians Richard Schoen and Shing-Tung Yau proved an important version of the hoop conjecture by showing how much matter must be crammed into a given volume to induce the space-time curvature necessary to create a closed trapped surface.

What are the next steps for mathematicians to prove the existence of black holes?
The next steps for mathematicians to prove the existence of black holes is to prove black hole existence based on “quasi-local mass,” which includes the energy coming from both matter and gravitational radiation, rather than from matter alone. They also need to prove whether compression in two directions or even just one is enough to create a black hole of three spatial dimensions.

AI Comments

👍 This article is an interesting and informative exploration of the hoop conjecture and how it relates to black hole formation.

👎 The article is too technical and difficult to understand for most readers.

AI Discussion

Me: It's about a math proof that draws new boundaries around the formation of black holes. It provides a new way of measuring the size of a black hole and proves that higher dimensional black holes can exist.

Friend: Wow, that's really fascinating! It's great that mathematicians and physicists are exploring the properties of these objects from both a theoretical and experimental perspective. What are the implications of this article?

Me: Well, the article provides a more practical way to determine if a region of space is likely to collapse to form a black hole. It also shows that the hoop conjecture, which suggests that a certain amount of mass must be confined within a certain region, is true and could be used to predict black hole formation. Lastly, it also demonstrates that higher dimensional black holes can exist, which was not something that could be confidently said before.

Action items

Research the mathematical physics and theories behind black hole formation.
Explore the implications of the hoop conjecture and the black hole existence theorem.
Read more about the Jang equation and the cube inequality to gain a better understanding of the recent paper.

Technical terms

Mathematical Physics: The study of the application of mathematics to physical problems.
Black Holes: A region of space in which the gravitational field is so strong that no matter or radiation can escape.
General Relativity: A theory of gravitation developed by Albert Einstein in 1915.
Gravity: The force of attraction between two objects with mass.
Mathematics: The study of numbers, shapes, and other abstract concepts.
Physics: The study of matter, energy, and the interactions between them.
Singularity: A point in space-time at which the curvature of space-time is infinite.
Hoop Conjecture: A conjecture proposed by Kip Thorne in 1972 that states that if matter is confined within a hoop of a certain size, a black hole is sure to form.
Schwarzschild Radius: The radius of a sphere of a given mass, such that if all the mass were compressed within that sphere, a black hole would form.
Closed Trapped Surface: A surface whose curvature is so extreme that outward-going light gets wrapped around and turned inward.
Jang Equation: An equation devised by the physicist Pong Soo Jang that can “blow up” — go to infinity — at certain points in space.
Cube Inequality: A relationship developed by the mathematician Mikhail Gromov that connects the size of a cube to the curvature of space in and around it.
Quasi-Local Mass: The energy coming from both matter and gravitational radiation, rather than from matter alone.