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Black holes are strange celestial bodies whose gravity is so strong that even light cannot escape. At the center of them is a singularity, where the density is infinite and the laws of physics fail. Black holes are one of the predictions of Einstein's general theory of relativity, and they play an important role in the universe.

But how do we know if a black hole exists? We can't observe them directly because they don't emit any light. We can only infer their existence from their effects on the matter and light around them. For example, we can see the strong radiation from the accretion disk around the black hole, or the gravitational lens effect of the black hole on the background starlight.

However, these methods require that we already have a candidate for a black hole. If we want to theoretically judge the existence of a black hole, we need some more basic conditions. In other words, we need to know what kind of matter distribution leads to the formation of a black hole.

This problem has plagued mathematicians and physicists for more than half a century. The earliest attempt was the singularity theorem proposed by Penrose in 1964. He proved that if there is a closed trapped surface in space-time, then there must be a singularity in space-time. A closed trapped surface is a surface with such extreme curvature that the light emitted outward is bent and turned inward. This means that nothing inside the surface can escape, thus forming a black hole.

However, Penrose's theorem does not tell us how to produce a closed trapped surface. In 1972, Thorne put forward a conjecture called the "circular conjecture". He believes that if enough mass is compressed into a ring of a certain size, a black hole must be formed. In other words, if you have an object with a mass of M and you can put it in a ring with a radius of RenewM / 2 π c ², then the object will collapse into a black hole.

This conjecture is enlightening, but it also has some problems. First of all, it only applies to the case of perfect spherical symmetry, that is, the object must be a uniformly distributed sphere. However, in reality, objects may have various irregular shapes and density distributions. Secondly, it is only applicable to black holes in three-dimensional space, that is, there are only three directions in space: up and down, left and right, front and back. However, in mathematics and physics, we sometimes consider higher dimensions, such as four, five, or more dimensions. Do black holes also exist in these spaces? If they exist, what kind of nature do they have?

In order to answer these questions, we need some more general and precise conditions to judge the existence of black holes. This is the main contribution of a recent article. The author of this article mathematically proved a surprising conclusion: if you have an object with a mass of M, and you can put it into a cube with sides of Rend2M / 3c ², then the object will collapse into a black hole. This condition applies not only to black holes in three-dimensional space, but also to black holes in arbitrary dimensions. In other words, no matter how many dimensions you are in, as long as you have a cube small enough, you can use it to create a black hole.

What is the significance of this conclusion? First of all, it gives us a simpler and more general condition for judging the existence of black holes. We don't need to consider the shape or density distribution of the object, we just need to consider whether it can be put into a cube.

Secondly, it gives us a way to prove the existence of black holes in high-dimensional space. We know that black holes exist in three-dimensional space, and there is a lot of observational evidence to support them. But do black holes exist in high-dimensional space? This is an interesting theoretical question because some physicists think that high-dimensional space may be a way to describe our universe. This article tells us that if the high-dimensional space is real, then black holes can also exist there.

How does this article prove these conclusions? It uses some complex and sophisticated mathematical tools, such as differential geometry, topology and partial differential equations. I will not elaborate on them here because they require a lot of a priori knowledge and skills. However, I can give you some intuitive ideas that use two main steps.

The first step is to prove that if you have an object of mass M and you can put it into a cube with sides of Rend2M / 3c ², then the object will form a closed trapped surface. This step uses a technique called "reverse light cone". The reverse cone is a way to describe how light propagates in space-time. You can imagine a point light source emitting a beam of light that spreads in all directions to form a cone. If you turn the time upside down, the light will shrink back from all directions and reconverge to the point light source. This is the reverse cone of light.

A characteristic of the reverse light cone is that its vertex always appears first in space-time, while its bottom surface always appears last in space-time. In other words, the reverse cone never propagates in the past. This is important because it means that if you find a reverse cone of light in space-time and it completely surrounds an object, then the object cannot escape to the outside. Because no matter how the object moves, it can only move along the inside of the reverse cone, which always appears earlier than the outside. It's like you're stuck in a place where you can never reach the future.

The author uses a reverse light cone to prove the existence of a closed trapped surface. He first assumed the simplest case, that is, the object is stationary and the density is uniformly distributed. He then constructed a special reverse light cone whose vertex is exactly in the center of the cube and its bottom surface is tangent to the surface of the cube. He proved that the reverse light cone satisfies the Einstein field equation and completely surrounds the object. Therefore, he came to the conclusion that the object formed a closed trapped surface.

He then considered a more general situation, that is, objects may have various irregular shapes and density distributions and may be in motion. He proved that as long as an object can be placed in a cube with sides of Rim2M / 3C ², then he can surround it with a similar reverse cone of light, and come to the same conclusion: the object forms a closed trapped surface. This completes the proof of the first step.

The second step is to prove that if an object forms a closed trapped surface, it will collapse into a black hole. This step uses Penrose's singularity theorem, which says that if there is a closed trapped surface in spacetime, then there must be a singularity in spacetime. The singularity is the core of the black hole, which is surrounded by the boundary of the black hole. Therefore, if an object forms a closed trap surface, it is surrounded by the event horizon and collapses into a black hole. This completes the proof of the second step.

To sum up, the author proves the cube theorem: if you have an object of mass M and you can put it into a cube with sides as long as Rend2M / 3c ², then the object will collapse into a black hole. This theorem applies not only to black holes in three-dimensional space, but also to black holes in arbitrary dimensions.

This article comes from the official account of Wechat: Vientiane experience (ID:UR4351), author: Eugene Wang

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