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Albert Einstein put forward the general theory of relativity in 1915, which is mainly based on the fact that mass and energy distort the structure of four-dimensional space-time, thus completely changing the concept of gravity.

Its potential geometric or mathematical formula is attributed to a mathematician named George Friedrich Bernhard Riemann, who constructed a class of geometry (or elliptic geometry), which is different from Euclidean geometry (or plane geometry). It is used to deal with high-dimensional hypersurfaces.

Riemann is a German mathematician who studies and works at the University of Gottingen. In Gottingen, he met a great teacher, Carl Friedrich Gauss. He also spent some time at the University of Berlin, where he met other outstanding mentors, including Jacobi, Dirichley and Eisenstein. He studied elliptic function theory from Eisenstein, and the one who influenced him most was Dirichlet.

According to Felix Klein:

Riemann's way of thinking is very similar to that of Dirichlet. Dirichley likes to make a keen logical analysis of basic problems and avoid lengthy calculations as much as possible. His way is very suitable for Riemann, Riemann adopted his method, and adopted Dirichlet's method of work.

Gauss began to think about the effectiveness of Euclidean geometry many years before Riemann. Gauss shared his curiosity with Riemann and asked him to reformulate the basis of Euclidean geometry in a way that would include surfaces beyond the usual three dimensions.

At that time, thinking about the fourth dimension was considered absurd. In his Algebra Theory, John Wallis describes the fourth dimension as "a monster in nature". At the same time, Polyo and Lobachevsky tried to promote their non-Euclidean geometry in the form of hyperbolic geometry, which provided inspiration for Riemann.

Riemann's groundbreaking speech

In order to win a permanent position at Gottingen University, Riemann had to write a long paper and give a speech. It took him two and a half years to finish a paper on the representability of trigonometric series of functions.

In 1854, he delivered a speech entitled "assumptions about the basis of Geometry", which was so extraordinary that it became the cornerstone of a new kind of geometry, Riemannian geometry.

At the beginning of the lecture, he focused on defining n-dimensional space, geodesic, curvature tensor and so on. In the last part, he tries to connect his geometry with the real world. Many scientists in the audience could not understand the lecture because it was so advanced that not everyone could understand and appreciate it. However, only Gauss fully understood the geometric nature of Riemann.

M. Monastyrsky mentions in a paragraph in Riemann, Topology and Physics

Among Riemann's audience, only Gauss can appreciate Riemann's ideas. The lecture took him by surprise and surprised him. Back at the teachers' meeting, he told William Weber about Riemann's ideas with great praise and rare enthusiasm. "

Calais Daoxiong wrote in his book Hyperspace

In retrospect, there is no doubt that this was one of the most important public speeches in the history of mathematics. Riemann completely broke the limits of Euclidean geometry that ruled mathematics in 2000, and the news soon spread throughout Europe. His speech was translated into several languages, which caused a sensation in the field of mathematics.

A comparison between Euclidean Geometry and Riemannian Geometry

Euclidean geometry is suitable for plane space, such as point, line, plane, etc. Riemannian geometry is suitable for curved surface space, such as cylinder, sphere, torus and so on.

Euclid's fifth axiom, the parallel axiom, is completely denied in elliptic geometry. Because it says, "through a point, not on a given line, only one line is parallel to the given line." In curve geometry, there are no parallel lines.

In plane space, the sum of the three angles of triangles is always equal to 180 °, while in curved space, the sum of triangles is either greater or less than 180 °, because the edges of triangles bend outward on the sphere and inward on the hyperbola.

In flat space, the shortest distance between two points is a straight line, which can be calculated by distance formula, while in curved space, the straight line is called geodesic, which represents the path of local distance minimization. Sometimes there are multiple geodesic lines between two points.

In his surface theory, Gauss describes that a vector has two components in one dimension, namely, size and direction. Riemann extended this concept to higher dimensions and proved that a vector would have six independent components to describe the curvature of any point in three-dimensional space. Similarly, in four-dimensional space, there are 20 independent components.

A tensor is a high-dimensional extension of a vector.

How does Riemannian geometry help Einstein popularize his theory of gravity?

When Einstein proposed the special theory of relativity, he was mainly concerned with its physical properties and interpretation, rather than any mathematical structure. His former teacher, Herman Minkowski, proposed the geometry of special relativity and described space and time as a single entity.

Therefore, for his general theory, Einstein did not realize the mathematical laws that control the influence of the gravitational field around a massive object.

Recently, I have been trying to study the problem of gravitation. Now that I have reached a stage, I have prepared static data. I know nothing about the dynamic field, which must follow the next.. .. Every step is extremely difficult.

He gradually realized that he had to abandon the use of a single scalar field to describe gravity and needed a new geometric language. To this end, he asked his mathematician friend Marcel Grossman, who works at the Zurich Institute of Technology, for help. He said, "Grossman, you have to help me, or I'll go crazy." Grossman instructed him to look for the new geometry proposed by Riemann.

Riemann's new mathematical framework was an unexpected blessing for Einstein because it led him to the conclusion that gravity is actually the result of the curvature of space-time. The greater the curvature of space-time, the greater the gravitational pull.

As Missner, Thorne and Wheeler summed up

Matter tells how space-time bends, and space-time tells matter how to move.

When Einstein realized that Riemannian geometry was the precise mathematical tool of general relativity, the next three years were the most difficult period of his research career. "I'm exhausted. But success is glorious," Einstein said in 1915.

Einstein was full of praise for Riemann's contribution

Physicists are still far away from this way of thinking: for them, space is still a rigid, homogeneous thing, unaffected by any changes or conditions. Only the genius Riemann, lonely and ununderstood, has put forward a new concept of space in the middle of the last century, in which space is deprived of its rigidity and its power to participate in physical events is recognized as much as possible.

Hans Freudenthal wrote in his biography:

General relativity strongly proves that his work is correct. In mathematics, which evolved from Riemann's lecture, Einstein found a framework consistent with his physical ideas, cosmology and cosmology. And the spirit of Riemann's speech is exactly what physics needs: a measurement structure determined by data.

Riemann has made valuable contributions in number theory, analysis, functions, topological properties of surfaces and so on.

In 1862, he caught a bad cold and ended up with tuberculosis. Finally, he died at the age of 39 in Seraska, Lake Maggiore, Italy, in 1866.

Dedeking wrote:

His physical strength declined rapidly, and he felt that his end was coming. But the day before he died, he was resting under a fig tree. His soul was filled with joy in the magnificent scenery. He was still working on his last work. Unfortunately, he didn't finish it.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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