★ What are the specific differences between space, time and space in physics and space in mathematics? ★

Extension and abstraction of the concept of physical space in mathematics. Such as Euclidean space, hyperbolic space, Riemannian space, various function spaces, topological spaces and so on. They reflect the development of people's understanding of various attributes of spatial structure.

The earliest concept of mathematical space is Euclidean space. It comes from the intuition of space, and reflects the preliminary understanding of the linearity, uniformity, isotropy, inclusiveness, positional relationship (distance), stereoscopicity, and even infinite extensibility, infinite separability and continuity of space. However, for a long time, people's understanding of space is limited to the scope of Euclidean geometry, thinking that it has nothing to do with time. 1In the 1920s, the appearance of non-Euclidean geometry broke through the traditional concept that Euclidean space is the only mathematical space. The concept of space in non-Euclidean geometry is more abstract and unified with Euclidean space into a constant curvature space, which is a special form of Riemannian space. /kloc-In the mid-9th century, G.F.B Riemann also introduced the concept of manifold. These concepts not only play a great role in the understanding of physical space, but also greatly enrich the concepts of space in mathematics.

1At the end of the 9th century and the beginning of the 20th century, people gave the topological definition of dimension, and made a deep study on the metric properties of function space, resulting in a series of important concepts of mathematical space, especially the general concept of topological space. Since 1930s, all kinds of spaces in mathematics have been unified on the basis of mathematical structure, and people have a better understanding of all kinds of mathematical spaces. With the deepening of the understanding of physical space and the development of mathematical research, various concepts of mathematical space have been popularized from algebra, geometry and topology. The popularization of the concept of space in algebra mainly comes from the emergence and development of analytic geometry. Geometric objects (points, lines, etc.). ) and the array form a corresponding relationship, so that people can accurately and quantitatively describe the space. In this way, it is easy to generalize the coordinate three array to the coordinate n array (vector), and the corresponding space is the n-dimensional linear space or vector space. This kind of space expands Euclidean space in dimension, but omits the concept of distance in Euclidean space. The linear space on real number field can usually be extended to general field, especially the linear space on finite field has become a space with only a limited number of points, and its spatial continuity has been abandoned. Algebraically and geometrically, space can be extended to affine space and projective space. Projective space can contain infinity points and infinity lines by geometric method or coordinate method. In addition, we can also make all kinds of spaces become intuitive models for physics and even other sciences to deal with motion through arrays, phase spaces, state spaces and so on.

The more abstract form of space is topological space. Because the topological structure reflects the distance relationship between points, the concepts of distance in Euclidean space and vector length in vector space are abandoned in topological space.

People's research on various mathematical spaces reflects the process from local and superficial intuition to a deeper understanding of various attributes of space. For example, with the development of topology, people have a deeper and more essential understanding of the dimension, continuity, opening and closing, bounded and unbounded, and the orientation of space. The study of manifold has also made people's understanding of the finiteness and infinity of space, the local and the whole leap. Manifold concept is an important development of space concept. It is Euclidean space locally, but it can have various forms as a whole. Can be opened and closed, can have edges, can have no boundaries. This profound understanding can promote the study of physical space. For example, Minkowski space is the mathematical model of special relativity, while Riemann space becomes the mathematical model of general relativity (see relativity).

Space in mathematics

Mathematically, space refers to a set with special properties and some extra structures, but there is no mathematical object simply called "space". In primary or middle school mathematics, space usually refers to three-dimensional space. Common space types in mathematics:

affine space

topological space

Uniform space

hausdorff space

Banach space

Vector space (or linear space)

Normed vector space (or linear normed space)

Inner product space

metric space

Complete metric space

Euclidean space

Hilbert space

projective space

Functional space

Sample space

probability space

Time and space in physics

Cai Zongru

introduce

Living in this vast universe, we naturally have two concepts: time and space. We saw everything in the universe. If there is no space, how can we arrange everything in the universe? We saw everything in the universe. If there is no space, how can we arrange everything in the universe? The change of everything, the success, survival and badness of events are different from the past, present and future. The change of everything, the success, survival and badness of events are different from the past, present and future. So time and space are used to arrange or sort everything. So time and space are used to arrange or sort everything. In our daily life, the importance of time and space is beyond words. In our daily life, the importance of time and space is beyond words. Moreover, when we try to describe, know and understand nature through science, time and space are more important. Moreover, when we try to describe, know and understand nature through science, time and space are more important. In physics, there is no physical equation that does not need time and space. In physics, there is no physical equation that does not need time and space. Therefore, this paper will give a brief introduction to the spacetime mentioned in physics, including Newton's spacetime and relativistic spacetime. Therefore, this paper will give a brief introduction to the spacetime mentioned in physics, including Newton's spacetime and relativistic spacetime.

Newton's Time and Space

Newton thought that space is absolute, time is absolute, and time and space exist independently. In Newton's book Mathematical Principles of Natural Philosophy, he defined absolute space: absolute space, by its nature, has nothing to do with any external things, and always remains similar and impossible. "Absolute space, in essence, has nothing to do with external objects, and it will never change or move. In other words, Newton thought that absolute space has nothing to do with the existence of matter and the characteristics of existing matter, but a three-dimensional space, following the framework of Euclidean geometry. In physics, the physical quantities of space are length, area, volume and so on. Because space is absolute, the distance between two points A and B in space measured by an observer who is stationary relative to the ground is the same as that measured by an observer who is moving relative to the ground (such as on a train or a car). In other words, if there is a stick on the ground, the length of the stick measured by the observer who is stationary relative to the ground must be the same as that of the same stick measured by the observer who is moving.

Newton also defined absolute time: absolute, real and mathematical time, which flows smoothly from its own nature and has nothing to do with any external things. "Absolute, real and mathematical time is essentially stable and has nothing to do with external objects. If time is absolute, the time interval between event A and event B measured by an observer stationary relative to the ground is the same as that measured by an observer moving relative to the ground. In other words, if an observer stationary relative to the ground measures events A and B at the same time, then an observer moving relative to the ground must measure events A and B at the same time.

Newton's view of time and space is "unaffected", so it is absolute. Because it is absolute, it has * * * generality and consistency, that is to say, the universe has only one time and one space, and time and space are completely irrelevant. Time and space have nothing to do with everything, but everything exists in time and space.

Time and space of relativity

Einstein put forward the special theory of relativity in 1905, which completely subverted Newton's concept of absolute time and space. A basic assumption of special relativity is that the speed of light in vacuum is constant. That is to say, the speed of light measured by an observer at rest relative to the ground is the same as that measured by an observer moving relative to the ground. At that time, physicists were very confused about the experimental results of the constant speed of light, because this result violated Newton's absolute time and absolute space. Einstein accepted the experimental result that the speed of light is constant and regarded it as the basic hypothesis. Under this assumption, he established the special theory of relativity. Special relativity tells us that the so-called two events A and B are "simultaneous" and relative, not absolute as Newton said. That is, an observer stationary relative to the ground measures two events A and B at the same time, while an observer moving relative to the ground measures the same two events A and B at the same time. Special relativity tells us that if there are two identical clocks, one of which is stationary relative to us and the other is moving relative to us, the moving clock will go slower than the stationary clock. In other words, a moving clock takes one second longer than a stationary clock. In other words, the second a person flies in the air is different from the second a person walks on the ground. Even on the same plane, the seconds of sitting people are different from those of walking people. Special relativity calls it time dilation. So far, time is no longer absolute, but relative. In space, the special theory of relativity leads to the movement of length contraction. What is the length contraction of ruler in motion? If there is a ruler standing on the ground, the length of this ruler is measured as L 0 by an observer who is stationary relative to the ground, and the length of the same ruler is measured as L by another observer who moves in the direction pointed by the ruler, then L will be less than L 0. In other words, the length of a ruler in motion will be shorter than that of the same ruler at rest. The distance between two different points in space is different from the distance measured by observers in different coordinate systems, so the space is not absolute, but relative. Special relativity ended Newton's absolute time and absolute space. The second influence of special relativity on time and space is that space and time are combined by the constant speed of light, and time and space cannot and are not independent of each other.

Einstein's special theory of relativity is called special because the category of material motion studied by special theory of relativity does not involve gravity and does not consider acceleration. However, in nature, any substance is bound to be affected by gravity. Einstein put forward the general theory of relativity in 19 16, studying the motion of gravity, space-time and matter. General relativity holds that space-time is not flat, and space-time will bend because of the distribution of mass and energy in space-time. Gravity is only the result of uneven space-time. The space-time of general relativity is curved, and the degree of curvature depends on gravity. In other words, as long as there is gravity, four-dimensional space-time is curved. The stronger the gravity, the more serious the space-time curve, and this curved space does not follow the framework of Euclidean geometry. General relativity also tells us that the stronger the gravity, the slower the clock goes. And gravity is related to the quality of matter. Therefore, in general relativity, four-dimensional space-time and matter are closely related. Before the publication of the general theory of relativity, time and space were considered as the stage for various events, but these events did not affect time and space. In general relativity, time and space are bound to be related to matter, and the movement of matter affects time and space; Conversely, time and space will also affect the movement of matter.

In addition to the theory of relativity, another great development of physics in the twentieth century is quantum mechanics. Quantum mechanics tells us that elementary particles (such as electrons and quarks) have the duality of particle fluctuation. We can't get the position and velocity of tiny particles at the same time, which is the uncertainty principle. So how can relativity and quantum mechanics be integrated in the world of tiny particles? To solve this problem, physicists are developing the theory of quantum gravity.

Physicists want to develop a theory that can describe the whole universe. The way physicists take is to divide the problems of the whole universe into many small parts (define research fields) and invent theories in these research fields. Each theoretical description and prediction has its scope limitation. It's like a blind man touching an elephant, and some theories have to be rearranged. Moreover, if every event in the universe is interrelated and inseparable, then the method adopted by physicists may be wrong. Let's go back to the time and space of physics. We should pay attention to the physical quantities used in physics (such as length, mass, time, etc. ) are all operational definitions, which means that these quantities are defined after various conditions (operations). What is the nature of space-time for physicists? Physicists are more interested in why the speed of light is constant. Physicists believe that time and space are used to arrange or sort everything. Time and space are relative, there is neither absolute time nor absolute space. Time and space are not independent of each other, but related, so they are called time and space. Time and space are relative, not absolute, that is to say, there are infinite time and space, and every object has its own time and space. In addition, time and space are closely related to matter, so it is meaningless to talk about time and space without matter.

From zero-dimensional space to four-dimensional space

-research on pure concepts in geometry

(Ma Lijin, Department of Mathematics, Longdong University, Qingyang, Gansu 745000)

abstract

Geometry is not necessarily a description of real phenomena, and geometric space and natural space cannot be treated equally. The study of pure concept is a milestone in the field of mathematics. The development from zero-dimensional space to three-dimensional space, especially from three-dimensional space to four-dimensional space, is a revolution in geometry.

key word

Zero dimension; One dimension; Two-dimensional; Three dimensions; Four dimensions; N dimension; Geometric elements; Point; Straight line; Airplane.

main body

The concept of N-dimensional space was put forward with the development of analytical mechanics in18th century. In the works of D'Alembert Euler and Lagrange, the concept of the fourth dimension appears irrationally, and D'Alembert proposed to imagine time as the fourth dimension in the entry about dimension in the Encyclopedia. /kloc-in the 0/9th century, geometry higher than three dimensions was still rejected. Karl august mobius (1790- 1868) pointed out in his calculation of center of gravity that two mutually mirrored images cannot overlap in three-dimensional space, but can overlap in four-dimensional space. But then he said: such a four-dimensional space is hard to imagine, so superposition is impossible. This happens because people treat geometric space and natural space exactly the same. Even until 1860, Ernst Eduard Kummer (18 10- 1893) was still mocking four-dimensional geometry. However, with mathematicians gradually introducing some concepts with no or little direct physical meaning, such as imaginary number, mathematicians learned to get rid of the concept that "mathematics is a description of real phenomena" and gradually embarked on a purely conceptual research method. Imaginary number is puzzling because it has no reality in nature. Taking imaginary number as the directed distance on the straight line and complex number as the point or vector on the plane, this interpretation pioneered four elements, non-Euclidean geometry, complex elements in geometry, N-dimensional geometry and various singular functions, and directly ushered in N-dimensional geometry for the concept of physical service.

1844, inspired by quaternion, grassmann made a greater generalization, published linear expansion, and revised it into expansion theory in 1862. He first involves the general concept of N-dimensional geometry. In an article in 1848, he said:

My extended calculus established the abstract foundation of space theory, that is, it broke away from all spatial intuition and became a pure mathematical science, and only formed geometry when making special applications to (physical) space.

However, the promotion of theorems in calculus is not only to translate geometric results into abstract language, but also of great universal importance, because ordinary geometry is limited by (physical) space. Grassmann emphasized that geometry can be applied in physics to develop the research of pure intelligence. Since then, geometry has cut off its connection with physics and developed independently.

After many scholars' research, after 1850, N-dimensional geometry was gradually accepted by the mathematical community.

The above is the tortuous course of the development of N-dimensional geometry, and the following are some concrete processes of the development of N-dimensional geometry.

First, we regard a point as a zero-dimensional space, a straight line as a one-dimensional space and a plane as a two-dimensional space, and observe the following postulates:

Two points belonging to a straight line determine this straight line. 1. 1

Two planes belonging to a straight line determine this straight line. (Compare this postulate with the postulate 1. 1). 1.2

Two straight lines belonging to the same point also belong to the same plane. (Inference of postulate 1.2) 1.3

Two straight lines that belong to the same plane also belong to the same point. 1.4

It can be inferred that:

Two spaces with the same 1. dimension determine that the other space is one dimension higher under some conditions. For example, two points (we regard them as two zero-dimensional spaces) determine a straight line (one-dimensional space). Two straight lines (two one-dimensional spaces) belonging to the same point (specified conditions) also belong to the same plane (two-dimensional space).

2. Two spaces with the same dimension can also determine a space with a lower dimension under certain conditions. For example, two planes (two two-dimensional spaces) define a straight line (one-dimensional space) belonging to them. Two straight lines (two one-dimensional spaces) belonging to the same plane (finite condition) determine a point (zero-dimensional space).

3. Conclusion 2 does not include two planes to define a space one dimension higher. It just assumes that they define a straight line, which is a space one dimension lower than the plane. This leaves a gap, allowing our thoughts to extend to high-dimensional space. This gap can be eliminated by inference 1.3 "Two straight lines belonging to the same point also belong to the same plane", and geometric element points, straight lines and planes are replaced by geometric element lines, planes and three-dimensional spaces in turn.

The following inference is the result of substitution. Two planes belonging to the same straight line also belong to the same three-dimensional space.

With this new inference, we include the geometric element directly corresponding to other geometric elements-three-dimensional space.

The next step is to apply the duality principle to this reasoning and get some internal conclusions from these newly expanded reasoning. The principle of existence duality will be applied by exchanging the positions of geometric elements-plane and space. Then we get the following inference:

Two three-dimensional spaces belonging to the same line also belong to the same plane. 1.5

From the inference 1.5, we can get the following postulate:

Two three-dimensional spaces belonging to a plane determine this plane. 1.6

On the basis of the above 1.5 and 1.6, the following viewpoints can be put forward:

The geometric condition of 1. four-dimensional space is obvious, because two known spaces with the same dimension can only exist in a space one dimension higher than them. For example, two different lines (one dimension) lie on the same plane (two dimensions); Two different * * * planes (two dimensions) (along a straight line * * *) are located in a three-dimensional space; Two different three-dimensional spaces (along a plane) are located in a four-dimensional space.

2. Geometrically, two planes that do not belong to the same straight line but intersect at a point belong to different three-dimensional spaces.

The concept of four-dimensional space can also be studied by analytic geometry. We can use algebraic equations to express geometric concepts. In order to observe and understand the four-dimensional space in this way, we will study the three geometrical features equations of point, line and surface in the three-dimensional space system. Using Cartesian system representation, we can write:

Equation of point: ax+b = 0 (coordinate system: a point on a straight line).

Equation of straight line: ax+by+c = 0 (coordinate system: two orthogonal straight lines on the plane).

Equation of plane: ax+by+cz+d = 0 (coordinate system: three mutually perpendicular planes in three-dimensional space).

From the above research, we can see that:

The number of variables in the equation of each geometric element (or space) is equal to the dimension of this space plus 1.

The geometric elements in the coordinate system have the same dimensions as those in the represented geometric space.

In this coordinate system, the number of geometric elements is equal to the dimension of the represented space plus 1. In the coordinate system, this geometric element number is the minimum requirement.

The coordinate system used to represent geometric elements is located in a space one dimension higher than the geometric elements it contains.

According to the above observation, we can write the following equations of three-dimensional space. It should be noted that this equation has four variables (x, y, z, u).

ax + by + cz + du + e = 0

Now we can conclude that:

1. The geometric elements of this coordinate system are three-dimensional, that is, they are three-dimensional spaces.

This coordinate system has four three-dimensional spaces.

This coordinate system is located in four-dimensional space.

Our research on four-dimensional space or even higher-dimensional space is not based on experimental summary. In reality, it is difficult for us to find and deduce their general laws. For these problems, we can adopt a new research method. Namely: the study of pure concepts. In this way, we can easily deduce these important but unimaginable new contents in reality.