Beginners often think that the biggest obstacle in the process of general relativity learning is unfamiliar tensor calculation, but in fact the bigger problem lies in the lack of intuitive geometric understanding of general relativity. In any case, mathematics must go. In fact, the biggest confusion for beginners is the abstraction of mathematics, so that they have to accept some existing definitions. The understanding of mathematical definitions and physical meaning is not profound. Here are some introductions to my teaching process and some of my own learning processes, providing some more intuitive understanding of the introduction and extension of various concepts.
- Why introduce the concept of manifold? General relativity is a discipline about time and space and the laws of physics. Time and space itself is a collection of events, and we usually believe that time and space are not only continuous, but also smooth. The continuity of the collection needs to introduce the concept of topological space, while the set that defines the smoothness is the manifold.
- Why discuss the tensor field on the manifold? The answer to this question has two layers:
- First, we don't have any information about the metric after introducing the manifold concept, that is, we can't talk about the length of a given manifold. But how do we define metrics on curved spaces? We can compare the concept of surfaces in 3D space. If we only focus on a very small area of the area in 3D space, then this area will be very close to the plane. Imagine that the ground around us that we observe on Earth is flat. On the plane we know that the length of a line is satisfied , that is, the Pythagorean theorem. If we choose another set of systems , there is , therefore, in specifying the line length on a surface, we can actually use two options, one is to find the local orthogonal coordinate system , the other is the more general form . This actual description, metric manifold may be formed on a tensor type to express.
- Second, the covariance of physical laws. This is actually the symmetry of the hidden theory itself. High school mechanics in high school did not depend on any inertial system. This universality is the deepest motive of Einstein's discovery of general relativity. Assuming that the laws of physics do not depend on any frame of reference (the reference system is simply equated with the coordinate system), one of the most economical methods requires that the laws of physics be all written as tensor expressions. The tensor is a linear map that does not depend on the coordinate system. The expression of its components under the coordinate transformation can be done through a linear talk. If a physics law is written as a tensor expression, then both sides of the equation follow the same linear transformation under arbitrary coordinate transformation. In other words, the regular expressions before and after the linear transformation are the same. This is the mathematical representation of the covariance of physical physics.
- General relativity expresses gravity as an expression of curved space-time. So how do you understand the curvature of the space? We still understand the comparison of two-dimensional surfaces and planes. The concept of translation of the vector on the plane has been involved in the physics of the university. Specifically, in the Euclidean coordinate system, the translation of the vector is to completely copy the components of the vector in the coordinate system to another point. Given two points can be found if they are two different lines coupled together, the results of the translation vector along the two lines are coincident. But on the surface, the situation becomes subtle. Let's take the surface of the earth as an example. If you stand on the equator and look at the North Pole, the direction of the vector that your gaze emits is perpendicular to the equator. If you are panning along a warp to the North Pole, you will find that the direction of the vector is the direction of the largest circle that is wound along the warp. If you go along the equator for a while and then walk along the meridian to the North Pole, you will find that the vector at this time is inconsistent with the direct travel along the meridian to the North Pole. This means that the same vector on the surface can be misaligned when it is translated to the same point along different paths. This is the most essential difference between straight and curved spaces.
- With the concept of translation, it is natural to define the derivative. In fact, it is natural to define the translation after the concept of the derivative.
- The concept of panning to derivative: If we know how to translate, we can get the concept of translating the vector along a curve, thus defining the derivative of the vector field. Again, the concept of the derivative of the tensor can be defined accordingly.
- If you have the concept of a derivative, how do you define translation? The core definition of the derivative is actually to link the information of different points, which is why the derivative is also called the contact. If we have the concept of a derivative, we can get the concept of the derivative along that direction along a curve direction. With this concept in fact, if you translate in a certain direction, the derivative along that direction is zero. There is also the concept of translation.
- The above is only a general discussion of the concept of translation, it should be noted that translation does not require measurement information.
- In fact, the geometry in our space has taught us how to derivate and translate. Imagine that we are in a curved space, and in a small area, the area can be approximated as straight. The translational concept in flat space is known, and the translational facts in the curved space are formed in each approximately flat local translation. As mentioned earlier, translation can naturally construct the concept of derivation. In fact, the geometric information of the curved space has given a definition of a natural derivative. In the straight space, we already have operators , which is also a derivative operator that is often exposed in university physics. The derivative operator in the curved space in each small local area that looked like a small flat space ! The curved geometry is very intuitive, because your gaze is small enough to be the geometry of a flat space. The overall curved space-time is such a structure that is spliced together by a small flat space. In terms of mathematical definition, it is the derivative operator that is adapted to the metric .
- As mentioned earlier, the important difference between curved space and flat space is the translation problem of the vector. So is there a quantity that can be used to measure the degree of bending of the space? This problem is actually very intuitive. In three-dimensional space, we know that if the radius of a ball is smaller, the degree of bending is higher, and the radius of curvature is its radius of curvature. The more curved the space, the greater the difference after the translation along the curve. For example, the vector from the point along the curve translational point after the results is that , along the curve moves to a flat result after point , then we can use the degree of bending to metric spaces. When the curve is taken very small, the result can be shrunk to a tensor at a point called the Riemann curvature tensor. This tensor is a measure of the degree of bending!