General relativity: Einstein's theory that all observers (no matter how they move) must have the same idea based on scientific laws. It explains gravity according to the curvature of four-dimensional space-time
General relativity? ) is Einstein's theory of gravity established in geometric language in 19 15. It combines special relativity with Newton's law of universal gravitation, and changes gravity to describe space-time bent by matter and energy to replace the traditional view that gravity is a force. Therefore, special relativity and the law of universal gravitation are only special cases of general relativity under special circumstances. Special relativity is the case that there is no gravity; The law of universal gravitation is the situation that the distance is close, the gravity is small and the speed is slow.
[Edit this paragraph] background
1907, Einstein published a paper on the influence of gravity and acceleration on light in special relativity, and the prototype of general relativity began to take shape. 19 12 years, Einstein published another paper to discuss how to describe gravity field in geometric language. At this point, the kinematics of general relativity appeared. 19 15 years, Einstein's field equation was published, and the dynamics of the whole general theory of relativity was finally completed.
After 19 15, the development of general relativity mostly focused on solving the field equation, and the physical explanation of the solution and the search for possible experiments and observations also accounted for a large part. However, because the field equation is a nonlinear partial differential equation, it is difficult to solve, so only a few solutions were solved before the computer was applied to science. There are three most famous solutions: Schwarzschild solution (19 16), Reissner-Nordstr? M solution and Kerr solution.
The observation of general relativity has also made a lot of progress. The precession of mercury is the first evidence to prove the correctness of general relativity. It was measured before the appearance of relativity, and it was not explained theoretically until Einstein discovered it. In the second experiment, 19 19 Eddington measured the starlight deflection caused by the solar gravitational field during the solar eclipse in Africa, which was completely consistent with the prediction of general relativity. At this time, the general theory of relativity has been widely accepted by the public and most physicists. After that, there were many experiments to test the theory of general relativity and confirm its correctness.
In addition, the expansion of the universe has also created another climax of general relativity. From 1922, researchers found that the solution of the field equation would be an expanding universe. Einstein naturally did not believe that the universe would rise or fall at that time, so he added a cosmological constant to the field equation, which made it possible to solve a solution that implicitly defined the universe. But there are two problems with this solution. Theoretically, the solution of the implicit universe is mathematically unstable. In addition, in 1929, Hubble discovered that the universe is actually expanding. This experimental result made Einstein give up the cosmological constant and declared that it was the biggest mistake in my career.
But according to the recent observation of a supernova, the expansion of the universe is accelerating. So the cosmological constant seems to have the possibility of resurrection, and the dark energy in the universe may be explained by the cosmological constant.
[Edit this paragraph] Basic assumptions
Equivalence principle: Gravity and inertia force are completely equivalent. Now some scholars have found that gravity and inertia force are not equivalent.
Principle of general relativity: the form of physical laws is constant in all reference systems. From a distance of 600 kilometers, we can see the simulation diagram of a black hole whose mass is ten times that of the sun.
In general physics (college textbooks), these two principles are described as follows:
Equivalence principle: all physical phenomena in inertial system under the action of uniform and constant gravitational field can be exactly the same as those in non-inertial system that is not affected by gravitational field but moves with constant acceleration.
Relativity principle of general relativity: all non-inertial systems and inertial systems existing in the gravitational field are equivalent, which are used to describe physical phenomena.
[Edit this paragraph] Basic concepts of general relativity
General relativity is based on special relativity. If the latter is proved wrong, then the whole theoretical building will collapse.
In order to understand general relativity, we must know how mass is defined in classical mechanics.
Two different expressions of quality:
First of all, let's think about what quality stands for in our daily life. "Weight"? In fact, we think that mass is something that can be weighed, just like we measure it: we put the object that needs to be measured on the balance. What kind of quality do we use to do this? It is the fact that the earth and the measured object attract each other. This mass is called "gravitational mass". We call it "gravity" because it determines the motion of all stars in the universe: the gravitational mass between the earth and the sun drives the earth to move around the latter in a nearly circular motion.
Now, try to push your car on a flat ground. You can't deny that your car strongly resists the acceleration you want to give it. This is because your car is of great quality. It is easier to move light objects than heavy ones. Mass can also be defined in another way: "It opposes acceleration". This mass is called "inertial mass".
So we come to the conclusion that we can measure quality in two ways. Either we weigh it (very simply) or we measure its resistance to acceleration (using Newton's law).
Many experiments have been done to measure the inertial mass and gravitational mass of the same object. All experimental results come to the same conclusion: inertial mass is equal to gravitational mass.
Newton himself realized that this kind of mass equivalence was caused by some reason that his theory could not explain. But he thinks this result is a simple coincidence. On the contrary, Einstein found that there is a channel in this equation that can replace Newton's theory.
Everyday experience proves this equivalence: two objects (one light and one heavy) will "fall" at the same speed. However, heavy objects are subject to greater gravity than light objects. So why didn't it "fall" faster? Because it is more resistant to acceleration. The conclusion is that the acceleration of an object in the gravitational field has nothing to do with its mass. Galileo was the first person to notice this phenomenon. It is important for you to understand that all objects in the gravitational field "fall at the same speed" are the result of the equivalence of inertial mass and gravitational mass.
Now let's pay attention to the expression "whereabouts". Objects "fall" because the gravitational mass of the earth produces the gravitational field of the earth. Two objects have the same speed in the same gravitational field. Whether it's the moon or the sun, their acceleration speed is the same. In other words, their speed increases by the same amount every second. (Acceleration is the increment of speed per second)
The third hypothesis in Einstein's argument is that gravitational mass and inertial mass are equal.
Einstein has been looking for the explanation that "gravitational mass equals inertial mass". To this end, he put forward a third hypothesis, called "equivalence principle". It shows that if an inertial system is uniformly accelerated relative to a Galileo system, then we can consider it (inertial system) to be stationary by introducing a uniformly accelerated gravitational field relative to it.
Let's examine an inertial system K', which has a uniform acceleration motion relative to Galileo system. There are many objects around k and k'. This object is stationary with respect to K, so these objects have the same accelerated motion with respect to K'. This acceleration is the same for all objects, contrary to the acceleration direction of K' relative to K. As we have said, the acceleration of all objects in a gravitational field is the same, so the effect is equivalent to that K' is static and has a uniform gravitational field.
So if the equivalence principle is established, it is only a simple inference that the masses of two objects are equal. This is why (quality) equivalence is an important argument to support the principle of equivalence.
Assuming that K' is static and the gravitational field exists, we can understand K' as a Galileo system, in which we can study the laws of mechanics. Therefore, Einstein established his fourth principle.
[Edit this paragraph] Main contents
Einstein put forward the "equivalence principle", that is, gravity and inertia force are equivalent. This principle is based on the equivalence of gravitational mass and inertial mass. According to the principle of equivalence, Einstein extended the principle of relativity in a narrow sense to the principle of relativity in a broad sense, that is, the form of physical laws is unchanged in all reference systems. The motion equation of an object is the geodesic equation in the reference system. Geodesic equation has nothing to do with the inherent properties of the object itself, but only depends on the local geometric properties of time and space. And gravity is the expression of local geometric properties of time and space. The existence of material mass will cause the bending of time and space. In curved space-time, objects still move along the shortest distance (that is, along the geodesic-in Euclidean space). For example, the geodesic movement of the earth in curved space-time caused by the sun actually revolves around the sun, resulting in a gravitational effect. Just like on the surface of the earth, if you move in a straight line, you actually walk around the great circle on the surface of the earth.
Gravity is the expression of local geometric properties of time and space. Although the general theory of relativity was founded by Einstein, its mathematical basis can be traced back to the axiom of Euclid geometry and the efforts made for centuries to prove Euclid's fifth postulate (that is, parallel lines are always equidistant). This effort culminated in the work of Lobachevsky, Bolyai and Gauss, who pointed out that Euclid's fifth postulate could not be proved by the first four postulates. Gauss's student Riemann developed the general mathematical theory of non-Euclidean geometry. So it is also called Riemannian geometry or surface geometry. Before Einstein developed general relativity, people thought that non-Euclidean geometry could not be applied to the real world.
In general relativity, the function of gravity is "geometrically"-that is, in special relativity, the physical picture of Min's space background and gravity becomes the physical picture of free movement in Riemannian space background without force (assuming there is no electromagnetic interaction), and its dynamic equation has nothing to do with its own mass and becomes geodesic equation;
The law of gravity is replaced by Einstein's field equation;
Where g is Newton's gravitational constant.
The equation is a second-order hyperbolic partial differential equation with elliptic constraints, with time and space as independent variables and measurement as dependent variables. It is famous for its complexity and beauty, but it is not perfect, and only approximate solutions can be obtained in calculation. Finally, people get the exact solution of true spherical symmetry-Schwartz solution.
The field equation after adding the cosmological constant is:
& lt Mathematics & GTR _-\ fracg _ r+\ lambda g _ =-8 \ pi {g \ over c 2} t _ < /math & gt；
Universe phenomenon and scientific research application of general relativity
According to general relativity, there is no gravity in the local inertial system, and one-dimensional time and three-dimensional space form a four-dimensional flat Euclidean space. In any frame of reference, there is gravity, which leads to the curvature of space-time, so space-time is a four-dimensional curved non-Euclidean space. Einstein discovered the gravitational field equation that the distribution of matter affects the space-time geometry. The bending structure of time space depends on the distribution of material energy density and momentum density in time space, and the bending structure of time space determines the motion trajectory of objects. When gravity is weak and curvature of spacetime is small, the predictions of general relativity tend to be consistent with those of Newton's law of universal gravitation and Newton's law of motion. But there is a difference between strong gravity and large curvature of spacetime. Since the general theory of relativity was put forward, the abnormal precession of Mercury's perihelion, gravitational redshift of light frequency, gravitational deflection of light and delay of radar echo have been predicted, which have been confirmed by astronomical observations or experiments. The observation of pulse binary stars in recent years also provides strong evidence for general relativity to predict the existence of gravitational waves.
The general theory of relativity is quickly recognized and appreciated by people because of its amazing confirmation and theoretical beauty. However, because Newton's gravity theory is accurate enough for most gravity phenomena, general relativity only provides a very small correction, which people do not need in practice. Therefore, the general theory of relativity has not been fully valued and developed rapidly in the half century after its establishment. In the 1960s, the situation changed, and the background radiation of strong gravitational celestial bodies (neutron stars) and the 3K universe was discovered, which made the research of general relativity flourish. General relativity is of great significance for studying the structure and evolution of celestial bodies and the structure and evolution of the universe. The formation and structure of neutron stars, black hole physics and black hole detection, gravitational radiation theory and gravitational wave detection, big bang cosmology, quantum gravity and topological structure of large-scale spacetime are being deeply studied, and general relativity has become an important theoretical basis for physical research.
[Edit this paragraph] Einstein's fourth hypothesis
Einstein's fourth hypothesis is a generalization of his first hypothesis. It can be said that the laws of nature are the same in all departments.
It is undeniable that it sounds more "natural" to claim that the laws of nature are the same in all departments than to claim that the laws of nature are the same only in Galileo. But we don't know whether there is a Galileo system.
This principle is called "the principle of general relativity"
Death elevator
Let's imagine an elevator falling freely in a skyscraper with a fool inside. The man left his watch and handkerchief at the same time. What will happen? For a person who takes the earth as the reference frame outside the elevator, watches, handkerchiefs, people and elevators all fall at exactly the same speed. Let's review: according to the principle of equality, the motion of an object in the gravitational field does not depend on its mass. ) So the distance between watch and floor, handkerchief and floor, man and watch, man and handkerchief is fixed. So for the person in the elevator, the watch and handkerchief will be left where he just threw them.
If this person gives his watch or handkerchief a certain speed, they will move in a straight line at a constant speed. The elevator behaves like Galileo system. However, this will not last forever. Sooner or later, the elevator will collapse, and the observers outside the elevator will go to an unexpected funeral.
Now do the second idealized experiment: our elevator is far away from any mass object. Like the depths of the universe. Our big fool escaped from the last accident. After staying in the hospital for several years, he decided to return to the elevator. Suddenly a creature began to drag the elevator. Classical mechanics tells us that constant force will produce constant acceleration. This rule does not apply to very high speed situations. Because the mass of an object increases with the increase of speed. In our experiment, we assume that it is correct. Therefore, the elevator will accelerate in Galileo.
Our genius fool stayed in the elevator and let his handkerchief and watch fall. People in Galileo's department outside the elevator thought that watches and handkerchiefs would hit the floor. This is because the floor hits them (handkerchiefs and watches) because of its acceleration. In fact, people outside the elevator will find that the distance between the watch and the floor and the distance between the handkerchief and the floor are decreasing at the same speed. On the other hand, the person in the elevator will notice that his watch and handkerchief have the same acceleration, which he will attribute to the gravitational field.
These two explanations seem to be the same: on the one hand, accelerated motion, on the other hand, consistent motion and gravitational field.
Let's do another experiment to prove the existence of gravitational field. A beam of light shone through the window on the opposite wall. Our two observers explained it this way:
People outside the elevator told us that light passes through the window at a constant speed (of course! ) shoots horizontally into the elevator along a straight line and shines on the opposite wall. But because the elevator is running upwards, the irradiation point of light should be slightly lower than this incident point.
The people in the elevator said, we are in the gravity field. Because light has no mass, it will not be affected by the gravitational field, and it will just fall on the point opposite the incident point.
Oh! The problem has arisen. Two observers disagree. However, the people in the elevator made a mistake. He said that light has no mass, but it has energy, and energy has mass (remember that the mass of one joule of energy is: m = e/c 2), so light will have a trajectory that bends to the floor, just as external observers say.
Because the mass of energy is very small (C 2 = 300,000,000× 300,000,000), this phenomenon can only be observed near a very strong gravitational field. It has been proved that light will bend when it is close to the sun because of its great mass. This experiment is the first demonstration of Einstein's theory (general relativity).
From all these experiments, we draw the conclusion that by introducing the gravitational field, we can regard an acceleration system as a Galileo system. By extension, we think it is suitable for all motions, whether it is rotating motion (centripetal force is interpreted as gravitational field) or non-uniform acceleration motion (gravitational field that does not meet Riemann condition is transformed by mathematical method). You see, general relativity conforms to practice everywhere.
The above example is taken from The Evolution of Constitution by Einstein and leopold infeld.