Inference relation
A very important inference of relativity is the relationship between mass and energy. Einstein said that the speed of light should be the same for everyone. This means that nothing moves faster than light. When people use energy to accelerate any object, whether it is a particle or a spaceship, what actually happens is that its mass increases, and further acceleration becomes more difficult. It is impossible to accelerate a particle to the speed of light, because the energy it consumes is infinite. As Einstein's famous formula E = MC 2 summed up, mass and energy are equivalent. In addition to quantum theory, an article entitled "On Electrodynamics of Moving Objects" published by Einstein in 1905 triggered another revolution in physics in the 20th century. This paper studies the influence of object motion on optical phenomena, which is another difficult problem faced by classical physics at that time.
electromagnetic field theory
/kloc-in the mid-9th century, Maxwell established the electromagnetic field theory and predicted the existence of electromagnetic waves propagating at the speed of light C. By the end of19th century, Maxwell's theory was completely confirmed by experiments. What is electromagnetic wave? To whom is its propagation speed c? At that time, the popular view was that the whole universe was full of a special substance called "ether", and electromagnetic waves were the propagation of ether vibration. But people find that this is a theory full of contradictions. If we think that the earth is moving in the static ether, then according to the principle of velocity superposition, the speed of light propagating in different directions on the earth must be different, but the experiment denies this conclusion. If we think that the ether was taken away by the earth, it is obviously inconsistent with some astronomical observations. Schematic diagram of Michelson Morey's experiment
1887, Michelson and Morey made a very accurate measurement by using the interference phenomenon of light, but they still didn't find any movement of the earth relative to the ether. In this regard, H.A. Lorenz put forward a hypothesis that all objects moving in the ether should contract along the direction of movement. From this, he proved that even if the earth moves relative to the ether, Michelson could not find it. Einstein studied this problem from a completely different way of thinking. He pointed out that all difficulties can be solved as long as Newton's concepts of absolute space and absolute time are abandoned, and there is no need for ether at all. (Ether: proposed by Greek scholars and considered as the medium of light propagation)
Put forward the basic principle of optics.
Einstein put forward two basic principles as the basis for discussing the optical phenomena of moving objects. The first is called the principle of relativity. That is to say, if the coordinate system K' moves at a constant speed relative to the coordinate system K without rotating, it is impossible to distinguish which coordinate system is K and which coordinate system is K' in any physical experiment made relative to these two coordinate systems. The second principle is called the principle that the speed of light is constant, which means that the speed of light c (in vacuum) is constant, and it does not depend on the moving speed of the luminous object. On the surface, the constant speed of light seems to conflict with the principle of relativity. Because according to the classical law of mechanical speed synthesis, the speed of light should be different for the two coordinate systems, k' and k, which move at a relatively uniform speed. Einstein thought that if we want to admit that these two principles do not conflict, we must re-analyze the physical concepts of time and space.
Two hypotheses
The law of velocity composition in classical mechanics actually depends on the following two assumptions: 1. The time interval between two events has nothing to do with the motion state of the clock used to measure time; 2. The spatial distance between two points has nothing to do with the motion state of the ruler used to measure the distance. Einstein found that if the principle of light speed invariance and the principle of relativity are recognized to be compatible, then both hypotheses must be abandoned. At this time, the simultaneous events of one clock are not necessarily simultaneous for another clock, and they are relative at the same time. In two coordinate systems with relative motion, the values obtained by measuring the distance between two specific points are no longer equal. Distance is also relative. If an event in the K coordinate system can be determined by three spatial coordinates X, Y, Z and a time coordinate T, and the same event in the K coordinate system is determined by X', Y', Z' and T', Einstein found that X', Y', Z' and T' can be solved by a set of equations. The relative velocity of the two coordinate systems and the speed of light c are the only parameters of the equation. This equation was first derived by Lorentz, so it is called Lorentz transformation. Using Lorentz transformation, it is easy to prove that the clock will slow down because of movement, the ruler will be shorter when it is moving than when it is at rest, and the sum of speeds satisfies a new law. The principle of relativity is also expressed as a clear mathematical condition, that is, under the Lorentz transformation, the space-time variables X', Y', Z' and T' with apostrophes will replace the space-time variables X, Y, Z and T, and any expression of natural laws will still take the same form as before. What people call the universal law of nature is covariant for Lorentz transformation. This is very important for us to explore the universal laws of nature.
The connection between time and space
Besides, in classical physics, time is absolute. It has always played an independent role different from the three spatial coordinates. Einstein's theory of relativity involves time and space. It is believed that the real world of physics is composed of various events, and each event is described by four numbers. These four numbers are its space-time coordinates T and X, Y and Z, which form a four-dimensional continuous space, usually called Minkowski four-dimensional space. In relativity, it is natural to examine the real world of physics in a four-dimensional way. Another important result caused by special relativity is about the relationship between mass and energy. Before Einstein, physicists always thought that mass and energy were completely different and were separately conserved quantities. Einstein found that in the theory of relativity, mass and energy are inseparable, and the two conservation laws are combined into one. He gave a famous formula of mass and energy: e = MC 2, where c is the speed of light. So quality can be regarded as a measure of its energy. Calculations show that tiny masses contain enormous energy. This wonderful formula has laid a theoretical foundation for mankind to obtain huge energy, make atomic bombs and hydrogen bombs, and use atomic energy to generate electricity. Most physicists, including Lorenz, the founder of relativistic transformation relation, find it hard to accept these new concepts introduced by Einstein. The obstacle of the old way of thinking made this new physical theory not familiar to physicists until a generation later. Even in 1922, when the science prize was awarded to Einstein by the Royal Swedish Academy, it only said, "Because of his contribution to theoretical physics and because he discovered the law of photoelectric effect." Not a word about relativity.
Establish the theory of relativity
Einstein further established the general theory of relativity in 19 15. The principle of relativity in a narrow sense is limited to two coordinate systems with uniform motion, while the principle of relativity in a broad sense cancels the restriction of uniform motion. He introduced an equivalence principle, arguing that it is impossible for us to distinguish between gravitational effect and non-uniform motion, that is, non-uniform motion and gravity are equivalent. He further analyzed the phenomenon that light will be bent by gravity when passing near a planet, and thought that the concept of gravity itself was completely unnecessary. It can be considered that the mass of the planet makes the space around it curved, and the light takes the shortest path. Based on these discussions, Einstein derived a set of equations, which can determine the curved space geometry caused by the existence of matter. Using this equation, Einstein calculated the displacement of the perihelion of Mercury, which was completely consistent with the experimental observation, and solved a long-term unexplained problem, which made Einstein excited. In his letter to Erenfest, he wrote that this equation gives the correct value of perihelion. You can imagine how happy I am! For days, I was so happy that I didn't know what to do. "
universal gravitation
1915165438+1On October 25th, Einstein submitted a paper entitled "Equation of Gravitation" to the Prussian Academy of Sciences in Berlin, which fully discussed the general theory of relativity. In this article, he not only explained the mystery of the perihelion motion of Mercury's orbit found in astronomical observation, but also predicted that the starlight would deflect after passing through the sun, and the deflection angle was twice that predicted by Newton's theory. The first world war delayed the determination of this value. 19 19 The total solar eclipse on May 25th provided people with the first observation opportunity after the war. Eddington, an Englishman, went to principe island on the west coast of Africa and made this observation. 165438+1On October 6th, Thomson solemnly announced at the joint meeting of the Royal Society and the Royal Astronomical Society that Einstein, not Newton, had proved this result. He praised "this is one of the greatest achievements in the history of human thought." Einstein discovered not an island, but a brand-new continent of scientific ideas. "The Times reported this important news under the title of" Revolution in Science ". The news spread all over the world, and Einstein became a world-famous celebrity. General relativity has also been elevated to a mythical sacred position. Since then, people have shown more and more interest in the experimental test of general relativity. However, because the gravitational field of the solar system is very weak and the gravitational effect itself is very small, the theoretical result of general relativity deviates very little from Newton's gravitational theory, which makes the observation very difficult. Since 1970s, due to the progress of radio astronomy, the observation distance has far exceeded the solar system, and the accuracy of observation has been greatly improved. Especially in September of 1974, Taylor of MIT and his student Hall observed with a large radio telescope with a diameter of 305 meters and found a pulse binary star. It is a neutron star and its companion star revolve around each other under the action of gravity, with a period of only 0.323 days. The gravity on its surface is100000 times stronger than that on the surface of the sun. This is a laboratory where it is impossible to test the theory of gravity on earth or even in the solar system. After more than ten years of observation, they got a very good result, which is in line with the prediction of general relativity. Because of this great contribution, Taylor and Hall won the 1993 Nobel Prize in Physics.
Narrow proof
Relativity formula and proof symbol unit symbol unit coordinate (x, y, z): m force f (f): n time t (t): s mass m(M): kg displacement r: m momentum p: kg*m/s velocity v (u): m/s energy E: J acceleration a: m/s 2 impulse: n.
A, Newtonian mechanics (preparatory knowledge)
(1): basic formula of particle kinematics: (1) V = DR/DT, R = R0+∫ RDT (2) A = DV/DT, V = V0+∫ ADT (Note: of the two formulas, the left formula is in differential form. As long as we know the motion equation of a particle r=r(t), we can know all its motion laws. (b): Particle Dynamics: (1) Niu Yi: All objects are always in a state of static or uniform linear motion when they are not acted by force. (2) Niu Er: The acceleration of an object is directly proportional to the force it receives and inversely proportional to the mass. F=ma=mdv/dt=dp/dt (3) Niu San: Two forces acting on the same object are balanced if they act on the same straight line and in the same direction. (4) Gravitation: The force between two particles is directly proportional to the product of mass and inversely proportional to the square of distance. F = GMM/r 2, g = 6.67259 *10 (-1) m3/(kg * S2) momentum theorem: I=∫Fdt=p2-p 1. Kinetic energy theorem: W=∫Fds=Ek2-Ek 1 (the work of external force combination is equal to the change of kinetic energy) Conservation of mechanical energy: Ek 1+Ep 1=Ek2+Ep2 only when gravity does work (Note: the core of Newton's mechanics is Newton's second law: F=ma, Similarly, if we know the equation of motion r=r(t), we can find A according to the basic formula of kinematics, and then we can know the force of the object from Niu Er. )
Second, special relativity mechanics
(Note: "γ" is a relativistic factor, γ = 1/sqr (1-u 2/c 2), β=u/c, and u is the velocity of inertial system. ) 1. Basic principle: (1) Principle of relativity: All inertial systems are equivalent. (2) The principle of invariability of the speed of light: the speed of light in vacuum is a constant that has nothing to do with the inertial system. (Give the formula first and then give the proof) 2. Lorentz coordinate transformation: x = γ (x-ut) y = y z = z t = γ (t-UX/c 2) 3. Speed transformation: v (x) = (v (x)-u)/( 1-. (γ ( 1-V (x) u/c 2)) 4。 Scaling effect: △L=△l/γ or DL = DL/γ 5. Clock slowness effect: △t=γ△τ or DT = dτ/γ 6. Doppler effect of light: ν (a) 7. Momentum expression: P=Mv=γmv, that is, m = γ m8. Basic equation of relativistic mechanics: f = dp/dt 9. Mass-energy equation: e = MC 2 10. Energy-momentum relation: e 2 = (E0) 2+p 2c 2 (note:) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Third, three-dimensional proof
The axiom summarized by 1. experiment cannot be proved. 2. Lorentz transformation: Let the coordinate system (A system) at (x, y, z, t) be stationary, while the speed of the coordinate system (B system) at (x, y, z, t) be U and positive along the X axis. At the origin of series A, x=0, and the origin coordinate of series B is X=-uT, that is, X+uT=0. Let x=k(X+uT) (1). Also, because the positions of each point in the inertial system are equivalent, K is a constant related to U (in general relativity, due to the curvature of space-time, each point is no longer equivalent, so K is no longer a constant. ) Similarly, in the B series, the origin also has X=K(x-ut). According to the principle of relativity, the two inertial systems are equivalent, and the two formulas should take the same form, that is, K = K, so there is X=k(x-ut) (2). For y, z, y, z, regardless of speed, Y = can be obtained. That is t = kt+((kloc-0/-k 2)/(ku)) x (5). (1) (2) (3) (4) (5) The principle of relativity is satisfied, and the principle that the speed of light is constant is needed to determine k. When the origins of the two systems coincide, an optical signal is emitted from the coincidence point. For the two systems, there are x=ct and X=cT respectively. Substitute (1)(2) to get ct=kT(c+u). CT=kt(c-u)。 When two formulas are multiplied to eliminate t and t, k =1/sqr (1-u 2/c 2) = γ. Substitute γ into the coordinate transformation of (2) and (5): x = γ (x-ut) y = (dx/dt-u)/(1-(dx/dt) u/C2) = (v (x)-u)/(1. Scale effect: If there is a thin rod with length L parallel to the X axis in system B, it is obtained from X=γ(x-ut): △X=γ(△x-u△t), and △t=0 (measuring the coordinates of both ends at the same time), then △X=γ△x, that is, △ L = γ. △X=0 (to be measured at the same place), so △ t = γ△ T (Note: the length, mass and time interval of an object that is relatively stationary with the coordinate system are called intrinsic length, static mass and intrinsic time, which are objective quantities that do not change with the coordinate transformation. 6. Doppler effect of light: (Note: Doppler effect of sound is: ν (a) = ((U+V 1)/(U-V2)) ν (b). ) A light source at the origin of system B emits light signals, while the origin of system A has a detector, and the two systems have two clocks respectively. When two systems have two clocks, the frequency of the light source in system B is ν(b), the wave number is n, and the time measured by the clock in system B is △t(b). According to the clock slowness effect, the time measured by the clock in system A is △t(a)=γ△t(b) (1). The detector starts receiving at t 1 Then △t(N)=( 1+β)△t(a) (2). Relative motion does not affect the wave number of the optical signal, so the wave number emitted by the light source is the same as that received by the detector. That is, argument (b) △ t (b) = argument (a) △ t (n) (3). It can be obtained from the above three formulas: ν (a) = sqr ((1-β)/(1+β)) ν. β=v/c) Newton's second law remains unchanged under galilean transformation, that is, it holds in any inertial system, but under Lorentz transformation, the original concise form becomes messy, so Newton's law needs to be revised, and the requirement is to keep the original concise form under coordinate transformation. In Newtonian mechanics, the forms of v=dr/dt and R are invariant under coordinate transformation ((x, y, z) in the old coordinate system and (x, y, z) in the new coordinate system). As long as the denominator is changed into an invariant (of course, it belongs to dτ when it is not fixed), the concept of speed can be corrected. Let V=dr/dτ=γdr/dt=γv be the relativistic velocity. Newton's momentum is p=mv, and replacing v with v can correct the momentum, that is, p=mV=γmv. Define M=γm (relativistic mass) and then p=Mv. This is the basic quantity of relativistic mechanics: relativistic momentum. (Note: We generally use Newton's velocity instead of relativistic velocity to participate in the calculation) 8. The basic equation of relativistic mechanics: according to the expression of relativistic momentum, F=dp/dt, which is the definition of force. Although it is exactly the same as Newton's second law, its connotation is different. Mass is variable in relativity. Equation of mass and energy: ek = ∫ FDR = ∫ (DP/dt) * Dr = ∫ DP * DR/dt = ∫ VDP = PV-∫ PDV = MV 2-∫ MV/sq. -MC 2 = MV 2+MC 2 (1-V 2/C 2)-MC 2 = MC 2-MC 2, that is, E = MC 2 = ek+MC 210. Energy momentum relation: E = MC. Available: e 2 = (E0) 2+p 2c 2 * * * * * * * * * * * * * * * * * * * * * * * * * * * * *.
Fourth, four-dimensional proof
The axiom 1. cannot be proved. 2. Coordinate transformation: dl=cdt, that is, DX 2+DY 2+DZ 2+(ICDT) 2 = 0 holds in any inertial system. Define dS as a four-dimensional interval, dS 2 = DX 2+DY 2+DZ 2+(ICDT) 2 (1), which is constant for optical signal DS, but generally not constant for any two time points. ds^2>； 0 is called class space partition, DS 2