What is Special Relativity?
Special Theory of Relativity is a new flat space-time theory that is different from Newton's space-time theory, which was proposed by Albert Einstein in 1905, entitled "On the Electrodynamics of Moving Objects". "Narrow sense" means that it only applies to inertial reference frames. The starting point of this theory is two basic assumptions: the principle of special relativity and the principle of constant speed of light. The core equation of the theory is the Lorentz transformation (group) (see Inertial Coordinate Transformation). The special theory of relativity predicts some new effects (relativistic effects) not available in Newton's classical physics, such as time expansion, length contraction, lateral Doppler effect, mass-velocity relationship, mass-energy relationship, etc. Special relativity has become one of the foundations of modern physical theory: all micro-physical theories (such as basic particle theory) and macro-gravity theory (such as general relativity) meet the requirements of special relativity. These relativistic kinetic theories have been confirmed by many high-precision experiments.
Special relativity
- Special relativity
- Special Relativity Principle : Everything
- If at some
- Make Newton's first law of mechanics (
- Because the Lorentz transformation rate of space-time coordinates is different and the movement data of the reference clock is different, the synthesis speed is not a simple mathematical addition or subtraction.
- If the Lorentz transform
- The time (interval) recorded by the same standard clock is called
- Special relativity prediction (this is not a prediction, but the result of mathematical assumptions and mathematical logic) . The "pointer" of a moving clock walks at a slower rate than when the clock is at rest.
- Slower clocks directly lead to relativistic
- Lorentz transformation rate hypothesis
- Consider a long rod placed on the x ' axis of the K' system. Its length is called
- The mathematical expression of Newton's mechanics under special theory of relativity: the velocity data is modified, and the mass data must be modified to ensure F = m a; the mass data is modified, and the momentum and kinetic energy data are destined to be rewritten.
- According to the special relativity principle, the laws of mechanics remain unchanged under the Lorentz transformation. For this reason, the second law of Newtonian mechanics F = m a should be rewritten as:
- F = d p / d t , where p = m u is the momentum of the object and u is the velocity.
- The assumption of the Lorentz transformation rate already assumes the limit speed
- From the mass-energy relationship and mass-velocity relationship, if an object with a non-zero net mass moves at the speed of light c , its
- In special relativity, one-dimensional time sum
- According to the special theory of relativity, the static mass is
- Consider three inertial frames K , K ' , K' ' , of which K , K'
- Validating Special Relativity
- The establishment of the special theory of relativity
- Einstein's Philosophical Belief: The Whole
- In 1922 Einstein's quilt was invited to Japan for a six-week tour. At Kyoto University, K. Nishida, a professor of philosophy, asked Einstein to explain how he discovered relativity. Einstein added a temporary speech to answer Nishida's question. Einstein's speech was translated into Japanese on the spot. Later, physicist Yoshimosa Oono translated Japanese into English and published it in the journal Physics Today (see reference 10). The next two paragraphs are a summary of the discovery of special relativity in that speech.
- Einstein said that when he read the experiments of Michelson and Morey when he was in college, he knew that the concept of the ether was wrong. But he was convinced that Maxwell and Lorentz's formula was correct. In addition, there are experiments by Fizeau that prove that the speed of light is constant and not affected by other speeds. This is different from our usual addition and subtraction, and there is a contradiction. Why is there such a contradiction? Einstein spent a year and thought of various ways to explain this contradiction, but none of them had good results.
- On a nice day in Bern, Switzerland, Einstein went to his friend M. Besso to discuss this issue, and they discussed it from different points of view. Suddenly Einstein understood the key to this problem: time is different at different speeds, and time will change due to relative speed. This is a new concept that can resolve this contradiction. The next day Einstein met Besso and said, "Thank you, I have solved this problem." Besso was puzzled when he heard it. Five weeks later, special relativity was written.
- The research of the several scientists mentioned in the above summary should still be explained. The experiments of Michelson and Morey were mentioned in Chapter III. This experiment proves that the speed of light is not affected by the speed of the earth and is a constant constant. Lorentz and Fitzgerald proposed a length reduction hypothesis to explain the results of Michelson and Morey.
- Maxwell published the basic formula of electromagnetic theory in 1873. These formulas are still applied frequently until now. He proved that the speed of electromagnetic waves can be calculated from the electromagnetic permittivity (permittivity = capacitance / meter) and magnetic permeability (permittivity = inductance / meter) in space, and found that the speed of electromagnetic waves is exactly the same as the speed of light. The famous physicist L. Botzman praised Maxwell's formula. He borrowed a word from Goethe: "Are these articles and formulas written by a god?" (See reference 36, (P. 963).
- Fizzo measured the speed of light in a flowing liquid in 1859 and found that the speed of light was not affected by the speed of the fluid. The speed of light is constant. This is a puzzling mystery (see Reference 8, page 891).
- Einstein noticed the mystery for 7 years. He finally solved the mystery in 1905. The key point to solve is that time can be changed. He came up with mathematical formulas to represent different times. When he solved these formulas, he found that there were many wonderful good results in solving the formulas. This is a masterpiece of science and has a great impact on the world.
- The theory of relativity is derived by mathematics. This chapter mainly deduces special relativity with high school algebra. Because math is difficult to read, patience is required. Relativity studies the relationship between time and space in two places or coordinate systems with different speeds. Now use stations and streetcars as examples. Fig. 0013 (a) represents a station with fixed coordinates. The space inside the station is represented by the horizontal position x and the vertical position by y . The clock on the station is A and the time is t . Fig. 0013 (b) represents a street car in moving coordinates. The street car leaves the station horizontally at a speed v . The space in the street car is represented by x ' for the horizontal position and y' for the vertical position. The clock in the car is A ' and the time is t' .
- The relationship between these two different speed coordinate spaces and time is mathematically called a transformation equation. The traditional transformation formula is also called Galiean or Newton's transformation formula.
- 0049
- 0050
- Newton's formula 0049 has two coefficients A and B , which have been previously determined using two conditions, A = 1 and B = v . The formula 0050 indicates that the time is the same in two different regions. If Newton's formula 0049 and 0050 are used, the speed of light will change due to the speed of the light source. For example, a light source approaches an observer at a speed v . According to these two formulas, the speed of light measured by the observer should be c + v . However, according to experiments by Cieter, Michelson, and Morey, no matter how fast the light source moves, the speed of light is still c , and Newton's formulas 0049 and 0050 do not meet.
- In order to be consistent with the experiment, Einstein proposed a new transformation formula as follows:
- 0051
- 0052
- There are four coefficients D , E , F , and G in formulas 0051 and 0052. It is customary to use English letters.
- How is Newton and Einstein's transformation formula different? Newton's formula (0049)
- The four coefficients D , E , F and G require four physical conditions to be solved. Two of these conditions (1) and (2) are the same as the previous two conditions of Newton. Einstein came up with two new conditions (3) and (4) based on the fixed speed of light. These four conditions will be explained below. Conditions (1) and (2) are the same as Newton's two conditions.
- Figure 0014 shows these two conditions. Figure 0014 (a) shows the fixed station and the station manager. There is a street car in Fig. 0014 (b), leaving the station at speed v .
- Condition (1): The station master measures the position of the origin of the moving coordinate O 'of the street car.
- Condition (2): A point in the fixed station is changed to a point A 'when the vehicle is driven in the street. Then change back to the fixed station, you must return to the original point A, Fig. 0014. Otherwise, point A can be changed to another position, which is unreasonable.
- Einstein's new conditions (3) and (4).
- Einstein came up with two new conditions based on the speed of light being constant and not affected by the speed of the light source. Figure 0014 (a) is a fixed station with a second-hand flashlight in the station. It is a signal light. Figure 0014 (b) shows a moving street car. It drives at the speed + v to the right. There are observers in the street car. Their speed is the same as the street car, which is + v . The right direction is set to (+), and the left direction is set to negative (-). Conditions can be applied in any order. The four coefficients will be solved in the order of conditions (1), (3), (4) and (2).
- Condition (1): The stationmaster measures the position of the origin of the moving coordinate O 'of the street car, Fig. 0014.
- At the beginning, the fixed and moving coordinate origins O and O 'can be at the same position. After the car started, the station manager saw that the street car left the station and measured the position of the street car O 'point x equal to the street car speed v times the time, that is, x = vt .
- A person in a street car, his own origin O 'goes with the car and is constant, so x' = 0 often.
- The stationmaster saw the street car origin O 'at
- 0053
- People in the street car see O 'point at
- 0054
- Substituting formulas (0053) and (0054) into Einstein's formula (0051)
- 0055
- Substituting formula (0054) into Einstein's formula (0052)
- 0056
- Divide equation (0055) by equation (0056), that is, division on the left side of the equal sign is equal to division on the right side.
- 0057
- Both t and t ' are cancelled out in the above formula.
- 0058
- Multiply the above formula by G and swap the left and right sides.
- 0059
- So condition (1) determines the relationship between E and G.
- Condition (3): As shown in Fig. 0015 (a) A flashlight in the station goes to the right at the speed of light +
- The webmaster sees the light in
- 0060
- People in the street car see the light in
- 0061
- Substituting formula (0060) and formula (0061) into Einstein's formula (0051)
- 0062
- Substituting equation (0061) into Einstein's formula (0052)
- 0063
- Divide equation (0062) by equation (0063) to get
- 0064
- In the above formula, t and t ' cancel each other out again.
- 0065
- To
- 0066
- Rearrange the above items into
- 0067
- Condition (4): Fig. 0015 (a) A flashlight in the station is illuminated at the speed of light- c to the left. The speed of light in the station is
- The conditions (4) and (3) are the same except that the direction of the light is changed to left. and so
- The webmaster sees the light in
- 0068
- People in the street car see the light in
- 0069
- Substitute the above formula and (0068) into Einstein's formula (0051)
- 0070
- Substituting equation (0069) into Einstein's formula (0052)
- 0071
- Divide equation (0070) by equation (0071) to get
- 0072
- In the above formula, t and t ' cancel each other out again.
- 0073
- To
- 0074
- Rearrange the above items into
- 0075
- The above formula (0075) and formula (0067)
- 0076
- Rearrange the various items of the above formula, and some can be added, and Gc can be canceled to get
- 0077
- Divide the above formula by 2 c 2 and have
- 0078
- Substituting equation (0078) into (0075)
- 0079
- Divide the above formula by c and have
- 0080
- Substituting equation (0080) into (0059) E = vG , we get
- 0081
- Then substituting equation (0081) into equation (0078), there is
- 0082
- According to the formula (0080), G = D , formula (0081), E = vD, and formula (0082)
- Condition (2): The point A in the fixed station is changed to the point A 'in the moving middle street car, and then returned to the station, it must return to the original point A, as shown in Fig. 00014. Otherwise, point A can be changed to another position, which is unreasonable.
- This condition can prove mathematically that the four systems D , E , F , and G must have the following relationships:
- 1 = DG - EF
- 0083
- Let Equation (0080) G = D , Equation (0081) E = vD , and Equation (0082)
- 0084
- Divide the above formula by
- 0085
- Finally, square the two sides of the above formula to get
- 0086
- The coefficient D has been determined! The other three lines E , F , G are easy.
- According to condition (2), it can be proved mathematically that the determinant of the four coefficient systems in the Einstein transformation formula must be equal to 1, or point A will go to another position. Einstein's transformation formula is
- 0051
- 0052
- The transformation matrix in the above formula is [
- Its determinant is
- 0083
- A more detailed proof of the formula (0083) is available in linerar algebra.
- Substituting equation (0086) into equation (0081) E = vD , we have
- 0087
- Substituting formula (0086) into formula (0082)
- 0088
- Finally, substituting (0086) into (0080) G = D , we have
- 0089
- All four coefficients are available, and Einstein's transformation formula has been deduced.
- Substituting equations (0086) and (0087) into Einstein's formula (0051)
- 0090
- Substituting equations (0088) and (0089) into Einstein's formula (0052)
- 0091
- Where x and t are the position and time within the station or the observer's location;
- x ' and
- v is the speed of the street car;
- c is the speed of light.
- Because the speed v of the street car is in the x direction, the other two directions ( y and z ) have no speed. y is perpendicular to x , and z is perpendicular to the plane of xy . From formula (0090), if y is substituted for x , y 'is substituted for x' and v = 0, then formula (0090) becomes y = y ' . Similarly, if x is replaced by z , z ' substitutes x' and v = 0, then formula (0090) becomes z = z ' . So have
- y = y ' and z = z'
- 0092
- Einstein used an interesting word to describe the formula (0092). His article wrote that there should be no "cynical" or strange things in the yz direction.
- Formulas (0090) and (0091) show that the observer saw the street car moving away in the station. If the observer is on a street car, the time in the car is t , the position is x and the time on the station is t ' , and the position is x' . People in the street car saw the station stand back at speed -v . Equations (53) and (54) can be rearranged into
- 0093
- 0094
- The formulas (0093) and (0094) are actually the same as the formula (0090), but the shapes are different. Formulas (0090) to (0094) are the main results of the special theory of relativity. A key step in this important theory is Einstein's formula (0052)