General Physics I. Lecture 16: The Principles of the Theory of Relativity. Prof. WAN, Xin 万歆.

Transcription

1 General Physics I Lecture 16: The Principles of the Theory of Relativity Prof. WAN, Xin 万歆 xinwan@zju.edu.cn

2 The Train of Reasoning You have been in Afghanistan, I perceive. Sherlock Holmes The train of reasoning ran: "Here is a gentleman of a medical type, but with the air of a military man. Clearly an army doctor then. He has just come from the tropics, for his face is dark, and that is not the natural tint of his skin, for his wrists are fair. He has undergone hardships and sickness, as his haggard face says clearly. His left arm has been injured. He holds it in a stiff and unnatural manner. Where in the tropics could an English army doctor have seen much hardship and got his arm wounded? Clearly in Afghanistan."

3 Outline Electromagnetic waves The existence of ether or not? Fizeau experiment Michelson-Morley experiment The constancy of the speed of light The principle of relativity Combining velocities

4 With Respect to What? The speed of light in empty space is 299,792,458 m/s with respect to what? The first obvious answer: To the source of the light. When we specify the speed of a bullet, we mean its speed with respect to the gun from which it has emerged. This answer is wrong! We discussed in the previous lecture that the speed of light is independent of the speed of the light source. Remember the 1964 experiment?

5 Depends on the Source Motion v stationary Alice notes a larger speed of the bullet than Bob does.

6 Independent in Vacuum v stationary Although Bob's flashlight moves with speed v, Alice notes the same speed of light from Bob's flashlight and her flashlight.

7 Maxwell's Theory In addition, the first reasonable answer is contradicted by our understanding of the electromagnetic character of light. Maxwell completed the unification of the laws of electricity and magnetism. His theory predicted the electromagnetic waves traveling at a speed of 300,000,000 m/s, numerically indistinguishable from the speed of light. Maxwell's theory implied that the speed of the E&M waves does not depends on the speed of the source of the radiation.

8 Electromagnetic Waves And Maxwell said, and there was light. 1 c= ε μ 0 0 Vacuum permittivity ε0= F /m Vacuum permeability μ0= H /m

9 Maestro Maxwell Was Right Heinrich Hertz ( ) proved Maxwell's theory by engineering instruments to transmit and receive radio pulses (in the ultra high frequency range). "It's of no use whatsoever[...] this is just an experiment that proves Maestro Maxwell was right we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there." Gee, he was completely wrong about its use!

10 The E&M Spectrum

11 Low-Frequency Spectrum Can you here the E&M wave with a frequency 1 khz?

12 Does Light Have a Medium? Physicists in the late 19th century believed Surface water waves must have a supporting substance, i.e. a "medium", to move across (in this case water). Audible sound requires a medium to transmit its wave motions (such as air or water). So light must also require a medium, the "luminiferous ether", to transmit its wave motions. Ether must have highly unusual properties The speed of light is so great. Material bodies pass through the ether without obvious friction or drag.

13 Speeds of Sound in Media

14 With Respect to What? Light travels with a speed of 299,792,458 m/s in vacuum. Light travels significant slower in transparent media like water or glass, and a little bit slower in air. The speed of sound does not depend on the speed of the source of the sound. By analogy, the light may be a vibration of something historically called the ether. The speed of light is then with respect to that ether. As Einstein later noted, the introduction of a 'luminiferous ether' will prove to be superfluous because there would be no way to determine the rest frame of the ether by any physical experiment involving electromagnetic phenomena.

15 Fizeau Experiment (1851) If n is the index of refraction of water, so that c/n is the velocity of light in stationary water. Now consider that water flows in the pipes at velocity v. Naively, one expects v' = However, experiment found c + v n c 1 v' = + v 1 2 n n ( ) This appears to suggest that it cannot be with respect to water.

16 Michelson-Morley Experiment Early belief was that light must also require a medium, the "luminiferous aether", to transmit its wave motions. Because light can travel through a vacuum, it was assumed that even a vacuum must be filled with aether.

17 Most Famous "Failed" Experiment The Experiments on the relative motion of the earth and ether have been completed and the result decidedly negative. Albert Abraham Michelson, 1887

18 Constancy of the Speed of Light stationary v The speed of light is 299,792,458 m/s in vacuum, regardless of the motion of its source or observer.

19 With Respect to What? The answer seems to be with respect to any inertial frame you like. But how can this even be possible? It is highly counterintuitive. No, it seems impossible. Bob notes in each second the light moves 299,792 km to his left, and Alice is moving 208 km (we assume this for simple illustration) to his right. Light is moving closer to Alice at 300,000 km/s, isn't it? But Alice should measure the light coming at her with a speed 299,792 km. Who is wrong? This appears to violate the principle of relativity!

20 The Principle of Relativity In Einstein's words, In electromagnetism as well as in mechanics, phenomena has no properties corresponding to the concept of absolute rest. The principle of relativity should now be a wellknow feature of mechanics to you. In everyday language, this means that all other things being the same, it does not matter how fast you are going if you are moving with fixed speed along a straight line. (First enunciated by Galileo.)

21 Newton's First Law of Motion In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity (that is, with a constant speed in a straight line). In simpler terms, when no force acts on an object, the acceleration of the object is zero. Any isolated object (one that does not interact with its environment) is either at rest or moving with constant velocity.

22 Significance of Newton's 1st Law Identifies a set of special reference frames in which we can apply the laws of classical mechanics. The tendency of an object to resist any attempt to change its velocity is called the inertia of the object. Newton's first law is often called the law of inertia. The reference frames to which it applies are called inertial frames.

23 The Application of the Principle Take a situation that you do not fully understand. Find a new frame of reference in which you do understand it. Examine it in the new frame of reference. Translate your understanding in the new frame back into the language of the old one.

24 Ex. 1: Perfectly Inelastic Collision Before Known 5 m/s 10 m/s Unknown After 5 m/s?

25 Ex. 1: Perfectly Inelastic Collision Before Train 5 m/s 10 m/s After 5 m/s 5 m/s Track Assuming the train is moving at 5 m/s to the right.

26 Ex. 2: Elastic Collision Before Known 5 m/s 10 m/s Unknown 5 m/s After 5 m/s 5 m/s?

27 Ex. 2: Elastic Collision Before Train 5 m/s 10 m/s 5 m/s After 5 m/s 5 m/s 10 m/s Track Assuming the train is moving at 5 m/s to the right.

28 Ex. 3: Very Small vs Very Big Before Known Unknown After 10 m/s 10 m/s 10 m/s?

29 Ex. 3: Very Small vs Very Big Before Train Track After 10 m/s 10 m/s 10 m/s 20 m/s 10 m/s Assuming the train is moving at 10 m/s to the left.

30 Ex. 4: Slow Down the Train Before Train Track 10 m/s 10 m/s 5 m/s After 5 m/s 15 m/s 5 m/s Assuming the train is moving at 5 m/s to the left.

31 Implicit Assumption Besides using the principle of relativity, we repeatedly made implicit use of the nonrelativistic velocity addition law v XZ = v XY + v YZ vxz is the velocity of X with respect to Z. Aha! Here comes the with respect to. How many of you find the addition law not so obvious? Why?

32 Innocent or Guilty? A train moves to the right at 5 m/s in the track frame. In the train frame a ball moves to the right at 5 m/s. Hence, the ball is moving 15 m/s to the right in the track frame. Any doubt? 5 m/s 10 m/s

33 Now the Cumbersome Version If the ball moves to the right in the train frame at 5 m/s, then in one second according to the train time its gets 5 meters further down the train. If the train moves to the right at 10 m/s in the track frame, then in one second according to the track time it gets 10 meters further right along the track. So in one second the ball gets 15 meters further right along the track the 5 it gains on the train and the additional 10 the train gains on the track. But what does in one second mean? It makes no sense unless the track time and the train time are the same.

34 From Newton to Einstein Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external. Isaac Newton It came to me that time was suspect! Albert Einstein

35 Combing Velocities The nonrelativistic velocity addition law w = u + v should be replaced, when the velocities is comparable with the speed of light, by the relativistic velocity addition law u + v w = u v 1 + c c ( )( ) In fact, the frame independence of the speed of light c and the nonrelativistic velocity addition law turn out to be special cases of a very general rule for combining velocities whether or not the speeds involved are small compared to the speed of light. We now derive on the general ground.

36 A Tale of Two Races Now we derive the velocity addition law by taking advantage of the fact that we do know, at least, the speed of light. We avoid using clocks, rulers, etc. We run two races between, say, a ball with a pulse of light (a photon in quantum mechanics), one with the train at rest on the track and another with a moving train.

37 The First Race c u c u c 1- f u u 1 f = c 1 + f f Solve for f.

38 The Second Race c w v L T0 c w v D c T1 w v fl

39 The Second Race c w v L T0 c w v D D = c T 0 w T0 L = ct0 v T0

40 The Second Race D = ct1 + wt1 fl = c T 1 + v T 1 c w v D c T1 w v fl

41 Relativistic Velocity Addition Law D = c T 0 w T0 D = ct1 + wt1 L = ct0 v T0 fl = c T 1 + v T 1 All the times and distances are unknown track-frame values, but since the problematic quantities all drop out in the final result, this should not cause any difficulty. Now solve for f again. Together with the result from the first race, you should conclude c w c u c v = c + w c + u c + v ( )( )

42 Comments The nonrelativistic velocity addition law is recovered when both u and v are small compared to the speed of light. If u = c, then w is required to be c, whatever the value of v may be. The speed the light remains constant in all inertial frames of reference. How to combine three (and more) velocities? How to include negative velocities?

43 Additional Comments Objects moving at the speed of light behave in some strange (meaning beyond classical mechanics) ways, the behavior of objects moving at speeds comparable to the speed of light can be just as peculiar. The peculiarity of motion at the speed of light is just a special case of a more general peculiarity of all motion, which becomes prominent only at extremely high speeds. The general peculiarity is an elementary but precise rule.