Electrodynamics

1. Electrodynamics REN Xincheng, Postdoctoral , Associate Professor Tel：2331505; 13310918078 Email:ydxxxyrenxch@163.com

2. Chapter 6. Special theory of relativity

3. In the nineteenth century, Newton’s theory has been developed quite perfect, in the classical theory, it has been dominant. Physics community believe that no new work to do, the remaining problems of physics are the application of Newton’s theory. It is in this case, the clear sky in physics appeared two small black clouds (The zero results of Michelson - Morley experiment and the blackbody radiation). However, in the early of 20th century, the poly rain storm of physics community was caused by the two small black clouds, major changes in physics have taken place by it, the theory of relativity and quantum theory was developed. And the two theories have become the two pillars of modern physics, they are the foundation of modern physics. • Theory of relativity is divided into special theory of relativity and general theory of relativity. Special theory of relativity was created by Einstein in 1905, it is a theory that limited to the inertial reference system; general theory of relativity was created by Einstein in 1915, it applies to any reference system, and it is a theory that including the gravitational field. In this course, only special theory of relativity is introduced.

4. Main content： Historical background and experimental basis The basic assumptions of special theory of relativity Relative property of simultaneity The relationship of Lorentz coordinate transformation Moving clocks running slower and contraction in length Lorentz velocity transformation The four-dimensional covariant form of the relativity theory Relativistic invariance of electrodynamics Relativistic mechanics

5. There are inertia system S and S’ moving relative to S §1The difficulties of classical theory and the experimental basis of theory special relativity The same physical event is investigated in the two inertia systems • 一. Galilean transformation t time, the object reaches point P

6. Direct transformation Inverse transformation

7. v is a constant Velocity transformation and acceleration transformation Direct transformation Both are inertial systems Inverse transformation

8. In the two inertial systems • 二. Newton’s principle of relativity • In Newtonian mechanics • Force has nothing to do with the reference system, mass has nothing to do with motion.

9. Mechanism of a macroscopic and low speed objects is in the same form in any inertial system. Or, Law of Newtonian mechanics is in the form of unchanged under a Galilean transformation. For example：law of conservation of momentum

10. 三、The contradiction of principle of relativity in electromagnetic theory • We have already discussed the basic content of electrodynamics. We know that electromagnetic fields is in the form of waves in the case of prompt varying, and propagate in space at the speed of light c. So far, we have not related to the coordinate system problem. If considering the reference system, under the Galilean transformation, that the electromagnetic wave propagate at the speed of light c is tenable only to a certain inertial system, light travels in all directions to differ relative to other inertial system. Further, electromagnetic theory (Maxwell’s equation) can only set up to this certain inertial system, so that relativity principle in mechanics—all inertial systems are equivalent is no longer applies to the electromagnetic phenomena, that is, the absolute resting reference system is existent to electromagnetic theory.

11. How do we explain this? Electromagnetic theory (Maxwell’s equation) are proved to be correct by a large number of experiments, then we are the only two ways to explain: • 1）Galilean transformation is applicable for electromagnetic theory, but there is a favorably of reference frame in electromagnetic theory, that is, there is an absolute resting reference frame. Relativity principle—all inertial frames are equivalent is no longer applicable for electromagnetic theory; • 2） Relativityprinciple (Covariance) is applicable for mechanical, and is also applicable for electromagnetic theory, then the Galilean transformation need to be modified, which means that laws of Newton's classical mechanics need to be modified.

12. Most people tend to accept the first way at that time, that the Galilean transformation is applicable for the electromagnetic theory, but there is an absolute resting and favorably of reference frame in electromagnetic theory. In addition, mechanistic philosophy is in fashion in physics community at that time, physics is considered that it can use a single image of classical mechanics to describe, its outstanding performance is the "ether" hypothesis. The hypothesis believe that the "ether" is a elastic medium delivering all the electromagnetic waves including light passed, including all the electromagnetic waves, it fills the whole universe. Electromagnetic wave is the mechanical motion state of "ether" medium, vibration of charged particles can cause deformation of the "ether", and propagation of this deformation as the form of elastic wave is electromagnetic waves.

13. So people will think the absolutely stationary reference frame relative to the light "ether" is stationary. And the absolute reference system relative to the movement is called the absolute motion. Earth walk through the "ether" , measure the Earth relative to the "ether" of absolute motion, naturally became a problem at that time people first concern. This measurement is the earliest known Michelson - Morley experiment. However, a large number of experiments have not measured the Earth's motion relative to this particular reference frame, which results are the following explanation:

14. A）The Earth is a better frame of reference than the sun. • ——But geocentric has been denied at that time. • B）Earth led movement of "ether" (or "ether" dragged completely by the Earth); • ——But it is Negated by flume experiment of Faissault. • C）There is no "ether" particular reference system； • ——Can be attributed to a second route.

15. D）On the basis of the "ether", the Lorentz use electronic theory and made some assumptions, at all levels effects. Proved that electromagnetic interaction are nothing to do with the movement of the system. • And proposed space-time transformation law - Lorentz transformation. He took advantage of this transformation. From the "ether" line changed to any other frame of reference, as long as the speed is less than c, electron theory equations of motion were held constant, the speed of light in all inertial frames are the same. Thus explaining the zero result of Michelson - Morley experiment , and he has got the right formula - relativistic space-time transformation. Nevertheless, it remains that there are fundamental flaws:

16. First, his theory attached to an Ethernet, so the principle of relativity does not apply to electromagnetic phenomena. But Visibility from the results discussed above, which itself leads "ether drift" can not detect ,and a logic error in conclusionthe speed of light is the same in all inertial frames. We can say that Lorentz theory drive away the "principle of relativity" from the front door , but from the back door to put it in. It shook as the absolute reference system of "ether" concept radically. • Second, in his transformation, each one have no specific physical significance. • Thus it seems that the theory itself, its relying and the starting point is wrong, it seems a bit far-fetched. • In summary, the first way is a dead end, then only take the second route. Namely the principle of relativity (covariance) is suitable for mechanical, it also applies to electromagnetic theory. Then Newton's laws of classical mechanics (Galilean transformation) need to be modified.

17. §2 The basic assumption of special relativity • On the basis of the theory of Lorentz and others, along the right path is a decisive step towards • Albert Einstein (A. Einstein). He summed up the fact that a large number of experiments, to get rid of the influence of the old machinery and mechanical concept of time and space. Fundamentally ,Bold discarded "ether" speaking, critical inherited and developed the results of Lorentz et al , proposed two basic assumptions of relativity, thereby establishing a new space-time theory - special relativity.

18. Discussion All the laws of physics Mechanical laws The basic assumption of Einstein's special theory of relativity • 1. Principle of Relativity：All the laws of physics is in the same form in any inertial. • 2. Constancy principle：The speed of light in vacuum for any inertial in either direction are c, and is independent of the movement of the light source. • The speed of light has nothing to do with sources Movement ,described by binary trajectory distortion not occur Relativity theory of A.Einstein is the development of thetheory ofNewton

19. Revolutionary Time scaleLength scaleMeasuring the quality of The speed of light is not change B. Constant speed of light and the Galilean transformation diametrically opposed to Galileo's principle of adding speed C. Conceptual change Regardless of the reference frame Newtonian mechanics Speed ​​and reference system related (Relativity) length time quality related to reference department Special relativity mechanics (Relativity)

20. §3 Relativity of simultaneity • 一. Spacetime coordinates • 1. Event space-time coordinates 2. synchronized clocks • 二. Relativity of simultaneity • -- A direct result of constancy principle • Einstein train for example

21. Experimental device Send a light signal Einstein train Reference System of Ground On the train Place signal receiver respectively Midpoint Place the optical signal generator

22. Received flash Received flash ？ ？ Flash emitted Simultaneously receive the optical signal Event1、Event2 Occur simultaneously Event1 Event 2 Research questionsThe time interval between two events Speed ​​of light is

23. Flash Facing the light Discussion how observerto see it in S system? Speed ​​of light is also Move with Received light earlier than Event1、Event2 not Occur simultaneously Event 1 occurs first 1）Relativity of simultaneity is a direct result of constancy principle 2）Relative effect3）When the speed is much smaller than c, the result of two inertial is the same.

24. Coincide The corresponding relationship between the two reference values Search • §４Lorentz transformation • 一.Export the (special) Lorentz transformation relative to the S series ,Sˊseries speed is υ. Simultaneously ,issued a flash, the light reached the point P after some time

25. By the constancy principle： • The uniformity of space, t ' independent of y and z , it can be assumed • Known, the space homogeneous and isotropic of transformation is linear, Namely: and

26. relationship is a linear relationship, Namely • The next task is to determine the coefficient using the above comparison coefficient Medley

27. Put (3), (4) into (2) and arrange, we have: • compare (5) and (1), can get （6）

28. Further discussion of the movement of S 'system origin • Substitute (7) into(6), we get • Thereby have

29. 二.Result Coordinate transformation (transformation)

30. Let Inverse transformation Then Transformation

31. Discussion 2）Ruler and clock of each lines Back to the Galilean transformation Lorentz transformation is the development of the Galileo transformation

32. C is the speed of light limit transformation meaningless 5)By the Lorentz transformation, we see the relativity of simultaneity Event 1 Event 2 Two events occur simultaneously ？

33. If Known • (Special) Lorentz transformation can be derived by other methods Relativity of simultaneity

34. Example 1(p.241)：In Figure Let the flash emitted from O points, S system on the observation of a second after the optical signal are received by simultaneously, • Let system speed is 0.8c relative to the S system , • Find receiving the optical signal in time and position on the system 解: S receive the optical signal P1 (c,0,0,1)；P2（-c,0,0,1) In S two events occur simultaneously

35. receive the optical signal Task（P290） ：4

36. §5 Space-time theory of relativity • In the theory of relativity，Space interval and time interval is changed due to different reference system. But space-time interval • 1. Interval invariance is Lorentz invariant. It can be proved by homogeneous and isotropic of space and the basic assumptions of relativity. Lorentz transformation can also be proved that ,the previous example also illustrates this point.

37. 2. Space-time structure • Interval can be divided into the following three conditions: • 1）Two events use optical signal contact • 2）Two events contact with less than the speed of light signal • 3）Two events without contact (spatial distance exceeds light can propagate in the time t distance)

38. It is known by the interval invariance. The above the division of the space-time interval is absolute. It is not changed by change of reference （Although in different reference lines, the same event at which the point is not the same, but in which the interval is constant.).Here we analyze the geometric meaning of this division： Three-dimensional space and one dimension time constitute a four-dimensional time and space. For facilitate and intuitive of discussion,as shown, we take the discussion of Three-dimensional space-time space, P stands for any one events

39. The projection of P on the xy plane represents the place of the incident；projection of ct-axis means that the time the event occurred multiplied by the speed of light c. • Corresponding event point on the vertex O of the cone surface (called light cone), Any point on the light cone can use light signals to associate with the O-point events, called class light interval.

40. P point is located outside the light cone, in this region, the event P is not related to and event O , they can not contact each other. This area is called spacelike region (interval). • Within the cone, the event link with less than the speed of light signals. This area is called timelike region(interval). It also can be divided into absolute past (ct <0, the lower cone) and the absolute future (ct> 0, the inner cone)

41. 3. Causal rate and the maximum transmission speed of interaction • In the classic range, the causal rate is absolute .In an inertial reference frame that event A precedes event B, • event A is the cause of event B, event B is the result of the event A .So in another inertial system the event A must also precede event B. Within the scope of the absolute nature of special relativity whether this causal rate can exist? • All things sports development, but there are certain causal relationship of things the development. By contacting the material movement,the first event as cause leads to second event as a result .This causality is absolute. It should not be transferred in different reference systems.

42. Concept of time is abstracted out of development movement. Correct time and space should reflect the absolute rate of causality. Here's to analyze the causal rate how to embodied in relativistic spacetime . • From the above discussion illustrates the structure of space-time, if the events light cone P is in the upper half of the O.（Including the cone of light cone），In any inertial, P held within the upper half of the light cone O, namely P is the absolute future of O . This interval feature is that P and O are linked by interacting of available speed of light or below the speed of light. So if superluminal interaction does not exist, a necessary condition for the occurrence of two events P and O is P within O cone of light. Causal rate in this region is absolute.

43. Since outside the light cone (namely class space separated), two events can not be linked to below the speed of light signals, without the existence of superluminal effect. So, any two events are not related in spacelike interval . • Therefore, a causal relationship does not exist in spacelike interval. The above conclusions can be directly illustrate through the Lorentz transformation: • For two events

44. If we discuss is timelike (lightlike) region, Then u represents the signal transmitted from event 1 to event 2, it can only be less than or equal to the speed of light. υ represents relative velocity of an inertial frame with another inertial frame.It is also the velocity of matter, from the existing experiments have proved that it can not be equal to the speed of light, namely • So, if have This proves that causality rate is absolute for the timelike space .Conversely, if the causal rate is absolute, then naturally introduced that interaction propagation speed of light c is the speed limit.

45. We've already discussed the relative simultaneously, and are described by the Lorentz transformation. • Relativity of simultaneity is two events occur simultaneously in a reference system in different locations. In another reference frame is not simultaneous. Obviously, spacelike interval, can not occur causality .The temporal order of successively or simultaneously has no absolute meaning, because it is different in different frames of reference. • 4 . Relativity of simultaneity

46. As the specter of life? • 5. Time dilation (slow motion clock) • Mentioned Bell actually represent a specific physical processes. It as timing standard. • Research is： And consolidation process of the measurement time of reference（namely the time between two events have occurred in the same place at intervals, with the same bell measuring only).With the process and other relative movement of the inertial frame,measuring the time relationship （Were used to measure with two different locations in the bell) between the process.

47. Proper time：In a reference frame, the time between two events have occurred in the same place at intervals. That is the process of consolidation of the time reference system of the measurement process, also called proper time. • Time in two places: different locations clock. • Different reference systems can only point on the clock in the same space. In the course of inertial relative motion, the time measurement process to use a range of different locations in the alignment of the clock. Investigation of a bell Two events occur at the same location Time the original Time in two places

48.  By Inverse transform of Lorentz Original time is the shortest

49. Discussion • 1） Motion clock slow effect is space-time itself objective characteristics. Caused no sense of people.It has been confirmed by many experiments. • With μ sub in a limited lifespan through the atmosphere to illustrate . • 2）Clock delay effectis relative . This can be explained by the principle of inertial equivalence. In order to deepen our understanding can be analyzed from the perspective of measurement.

50. 3）In the case of acceleration, the clock delay will cause the absolute physical effects. If you start the alignment of the clock, One does uniformly accelerated motion around closed path . Another does not move in the same place. When the former motion one week with the latter meet, the total time it has experienced is less than the original location stationary clock elapsed time . If the above two bell instead of a pair of twins, will appear people travelling and returning along a closed path one week absolutely more younger than Stagnant man. There is a saying that the sky day on the ground for a year. This is the absolute effect of the time delay. It involves accelerating reference frame, are of general relativity. Introduction twin paradox.