General Physics I. Lecture 17: Moving Clocks and Sticks. Prof. WAN, Xin ( 万歆 )

Transcription

1 General Physics I Lecture 17: Moing Clocks and Sticks Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn

2 With Respect to What? The answer seems to be with respect to any inertial frame you like. But how can this een be possible? It is highly counterintuitie. No, it seems impossible. Frank notes in each second the light moes 299,792 km to his left, and Mary is moing 208 km (we assume this for simple illustration) to his right. Light is moing closer to Mary at 300,000 km/s, isn't it? But Mary should measure the light coming at her with a speed 299,792 km. Who is wrong? It came to me that time was suspect! Albert Einstein

3 Your Time Is Not My Time If the ball moes 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 moes 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.

4 Combing Velocities The nonrelatiistic elocity addition law should be replaced, when the elocities is comparable with the speed of light, by the relatiistic elocity addition law w = u ( u c ) ( c ) w = u + In fact, the frame independence of the speed of light c and the nonrelatiistic elocity addition law turn out to be special cases of a ery general rule for combining elocities whether or not the speeds inoled are small compared to the speed of light. We now derie on the general ground.

5 Mr Tompkins in Wonderland By George Gamow True or false? The streets grew shorter, the windows of the shops began to look like narrow slits, and the policeman on the corner became the thinnest man he had eer seen

6 Outline Eents and simultaneous eents Synchronizing clocks Moing clocks run slowly and moing sticks shrink

7 A Misconception Prior to Einstein in 1905 eeryone implicitly belieed that there is an absolute meaning to the simultaneity of two eents that happen in different places, independent of the frame of reference in which the eents are described. How do we know that they happen at the same time?

8 Eent Here we define an eent to be something that happens at a definite place at a definite time. It is the space-time generalization of the purely spatial geometric notion of point. An eent is an idealization. Instantaneous: zero extension in time Pointlike: zero extension in space In reality, its temporal and spatial extensions in a gien frame of reference are small compared with other quantities of interest.

9 Making Marks on the Tracks To be concrete, suppose one eent consists of quickly making a tiny mark on the tracks. This way, we can discuss whether two different eents, happening in different places simultaneous in the train frame are also simultaneous in the track frame. Alternatiely, one can also consider simultaneous eents in the track frame, ask whether they are simultaneously in the train frame.

10 Simultaneity in the Train Frame How does Mary, who uses the train frame, make sure that making a tiny mark on the tracks from the front of the train and doing the same from the rear of the train happen at the same time? Make sure both ends of the train hae a clock and the two clocks are synchronized. But how do you know they are properly synchronized?

11 Solution for Mary Mary brings two clocks together, synchronize them, and then hae them carried to the two ends of the train. But how does she know how fast the two clocks run as they get carried away? Well, start two clocks at the exact middle. Carry them to the ends in exactly the same way except for their opposite directions. (The same peculiarity will happen to both clocks if there is one.)

12 Using Light to Synchronize We can replace the clock moing with the use of light signals for the synchronization. After all, a light pulse traels at the speed of light regardless of the direction it is moing in and the frame of reference in which the speed is measured. c c

13 Synchronize with Light c c L M c c L M D M E L E R Simultaneous eents E L and E R in the train frame. Mary also concludes that the distance between the two marks on the tracks is D M = L M.

14 Obseration from Frank Would Frank, using the track frame, agree that the two clocks are properly synchronized? Frank would agree that the lamp is at the center of the train and the speed of light in either direction is c. But Frank must conclude then the light reaches the rear of the train before it reaches the front, as the train is moing forward. Hence the mark in the rear is made before the mark in the front.

15 In the Track Frame c c c E L c T 1 T 2 E R c T F D F

16 In the Track Frame c c c E L c T 1 ct 1 c T 1 = 1 2 T 1

17 In the Track Frame c c c T 2 = T 2 T 2 ct 2 c D F

18 In the Track Frame c T 1 = 1 2 T 1 c T 2 = T 2 c T F = c(t 2 T 1 ) = (T 2 + T 1 ) T 1 D F = c(t 2 + T 1 ) T 2 T F = D F /c 2 T F ct 1 ct 2 D F

19 The Tale of Two Frames Track Frame (Frank's) 0 -T M 0 0 Train Frame (Mary's) 0 0 L M 0 T F D F D M T M 0 T F = D F /c 2 T F T F D F

20 Mary and Frank Found About simultaneous eents: If eents EL and E R are simultaneous in one frame of reference (M), then in a second frame (F) that moes with speed in the direction pointing from E R to E L, the eent E R happens at a time D/c 2, in the second frame (F), earlier than the eent E L, where D is the distance between the two eents also in the second frame (F). How big an effect is this in eeryday life?

21 The Peculiar & the Familiar If we measure time in nanoseconds and distance in feet (as Washington still insist), we hae approximately c = 1. What we hae learned is that if two eents happen at the same time in the train frame, then in the track frame the time between them in nanoseconds is equal to the distance between them in feet, multiplied by the speed of the train along the tracks in feet per nanoseconds. If we interchange time and space in the aboe sentence, we obtain that if two eents happen at the same place in the train frame, then in the track frame the distance between them in feet is equal to the time between them in nanoseconds, multiplied by the speed of the train along the tracks in feet per nanoseconds. Familiar? (Relatie elocity of frames)

22 Surely You're Cheating, Professor! Shouldn't you object, You are cheating by using the highly peculiar light to synchronize the clock? Am I? Are the rule for simultaneous eents and the one for synchronized clocks just illusions that should disappear once we use, say, sound to synchronize the clocks? Well, let's see. Now we blow a whistle in the middle to synchronized the clocks.

23 Synchronize with Other Signals w L w R Frank obseres in the track frame. T L w L w R T R T w R D

24 Synchronize with Other Signals w L w R Frank obseres in the track frame. T L w L w R w L T L = 1 2 L T F L T L = /2 w L +

25 Synchronize with Other Signals w L w R Frank obseres in the track frame. w R T R = T R Together with the equation at the preious page T R = /2 w R T R D = w R T R + w L T L D

26 Synchronize with Other Signals T L = /2 w L + T = 1 2 ( D = 1 2 ( T R = /2 D = w R T R w R 1 w R 1 ) w L + w R w R + w L ) w L + T D = 2 (w R w L ) 2w R w L + (w R w L ) + w L T L T L T R T D

27 Synchronize with Other Signals w L w R u u In the classical mechanics, w R = u + w L = u T D = 2 (w R w L ) 2w R w L + (w R w L ) = 0 This is the nonrelatiistic rule: two eents that are simultaneous in the train frame are also simultaneous in the track frame.

28 Synchronize with Other Signals w L u u w R With the relatiistic elocity addition law, w R = u ( u c )( c ) w L = u 1 ( u c )( c ) T D = 2 (w R w L ) 2w R w L + (w R w L ) = c 2 We recoer the result obtained with light, D = D F and T = T F.

29 Rule for Synchronized Clocks If two clocks are synchronized and separated by a distance D in their proper frame (in which they do not moe), then in a frame in which the clocks moe along the line joining them with speed, the reading of the clock in front is behind the reading of the clock in the rear by D/c 2, i.e., rear clock ahead. While the readings were made in the frame that the two clocks are not synchronized, the time difference is still defined in the frame that they are synchronized.

30 The Tale of Two Frames Track Frame (Frank's) 0 -T M 0 0 Train Frame (Mary's) 0 0 L M 0 T F D F D M T F T M 0 T F D F T F = D F /c 2 By the principle of relatiity, T M = L M /c 2

31 Compare Time & Length The principle of relatiity guarantees that the rule for synchronized clocks must be alid in any inertial frame of reference. So, we can compare clocks synchronized in Mary's frame and clocks synchronized in Frank's frame. We shall deduce that moing clocks must run slowly. Likewise, moing trains (or tracks) must shrink along the direction of their motion.

32 The Slowing-Down Factor Track Frame (Frank's) 0 -T M 0 0 Between the two pictures, both Frank s (track) clocks adance by a time T F and both Mary s (train) clocks adance by a time T M. Therefore, the slowingdown factor is D F T M 0 T M T F = L M D F = D M D F = s T F T F Note that the train-frame picture is a single-moment snapshot (as is eident from the same reading of the train clocks), hence L M = D M. D F

33 The Shrinking Factor Track Frame (Frank's) 0 -T M 0 0 Train Frame (Mary's) 0 0 L M 0 T F D F D M The shrinking factor for a moing object L M = D M D F s

34 Soling the S-Factor Track Frame (Frank's) 0 -T M D F = + T F = + 2 D F c D F = 1 2 /c 2 D F From the preious page T M 0 = s L M = s D M = s 2 D F s = 1 2 /c 2 T F T F Note that s < 1 justifies the names of the slowing-down and shrinking factor. D F

35 Nomenclature Time dilation refers to the slowing down of the moing clock, although time itself (or whateer that might be) is not really expanding it is the relation between moing clocks that changes. The shrinking of moing objects along the direction of their motion is called the Lorentz- Fitzgerald contraction (in the context of moing relatiely to the ether). In the literature g = 1/s is quite often used.

36 New Puzzles How can Mary conclude that Frank's clocks are running slowly, while Frank concludes that it is Mary's clocks that are running slowly? How comes the slowing-down factor s for moing clocks is the same as the shrinking factor s for moing sticks? Note that in the theory of relatiity consistency, rather than the eeryday experience, is more important. Do you see any inconsistency?

37 Both Are Correct To determine the rate of a moing clock, it is necessary to compare two of the readings of the moing clock with the readings of two stationary clocks that are next to it when it shows those readings. This requires one to use two synchronized clocks in two difference places. To determine the length of a moing stick, one must determine where its two ends are at the same time. This requires the judgment about whether spatially separated eents at the two ends of the stick are simultaneous. Both Mary and Frank can point out that the other fails to synchronize her/his clocks properly. So both are correct.

38 Fundamental Disagreement The fundamental disagreement is on whether two eents in different places happen at the same time or, equialently, on whether two clocks in different places are synchronized. Time dilation and length contraction are merely conentions about how we describe the behaior of clocks and measuring sticks.

39 What Mary and Frank Agree Track Frame (Frank's) 0 -T M 0 0 Train Frame (Mary's) 0 0 L M 0 T F D F D M T M 0 T F = D F /c 2 T F T F T M = L M /c 2 D F

40 Both Factors Are the Same Stick Frame Clock Frame 0 L 0 sl L T sl T T = s'(l/) time dilation T = (sl)/ length contraction Time T on the clock must be agreed since it is about things at a single place and time.

41 Take-Home Message Rule for synchronized clocks If two clocks are stationary, synchronized, and separated by a distance D in one frame of reference, then in a second frame, in which they are moing with speed along the line joining them, the clock in the front lags behind the clock in the rear by T = D /c 2 Rule for shrinking of moing sticks or slowing down of moing clocks The shrinking (or slowing-down) factor s associated with a speed is gien by s = 1 2 /c 2

42 An Educated Protocol Mary prepares two identical stationary clocks in the same place, say at the rear of the train. She synchronizes them to 0 by direct comparison in the same place. She sets one of them into uniform motion with speed u toward the front of the train. She stops the motion of the clock at the front of the train, at which moment the clock reads t. Mary resets the clock to t / 1 u 2 /c 2

43 Questions Are the two clocks synchronized in Mary's frame? Assume, again, Mary and the train is moing with respect to Frank on the ground with a speed. Does Frank agree that Mary's clocks are synchronized? If not, how much does the two clocks differ in Frank's frame? Show explicitly, through the slowing-down factors s u, s, s w, that the result can be calculated in Frank's frame. Here, w refers to the combined elocity of u and.