General Physics I. Lecture 18: Lorentz Transformation. Prof. WAN, Xin ( 万歆 )

Transcription

1 General Physics I Lecture 18: Lorentz Transformation Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn

2 Outline Experimental erification of the special theory Lorentz transformation The inariant interal

3 Muon Decay Cosmic rays interact with particles in the earth's outer atmosphere and create particles called muons. Muons are unstable with half-life 1.52 ms in their own rest frame, i.e. N = N 0 2 t /t 1/2 Detector on top of a 2000-m mountain counts 1000 muons traeling at = 0.98c oer a gien period of time. Now at the sea leel, how many muons do you expect to detect at = 0.98c oer the same period of time? See B. Rossi and D. B. Hall, Phys. Re. 59, 223 (1941).

4 Nonrelatiistic View Classically, muons trael 2000 m in 6.8 ms. Expect only 45 muons to surie muons 2000 m m/s = 6.8 μ s 2000 m /1.52 = ms 45 muons Experimental measurement indicates 542 muons surie, a factor of 12 more. Why? See B. Rossi and D. B. Hall, Phys. Re. 59, 223 (1941).

5 Relatiistic, Earth Frame At the high speed = 0.98c of the muons relatie to the experimenters fixed on the earth, we conclude that they obsere the muons' clock to be running slower by a slowing-down factor s = 0.2. In the earth frame, the halflife of a muon is not 1.52 ms, but 7.6 ms /7.6 = 538 The radioactie decay law predicts that 538 muons surie. Hence the measurement (542 muons) erified the special theory of relatiity within experimental uncertainties.

6 Relatiistic, Muon Frame In the muons' rest frame, the mountain is moing at the high speed = 0.98c relatie to the muons. We conclude that muons obsere the mountain s height shrinks by a shrinking factor 0.2. So the height of the mountain is not 2000 m, but 400 m. The flight time oer 400 m is 1.36 ms. Again, we obtain /1.52 = 538

7 Comparison Relatiistic, Muon Frame Relatiistic, Earth Frame Nonrelatiistic Distance 400 m 2000 m 2000 m Time 1.36 ms 6.8 ms 6.8 ms Halflies Suriing The relatiistic approach yields agreement with experiment and is greatly different from the non-relatiistic result. Note that the muon and ground frames do not agree on the distance and time, but they agree on the final result. One obserer sees time dilation, the other sees length contraction, but neither sees both.

8 Atomic Clock Time Measurement Cerium-133 atom has a well-defined transition at a frequency of 9,192,631,770 Hz and can be used as an accurate measurement of a time period. In October 1971, Joseph C. Hafele and Richard E. Keating took four cesiumbeam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the US Naal Obseratory. J.C. Hafele and R. E. Keating, Science 177, 166 (1972)

9 Atomic Clock Time Measurement General relatiity predicts an increase in graitational potential due to altitude further speeds the clocks up. That is, clocks at higher altitude tick faster than clocks on Earth's surface. J.C. Hafele and R. E. Keating, Science 177, 166 (1972)

10 Galilean Transformation The two inertial obserers agree on measurements of acceleration.

11 Lorentz Transformation Expect to find a linear map t = at ' + b x ' y y ' x = c t ' + d x ' where a, b, c, and d depend on. z Frank x z ' Mary x ' How to determine a, b, c, and d? The Galilean transformation: t = t ' x = x ' + t '

12 Lorentz Transformation Expect to find a linear map t = at ' + b x ' y y ' x = c t ' + d x ' where a, b, c, and d depend on. z Frank x z ' Mary x ' 1. Relatie speed : For the origin in Frank's frame, x = 0 x ' = t ' c/ d = x = d(x ' + t ')

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

14 Lorentz Transformation Expect to find a linear map t = at ' + b x ' y y ' x = c t ' + d x ' where a, b, c, and d depend on. z Frank x z ' Mary x ' 2. Rear clock ahead t = 0 t ' = x ' /c 2 b/a = /c 2 t = a(t ' + x ' /c 2 )

15 Length Contraction Track Frame (Frank's) 0 -T M L F 0 0 Train Frame (Mary's) 0 0 L M 0 T F D F D M T M 0 L F L M = D M = s D F T F T F where s = 1 2 /c 2 D F

16 Lorentz Transformation Expect to find a linear map t = at ' + b x ' y y ' x = c t ' + d x ' where a, b, c, and d depend on. z Frank x z ' Mary x ' 3. Length contraction t ' = 0 x ' = sx d = 1/ s

17 Time Dilation Stick Frame Clock Frame L 0 sl T' T' T L T sl In the stick frame (at rest), T = L/ for clock to arrie at the right end. In the clock frame, T ' = s( L/) (time dilation)

18 Lorentz Transformation Expect to find a linear map t = at ' + b x ' y y ' x = c t ' + d x ' where a, b, c, and d depend on. z Frank x z ' Mary x ' 4. Time dilation x ' = 0 t ' = st a = 1/ s

19 Lorentz Transformation Expect to find a linear map t = at ' + b x ' y y ' x = c t ' + d x ' where a, b, c, and d depend on. z Frank x z ' Mary x ' Relatie speed : Rear clock ahead: Length contraction: Time dilation: c/ d = b/a = /c 2 d = 1/ s a = 1/ s t ' + x '/c2 t = s x = t ' + x ' s

20 Lorentz Transformation y y ' t = t ' + x '/c2 1 2 /c 2 x = x ' + t ' 1 2 /c 2 z Frank x z ' Mary x ' t ' = x ' = t x /c2 1 2 /c 2 x t 1 2 /c 2 When c t ' = t x ' = x t recoer the Galilean transformation.

21 Lorentz Transformation The complete transformation 1) t ' = γ(t x/c 2 ) y y ' 2) x ' = γ(x t) 3) 4) y ' = y z ' = z where γ = /c 2 z Frank x z ' Mary We went all the way from the postulates to the fundamental effects, and finally to the Lorentz transformation. One may also start from the postulates to derie the Lorentz transformation first, then to the relatiistic effects. See, e.g., Classical Dynamics of Particles and Systems by S. T. Thornton and J. B. Marion. x '

22 Example: Length Contraction Consider a rod of length l lying along the x-axis of an inertial frame K. An obserer in system K' moing with uniform speed along the x-axis measures the length of the rod in the obserer's own coordinate system by determining at a gien instant of time t' the difference in the coordinates of the ends of the rod, x'(2) x'(1). According to Lorentz transformation (Eq. 2) x '(2) x '(1) = [ x(2) x(1)] [t (2) t (1)] 1 2 /c 2 where x(2) x(1) = l. Because t'(2) = t'(1), Eq. 1 leads to t (2) t (1) = [ x(2) x(1)]/c 2

23 Example: Length Contraction The length l' as measured in the K' system is therefore l ' = x '(2) x '(1) = [x(2) x(1)] 1 2 /c 2 = sl where s = 1 2 /c 2 Therefore, to the obserer in K', objects in K appear contracted. Similarly, to a stationary obserer in K, objects in K' also appear contracted. Put together, to an obserer in motion relatie to an object, the dimensions of objects are contracted by a factor s.

24 The Inariant Interal Under Galilean transformation the distance between two eents is inariant, and Newton's laws are inariant. (Δ s) 2 = (Δ x) 2 + (Δ y ) 2 + (Δ z) 2 Under Lorentz transformation The interal between eents is inariant, and We will consider dynamics in the coming lecture. (Δ s) 2 = c 2 (Δ t) 2 (Δ x) 2 (Δ y) 2 (Δ z) 2

25 Proof of the Inariant Interal For simplicity we drop D and ignore y and z directions. According to the (inerse) Lorentz transformation c 2 t 2 x 2 = c2 (t ' + x '/c 2 ) /c 2 (x ' + t ')2 1 2 /c 2 = t '2 (c 2 2 ) x ' 2 (1 2 /c 2 ) 1 2 /c 2 = c 2 t ' 2 x ' 2 s 2

26 Timelike, Spacelike & Lightlike Timelike separation: s 2 > 0 It is possible to find a frame in which the two eents happen at the same place. s/c is called the proper time. It is possible for a particle to trael from one eent to the other. Spacelike separation: s 2 < 0 It is possible to find a frame in which the two eents happen at the same time. s is called the proper distance, or proper length. Lightlike separation: s 2 = 0 In eery frame a photon emitted at one of the eents will arrie at the other.

27 Ex: Time Dilation Let frame K' moe at speed with respect to frame K. Consider two eents at the origin of K', separated by time t'. The separation between the eents is in K': (x', t') = (0, t'), in K: (x, t) = (t, t). The inariant interal implies c 2 t ' 2 0 = c 2 t 2 2 t 2 t = Note the assumption x' = 0. t ' 1 2 /c 2