1. EPPT M2 INTRODUCTION TO RELATIVITY K Young, Physics Department, CUHK  The Chinese University of Hong Kong

2. CHAPTER 3 MOVING REFERENCE FRAMES

3. Objectives  Galilean transformation  M-M experiment  Derive L transformation from c = const  Explicit form of L transformation  Inverse transformation

4. Basic subject is an event e.g. E = bullet shot from rifle E = (t, x, y, z) E = (t, x)

5. S V S' Vt  S' moving relative to S with velocity V along x  Basic object is an event E E

6. Galilean Transformation

7. Galilean transformation y x x' P Vt V y' x' x

8. Galilean transformation Velocities "add"

9. Michelson-Morley Experiment

10. L Galilean c c c  V c + V V

12. MM experiment  No effect found  Galilean transformation wrong

13. Details of MM experiment

14.  "Train" = Earth V ~ 3  10  4 m s -1 V/c ~ 10  4  Difference is

15.  There is no way to stop this "train" and compare with the case V = 0  Instead, compare rays parallel and perpendicular to direction of motion

16. Say  ~

17. How did MM measure such a small difference?

18. A sketch of the Michelson-Morley experiment L L V A S D C B

19.  Interference  ~ ?

20.  No effect found  Speed of light is same in all reference frames

21. No absolute motion NO!! YES!! cc c - Vc + V cc V V

22. Derivation of Lorentz Transformation

23. Derivation of Lorentz transformation  Basic object is an event E  Linear assumption [x'] = [L] [x]  Identify an invariant  Condition on transformation matrix

24. Basic object An event E V stone hitting ground atom emits photon e.g.

25. Linear assumption 16 coefficients Simpler notation

26. Consider the 4-dimensional coordinate Express as column vector

27. In 2-D case

31. Identify invariant    2 can be negative  Claim: M-M experiment 

32. Emit at (0, 0, 0, 0) Why proportional? S M-M  S': same argument, same c   Proportional E receive at (t, x, y, z)

33.  Invariance = MM  Therefore (up to a sign)  Consider reverse transformation  Independent of direction  Proportional

34. Invariant interval 1 D space 3 D space Minkowski

35. Condition on transformation matrix

37. Compare chapter 2

38.  3 conditions 

41. Relate to relative velocity S S'

44. How to remember signs?

45. Galilean limit

47. Inverse Transformation

48. What you should not do

50. 1 Do not hide a genuine difference  Euclidean x w Closed Finite

51.  Minkowski Open Infinite t x

52. 2Genuine i / Fake i QM Rel Impossible to keep track! But only for the "genuine" i not for the "fake" i

53. Choice of Units

54.  Use same units  Choose c = 1 x y x in m y in km 1km = c 1m

55. Similarly  Choose  All formulas simpler  Can multiply / divide by c n

56. Example Time = 3.0 m ??? Time = 3.0 m = 10 -8 s

57. Example ??? Energy = 10 -10 kg J = 9.00 10 6 kg m 2 s -2

58. Actual units  time = year  distance = light year

59.  distance = 3.00 10 8 m s -1 10 -9 s Actual units  time = ns = 10 -9 s = 0.3 m

60. Standard of length and time Optical transition  1 tick = 1 period T  1 rod = 1 wavelength  Velocity of light because defined quantity

61. Combining two transformations

64. Claim Prove it!

65. Inverse transformation

66.  Invert algebraically 

67. Four Vectors

69. Objectives  Index notation  Galilean transformation  M-M experiment  Derive L transformation from c = const  Explicit form of L transformation  Inverse transformation

70. Acknowledgment =I thank Miss HYShik and Mr HT Fung for design