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

2. CHAPTER 4 APPLICATIONS OF THE LORENTZ TRANSFORMATION

3. Objectives  Length contraction  Concept of simultaneity  Time dilation –Twin paradox  Transformation of velocity  Adding velocities  Four-velocity

4. Length Contraction

5. Choice of Units

6. In this Chapter c =1

7. Example

8. Measure separation between 2 ends of a rod

9. Example

10. Length contraction  Formula for contraction  Concept of simultaneity  Paradoxes

11. Length contraction S x y V S' x' y' L0L0 What is length L as it appears to S?

12. Definition of length xAxA xBxB At the same time!

13. Use of Lorentz transformation  Both are correct  Which is more convenient? Rod is fixed in S',   x' = L 0 always   x = L when   t = 0 A moving rod appears contracted

14. What if we use the other equation?  Simultaneity is not absolute NOT simultaneous in S'  2 events are simultaneous in S (What are 2 events?)

15.  Generally  2 events which are –simultaneous in S (   t = 0) –but occurring in different places (   x  0) would not be simultaneous in S' (   t'  0)

16. 2L02L0 DB A E C Problem Seen by  S' co-moving with train  S on ground sees train moving at V =  c

17. Event B L ct Vt Sign?  0? Event D

18. Are they simultaneous? 2L02L0 DB A E C

19. Lack of symmetry?  All observers equivalent?  Symmetry S  S'?  L < L 0 ??? We're equivalent I'm special

20. Paradox

21.  Hole of length L 0  Rod of length L 0, moving at V  Push both ends of rod at the same time  Can rod go through? V

22.  At rest with hole  Rod contracted Goes throughDoes not go through  At rest with rod  Hole contracted Observer S Observer S'

23. Paradox V  Hole of length L 0  Rod of length L 0, moving at V  Push both ends of rod at the same time  Can rod go through?

24. At the same time in S At the same time in S' ? SS'

25. Time Dilation

26. Time dilation  What is time   t as it appears to S?   t is the time separation between 2 events. Which 2 events? S 1 2 V S' 1' 2'

27. =Both are correct =Which is more convenient? Clock is fixed to S' (co-moving frame),   x' = 0 Moving observer measures a longer time Proper Time

28. Lack of symmetry? We are equivalent I'm special

29. Twin Paradox

30. Twin paradox  Who is older? Is there symmetry? Motion (velocity) is relative  Acceleration is absolute — S' has travelled Clock shows shorter time S S'

31. Example P Q 10 ly  According to Q,  According to P,  Who has aged more?

32. Example  Who has experienced acceleration?  Who is the “moving observer”? P Q 

33. Experimental proof: elementary particle  

34.  Lifetime appears longer.  Clearly verified.

35. Other clocks? =Atomic clocks =Quartz watches =Biological clocks =Weak decays =Strong decays Do these all "slow down" when moving? =

36.  Analyze in detail  lnvoke Principle of Relativity Discrepancy not allowed  Study laws of physics (e.g. EM) rather than phenomena

37. Transformation of Velocity

38. Transformation of velocity  Galilean transformation  Relativistic transformation –Using Lorentz transformation directly –Using addition of "angles"

39. P Transformation of velocity 1. Galilean V Vt x x' "Addition of velocities" Same t !!

40. 2. Relativistic Note + A. Using Lorentz transformation

41.  Cannot add to more than c  If v' or V << c, the reduce to Galilean

43. "0.01 + 0.01" "0.9 + 0.9" Example

44. B. Using addition of angles SS'P =Easy to do multiple additions  Obvious that resultant  satisfies

45. Four Velocity

46. Four velocity  Velocity transforms in a complicated nonlinear manner V frame v, v' particle

47. Displacement is 4-vector Simple case:

48. 4-vector transforms as

49.  Velocity does not transform simply  because we divide by, and  is not an invariant, 

50.  transforms simply;  If we divide by a constant (e.g. 3.14), the result is still a 4-vector  Hint: Divide by a universal time

51.  called four -velocity

53. For relative motion along x:

54. Evaluation

56. Objectives  Length contraction  Concept of simultaneity  Time dilation –Twin paradox  Transformation of velocity  Adding velocities  Four-velocity

57. Acknowledgment  I thank Miss HY Shik and Mr HT Fung for design