EPPT M2 INTRODUCTION TO RELATIVITY

1. EPPT M2INTRODUCTION TO RELATIVITY K Young, Physics Department, CUHK The Chinese University of Hong Kong

2. CHAPTER 4APPLICATIONS 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. Example Measure separation between 2 ends of a rod

9. Example

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

11. y y' S S' V L0 x' x Length contraction What is length L as it appears to S?

12. At the same time! xA xB Definition of length

13. Use of Lorentz transformation • Both are correct • Which is more convenient? Rod is fixed in S', Dx' = L0 always Dx = L when Dt = 0 A moving rod appears contracted

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

15. Generally • 2 events which are • simultaneous in S(Dt = 0) • but occurring in different places (Dx 0) • would not be simultaneous in S'(Dt' 0)

16. A D B C E 2L0 Problem Seen by • S' co-moving with train • S on ground sees train moving at V = b c

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

18. A D B C E 2L0 Are they simultaneous?

19. I'm special We're equivalent Lack of symmetry? • All observers equivalent? • Symmetry SS'? • L <L0???

20. Paradox

21. V Paradox • Hole of length L0 • Rod of length L0, moving at V • Push both ends of rod at the same time • Can rod go through?

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

23. V Paradox • Hole of length L0 • Rod of length L0, moving at V • Push both ends of rod at the same time • Can rod go through?

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

25. Time Dilation

26. 2 2' S S' V 1 1' Time dilation • What is time Dt as it appears to S? Dt is the time separation between 2 events. Which 2 events?

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

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

29. Twin Paradox

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

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

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

33. Experimental proof: elementary particle

34. Lifetime appears longer. • Clearly verified.

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

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

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

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

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

44. S S' P • Obvious that resultant bsatisfies B. Using addition of angles • Easy to do multiple additions

45. Four Velocity

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

47. Displacement is 4-vector Simple case:

48. 4-vector transforms as

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

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