EPPT M2 INTRODUCTION TO RELATIVITY

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

2. CHAPTER 6VELOCITY, MOMENTUM and ENERGY

3. Displacement Velocity Momentum, Conservation Force Newton's second law

4. Objectives • Momentum • Collisions

5. Momentum

6. Momentum • Newtonian momentum is wrong • Should transform as 4-vector • Form of p and E

7. Four-velocity

8. Coordinates= ( time, space ) • Displacement = change of postion

9. Example • P travels to a star 5 ly away • At a speed 0.5c

10. Four velocity • Displacement per unit proper time

14. Example Particle is travelling at 300 m s -1

15. Example Particle is travelling at 0.6c

16. is just ordinary velocity • carries no information Case of low velocities

17. Time component carries no extra information • True in general

18. etc. Three spatial components

19. Four-Momentum

20. Momentum = mass velocity • Now is more convenient

21. Explicit expression

22. If , = ordinary expression • If • as • Recover Newtonian physics

23. px p = mv v v = c Do not call this effective mass M! Spatial component

24. Time component

25. Assuming mass does not change

26. Apart from additive constant, which does not matter

28. E E0 = m c2 v v = c • Provided m 0, takes E = to reach v = c • Therefore can never attain v = c

29. Faster than light? There was a young fellow named Bright Who travelled much faster than light. He set off one day, in a relative way And come back the previous night!

30. Kinetic energy

31. Application to collisions "Classical" collisions / Elastic collisions

32. Nuclei / Elementary particles Mass is "converted" to energy

33. Analogy

34. Relation between E and p Newtonian Relativistic

35. System of units E :eV MeV GeV pc : eV MeV GeV p : eV/c MeV/c GeV/c mc2 : eV MeV GeV m : eV/c2 MeV/c2 GeV/c2

36. Particle Mass (MeV/ c2) electron 0.5110 muon 105.7 proton 938.3 neutron 939.6

37. Conservation of four -momentum

38. The four-momentum • Recall • Contains energy + momentum

39. Conservation law For an isolated system, the total 4 – momentum is conserved.

40. Collisions

41. 1/3 0 u v 2 2 1 1 Example

42. Better to analyze in terms of p • p, E directly measured and quoted • v = 0.999… inconvenient • formulas apply to massless particles (photons)

43. known

44. known Example Production of p at threshold

45. e Q q Z P = 150 GeV M = 90 GeV Q e+ Example

46. Energy in the CM frame

47. In a collision, much of the energy of the projectile is used to carry the whole system forward; only a small fraction is used to produce new particles

48. E M Example Both of mass M E* in CM = ? Fixed target experiments are inefficient Colliding beams much better

49. Principle of Relativity

50. Conservation