Chapter 32 Special Theory of Relativity

1. Chapter 32 Special Theory of Relativity

2. Classical physics Theory of relativity Quantum theory Albert Einstein （1879 - 1955） Modern physics Special theory of relativity: inertial frames Relationship between space, time & motion

3. Galilean-Newtonian relativity Relativity principle: The basic laws of machanics are the same in all inertia reference frames. or:All inertial reference frames are equivalent for the description of mechanical phenomena. No one inertial frame is specialin any sense. There is no experiment to tell which frame is “really” at rest and which is moving.

4. M-equations & speed of light Maxwell: light → electromagnetic wave Speed of light from M-equations: c=3.00×108m/s In what reference frame is this valid? Speed relative to the medium —— “ether” But it does not satisfy the relativity principle Reference frame of ether is a special frame! 4

5. < M’ < S C M v *Michelson-Morley experiment Speed of earth relative to ether? Rotating < No significant fringe shift! 5

6. Einstein’s two postulates First postulate (the relativity principle): The laws of physics have the same form in all inertial reference frames. Second postulate (constancy of the speed of light): Light propagates through empty space with a definite speed c independent of the speed of the source or observer. Give up commonsense notions of space and time! 6

7. O1 . A1 B1 . A2 B2 O2 Simultaneity Time is no longer an absolute quantity! Simultaneity of events depends on the observer Same time at same place / timing by light v (a) Observer O1 (b) Observer O2 7

8. Time dilation (1) (b) Observer C on Earth (a) Observer B on spaceship 8

9. Time dilation (2) 9

10. Time dilation (3) Clocks moving relative to an observer are measured by that observer to run more slowly (as compared to clocks at rest). 1) Proper time (events occur at same point) 2) Relativity factor 3) Relativistic effect & universality 4) Space travel & Twin paradox 10

11. Lifetime of a moving muon Example1: The mean lifetime of a muon at rest is 2.2×10-6s, and it is traveling at v=0.6c relative to the lab. a) What is the mean lifetime measured in the lab? b) How far does it travel before decaying? Solution: a)Mean lifetime at rest: proper time b)Distance travel before decaying: 11

12. Time dilation of space station Example2: A space station moves around the Earth with v=7700m/s. A spaceman stays in the station for 100 days, how much younger will he become? Solution: Time dilation: Ignorable for human Correction of GPS time 12

13. v Earth Mars L0 v v Earth Mars L Length contraction (1) 13

14. Length contraction (2) The length of an object is measured to be shorter when it is moving relative to the observer than when it is at rest. 1) Proper length (measured at rest) 2) Not noticeable in everyday life 3) Relativistic effect & universality 4) Occurs only along the direction of motion 14

15. Painting’s contraction Example3: A painting (1.5m×1.0m) is hanging on a spaceship with v=0.9c relative to Earth. What are the dimensions as seen (a) in spaceship; b) on Earth? Solution: (a) In spaceship 1.0m Looks perfectly normal! 1.5m (b) On Earth: 0.9c So it has dimensions 0.65m×1.0m 15

16. v *Lorentz contraction 1 2 L L No time difference for any observer! Lorentz contraction Lorentz transformation 16

17. Lorentz transformation (1) S’ moves with v in x direction relative to S Space and time of p in different frames: S S y y v .p x x O O z z Lorentz transformation

18. Lorentz transformation (2) 1) Space-time relationship in relativity 2) v << c → Galilean transformation 3) Space and time are relative, 4-D vectors 18

19. Velocity transformation (1) 19

20. Velocity transformation (2) u, v << c → Classical 1) y, z components are also affected by v 2) Speed of light is independent of observer 3) c is the ultimate speed 20

21. The ultimate speed Example4: Two spaceships move with same speed 0.9c relative to Earth, but in opposite direction. What is the speed of one ship relative to the other? Solution: Example5: Show that the speed of light is always c in two different frames with relative speed v. Solution: How about light traveling in other direction? 21

22. S y v A B m m0 u A B o x Relativistic momentum Momentum is defined by In relativity, same expression with modified m Inelastic collision of two identical balls Reference frame S: Ball A with mass m moves at v to the right Ball B with mass m0 stays at rest 22

23. Conservation of momentum Conservation of momentum: mv=(m+m0)u(1) Reference frame S’: Ball A with m0 stays at rest; Ball B with mass m moves at v to the left Conservation of momentum: -mv=(m+m0)u’(2) 23

24. Cyclotron Relativistic mass Solving Eq.(1)(2)(3): m: relativistic mass; m0: rest mass Mass of an object increases with speed! 24

25. Relativistic dynamics Newton’s second law in relativity: Equation F=ma is not valid in relativity Ultimate speed c: 1) infinite mass & momentum 2) F→a 25

26. Kinetic energy Assume work-energy principle is still valid Work done to accelerate a particle from rest: Relativistic kinetic energy 26

27. E=mc2 Kinetic energy: Rest Energy: Total Energy: —— Mass-energy equation One of the most famous equations in physics It relates the concepts of energy and mass Confirmed in nuclear/particle experiments 27

28. Nuclear reactions Example: In a nuclear fission reaction: 1mol: 236.133 235.918 Mass decrease: m=236.133-235.918=0.215g Energy released: E=c2m=1.93×1013J =5.37×106 KWH=4600T TNT-equivalent Comparing with chemical reactions 28

29. Relationship of quantities Only when v<<c: Lorentz invariant Photon: m0=0 →E=pc 29

30. High speed pion Example6: A pion (m0=2.4×10-28 kg) travels at v=0.8c. What is its momentum and kinetic energy? Solution: Relativity factor Momentum: Kinetic energy: Comparing with classical kinetic energy: 30

31. High energy electron Example7: Determine: a) rest energy of an electron (m=9.0010-31kg, q=–e=–1.6010-19C); b) speed of electron accelerated from rest by electric potential 20kV (teletube) or 5.0 MV (X-ray machine). Solution: a) b) Total energy: 31

32. Energy in collision Example8:Two identical particles of rest mass m0 move oppositely at equal v, then a completely inelastic collision occurs and results a single particle. What is the rest mass of the new particle? Solution: Conservation of momentum 0=mv–mv=MV→ V=0 Conservation of energy: No energy loss! 32

33. Brief review of previous chapters 33

34. Ch10-11: Rotational Motion Rigid body & angular quantities Torque = force × lever arm: Angular momentum: Vector form: Rotational theorem: Conservation of angular momentum Kinetic energy: Inertial forces 34

35. Ch12: Oscillations Simple harmonic motion D-equation: Force: Parameters: Phase & rotational vector method Total energy is conserved in SHM: Superposition; Damped & forced oscillations 35

36. Ch13: Wave Motion Wave motion is a propagation of oscillation Wave velocity: Energy of wave, intensity: Intensity of spherical wave: Superposition & interference, check Reflection, standing wave Node / antinode 36

37. Ch14: Sound Loudness → energy; pitch → frequency Sound level: Logarithmic scale: Doppler effect: 37

38. Ch30: Wave Nature of Light; Interference Wave nature of light; Huygens’ principle Young’s double-slit : Optical path difference: Thin films: ? Wedge-shaped, Newton’s rings, interferometer 38

39. Ch31(A): Diffraction Fresnel’s wave theory: interference of wavelets Single slit: Circular hole, Rayleigh criterion, resolving power Diffraction grating: combination of two effects Principal maxima: Position → interference; intensity → diffraction 39

40. Ch31(B): Polarization Polarization: direction of electric field vector 3 status of polarization, Polaroid Malus’ law: Brewster’s law: 40

41. Ch32: Relativity Einstein’s two postulates; Simultaneity Time dilation: Length contraction: Velocity transformation: Relativistic mass: Kinetic energy: Total Energy: 41

42. End of this semester Good luck!