1. Chapter 32 Special Theory of Relativity

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

3. Galilean-Newtonian relativity 3 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 special in any sense. There is no experiment to tell which frame is “really” at rest and which is moving.

4. M-equations & speed of light 4 Maxwell: light → electromagnetic wave Speed of light from M-equations: c=3.00×10 8 m/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!

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

6. Einstein’s two postulates 6 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!

7. Simultaneity 7 Time is no longer an absolute quantity! Simultaneity of events depends on the observer A1A1 B1B1 O1O1. A2A2 B2B2 O2O2. v (a) Observer O 1 (b) Observer O 2 Same time at same place / timing by light

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

9. Time dilation (2) 9

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

11. Lifetime of a moving muon 11 Example1: The mean lifetime of a muon at rest is 2.2×10 -6 s, 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:

12. Time dilation of space station 12 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 humanCorrection of GPS time

13. Length contraction (1) 13 EarthMars v L0L0 EarthMars v L v

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

15. Painting’s contraction 15 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 0.9c 1.5m 1.0m Looks perfectly normal! (b) On Earth: So it has dimensions 0.65m×1.0m

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

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

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

19. Velocity transformation (1) 19

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

21. The ultimate speed 21 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?

22. Relativistic momentum 22 Momentum is defined by In relativity, same expression with modified m Inelastic collision of two identical balls B m0m0 A B u x y S A m v o Reference frame S: Ball A with mass m moves at v to the right Ball B with mass m 0 stays at rest

23. Conservation of momentum 23 Conservation of momentum: m v =(m+m 0 )u (1) Reference frame S’: Ball A with m 0 stays at rest; Ball B with mass m moves at v to the left Conservation of momentum: - m v =(m+m 0 )u’ (2)

24. Relativistic mass 24 Solving Eq.(1)(2)(3): m: relativistic mass;m 0 : rest mass Mass of an object increases with speed! Cyclotron

25. Relativistic dynamics 25 Newton’s second law in relativity: Ultimate speed c: 1) infinite mass & momentum2) F → a Equation F = ma is not valid in relativity

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

27. 27 E = mc 2 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

28. Nuclear reactions 28 Example: In a nuclear fission reaction: Mass decrease:  m = 236.133-235.918 = 0.215g 1mol: 236.133 235.918 Energy released:  E = c 2  m =1.93×10 13 J = 5.37 × 10 6 KWH = 4600T TNT-equivalent Comparing with chemical reactions

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

30. High speed pion 30 Example6: A pion (m 0 =2.4 × 10 -28 kg) travels at v=0.8c. What is its momentum and kinetic energy? Solution: Relativity factor Kinetic energy: Comparing with classical kinetic energy: Momentum:

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

32. Energy in collision 32 Example8: Two identical particles of rest mass m 0 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 = m v – m v = MV → V = 0 Conservation of energy: No energy loss!

33. Brief review of previous chapters 33

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

35. Ch12: Oscillations Simple harmonic motion Parameters: Phase & rotational vector method Total energy is conserved in SHM: Superposition; Force: D-equation: 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 waveNode / 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 : Thin films: Optical path difference: 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 Position → interference; intensity → diffraction Diffraction grating: combination of two effects Principal maxima: 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!