The Theory of Special Relativity

1. The Theory of Special Relativity Ch 26

2. Two Theories of Relativity Special Relativity (1905) • Inertial Reference frames only • Time dilation • Length Contraction • Momentum and mass (E=mc2) General Relativity • Noninertial reference frames (accelerating frames too) • Explains gravity and the curvature of space time

3. Classical and Modern Physics Classical Physics – Larger, slow moving • Newtonian Mechanics • EM and Waves • Thermodynamics Modern Physics • Relativity – Fast moving objects • Quantum Mechanics – very small

4. 10% c Speed Atomic/molecular size Size

5. Correspondence Principle • Below 10% c, classical mechanics holds (relativistic effects are minimal) • Above 10%, relativistic mechanics holds (more general theory)

6. Inertial Reference Frames • Reference frames in which the law of inertia holds • Constant velocity situations • Standing Still • Moving at constant velocity (earth is mostly inertial, though it does rotate)

7. Basic laws of physics are the same in all inertial reference frames • All inertial reference frames are equally valid

8. Speed of Light Problem • According to Maxwell’s Equations, c did not vary • Light has no medium • Some postulated “ether” that light moved through • No experimental confirmation of ether (Michelson-Morley experiment)

10. Two Postulates of Special Relativity Einstein (1905) • The laws of physics are the same in all inertial reference frames • Light travels through empty space at c, independent of speed of source or observer There is no absolute reference frame of time and space

14. Simultaneity • Time always moves forward • Time measured between things can vary Lightning strikes point A and B at the same time O will see both at the same time and call them simultaneous

16. Moving Observers 1 On train O2 - train O1 moves to the right On train O1 – train O2 moves to the left

17. Moving Observers 2 • Lightning strikes A and B at same time as both trains are opposite one another

18. Train O2 will observe the strikes as simultaneous • Train O1 will observe strike B first (not simultaneous Neither reference frame is “correct.” Time is NOT absolute

19. Time Dilation • Consider light beam reflected and observed on a moving spaceship and from the ground

21. Distance is shorter from the ship • Distance is longer from the ground • c = D/t • Since D is longer from the ground, so t must be too.

22. On Spaceship: c = 2D/Dto Dto = 2D/c On Earth: c = 2 D2 + L2 Dt v = 2L/Dt L = vDt 2

23. c = 2 D2 + v2 (Dt)2/4 Dt c2 = 4D2 + v2 Dt2 Dt = 2D c 1 –v2/c2 Dt = Dto 1 - v2/c2

24. Dt = Dto √ 1 - v2/c2 Dto • Proper time • time interval when the 2 events are at the same point in space • In this example, on the spaceship

25. Is this real? Experimental Proof • Jet planes (clocks accurate to nanoseconds) • Elementary Particles – muon • Lifetime is 2.2 ms at rest • Much longer lifetime when travelling at high speeds

29. Time Dilation: Ex 1 What is the lifetime of a muon travelling at 0.60 c (1.8 X 108 m/s) if its rest lifetime is 2.2 ms? Dt = Dto √ 1 - v2/c2 Dt = (2.2 X 10-6 s) = 2.8 X 10-6 s 1- (0.60c)2 1/2 c2

33. Time Dilation: Ex 2 If our apatosaurus aged 10 years, calculate how many years will have passed for his twin brother if he travels at: • ¼ light speed • ½ light speed • ¾ light speed

34. Time Dilation: Ex 2 • 10.3 y • 11.5 y • 10.5 y

35. Time Dilation: Ex 3 How long will a 100 year trip (as observed from earth) seem to the astronaut who is travelling at 0.99 c? Dt = Dto 1 - v2/c2 Dto = Dt 1 - v2/c2 Dto = 4.5 y

36. Time Dilation: Ex 3 If our apatosaurus aged 10 years, and his brother aged 70 years, calculate the apatosaurus’ average speed for his trip. (Express your answer in terms of c). ANS: 0.99 c

37. Length Contraction • Observers from earth would see a spaceship shorten in the length of travel

38. Only shortens in direction of travel • The length of an object is measured to be shorter when it is moving relative to an observer than when it is at rest.

39. Dto = Dt √ 1 - v2/c2 v = L Dto = L/v (L is from spacecraft) Dto Dt = Lo/v Lo = L v v √ 1 - v2/c2 L = Lo √ 1 - v2/c2

40. L = Lo √ 1 - v2/c2 Lo = Proper Length (at rest) L = Length in motion (from stationary observer)

41. Length Contraction: Ex 1 A painting is 1.00 m tall and 1.50 m wide. What are its dimensions inside a spaceship moving at 0.90 c?

42. Length Contraction: Ex 2 What are its dimensions to a stationary observer? Still 1.00 m tall L = Lo √ 1 - v2/c2 L = (1.50 m)(√ 1 - (0.90 c)2/c2) L = 0.65 m

45. Length Contraction: Ex 3 The apatosaurus had a length of about 25 m. Calculate the dinosaur’s length if it was running at: • ½ lightspeed • ¾ lightspeed • 95% lightspeed

46. 21.7 m • 15.5 n • 7.8 m

47. Four-Dimensional Space-Time Consider a meal on a train (stationary observer) • Meal seems to take longer to observer • Meal plate is more narrow to observer

48. Move faster – Time is longer but length is shorter • Move slower – Time is shorter but length is longer • Time is the fourth dimension

49. Momentum and the Mass Increase p = mov 1 - v2/c2 Mass increases with speed mo = proper (rest) mass m = mo 1 - v2/c2

50. Mass Increase: Ex 1 Calculate the mass of an electron moving at 4.00 X 107 m/s in the CRT of a television tube. m = mo 1 - v2/c2 m = 9.11 X 10-31 kg = 9.19 X 10-31 kg 1 - (4.00 X 107 m/s)2/c2