Special Theory of Relativity

1. Albert Einstein Special Theory of Relativity Micro-world Marco-world Lect 11

2. watching a light flash go by v c 2kk The man on earth sees c = (& agrees with Maxwell)

3. watching a light flash go by v c If the man on the rocket sees c-v, he disagrees with Maxwell

4. watching a light flash go by It doesn’t works for this guy, v c If Maxwell’s theory works for this guy, or vice versa

5. Do Maxwell’s Eqns only work in one reference frame? If so, this would be the rest frame of the luminiferous Aether.

6. If so, the speed of light should change throughout the year upstream, light moves slower downstream, light moves faster “Aether wind”

7. Michelson-Morley No aether wind detected: 1907 Nobel Prize

8. Einstein’s hypotheses: 1. The laws of nature are equally valid in every inertial reference frame. Including Maxwell’s eqns 2. The speed of light in empty space is same for all inertial observers, regard- less of their velocity or the velocity of the source of light.

9. All observers see light flashes go by them with the same speed v No matter how fast the guy on the rocket is moving!! c Both guys see the light flash travel with velocity = c

10. Even when the light flash is traveling in an opposite direction v c Both guys see the light flash travel past with velocity = c

11. Gunfight viewed by observer at rest He sees both shots fired simultaneously Bang! Bang!

12. Viewed by a moving observer

13. Viewed by a moving observer He sees cowboy shoot 1st & cowgirl shoot later Bang! Bang!

14. Viewed by an observer in theopposite direction

15. Viewed by a moving observer He sees cowgirl shoot 1st & cowboy shoot later Bang! Bang!

16. Time depends of state of motion of the observer!! Events that occur simultaneously according to one observer can occur at different times for other observers

17. Light clock

18. Seen from the ground

19. Events y (x2,t2) (x1,t1) x x x1 x2 x t

20. Prior to Einstein, everyone agreed the distance between events depends upon the observer, but not the time. Same events, different observers y’ y’ y (x2,t2) (x1,t1) x x (x1’,t1’) (x2’,t2’) t’ t’ x1’ x1’ x2’ dist’ x’ x’ x1 x2 x t dist

21. Time is the 4th dimension Einstein discovered that there is no “absolute” time, it too depends upon the state of motion of the observer Einstein Space-Time Newton Space & Time completely different concepts 2 different aspects of the same thing

22. How are the times seen by 2 different observers related? We can figure this out with simple HS-level math ( + a little effort)

23. Catch ball on a rocket ship Event 2: girl catches the ball wt v= =4m/s w=4m t=1s Event 1: boy throws the ball

24. Seen from earth V0=3m/s V0=3m/s Location of the 2 events is different Elapsed time is the same The ball appears to travel faster d=(3m)2+(4m)2 =5m w=4m v0t=3m dt v= = 5m/s t=1s

25. Flash a light on a rocket ship Event 2: light flash reaches the girl wt0 c= w t0 Event 1: boy flashes the light

26. Seen from earth V V Speed has to Be the same Dist is longer Time must be longer d=(vt)2+w2 w vt dt c= = (vt)2+w2 t t=?

27. How is t related to t0? t= time on Earth clock t0 = time on moving clock wt0 c = (vt)2+w2 t c = ct = (vt)2+w2 ct0 = w (ct)2 = (vt)2+w2 (ct)2 = (vt)2+(ct0)2  (ct)2-(vt)2= (ct0)2  (c2-v2)t2= c2t02 1 1 – v2/c2 c2 c2 – v2  t2 = t02  t2 = t02 1 1 – v2/c2  t= t0  t= g t0 this is called g

28. 1 1 – v2/c2 Properties of g = Suppose v = 0.01c (i.e. 1% of c) 1 1 – (0.01c)2/c2 1 1 – (0.01)2c2/c2 g = = 1 1 – 0.0001 1 1 – (0.01)2 1 0.9999 = = g = g = 1.00005

29. 1 1 – v2/c2 Properties of g = (cont’d) Suppose v = 0.1c (i.e. 10% of c) 1 1 – (0.1c)2/c2 1 1 – (0.1)2c2/c2 g = = 1 1 – 0.01 1 1 – (0.1)2 1 0.99 = = g = g = 1.005

30. Let’s make a chart

31. 1 1 – v2/c2 Other values of g = Suppose v = 0.5c (i.e. 50% of c) 1 1 – (0.5c)2/c2 1 1 – (0.5)2c2/c2 g = = 1 1 – (0.25) 1 1 – (0.5)2 1 0.75 = = g = g = 1.15

32. Enter into chart

33. 1 1 – v2/c2 Other values of g = Suppose v = 0.6c (i.e. 60% of c) 1 1 – (0.6c)2/c2 1 1 – (0.6)2c2/c2 g = = 1 1 – (0.36) 1 1 – (0.6)2 1 0.64 = = g = g = 1.25

34. Back to the chart

35. 1 1 – v2/c2 Other values of g = Suppose v = 0.8c (i.e. 80% of c) 1 1 – (0.8c)2/c2 1 1 – (0.8)2c2/c2 g = = 1 1 – (0.64) 1 1 – (0.8)2 1 0.36 = = g = g = 1.67

36. Enter into the chart

37. 1 1 – v2/c2 Other values of g = Suppose v = 0.9c (i.e.90% of c) 1 1 – (0.9c)2/c2 1 1 – (0.9)2c2/c2 g = = 1 1 – 0.81 1 1 – (0.9)2 1 0.19 = = g = g = 2.29

38. update chart

39. 1 1 – v2/c2 Other values of g = Suppose v = 0.99c (i.e.99% of c) 1 1 – (0.99c)2/c2 1 1 – (0.99)2c2/c2 g = = 1 1 – 0.98 1 1 – (0.99)2 1 0.02 = = g = g = 7.07

40. Enter into chart

41. 1 1 – v2/c2 Other values of g = Suppose v = c 1 1 – (c)2/c2 1 1 – c2/c2 g = = 1 0 1 1 – 12 1 0 = = g = g =  Infinity!!!

42. update chart

43. 1 1 – v2/c2 Other values of g = Suppose v = 1.1c 1 1 – (1.1c)2/c2 1 1 – (1.1)2c2/c2 g = = 1 1-1.21 1 1 – (1.1)2 1 -0.21 = = g = g = ??? Imaginary number!!!

44. Complete the chart

45. Plot results:  1 1 – v2/c2 g = Never-never land x x x x x v=c

46. Moving clocks run slower v t0 1 1 – v2/c2 t= t0 t t= g t0 g >1  t > t0

47. Length contraction v L0 time=t L0 = vt Shorter! man on rocket Time = t0 =t/g Length = vt0 =vt/g =L0/g

48. Moving objects appear shorter Length measured when object is at rest L= L0/g g >1  L < L0 V=0.9999c V=0.1c V=0.86c V=0.99c

49. Length contraction

50. change inv time mass: F=m0a =m0 t0 a time=t0 m0 Ft0 m0 change inv = Ft0 change inv m0 = mass increases!! gFt0 change inv Ft change inv = gm0 m= = m = g m0 t=gt0 by a factor g